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Using nonlinear feedback control to model human landing mechanics

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Using nonlinear feedback control to model human landing mechanics

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Using Nonlinear Feedback Control to Model
Human Landing Mechanics

Authored by Edward V. Wagner

A Dissertation Presented to the
Department of Aerospace and Mechanical Engineering
Faculty of the University of Southern California Graduate School

In Partial Fulfillment of the Requirements for the Degree
Doctor of Philosophy
Mechanical Engineering

University of Southern California

May 2018

Copyright 2018 Edward V. Wagner

i

Dedicated to my parents,

Tim and Liz,
Thank you for your boundless love and support
without which I could never have pushed the bounds of science.

“Science is a co-operative enterprise, spanning the generations. It's the passing of a torch from teacher to student to
teacher. A community of minds reaching back from antiquity and forward to the stars.”
- Neil deGrasse Tyson

… And then I woke up.
ii

ACKNOWLEDGEMENTS

I wish to express my sincerest thanks to:
 My advisors Dr. Henryk Flashner and Dr. Jill McNitt-Gray for their wisdom and guidance
 Dr. Veronica Eliasson for use of the ultra-high speed camera, Phantom v7.11
 The USC Biomechanics Lab (Ian, Korkut, Travis, Antonia, Nate, Sneha, Sam, Casey, and Marisa) for their
support and friendship
 The BELA GK12 Fellowship for the financial support and teaching experience
 The ARCS Foundation for their financial support and belief in the value of science and education

iii

ABSTRACT

Human beings interact with a variety of terrains on a regular basis, adapting their joint coordination strategies to
account for these diverse mechanical conditions[1]. Because these adaptive processes are always engaged, they play
an integral role in driving the dynamics of the body. Attempts to model macroscale human body landing dynamics
have, until recently, largely neglected this active aspect, focusing instead on describing the system in terms of
mechanical elements with predefined parameters. Thus, the aim of this research is to develop and validate an
experimentally based dynamic model of the human body that captures this ability to modulate ground reaction forces
to achieve goal oriented task objectives. The goal of modeling, as Weyand[2] puts it, is to “bridge the gap between
overly simplistic models which provide little insight and overly complex models which are not broadly applicable.”

During foot-first landings human subjects attempt to satisfy multiple, sometimes competing, demands: impact
preparation, total body vertical momentum reduction, and system stabilization. All of these objectives are subject to
constraints imposed by the limits of the physical system: kinematic manifolds steering away from joint extension
limits and kinetic upper bounds on the ground reaction forces to avoid overloading segments or over exerting joints.
To characterize the motor control strategies by which humans accomplish these objectives, foot-first drop-landing
experiments are conducted with human subjects. In the task a subject is asked to step off a platform and land on force
plates embedded in the floor under three conditions: self-selected normal, harder-than-normal, and softer-than-normal.
The position of retroreflective markers on each segment are tracked in the sagittal plane of motion using an ultrahigh-
speed camera to ensure all kinematic impact phenomena are properly documented. Due to the bifurcation in dynamic
conditions during landings with impact, a novel time-dependent filtering method is developed and implemented to
provide second order time-derivatives of these kinematics across the different phases of landing. Subjects exhibit
phase specific control strategies during landing including segment pull-up during the flight phase which reduces the
downward velocity of their end effectors, stiffness regulation during impact phase modulating the peak vertical ground
reaction force, and subject specific joint coordination manifolds during the post-impact phase.

This research proposes a 2D experiment-based model, composed of 4 rigid links connected by hinge joints and driven
by nonlinear feedback controlled net joint moment actuators is complex enough to capture the dynamics of the human
body during multiphase drop-landing tasks involving foot-first impact, as measured by the vertical ground reaction
forces. The phase-specific feedback control architecture changes according to the changing mechanical objectives of
each phase of landing: joint angle following control during flight phase, impedance control during the impact phase,
and energy-shaping control during the post-impact phase. Simulations revealed that the proposed model accurately
(±10⁰) follows joint angles during the flight phase, accurately (±10%) predicts both the peak vertical ground reaction
force and total vertical impulse during the impact phase, and sufficiently dissipates the remaining kinetic energy to
stabilize the system during the post-impact phase of a drop landing for all three drop landing conditions.

Keywords: control, nonlinear, multiphase, impact, filtering, biomechanics, 2D kinematics, net joint moments,
impedance, energy shaping, passivity-based, forward dynamics, simulation, reaction force, landing, lower-extremity
iv

PUBLICATIONS

PAPERS
1. An Approach for Characterizing Technique Modification through Analysis of Time-Dependent Functional Joint
Axis Components (in Progress)

2. Multiphase Continuous Piecewise Weighted Cubic Spline (PWCS) Smoothing (in Progress)

3. Modelling the effects of Technique Modification on Human Body Dynamics during the Impact Phase of Landings
through Impedance Based Multiphase Control (in Progress)

4. Modelling the effects of Technique Modification on Human Body Dynamics during the Post-impact Phase of
Landings through Energy Based Multiphase Control (in Progress)

CONFERENCE PROCEEDINGS
1. Muller-Karger, C., Wagner, E.V, et al. (2013). Representation of Shoulder Kinematics during Multiplane Tasks
Performed by Manual Wheelchair Users, 24
th
Congress of the International Society of Biomechanics, Natal,
BRZ

2. Wagner, E.V, et al. (2014). Dual-quaternion Analysis of Shoulder and Upper Extremity Motion for Calculation
of Angular Velocity Joint Axis, 7
th
World Congress of Biomechanics, Boston, MA.

3. Wagner, E.V, Flashner, H, & McNitt-Gray, J.L, Eliasson, V. (2015). Modeling Multijoint Control for Regulation
of Reaction Forces during Impact Phase of Landings, 25
th
Congress of the International Society of
Biomechanics, Glasgow, SCO

4. Wagner, E.V, Flashner, H, & McNitt-Gray, J.L. (2016). An Approach for Characterizing Multiplanar Upper
Extremity Motion through Parsed Angular Velocity Vector Components, 40
th
Annual Meeting of the American
Society of Biomechanics, Raleigh, NC.

5. Wagner, E.V, Flashner, H, & McNitt-Gray, J.L. (2017). Lower Extremity Feedback Control in Multiphase
Landings with Impact, 16th International Symposium on Computer Simulation in Biomechanics, Gold Coast,
AUS

6. Wagner, E.V, Flashner, H, & McNitt-Gray, J.L. (2017). Multijoint Impedance Control of Impulse during Impact
Phase of Foot-first Landings, 26
th
Congress of the International Society of Biomechanics, Brisbane, AUS

v

TABLE OF FIGURES
Figure 1. Phase definitions ............................................................................................................................ 3
Figure 2. Human anatomical modeling in da Vinci's art. .............................................................................. 4
Figure 3. Marey's chrono-photography ......................................................................................................... 4
Figure 4. Early ground reaction force time curves ........................................................................................ 5
Figure 5. McMahon effective leg-stiffness ................................................................................................... 7
Figure 6. Model complexity studies of decreasing degrees of freedom........................................................ 9
Figure 7. Gruber wobbling mass model ...................................................................................................... 10
Figure 8. Typical PID block diagram .......................................................................................................... 13
Figure 9. Impedance control ....................................................................................................................... 14
Figure 10. Impedance feedback control block diagram .............................................................................. 14
Figure 11. Energy shaping control .............................................................................................................. 15
Figure 12. Human body as a closed-loop control system ........................................................................... 18
Figure 13. Impedance control in humans .................................................................................................... 18
Figure 14. Energy shaping control in humans ............................................................................................ 19
Figure 15. Experimental marker placements .............................................................................................. 25
Figure 16. Typical impact force time time curve ........................................................................................ 26
Figure 17. Digital ankle trajectory .............................................................................................................. 28
Figure 18. Frequency domain analysis ....................................................................................................... 29
Figure 19. Signal windowing ...................................................................................................................... 30
Figure 20. FFT ............................................................................................................................................ 31
Figure 21. Raw signal FFTs vs. Low band pass ......................................................................................... 32
Figure 22. Kinetic spectrograms ................................................................................................................. 33
Figure 23. Kinematic spectrograms ............................................................................................................ 34
Figure 24. Phase based low band pass filtering .......................................................................................... 35
Figure 25. Smoothing factor (p) cutoff method .......................................................................................... 36
Figure 26. Residual as a function of p ........................................................................................................ 37
Figure 27. Exemplary smoothed vertical marker acceleration .................................................................... 38
Figure 28. Phase weighting ......................................................................................................................... 39
Figure 29. Piecewise weighted cubic spline technique ............................................................................... 40
Figure 30. PWCS robustness ...................................................................................................................... 41
Figure 31. Ankle velocity as a function of sampling frequency ................................................................. 41
Figure 32. Planar motion assumption ......................................................................................................... 44
Figure 33. Inverse dynamics free body diagrams ....................................................................................... 49
Figure 34. Marker centroid tracking noise .................................................................................................. 50
Figure 35. Jackson cutoff frequency method .............................................................................................. 51
Figure 36. Low bandpass filter outcomes ................................................................................................... 52
Figure 37. Comparing filtered signal to original data ................................................................................. 53
Figure 38. Shortcomings of single bandpass filtering ................................................................................. 53
Figure 39. GRF spectrogram ....................................................................................................................... 54
Figure 40. Ankle trajectory spectrogram .................................................................................................... 55
vi

Figure 41. Filtering boundaries discontinuity ............................................................................................. 56
Figure 42. Constant weighting CSAPS filtering ......................................................................................... 57
Figure 43. CSAPS filtered trajectories ........................................................................................................ 57
Figure 44. PWCS method outcomes ........................................................................................................... 58
Figure 45. PWCS coefficients ..................................................................................................................... 59
Figure 46. Comparison of raw digitized ankle marker vs. PWCS trajectory .............................................. 59
Figure 47. Spectrogram filtering method comparison ................................................................................ 60
Figure 48. Marker location spectrogram comparison ................................................................................. 61
Figure 49. PWCS coefficient variation by marker location ........................................................................ 62
Figure 50. PWCS method marker location spectrograms ........................................................................... 63
Figure 51. Marker vertical acceleration by location ................................................................................... 64
Figure 52. Downsampling average trajectory error vs. frequency .............................................................. 65
Figure 53. Downsampling maximum trajectory error vs. frequency .......................................................... 66
Figure 54. Instantaneous trajectory error vs. frequency .............................................................................. 67
Figure 55. Undersampled PWCS method average trajectory errror ........................................................... 68
Figure 56. Undersampled PWCS method maximum trajectory error ......................................................... 68
Figure 57. Instantaneous undersampled PWCS trajectory error vs. frequency .......................................... 69
Figure 58. Undersampled PWCS trajectories ............................................................................................. 70
Figure 59. Parsed functional joint axis components ................................................................................... 71
Figure 60. Peak vertical COM position ...................................................................................................... 72
Figure 61. Kinematic filmstrip .................................................................................................................... 73
Figure 62. Initial joint angle equivalence .................................................................................................... 73
Figure 63. Initial torso orientation equivalence .......................................................................................... 74
Figure 64. Evidence of vertical ground reaction force regulation .............................................................. 75
Figure 65. Flight phase pull-up technique .................................................................................................. 76
Figure 66. Flight phase segment specific pull-up analysis ......................................................................... 77
Figure 67. Flight phase NJM control .......................................................................................................... 78
Figure 68. Impact phase body compression ................................................................................................ 79
Figure 69. Impact phase vertical momentum reduction .............................................................................. 79
Figure 70. Impact phase effective subject stiffness .................................................................................... 80
Figure 71. Impact phase NJM control ......................................................................................................... 81
Figure 72. Post-impact phase vertical equilibrium ..................................................................................... 82
Figure 73. Post-impact phase horizontal equilibrium ................................................................................. 83
Figure 74. Post-impact phase empirical vertical stability point .................................................................. 84
Figure 75. Post-impact phase joint coordination ........................................................................................ 85
Figure 76. Post-impact phase NJM control ................................................................................................. 86
Figure 77. SimMechanics simulation environment / plant definition ......................................................... 97
Figure 78. Open loop control simulated VGRFs ........................................................................................ 99
Figure 79. Linear fit technique to VGRF prediction optimization ............................................................ 100
Figure 80. Open loop control block diagram ............................................................................................ 102
Figure 81. Closed loop control necessity .................................................................................................. 103
Figure 82. Block diagram: flight-phase following control ........................................................................ 104
vii

Figure 83. Simulated flight phase joint angles .......................................................................................... 105
Figure 84. Impedance control definition ................................................................................................... 107
Figure 85. Block diagram of impact phase impedance control ................................................................. 117
Figure 86. Impedance control mechanical component approximations .................................................... 119
Figure 87. Energy shaping control description ......................................................................................... 122
Figure 88. Evidence of subject specific joint coordination ....................................................................... 128
Figure 89. Chen’s weight definitions vs. modified weight definitions ..................................................... 134
Figure 90. Empirical relationship between peak GRF and degree of collapse ......................................... 137
Figure 91. Open-loop control simulated VGRF results ............................................................................ 141
Figure 92. Flight phase simulated shank orientation sufficiency .............................................................. 142
Figure 93. Flight phase simulated joint angle time history sufficiency .................................................... 143
Figure 94. Flight phase simulated pull-up ankle velocity ......................................................................... 144
Figure 95. Impact phase simulated VGRF sufficiency ............................................................................. 145
Figure 96. Impact phase simulated vertical body collapse ....................................................................... 146
Figure 97. Post-impact phase simulated COM horizontal stabilization .................................................... 147
Figure 98. Post-impact phase simulation joint coordination sufficiency .................................................. 148
Figure 99. Simulated VGRF waveform sensitivity to GRF coefficients .................................................. 150
Figure 100. Simulated VGRF sensitivity to impedance control coefficients ............................................ 154
Figure 101. Simulated COM stability sensitivity to constraint weightingsnstraint for stability ............... 155
Figure 102. Simulated COM stability sensitivity through energy analysis............................................... 156
Figure 103. Miscued VGRF impedance control NJM compensation ....................................................... 157
Figure 104. Effort redistribution during impact phase NJM modifications .............................................. 158
Figure 105. Effect of effort distribution on predicted VGRFs .................................................................. 158
Figure 106. Joint limit range effect on predicted NJMs............................................................................ 159
Figure 107. Ratio of vertical momentum to stabilization control effect on constraint mechanism .......... 160
Figure 108. Controls in biomechanics ...................................................................................................... 171
Figure 109. Spatial resolution of the kinematic data................................................................................. 186
Figure 110. Kinematic filmstrip of segments during all three landing conditions .................................... 187

viii

TABLE OF CONTENTS
ABSTRACT ............................................................................................................................................................. III
PUBLICATIONS ...................................................................................................................................................... IV
TABLE OF FIGURES ................................................................................................................................................. V
CHAPTER 1 INTRODUCTION / BACKGROUND ........................................................................................................ 1
CHAPTER OVERVIEW .................................................................................................................................... 1
THE AREA OF STUDY ..................................................................................................................................... 1
PREVIOUS LITERATURE ................................................................................................................................. 3
1.3.1 State of the Art of Biomechanics.............................................................................................................. 3
1.3.2 Modeling Human Motion ......................................................................................................................... 5
1.3.3 Control of Mechanical Systems .............................................................................................................. 12
STATEMENT OF PURPOSE ........................................................................................................................... 16
RESEARCH QUESTIONS ............................................................................................................................... 19
1.5.1 Data Collection Investigations ............................................................................................................... 19
1.5.2 Experimental Control Theory Investigations .......................................................................................... 20
1.5.3 Modeling Control Theory Investigations ................................................................................................ 21
CHAPTER ORGANIZATION .......................................................................................................................... 22
CHAPTER 2 DROP LANDING EXPERIMENTS ......................................................................................................... 24
CHAPTER OVERVIEW .................................................................................................................................. 24
METHODS ................................................................................................................................................... 24
2.2.1 Equipment and Setup ............................................................................................................................. 24
2.2.2 Task ........................................................................................................................................................ 25
2.2.3 Data Collection: Force and Kinematics .................................................................................................. 25
2.2.4 Data Processing: Force and Kinematics ................................................................................................. 26
2.2.5 Filtering and a Novel Smoothing Method .............................................................................................. 27
2.2.6 Kinematic Data Analysis ........................................................................................................................ 42
2.2.7 Parsed 4-Element Functional Joint Axis ................................................................................................. 45
2.2.8 Kinetic Data Analysis / Inverse Dynamics .............................................................................................. 47
RESULTS ...................................................................................................................................................... 49
2.3.1 Marker Filtering ..................................................................................................................................... 50
2.3.2 Lower Sampling Rates ............................................................................................................................ 64
2.3.3 Parsed 4-Element Functional Joint Axis Sample Study Results .............................................................. 70
2.3.4 Initial Conditions .................................................................................................................................... 72
2.3.5 Vertical Ground Reaction Force Variation ............................................................................................. 74
2.3.6 Observed Flight Phase Control: Kinematics ........................................................................................... 76
2.3.7 Observed Flight Phase Control: Calculated Joint Moments ................................................................... 77
2.3.8 Observed Impact Phase Control: Kinematics ......................................................................................... 78
2.3.9 Observed Impact Phase Control: Calculated Joint Moments ................................................................. 80
2.3.10 Observed Post-Impact Phase Control: Kinematics ............................................................................. 82
2.3.11 Observed Post-Impact Phase Control: Calculated Joint Moments ..................................................... 85
DISCUSSION ................................................................................................................................................ 87
2.4.1 Frequency Content during Impact Landings .......................................................................................... 88
2.4.2 PWCS Filter Comparison ........................................................................................................................ 88
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2.4.3 PWCS Filtering Method .......................................................................................................................... 89
2.4.4 Reduced Sampling Rates ........................................................................................................................ 90
2.4.5 Parsed 4-Element Kinetically Contextualized Functional Joint Axis ....................................................... 90
2.4.6 Initial Conditions .................................................................................................................................... 91
2.4.7 Vertical Ground Reaction Force Variation ............................................................................................. 91
2.4.8 Observed Flight Phase Control ............................................................................................................... 92
2.4.9 Observed Impact Phase Control ............................................................................................................. 92
2.4.10 Observed Post-Impact Phase Control ................................................................................................ 93
CHAPTER SUMMARY .................................................................................................................................. 93
CHAPTER 3 CONTROLS BASED MODELING AND SIMULATION ............................................................................. 95
CHAPTER OVERVIEW .................................................................................................................................. 95
MODEL DESIGN .......................................................................................................................................... 95
3.2.1 Mechanical Model Anatomy .................................................................................................................. 95
3.2.2 Ground-Foot Interaction ........................................................................................................................ 98
NONLINEAR CONTROL THEORY APPLIED TO HUMAN BODY CONTROL .................................................... 101
3.3.1 Open Loop Control: Kinematics and Kinetics ....................................................................................... 101
3.3.2 Flight Phase: Following Control ........................................................................................................... 103
3.3.3 Impact Phase: Impedance Control ....................................................................................................... 105
3.3.4 Post-Impact Phase: Passivity Based Energy Shaping Control .............................................................. 120
MODEL: VALIDATION ................................................................................................................................ 135
3.4.1 Open Loop Control ............................................................................................................................... 135
3.4.2 Phase Criteria ....................................................................................................................................... 135
MODEL: SENSITIVITY ................................................................................................................................ 138
MODEL: APPLICATIONS ............................................................................................................................ 138
3.6.1 Miscued Control Objectives.................................................................................................................. 139
3.6.2 Modifying Relative Joint Weight Distributions..................................................................................... 139
3.6.3 Joint Flexibility...................................................................................................................................... 139
3.6.4 Characteristic Stabilization Force Stiffness Study ................................................................................ 139
RESULTS .................................................................................................................................................... 140
3.7.1 Open Loop Control ............................................................................................................................... 140
3.7.2 Phase Metrics ....................................................................................................................................... 141
3.7.3 Sensitivity Analysis ............................................................................................................................... 148
3.7.4 Model Application Results ................................................................................................................... 156
DISCUSSION .............................................................................................................................................. 161
3.8.1 Model Limitations ................................................................................................................................ 161
3.8.2 Research Questions .............................................................................................................................. 163
CHAPTER SUMMARY ................................................................................................................................ 168
CHAPTER 4 CONCLUSIONS ................................................................................................................................ 170
DROP LANDING EXPERIMENT ................................................................................................................... 170
DROP LANDING SIMULATION ................................................................................................................... 170
FUTURE WORK ......................................................................................................................................... 171
REFERENCES ....................................................................................................................................................... 174
APPENDIX A FORWARD KINEMATIC END-EFFECTOR DESCRIPTIONS.............................................................. 177
x

APPENDIX B FORCE PLATE RESOLUTION ANALYSIS ....................................................................................... 185
APPENDIX C EVIDENCE OF PIXEL SPATIAL RESOLUTION DISCRETIZATION ..................................................... 186
APPENDIX D FILMSTRIP KINEMATICS FOR ALL TRIAL CONDITIONS ................................................................ 187

1

CHAPTER 1

INTRODUCTION / BACKGROUND

CHAPTER OVERVIEW

In this chapter an introduction to role of modeling in biomechanics is given. Previous literature on modeling landings
is discussed and the motivation of this research is outlined. Improvements to current modeling practices are proposed
with respect to previous research. Finally, an overview of this dissertation presented, providing a structural outline of
the work as a whole.

THE AREA OF STUDY

Throughout history, society has gained valuable knowledge by studying nature. In fact, paradigm shifting inventions
and innovations often result from discoveries of natural phenomena. The J-shaped prosthetic limb called “Flex-Foot
Cheetah”, for example, is designed to mimic the functionality of a cat's hind legs and has successfully helped
Paralympians regain (some even suggest enhance) their ability to run[3]. In an act of mechanical introspection, human
beings have long sought to create accurate dynamic models of themselves in order to optimize, and even supersede
their limited dynamic capacities.

Dynamic models are designed to provide predictive insight into the quantifiable characteristic differences in a
mechanical system’s behaviors when exposed to different stimuli, both internal and external. As such, the utility of a
model is not only defined by its predictive capacities, but also by the clarity of the insights it provides. In other words,
if a model is too complex, the information gained by executing it is diluted by an overabundance of sources for
observable variation. Thus, when attempting to model the human body, one starts with the simplest model first,
increasing complexity only when necessary or intrinsically relevant[4].

Researchers began studying landing impact mechanics of the human body as a point mass, and have since increased
the complexity to include several rigid bodies connected by ideal-hinge joints. With impact, it is similarly crucial that
the contact surface model provides accurate amounts of force when an object infringes upon its boundary. This is a
complex interaction involving large fluctuations in force in a short period of time. Thus, for simplicity in landing
tasks, the ground is approximated as a viscoelastic element. As models of the body increase in complexity, they are
better able to predict features like ground reaction forces (GRFs) or joint kinematics. However, certain features are
even more complex and nonlinear by nature, such as the coordinated use of muscles across joints resulting in net joint
moments (NJMs). While one can calculate the resultant NJM from modeled GRFs and segment kinematics using
2

inverse dynamics, these calculations have proven difficult to verify since kinematic tracking data often suffers from
oscillations of topical soft tissue. Even with clean kinematic data, experimental control validation would require
simulating the system with derived NJMs in an open-loop control architecture. Without feedback and any
corresponding stability analysis, a nonlinear system such as the human body very rarely achieves any type of active
stability. Of course, this also means that forward dynamic simulations driven by these calculated NJMs generate
similar uncertainty in predicting the corresponding measured GRFs.

The sources of these inaccuracies have been hypothesized to come from a broad range of experimental errors and
modeling assumptions. To account for these differences, some researchers have included wobbling-masses on each
model segment to simulate the loose tissue oscillation over rigid bone segments. While these additions serve to better
align the model calculated and experimentally measured GRFs, the significance of these changes becomes lost in the
number of free parameters included. Similarly, as the number of trial specific model parameters increases, the less
broadly the model’s results can be applied, rendering the utility of the predictive model trivial. Instead of increasing
the complexity of this passive model of masses, springs, and dampers to better match the measured GRFs, one
hypothesizes that the inclusion of a feedback based multijoint control will account for these differences.

Even at a young age, human beings attain skills such as balance and hand-eye coordination, effectively performing
system identification and designing the control effort for their muscles. This ability to stand and walk on two legs is
an example of the human system using feedback signals to accomplish the dynamically unnatural act of balancing a
center of mass over the base of support, akin to an inverted pendulum. This everyday feat demonstrates how careful
coordinated control of the system’s muscles, and consequently joints, is an inherent part of the human condition. With
this in mind, when attempting to study the human body, one must seek to understand the entity both as a plant,
comprised of segments and joints with natural dynamics, and in terms of its hierarchical controls-based feedback loop.

The main goal of this study is to develop an experimentally-validated, dynamic, multilink model of the human body
for the purpose of predicting the kinematic and dynamic effects of landing tasks with varying secondary objectives,
using only rigid-links, idealized hinge joints, and control theory based NJM coordination. The proposed control laws
are specific to each of the three phases of landing, flight phase, impact phase, and post-impact phase. Flight phase is
defined as the time before ground contact, impact phase ranges from contact until double the time from contact to the
peak ground reaction force, and post-impact is the time from impact phase until the total body momentum is brought
to zero, Figure 1.

3

Figure 1. The impact phase is defined as extending from contact until twice the time to peak vertical ground reaction
force (VGRF) is reached. Impact is often approximated as by a triangle which illustrates the why the phase is
understood to last twice the time to peak, with a symmetric triangle having a base of that duration precisely. Notice,
the impact phase impulse is also defined in this figure, shown here as the area shaded in blue.

PREVIOUS LITERATURE

Biomechanics is an engineering based subcategory of one of the original sciences, kinesiology which date back to
the ancient Greek traditions of studying motion. The word kinesiology itself derives its name from the Greek
'kinesis' meaning motion or movement, and 'logia' meaning study of.

1.3.1 State of the Art of Biomechanics

Regarded as one of the classical fathers of biomechanics, Aristotle first began the documentation of this study with
his book, "De Motu Animalium" (On the Movement of Animals). Though this work was largely philosophical in
nature, it led to the academic movement of studying how the world relates to willful biological motion[5]. A student
of both the arts and mechanics, modeling the human body is famously associated with the great Renaissance painter
and engineer of the 15th century, Leonardo da Vinci. As this was before the invention of photography, his artistic
abilities assisted his work mapping the muscles and their lines of action, and they laid the foundation for the scientific,
methodological study of biology within a mechanical framework, Figure 2.

4

Figure 2. Human anatomical modeling in da Vinci's art.

Until the industrial revolution, documentation of the human body was hampered by an inability to portray it without
artistic bias. In the late 1800's when French scientist, Étienne-Jules Marey developed the field of chrono-photography,
a photographic technique which captures movement by rapid succession of exposures, he paved the way for objective
documentation of biological motion, Figure 3. In a similar application, Eadweard Muybridge used a series of cameras
connected by tripwires to analyze equine gait. This work would later lead to the scientific use of cinematography for
tracking biological motion, a tool which is still being used by biomechanicians to track motion today.

Figure 3. Beginnings of biological kinematics are visible in Marey's landing crane chrono-photography, which
depicts subject position and orientation as a function of time.

Around the time of the First World War, Marey and some of his students collaborated to develop a pneumatic
mechanism to indicate ground reaction force, Figure 4. This work, while inventive, did not provide the necessary
quantifiable results for mechanical analysis of ground reaction forces and inverse dynamics. Finally, in the early 1950's
Cunningham and Brown were able to produce measurable force curves as a function of time, with the advent of the
5

strain-gauge[6]. These force-time curves, when coupled with kinematic information, provide the necessary
information to analyze the complex nonlinear interactions of multisegment human body.

Figure 4. With Carlet and Marey's pneumatic device built into the shoe[6] pressure vs. time curve could be taken
for the first time. Notice, even at these early depictions, one can clearly observes the variation in ground reaction
forces, here showing typical dual peak ground reaction force time curve of heel toe contact.

With improvements in wireless technology, some more modern inventions are now taking a role in the study of
biomechanics. Depending on the facet of interest, some scientists have employed tools such as electromyography
(EMG) to measure muscle activation as indicated by the electrical signals used by an active muscle[7]. Others have
begun to employ inertial measurement units (IMUs) to directly measure the orientation and acceleration of a body
segment[8]. While these methods are more direct in their measurement of muscle activation and kinematics,
respectively, they also have their drawbacks, such as any electrical interference or magnetic field distortion by ferrous
metals or other electrical sources. With these basic technologies in place, researchers are able to piece together both
sides of the equations of motion, providing a dynamic picture of the way human beings move. This macroscopic
perspective of the biological mechanisms which produce human motion is a branch of biomechanics known as
musculoskeletal systems analysis. Within this field, the research presented in the following chapters provides new
insights into the coordinated use of joint actuators to alter the effective dynamics of the body, musculoskeletal controls.

1.3.2 Modeling Human Motion

Many studies of different types, scales, and complexity have been conducted to obtain new insights into the
fundamental aspects of successful human motion. The following is a small survey of quintessential applications of the
modeling techniques employed throughout biomechanics, from the most basic combined point mass model to the more
recent wobbling-mass multilink models, and even multilink models involving control. It is focused on the goals,
methods, results, and shortcomings of each study in an attempt to provide a narrative explanation of the foundations
and current state of biomechanical modeling.

6

1.3.2.1 Understanding Mass Distribution: Point Mass Approximation
Due to the complex nature of human anatomy and the variance between each subject, current mass distribution
methods rely on one of two types of approximations: the geometric approach, where solids (usually conical frusta) of
uniform density shaped to match the subject[9]; or the proportional approach, where a subject's mass and height are
used to create a scaled version of the typical human[10]. In order to determine the effect of this mass distribution
assumption on the trajectory of the center of mass (COM), Kingma et al. used a point mass model of the human body
to calculate a subject specific optimal torso COM location[11], as compared to the proportional method suggested by
Plagenhoef[12].

In this study, Kingma had 5 separate subjects stand on a set of force plates holding their torsos at a 3 different angles
of flexion. By approximating the entire body as a point mass located over the measured center of pressure (COP), the
horizontal position of the body COM was determined as a function of the torso angle, providing an optimized estimate
of each subject's actual torso COM. Finally, the angular moment about the COM caused by the GRFs during a weight
lifting task was compared between a model using the optimized torso COM and one using the proportions based torso
COM. These results were compared to the rate of change of the angular momentum of the body, as measured by
kinematic tracking.

The calculated angular moments better matched the measured rate of change of the angular momentum of the model
with the optimized torso COM in all but one case. This proves the error associated with mass segment distribution
will have a measurable effect on the calculated NJMs. However, an improvement in the overall system COM location
does not guarantee that each segment's optimized COM would produce significantly more accurate NJM results. This
study provides little insight into the degree of improvement relative to the magnitude of measured GRFs, making the
exact margin of improvement unclear. Since this process is cumbersome to complete for each segment of each subject,
the effort to improvement ratio does not seem reasonable. However, it does remind researchers that there is a certain
level of uncertainty built into the model that must be taken into consideration when making claims based upon their
models.

1.3.2.2 Effective Leg-Stiffness: Mass-Spring
The mass-spring model is common for use in studying the lower extremities, particularly in running gait studies.
Studies like those by McMahon and Cheng[13] and Farley and González[14], focus on effective lower extremity
stiffness in different environmental or task driven conditions. McMahon and Cheng's study on the correlation between
segment stiffness and running speed, assumes motion is limited to the sagittal perspective, so the mathematical model
can be reduced to a 2D mass-spring system, in one of the early impact studies in running gaits. This study set out to
determine the quantitative relationships between leg-spring stiffness, gravity, and forward speed in running in a non-
dimensional form so that its results are widely applicable.

7

Figure 5. Model of the early impact model developed by McMahon to examine effective leg-stiffness during
running gait[13].

By assuming a linear relationship between the vertical displacement of the COM and the vertical GRF (as captured
by the acceleration of the COM), the research team was able to create an artificial stride pattern of the COM and a
pivoting spring-leg. The initial conditions (horizontal COM velocity and vertical COM velocity, and leg angle) were
used with a hypothesized leg stiffness. This model was integrated forward until the mass returned to its initial vertical
position. The final leg angle was used to characterize whether the leg stiffness was too high or low (as it should mirror
that of the initial angle for a smooth running gait). Finally, the predicted vertical displacement and accelerations were
compared to experimental results for several different species.

The non-dimensional model is a plausible representation of biological running gait based upon the agreement between
the simulated and experimental results. However, the model does not capture the early rise-fall-rise pattern in the
vertical GRF during impact. In addition, it assumes symmetry of gait landing and take-off which has been shown not
to hold true for all subjects, particularly humans.

1.3.2.3 Mass Trajectory to NJMs: Variable Length Pendulum
The kip is a fundamental component of any gymnast's parallel bars routine, in which the gymnast seeks to bring their
COM from below the parallel bar to above it by careful exchange of kinetic and potential energy. Due to the
symmetrical nature of this task, most studies simplify the model by restricting motion to 2D in the sagittal plane (side
view). In a study by Nakawaki et al, researchers attempted to identify key components of a successful kip in order to
provide a coaching tool for optimizing kip technique[15]. The team modeled this task by using a variable length
pendulum to simulate the COM trajectory. Nakawaki and team collected kinematic data of several expert and novice
gymnasts performing a kip. From these data sets, the team attempted to determine the optimal COM trajectory for
minimizing both external torque at the fulcrum, and the weighted muscle burden. This trajectory is used in conjunction
with a 3-link model (arms, torso, and legs) which follows expert gymnast segment motion with PD-control to optimize
kip task segment trajectories.

8

The resulting simulation shows there are three significant components to a successful kip: arching of the body during
forward swing, a quick pike motion in between forward and backward swing, and grip readjustment as the body swings
backwards. While matching expert gymnast methodologies closely, the results are heavily determined by cost
functions and multiple layers of optimization. It is important to note, that each optimization comes with its own set of
assumptions and should be experimentally verified.

1.3.2.4 Segmental Contribution to Vertical GRF in Running Gait Impact
Multilink models, are typically composed of a series of rigid links connected at their endpoints by ideal hinge joints,
a modeling assumption made to simplify biological joints which often contain complex sliding modes, or even coupled
rotation modes. In their study of segmental contribution to GRFs during running, Bobbert and Nigg proposed a 7-link,
rigid segment (2 feet, 2 shanks, 2 thighs, and single link (HAT) to represent the head, torso, and arms) model[16] with
mass distribution determined by the proportional method of Clauser[17]. The increase of running related injuries
inspired this them to focus on some factors which may contribute to these injuries. Bobbert and Nigg sought to
examine the contribution of each segment to the overall vertical GRF-time curve during a running stride. More
specifically, they wanted to determine if the sum of this segmental contribution was adequate for mechanical analysis
during impact.

Each segment position and orientation was tracked using 3D marker tracking techniques, which involve palpating
specific bony points on the body and attaching retro-reflective spheres to those points by Velcro or athletic tape. These
position measures are particularly important, as the double derivative of this data is what is used to determine segment
accelerations for the right hand side of Newton's second law. While 3D data was collected, the study, and hence model,
focused almost exclusively on the vertical motion. This type of study often runs into issues of marker oscillation due
to the high frequency force content. Their solution to this problem was to attach the markers to a hinged set of wooden
dowels that could be attached to the segments, reducing the degrees of freedom of the markers. The GRFs were
measured using force plates imbedded in the path of progression. Finally, both sides of Newton's second law were
compared for the total body during the contact phase of running (i.e. from the time of foot contact with the force plate
to the time of foot departure).

This study was successful in correlating the vertical GRFs with segmental contribution, however, the authors noted
“[despite] the precautions taken in this study, sinusoidal accelerations of markers relative to the mass centers of
segments still occurred after impact.” The study continued by reiterating the importance of a good signal to noise ratio,
such that the information of interest is retained. This was one of the first studies (along with Denoth[18] and
Gruber[19]) to clearly state these problems caused by loose tissue and adhesion of markers to the skin (which is not
rigidly attached to the bones). While the scope of this study was limited to the heel-toe impact phase dynamics of
running gait, the sensitivity analysis performed helped to determine that no more than one camera was needed to track
this type of sagittal plane motion.

9

1.3.2.5 Gymnast Layout: Flight phase
Many of the studies described thus far have used models as a tool to determine the fundamental mechanisms driving
human dynamics. A fundamental step in this process is verifying the accuracy of the model[4] based upon its
adherence to fundamental laws of motion. In the following study by Requejo[20], experimental data from an Olympic
gymnast during flight phase was used to create multilink rigid body mathematical models with 3- to 8-segments. The
results of the 4-link model were compared against a multilink rigid body mathematical model created from the
experimental data of a wooden 4-link physical model during flight phase in order to determine:
1. the contribution of error from the process of developing an experimentally verified model
2. the necessary level of model complexity to best satisfy the conservation of linear and angular momentum
laws, a expressed through the number of segments modeled
effectively measuring the accuracy of this process, and hence, the quality of the results obtained from the resulting
model. Since the utility of any model is defined by how closely its results mirror reality, one must have some
methodology of obtaining such a measure.

The first component of this study involved creating a wooden ¼ scale 4-link hinged model (WM) which was used to
provide an idealized experimental data set, where the rigid body assumptions are known to apply. Using the typical
optical kinematic data collection process, video was taken of the WM during the flight phase of an artificial layout
(aerial dismount) maneuver. This data was filtered using the Woltring method of 5th order splines at uniform knot
locations to provide smoothed second order derivatives for analysis[21]. A 2D mathematical model consisting of 4-
rigid links connected by hinge joints was created from the experimental WM according to the sagittal plane data
collected. Similarly, kinematic data for an Olympic athlete performing a layout was collected and processed. From
these experimental kinematics, a 2D mathematical model composed of the same 4-rigid links from the WM was
created. The adherence of each of model to the Laws of Conservation of Momentum, during flight phase, was used as
a performance index to suggest how accurate the model performed with respect to reality. The second stage of this
study, the kinematic data collected for an Olympic athlete performing a layout was used to create 2D mathematical
models composed of a varying number of rigid links as shown in Figure 6. The adherence of each of model to the
Laws of Conservation of Momentum, during flight phase, was used as a performance index to suggest how accurate
the model performed with respect to reality.

Figure 6. Models of decreasing degrees of freedom (ranging from 8 segments to 3 segments) are used to determine
which level of complexity best adheres to conservation of momentum laws[20].

10

The models composed of 5 links and 7 links were the most successful at reducing deviation from the angular and
linear momenta calculated at the beginning of the flight phase. Thus, the 5-link model is the best approximation for
this task because models should both retain the essential dynamics of a system and simplify the complex behavior as
much as possible. This study was successful in determining an unbiased methodology for examining the integrity of
a mathematical model. However, it is unclear whether or not the segment choice had influence on the necessary model
complexity, rather than just the amount of segments. For example, the model was first increased in complexity by
adding a shank segment. One is unable to determine the quality of the kinematic data obtained from digitization of the
shank segment, as compared to that of the head segment. If the complexity was increased in a different order, it may
have led to different results. Similarly, the filtering method selected is limited to one phase of kinematic data (flight
phase), assuming the same frequency content throughout the time range of interest which allows for more simplistic
filtering methods. Thus, if one is interested in the model quality across different phases of landing, the filtering
technique would need to hold for data of varying complexity.

1.3.2.6 Wobbling Mass Models
As mentioned in the studies by Bobbert, Nigg, and Pain[16], [22], studies often suffer from an inability to accurately
recreate the GRF from the predicted NJMs, with models typically composed of rigid segments connected by ideal
hinges. In 1987, Gruber and her team addressed this discrepancy by creating a wobbling mass model, Figure 7, to
analyze the potential modeling error resulting from oversimplification of the human body dynamics[23].

Figure 7. Wobbling mass model created by Gruber[19] in order to compensate for the differences between measured
GRFs and model predicted GRFs.

Gruber sought to quantify the error caused by the rigid mass assumption in multilink models, which ignore the loose
tissue oscillation captured in high speed data collections, as a warning against oversimplification of biological
11

dynamics. Gruber used predefined NJM-time curves to drive the wobbling mass model in a forward dynamic heel-
first drop-landing simulation, obtaining idealized “experimental” GRF-time curves and kinematic data. The kinematic
data is referred to as idealized due to the lack of digitization or data collection error, and it was assumed to represent
the dynamics and kinematics of a perfect data collection. Next, these measurements (GRF and rigid segment
kinematics) were used with inverse dynamics in order to calculate the NJM at each joint in the equivalent rigid body
3-link model. The differences in NJM-time curves between those that were input for the original forward simulation
and those that were calculated from these rigid body inverse dynamics calculations were identified as the modeling
error which would result if the wobbling mass was not considered.

As expected, the inverse dynamic NJMs show significant differences when using a rigid body model vs. a wobbling
mass model. Gruber points to large magnitude (5 times that of the wobbling mass model) oscillation in the resulting
rigid body NJMs as unrealistic, compared to the slowly increasing NJM of the wobbling mass model. Finally, Gruber
warns against neglecting wobbling mass as a potential oversimplification of the human body model, referencing papers
which the rigid body model assumption produced similar oscillatory results.

A significant shortcoming of this paper is the circular logic used to prove the necessity of the wobbling mass model
components. Using a wobbling mass forward dynamic simulation to calculate inverse dynamic solutions for a rigid
body model guarantees differences in the NJMs. In addition, this study only analyzes heel-first landings, which may
occur in running and gymnast landings, but it ignores an important aspect of impact force reduction in drop-landings,
NJM at the ankle[24]. In 2004, Pain and his team performed a sensitivity analysis of the wobbling mass model and
determined it shows high sensitivity to torque timing and heel stiffness. Pain also noted that the mass distribution
between rigid and soft tissue components was disproportionate to those reported in previous studies[22].

The foot-surface interaction model is an integral component to achieving accurate GRF from simulations. In fact, this
aspect of the modeling process has proven so difficult to model correctly that some research teams have employed
wobbling mass models[22], [23] to accounting for differences between predicted GRFs and the smaller experimentally
measured GRFs. Some studies have combined the complicated dynamics of the foot into the ground reaction force
model, to simplify the modeling process[13], [22], [23], while others have separated the two, using viscoelastic
components to represent the floor and hinged rigid-link segment for the foot[24]. The latter provides a more realistic
division of dynamics and will be discussed in following chapters of this dissertation. The mathematical form of this
nonlinear viscoelastic floor model is taken from the Gruber model, but the coefficients will be calculated by calibration
methods described in Section 3.2.2 Ground-Foot Interaction.

1.3.2.7 More Complex Models
One would like to take a moment to acknowledge the existence of several different models which include control
feedback as an integral part. Hicks states that a “search on Google Scholar for biomechanical or musculoskeletal
modeling or simulation produced fewer than 200 papers in 1990, about 500 papers in 2000, and nearly 2000 papers in
12

2013”[4]. In short, as technology lowers the cost of entry to the field of biomechanical simulation and computers
become more capable, the number of models begins to increase exponentially. These models can even be optimized
to multiple tasks, as parallel computing has reduced the turn-around time for iterative simulations. However, each
model tends to focus on a unique aspect of the human body, and they add complexity as necessary. The research
presented in this work, however, seeks to replicate the evidence of macroscopic total body control experimental results
with as simplistic a model as possible. Thus, while many of these studies have introduced control into their models
by optimizing the activation patterns of specific muscle sets in OpenSIM[25], one focuses instead on explaining the
human body control in terms of macroscale control objectives and outcomes. Studies which have focused on
macroscale control objectives like flight phase work by Requejo[20] or kip control by Nakawaki[15] are usually
limited in phase. Nakawaki limited the study to the time when the gymnast is in contact with a high bar, while Requejo
studied the time when the gymnast is in flight. Thus, both omit the effect of these continuous NJM-time curves on and
across the phases.

1.3.3 Control of Mechanical Systems

Previous research in the field of biomechanics has sought to characterize various aspects of human body landing
dynamics as a function of a set number of measurable parameters, showing how the mass distribution [11], the
kinematics of specific segments [2], [8], the vertical heel velocity just before impact[26], lower extremity-
stiffness[13], wobbling mass[19], affects the GRF waveform during impact. While these studies offer valuable insight
into mechanisms that can be used to describe the system via modeling, most have understated the responsive nature
of the human body opting to focus instead on the resulting dynamics of these systems with a fixed set of mechanical
parameters. Those models including controls in their dynamics generally look at a particular phase of a task, flight
phase [20], [26], [27] or impact/post-impact (stride) phase, [22], [23]. The modeling process has not yet adequately
captured what, in the opinion of a controls engineer, may be the most significant and influential aspect of human
dynamics, the multiphase musculoskeletal control laws.

By now it is well known that the central nervous system controls the human body via electrochemical signals sent to
the system's actuators, the muscles. In fact, many biomechanics studies even measure this electrical signal as a method
of indicating muscle activation. Thus, for models of the human body to portray the system with any accuracy, they
must include this active aspect. Some have begun to show the evidence of control, identifying the manipulation of
CM position with respect to the feet[26], segment orientation[28], and relative segment velocities[26] during the flight
phase. There have been attempts to identify the control strategies used in this preparatory manner, such as the use of
optimal control[29], or a type of learning inspired control called adaptive control[30]. But these do not attempt to
apply control across multiple phases of motion, as in the proposed land-and-stop model. The effective leg-stiffness
during contact phase (impact/post-impact) proposed by McMahon[13], demonstrates that control spans more than just
the flight phases and must be considered continuously throughout the motion. In order to illuminate the phase specific
13

stable control laws to be applied in the proposed model of the human body, the following background in control theory
is introduced:

1. Position-velocity based control used for segment position and orientation tracking during flight phase
2. Impedance control used to reduce momentum during impact phase
3. Passivity-based energy shaping control to dissipate energy while steering the COM during post-impact phase

At the most basic level, these control algorithms stem from the field of classical controls. Classical control techniques,
as suggested by their name, have been well documented throughout the history of the field of controls. The basic
controllers take the form of proportional (P), derivative (D), and integration (I) control, which each refer to gains
applied to an error signal, its derivative, and its integral, respectively, Figure 8 and Eq. 1. The traditional control
notation is used here, where 𝑞 𝑟 represents the generalized reference trajectory, 𝑞 is the generalized system coordinates,
and 𝐾 𝑃 , 𝐾 𝐷 , and 𝐾 𝐼 are the control gains.

𝑈 𝑃𝐼𝐷 = 𝐾 𝑃 ( 𝑞 𝑟 ( 𝑡 )− 𝑞 ( 𝑡 ) )+ 𝐾 𝐷 ( 𝑞 ̇ 𝑟 ( 𝑡 )− 𝑞 ̇( 𝑡 ) )+ 𝐾 𝐼 ∫( 𝑞 𝑟 ( 𝑡 )− 𝑞 ( 𝑡 ) )𝑑𝑡 ( 1 )

Figure 8. Typical block diagram showing feedback control of a system with PID control. Note, the Laplacian
transform of the PID controller shows how error signals, e, are converted into actuator inputs, u, by summing scaled
forms of the signal, its integral (1/s), and its derivative (s).

When it comes to its application to humans or robotics, the control objective describes reaching a desired segment
configuration and orientation by following some predetermined or optimized trajectory. Biomechanics research has
shown that preparation for the expected GRF begins even before contact, measuring muscle activation during flight
phase[7]. So this simple PID block diagram could describe the ability to track real-time segment configuration with
respect to a desired set of segment trajectories.

During an impact, many biomechanics papers describe measuring “effective segment stiffness” or damping from the
trajectory of the COM with respect to the point of contact[13], [14]. A parallel in the field of controls is the use of an
effective resistance to changes in velocity is sometimes referred to as impedance control. By employing forward
kinematics and inverse dynamics, the controller determines joint specific torques required to give the end-effector
(contact point, in this case the foot toe/heel) the effective dynamic characteristics of a desired mass-spring-damper
system, Figure 9.
Plant:
G
P
(s)
Controller:
𝐾
+ 𝐾
+
𝐾

+
-
Sensor:
H(s)
q
r
(s) q(s)
e u
Actuator:
G
A
(s)
F,τ
14

Figure 9. Impedance control takes a multilink system with complex dynamics and designs the inputs to each
actuator to achieve dynamic environment interactions equivalent to those of a desired mass-spring-damper system.

As Hogan describes it, this technique of control is used when the stiffness of the impending environmental interaction
is not known explicitly[31]. Thus, rather than only defining a position or velocity for the controller to follow, which
may result in actuator saturation and system instability, the control guides the system trajectory according to
interaction forces. In this way, one could apply exactly enough pressure to flexible surfaces to generate the desired
reaction forces, no matter the stiffness of the actual surface in question.

Figure 10. The impedance feedback control block diagram demonstrates how force feedback serves to adjust the
reference trajectory, such that external forces mirror those of the designed impedance. Note, K F describes a
transformation of the joint space measurements into expected environmental reaction forces. Similarly, K e describes
the transformation of these forces into joint space corrections which define a new reference trajectory to achieve
the desired level of force. Finally, the forces are also compensated for in the controller, using the Jacobian transpose,
J
T
, which uses forward dynamics to predict how the forces would affect the joint torques.

DeWit[32] established the existence of a solution for this type of control law, and Hogan showed the foundations of
its application. Since then, impedance control has become an important part of robotic control for safe interaction for
with humans. In his work, Hogan gives the form of this control law for position-force control[31] and reproduced here
for reference, Eq. 3, where 𝒒
is a virtual trajectory which the user desires their end-effector to follow, 𝒒 is the
generalized system coordinates, and 𝐾 𝑃 ,𝐾 𝐷 are the impedance stiffness and damping coefficients, respectively. Note,
the subscript, i, indicates that more than one line of impedance action may be defined for a single system.
K
F
K
e
Plant:
G(s)
Control Law:
K
p
+ K
d
s + K
i
/s
+
-
Sensor:
H(s)
q
e τ
+
-
q
d
F
e
+
-
J
T
q
d
'
15

𝐼 ( 𝒒 ) 𝒒 ̈ + 𝐶 ( 𝒒 ,𝒒 ̇)+ 𝐷 ( 𝒒 ̇)= 𝝉 𝑪 + 𝝉 𝒆 ( 2 )

𝜏 𝑐
= 𝐾
( 𝒒
− |𝒒
|)+ K
di
( 𝒒 ̇
− 𝒒 ̇
) ( 3 )

After impact, the mechanical objective becomes the stabilization and further kinetic energy reduction of the system,
bringing the total body momentum to zero. The control law for this phase must dissipate energy while maintaining the
COM over the base of support, a framework which lends itself well to energy-based descriptions of control. Grounded
in Lagrangian dynamics, energy shaping control is an algorithm suited to the control of nonlinear systems which takes
advantage of the inherent energy exchange within a system. It reduces the kinetic energy of the system, while placing
an end-effector attractor in the form of a potential energy equilibrium position. In a review of the energy shaping
control methodology, Koditschek explains that rather than relying on specific tracking based control algorithms for
which one has to define a specific trajectory from initial position to the goal position and subsequently define an
explicit tracking controller to pair with it, the method of energy shaping uses a cost function in the form of a potential
energy to define the goal position as an (ideally global) equilibrium point[33]. Functionally, this type of control is
identical to classical PD control with the exception that it controls the end-effector in the Cartesian task space directly
leaving the specific joint angle control to be handled internally by the Jacobian pseudo-inverse, J
#
, Figure 11.

Figure 11. Energy shaping control appears similar to classical PID control. Where PID trajectory following control
presents a set of joint angles time-series to follow, energy shaping merely defines a desired equilibrium point in
Cartesian task space and applies a virtual spring between that point and an end-effector in the form of proportional
control, K p. Similarly, the total kinetic energy of the system is dissipated by the damping element, K d, to ensure
that the system eventually comes to rest. Notice, in this version of the block diagram the Jacobian pseudo-inverse
is used to convert between Cartesian space control efforts and joint space plant inputs, which is only made possible
by the definition of the control law in terms of local task space coordinates. As shown later in the work, this
relationship is too narrow for the purposes of this research, and thus, it is modified to use the Jacobian transpose
(J
T
) instead of the pseudo-inverse (J
#
).

Applying this control coupled with an energy dissipation function, a grounded system’s end-effector will
asymptotically approach a desired stable equilibrium point, provided the trajectory does not include any zero velocity
state points, as LaSalle's theorem indicates. In his 1981 study[34], Arimoto and his team became the first to apply
these concepts of energy shaping to the field of robotics. Since that time, this concept has been applied to many
nonlinear robotic systems, including inverted pendulums in the form of controlled Lagrangians[35].

Deriving the equations motion via energy analysis means using the Lagrangian formulation, Eqs. 4-7. In terms of
generalized coordinates, 𝒒 , the first equation describes the natural kinetic energy of the system, 𝑇 , while the second
describes its natural potential energy, 𝑉 . The traditional notation is used, where 𝐿 represents the Lagrangian, 𝑴 is the
Plant:
G(s)
Control Law:
K
p
+ K
d
s
Sensor:
H(s)
x e F
+
-
x
d
J
#
τ
16

positive definite mass matrix, and 𝒈 is the vectorized form of gravity. On the other side of the equation, 𝑭 𝑐 represents
the control efforts, while 𝑭 𝑒 represents the external forces and torques. By using a potential function in the form of a
quadratic, positive-definite Hooke's law and a dissipative energy function as the controlled inputs the system, the
dynamics are altered such that the natural equilibrium position of the COM moves to a new supported position 𝒒

over the base of support (feet).

𝑇 =
1
2
𝒒 ̇ 𝑇 𝑴 ( 𝒒 ) 𝒒 ̇
( 4 )

𝑉 = 𝑴 ( 𝒒 ) 𝒈𝒉 ( 𝒒 ) ( 5 )

𝐿 = 𝑇 − 𝑉 ( 6 )

𝑑 𝑑𝑡
(
𝜕𝐿
𝜕 𝒒 ̇ )−
𝜕𝐿
𝜕 𝒒 = 𝑭 𝑐 + 𝑭 𝑒 ( 7 )

A generalized form of the potential function input shows how the system energy is dissipated by the damping element,
while the potential energy element sets a new task-space Cartesian minimum at 𝒑
, Eq. 8.

𝑭 𝐶 = −
1
2
( 𝒑
− 𝒑 )
𝑇 𝐾 𝑃 ( 𝒑
− 𝒑 )− 𝐾 𝐷 ( −𝒑 ̇)
( 8 )

In the final form for each of these control definitions, a forward kinematic matrix and its Jacobian are used to convert
between absolute Cartesian coordinates and the joint space. All three of these control methodologies are used through
phase-based control switching to result in a stable land-and-stop NJM coordination technique capable of manipulating
the GRF waveform according to secondary task objectives.

STATEMENT OF PURPOSE

Studies of the human body are made difficult by the responsiveness of the system to environmental stimuli. Whether
it be reactionary (pulling away from a hot surface) or intentional (reducing hand grip forces to catch an egg), human
beings are constantly adapting their behaviors to the mechanical requirements of their current objective. As there can
be no dexterous, intentional movement without sensory feedback, these inseparable behavior patterns must be
considered when attempting to capture the fundamental elements of human dynamics in the context of a task. With
this in mind, the research presented in this work investigates the sufficiency of an experimentally validated 2D forward
dynamic model of the human body capable predicting GRF-waveforms as a function of landing task objectives using
only rigid-body segments, idealized joints, and feedback-based, coordinated NJM control to capture the fundamental
control-based mechanisms which define the behavior of the human body during the three phases of landings: flight-
phase, impact-phase, and post-impact phase.
17

The Gruber wobbling mass impact study suggests a large part of these waveforms are the product of the loose body
tissue which oscillates over rigid bony segments during impact[19]. However, this research project proposes a
simplified model by combining the wobbling masses back into the rigid segments, reducing the number of
complexities. Instead, differences between modeled GRFs and those measured experimentally are hypothesized to be
better explained through active NJM coordination. While Gruber's model does include open-loop joint torques, they
are activated 5ms after impact which contradicts studies which show muscle activation measured prior to impact[16].
In another deviation from the Gruber study, the model presented in this work includes a foot segment to allow for
control of foot stiffness parameters by ankle NJMs, while the Gruber model forces these heel/foot stiffness parameters
to be absorbed into the ground reaction force floor model. Note, this is more physiologically accurate since the physical
floor does not, and thus the floor model should not, change for different GRF-waveforms resulting from different foot
stiffness parameters.

Similar to work by Bobbert, this research investigates the adequacy of a simple rigid segment model in the
reproduction of experimentally measured GRFs[16]. In addition, this work extends the Bobbert study by including
toe-heel landings to emphasize the importance of ankle joint moments and their capacity to affect the GRF waveform.
Perhaps most similar to the muscle activation modeling study by Gerritsen[24] which focuses on the effect of different
muscle activations on the GRF waveform, this work shifts the focus from specific muscle activation patterns to
investigate the effects of total body coordination control. In effect, one seeks to determine if nonlinear control
algorithms can adequately coordinate segment control in simulation to reproduce the dynamic effects of human body
control in experiment.

Development of this model begins with a 2D land-and-stop experiment with different GRF impact scale objectives,
from which the human capacity for engineered response can be explored. The model uses deLeva's[36] mass
distributions to scale it to a 50th percentile individual, an assumption which influences the predicted NJMs if the
subject’s dimensions are actually distributed differently[11]. Because the model employs closed loop control, the
predicted NJMs are shaped to the model such that the simulated outcomes are mechanically equivalent to those of the
experiments.

During flight phase, the model’s joint angles are tracked to those of the subject’s relative segment motion, Figure 12.
Similar to the variable pendulum model mentioned in the Previous Literature Section this project uses traditional
control methodologies to design the NJM coordination for position, orientation, and velocity of the foot prior to
impact, using the experimental kinematics of the subject as a guide[15].

18

Figure 12. The human body is a naturally occurring closed-loop control system. The body’s inertial parameters
define the plant dynamics. The central nervous system (CNS), on the other hand, can be thought of as continuously
designing motor control input signals to the muscles such that the plant moves a desired way. Specifically, the CNS
compensates differences between a desired segment trajectory (efferent copy) and those sensed by organs.

During the impact phase, the effective leg-stiffness concepts from McMahon are used in this model to determine the
NJM coordination, producing lower body stiffness in terms of impedance control[13]. The human form of this
impedance control block diagram shows the biological elements which correspond to these theoretical control
elements, Figure 13. While most models capture both the human body and muscle blocks, many fail to include the
central nervous system, which has large influence over the motions of the system.

(a)

(b)

Figure 13. In impedance control (a), a desired trajectory is tracked according to standard PD control until force
feedback modifies the reference trajectory to account for intended degree of environmental interaction. The result
is trajectory tracking which deviates in the presence of external forces to ensure the interaction mirrors that of a
user defined system impedance. Notice, the dotted line represents how the feedback may not be available fast
enough during an impact, as will be discussed in later chapters. Overall, this complex control loop simplifies the
human body-environment interaction to an effective mass-spring-damper system (b).

Finally, the energy-shaping control is used in this research to reduce the total kinetic energy of the system while
maintaining the COM over the base of support, Figure 14. Notice, a key difference between previous literature
formalizations and those presented in this work is that the base frame is not fixed to its environment in human
applications. Thus, this study is expanding the application of energy shaping control to illustrate its use in a local
reference frame to reject external disturbances, as well as incorporate human trends in joint coordination strategy.

Human Body:
Mass, Joints,
Tendons
Motor System:
Central Nervous
System
+
-
Proprioceptors
Desired
Motion
Actual
Motion
Perceived
Error
u
Muscles
F, τ
Perceived
Motion
Kinetic Sensors:
Mechanoreceptors
Force Compensation:
Soft, Normal, Hard
Human Body:
Mass, Joints,
Tendons
Motor Controls:
Nervous System
+
-
Kinematic Sensors:
Proprioceptors
Actual
Motion
Perceived
Error
u
Motors:
Muscles
F, τ
Perceived
Motion
External
Forces
+
-
Desired
Motion
Perceived
Forces
Corrective
Motion
+
-
K
T
[
T
M
19

Figure 14. Energy shaping control appears very similar to trajectory following control. However, the main benefit
of energy shaping control is that one need only define a total body center of mass location objective (potential
energy equilibrium) and an energy dissipation element (damping) to generate coordinated joint trajectories which
achieve this objective. Additionally, joint specific weighting allows one to define joint velocity distribution to take
advantage of the system’s null space.

RESEARCH QUESTIONS

The research presented in this work pursues three main lines of inquiry:

1. Experimental Analysis and its Effect on Biomechanical Outcomes
2. Experimental Evidence for Human Control
3. Using Closed Loop Control to Model the Effects of the Responsive Human Body

The first set of questions focus on how sampling rate and filtering can affect the measurement of biomechanical
phenomena. The second set investigates the evidence for active feedback control in human subjects. Finally, the third
set examines the efficacy of using control theory to capture the human ability to adapt their technique according to the
time variant goals of a task.

1.5.1 Data Collection Investigations

Q1. How does kinematic frequency content change as a function of time during a landing involving impact?
While much of the literature in biomechanics discusses filtering kinematic data by a single cutoff frequency, this
method may not be adequate for data sets which include bifurcations of dynamics. Proper filtering of associated
kinematic data must be flexible to any changes in frequency content in order to achieve a continuous, sufficiently
smooth trajectory. In the context of a landing, a bifurcation in behavior occurs at the boundary between flight and
impact phases. Digital evidence of such a bifurcation is expected to appear in the frequency content of the digitized
marker trajectories as a function of time, a spectrogram.

Q2. How does frequency content of a marker trajectory vary as a function of its position on the body during impact?
In an exploration of the frequency content, one expects to find variation in the degree to which digitized marker
trajectories are affected by the impact[37]. Proper filtering of associated kinematic data must be flexible to differences
20

in frequency content in order to prevent over-smoothing trajectories which aren’t visibly affected by impact. Similar
to the problem addressed in Q2, differences in frequency content between markers suggests the need for different
degrees of data smoothing for different markers. Accordingly, the frequency content of each marker is examined to
demonstrate the variation in proper smoothness.

Q3. How does the piecewise-weighted cubic spline method compare to conventional filtering techniques?
As a fundamental proof of the necessity of filtering/smoothing flexibility, smoothed trajectories from different filtering
techniques are compared. For consistency, Jackson’s filtering cutoff methodology is implemented in both types of
filtering[38]. One expects to find inadequate filtering across different phases of motion when using a single cutoff
frequency. Conversely, the piecewise weighted cubic spline method (described in the Filtering and a Novel Smoothing
Method section) uses kinetic indicators to inform the filtering technique, and it is expected to provide sufficient
filtering across behavior bifurcations.

Q4. How does sampling rate affect kinematic measures?
The equipment used in this research provided the investigators with ultra-highspeed data collection capabilities. Thus,
it enables investigation into minimum sampling rate requirements in collection of macroscale tasks involving impact.
Specifically, one uses motion of the ankle retroreflective marker and subsequent digitized data to investigate how
sampling rate affects the observed impact conditions.

1.5.2 Experimental Control Theory Investigations

Q5. Is an individual capable of modifying landing technique to different landing objectives?
An important assumption of this work is the idea that humans are capable of modifying the GRF waveform when
starting from the same initial conditions. The inference being that humans have internal mechanisms by which they
can adjust their dynamic interaction with their environment. These mechanisms are hypothesized to manifest in
kinematic modifications prior to touchdown, kinematic modifications during impact, and finally, changes in GRF-
waveform.

Q6. If subjects can modify their landing technique, what modifiers allow them this capacity?
The crux of this research hinges upon a subject’s ability to modify their technique according to mechanical objectives.
In this context, one seeks to understand the mechanisms by which this feat is accomplished. After identifying the
sources of dynamics modification, one defines uses mechanically equivalent control algorithms to simulate these
phenomena in a model of the human body.

Q7. How might one examine these mechanisms in tasks where motion is more complex, even multiplanar?
Questions proposed in Q5 are limited to the sagittal plane by assuming of mostly planar motion. However, this
limitation may be overlooking phenomena which manifest in higher dimensional motion. As the degrees of freedom
21

for human motion increase, methods for characterizing subject specific joint coordination techniques must also evolve
to capture the complex and overlapping functional spaces. Technique characterization of shoulder motion provides a
starting point for analyzing how joint axis might play a role in multiplanar segment control during a vehicle loading
task, allowing motion in some directions while preventing it in others.

1.5.3 Modeling Control Theory Investigations

Q8. Is a hinged rigid-body model of sufficient complexity to capture human body dynamics across multiple phases of
landing, when driven by nonlinear feedback control?
Model adequacy is defined in terms of its ability to replicate experimental phenomena. Thus, there are several phase
specific measures by which one determines the sufficiency of the model defined in this work.

During flight phase, the model’s joint angles are tracked to those of the subject’s relative segment motion because
kinematic landing preparation is the key objective during flight phase. Thus, measures of model success in this phase
are manipulation of CM position with respect to the feet, segment orientation, and relative velocities prior to
touchdown[28], [39].

During the impact phase, the model is assigned an effective impedance to absorb the momentum of the body at
touchdown. This phase is characterized by the degree to which this momentum is reduced. Therefore, the measure of
model success in this phase is how well the simulated impulse magnitude matches that of the experiment.

During the post-impact phase, the model is assigned an equilibrium point via energy shaping control. This phase is
characterized by the ability to stabilize the model over its base of support. It also serves to exhibit subject specific
joint coordination solution spaces. Thus, the measure of model success in this phase is how well the simulated impulse
magnitude matches that of the experiment.

Q9. How do changes in a subject’s control strategy affect the dynamic outcomes?
With control theory at its core, this model provides insight into goal oriented strategies a subject may employ to
successfully complete a task. To this end, it is used to examine how changes one’s internal objective schema affect
kinetic and kinematic outcomes. Control strategy modification is difficult to examine in human subjects because, as
previously mentioned, humans are inherently responsive. This means they naturally adapt their system dynamics, to
preserve body function, shying away from strategies which might result in injury. Instead, one can use a model to
isolate specific changes to the control strategy and examine their effect on the task. These studies are dual purpose, as
they also speak to the sensitivity of the model itself. In any modeling study, the results produced are only as insensitive
as the model.

22

Q10. How do differences between a learned control strategy and environment affect dynamic outcomes?
When there is a mismatch between expected environmental interaction[40], the human body executes recovery
maneuvers. Control strategy disconnects are difficult to examine in human subjects because, as previously mentioned,
humans are inherently responsive. This means they naturally adapt their system dynamics, to preserve body function,
shying away from strategies which might result in injury. Instead, one can use a model to examine how specific
changes to the environment might affect joint torque requirements or kinematic outcomes, in the event an individual
does not properly adjust for an impending impact. This may offer insight into injury mechanisms which could result
from such a mismatch. Similar to the Q6, conducting a sensitivity analysis of environmental parameters is an important
model validation step. If a model is extremely sensitive to the floor values, then it may not provide robust floor-related
results.

CHAPTER ORGANIZATION

Understanding the building blocks of human body dynamics ultimately sets the stage for their manipulation; restoring
capabilities, improving current techniques, or even providing alternative mechanical modalities for accomplishing
landing tasks. This work has been divided into two main sections: the Drop Landing Experiments (Ch. 2) and the
Controls Based Modeling and Simulation of Drop-Landings (Ch. 3).

Chapter 2 focuses on the drop landing experiment, describing the task, data collection, data processing techniques,
and how they can effect biomechanical results. Novel filtering methods are presented in the data processing sections
in order to handle the multiphase biomechanical kinematic data. A brief aside alludes to a novel functional joint parsing
technique that one developed for the purposes of describing 3D motions in a kinetic context to better understand task
motions in terms of underlying control mechanisms. Inverse dynamics are used to produce the empirical measures of
condition based human body control modifications, net joint moment (NJM) time-histories. Finally, the results are
presented demonstrating that humans are capable of control modifications. These results are further subdivided by
phase to describe these control modifications in terms of phase-specific mechanical objectives.

Chapter 3 describes the controls-based modeling and simulation process, from the definition of the model and
simulation environment to the phase specific control theory implemented to mirror the empirical joint control
strategies. Next, the model is validated against empirical measurements from two different subjects to ensure its
predictive capacity is sufficient for more than just the subject to which it is trained. The final model is tested for its
parameter sensitivity to ensure the model results are robust to small changes in the definition. Finally, the model is
applied to several different technique based human motor control questions to understand the potential implications
of this controls inclusive, experimentally-validated model. The implications of each of the previous sections are
discussed in this responsive system context.

23

Chapter 4 summarizes the research again to underscore the importance of including closed-loop feedback control in
models of the human body model. Specifically, models which use control laws to define system dynamics offer a
means of studying system dynamics throughout tasks with kinetically dissimilar phases, as well as being able to predict
complex system responses using rather simplistic plant (body) definitions. This model provides a framework for
analyzing any responsive system during tasks that have fundamental shifts in the state of their environmental
interactions. As such, future work in this field should extend these ideas to 3D models in addition to a broader range
of tasks.
24

CHAPTER 2

DROP LANDING EXPERIMENTS

CHAPTER OVERVIEW

The following sections detail the methods and techniques used during each stage of a drop-landing experiment
conducted for two subjects. Data on the drop landing task was collected to establish a human being’s capacity for
vertical ground reaction force (VGRF) modulation through simple technique modification. Two subjects were asked
to perform drop landings, stepping off of a short platform and landing feet first on force plates, under self-selected
normal, softer than normal, and harder than normal conditions. The following sections the methods of data collection
and analysis, the experimental results, and their implications.

METHODS

In this study, a healthy female subject (1.575m, 64.5kg - Subject 1) with a background in gymnastics performs three,
laterally symmetric, toe-first, drop landings of various self-selected hardness. The experiment is repeated with a
healthy male subject (1.829m, 84.3kg - Subject 2) with a background in collegiate sports. The participants provided
informed consent in accordance with the Institutional Review Board (IRB) of the University of Southern California.

2.2.1 Equipment and Setup

Subjects are observed performing foot-first, drop-landings, using an ultra-high-speed camera and force plates to
expose landing impact phenomena that have never been seen before. Kinematic data in the sagittal plane was captured
using the Phantom.v711 high-speed camera, collecting at a rate of 11000fps. The subject drops vertically from a
0.445m platform onto KISTLER 9281B force plates that sample at a rate of 1200Hz with force resolution of ±1.3N,
Appendix B. Using the platform drop height as the spatial calibration measurement, the kinematic data is captured at
a resolution of 1.5mm/pixel, aligned with the sagittal plane. Given this single perspective, all visual indicators are
placed on the subject’s right hand side, Figure 15, and symmetry of motion is assumed. For Subject 2, the kinematic
data sampling rate is reduced to 120fps, with all other conditions held constant.

25

(a)

(b)

Figure 15. (a) Sagittal view of Subject 1 showing the position of each set of 4 retroreflective markers per segment.
(b) Sagittal view of subject 2 showing the position of 4 retroreflective markers per segment. Note, with the
exception of the foot, these markers are attached to rigid cuffs to minimize the effects of local loose tissue vibrations

2.2.2 Task

Each subject was instructed to perform drop-landings for three different impact conditions; self-selected normal, softer
than normal, and harder than normal. These impact scale is understood to manifest as changes to the landing impulse,
with softer landings showing a reduction in the peak VGRF and an extension in the duration of impact. Each type of
landing is collected once to provide a baseline set for comparison. In addition, a static calibration trial is collected to
determine body mass and segment lengths.

2.2.3 Data Collection: Force and Kinematics

XYZ force-time series are collected from the KISTLER force plates sampling at 1200Hz. The force plates are zeroed
immediately prior to each trial to prevent the effects of drift. The subject lands with one foot on each force plate, so
the measurements from these two plates are combined for net vertical and horizontal GRF data series, Figure 16.
Eventually, key points in the vertical ground reaction force waveform serve to define the three phases of landing:
flight phase, impact phase, and post-impact phase.

26

Figure 16. Force time curve of subject's hard-intensity platform-to-ground impact landing illustrating phases of the
trial. F z is small, so planar XY motion is assumed. For analysis purposes, the task is subdivided into different phases
according to mechanical objective. Flight phase is defined as the time before contact with the ground. Impact phase
spans from first contact to twice the time from contact to peak vertical force, shown shaded in blue. Finally, post-
impact is defined as any time after the impact phase. Notice, the impact phase impulse (shaded in blue) is the
integration of the vertical reaction force as a function of time during the impact phase, and it serves to characterize
the degree to which vertical momentum is reduced during impact in preparation for post-impact.

Focusing on the force mitigation by the subject’s lower extremities, XY position data in the vertical plane of motion
is captured using reflective markers and the Phantom v711 ultra-high-speed camera sampling at 11000fps. The foot,
shank, and thigh on the right hand side are each tagged and tracked with a set of 4 reflective markers, and kinematic
symmetry is assumed. Rigid cuffs are attached to the upper and lower legs with Coban athletic tape to serve as stable
marker attachment points. These cuffs are used to prevent the effects of relative vibration of local markers on segment
orientation calculations[41]. Each marker is attached with Velcro and is supported by more Coban. Similarly, the foot
has 4 reflective markers at the Lateral Malleolus, Calcaneus, 5
th
Metatarsal, and the 3
rd
Phalange, respectively. Note,
while upper body markers are not included, hip angles are determined by manually digitizing the ear such that the
vector between the head and hip bisects the torso as best as possible.

2.2.4 Data Processing: Force and Kinematics

The large mass of the KISTLER force plates means they are relatively robust to external noise or disturbances.
Similarly, their analog piezo-electrics ensure that digital noise only enters the signal at the analog-to-digital signal
converter. For these reasons, it is not necessary to filter the reaction force time-series, with corruption errors occurring
on a scale much smaller than that of the true signal. However, for Subject 1, the force-time series is sampled at a much
lower sampling rate than that of the kinematic series. Thus, the force data is up-sampled by cubic spline interpolation,
Initial Contact
End of
Impact Phase
27

a tool available in MatlabR2016a explained in further detail later. Furthermore, the force-time signal is isolated from
other biomechanical measures during collection and must, therefore, be time-synced to the kinematic signals. With
first ground contact as a sync point, kinetic contact is identified as the first sample where vertical ground reaction
force exceeds 10N, while kinematic contact is identified manually.

The focus of this research is the identification and characterization of the human body’s musculoskeletal control
algorithms during foot-first landings involving impact. As such, investigators seek to understand the net expressions
of musculoskeletal control, experimentally derived net joint moments, in order to parse and model the mechanisms
which drive a subject’s task oriented kinematic techniques. Net joint moments and forces are calculated by the free-
body diagram / force-torque balance process of inverse dynamics, wherein measured reaction forces are systematically
applied at the distal end of each segment of the kinematic chain progressing proximally (see the Kinetic Data Analysis
/ Inverse Dynamics Section). Notice, this force-torque balance procedure requires segment linear and angular
accelerations as inputs, but these quantities are only measured indirectly through segment position tracking. The
indirect nature of this measurement creates complications due to the noise introduced by taking time-derivatives of
discrete data. In this study, a novel cubic spline smoothing algorithm is proposed to filter and smooth each marker
position data series across different phases of impact, resulting in a continuous, smooth second derivative of measured
position data spanning the trial duration.

2.2.5 Filtering and a Novel Smoothing Method

In studying human body dynamics during a planar drop landing involving impact, one directly measures the 2D spatial
location of segments via their associated retroreflective marker sets in terms of pixels. These videos are processed by
digitization using the DLTdv5/6[42] MATLAB marker tracking package, introducing both digital noise on the order
of 1.5mm and human error into the data set. Consequently, one must calculate the first and second time-derivatives of
these noisy positions to obtain the linear and angular accelerations of each segment. Discrete temporal derivatives of
the digitized kinematic trajectories are calculated as a starting point. The resulting time derivatives of each
retroreflective marker, velocity and acceleration, are dominated by digital noise on the order of 10
6
[m/s
2
].
Unfortunately, this is not as simple as directly taking discrete time derivatives of the raw kinematic data, as shown by
Figure 17. The process of numerical differentiation amplifies signal noise, producing oscillations that increase with
derivative order. As a result, investigators must determine other ways to artfully extract the essential content of
segment motion through filtering and smoothing retroreflective tracking marker trajectories.
28

(a) (b) (c)

Figure 17. The horizontal, X, and vertical, Y, coordinate time-histories (a) depict the path of Subject 1’s right ankle
during the soft landing condition, as digitized by the automatic marker centroid tracking algorithm the MATLAB
sub-function DLTdv5. The reader will note that a significant amount of noise is present in the signal arising from
the spatial discretization, as the algorithm calculates the marker centroid location. These oscillations are amplified
through the process of taking time-derivatives, resulting in unusable acceleration data.

The biomechanics community suggests the use of a variety of different techniques for removing this noise ranging
from filtering the frequency content of the signal (usually low band pass filter) to fitting splines to the signal[43].
Harmonic analysis tends to be most useful when the true signal frequency and the digital noise frequency are
sufficiently distinct to be separable by attenuation of the raw signal according to some low-bandpass cutoff or band-
stop filter. Thus, cyclic motion may be best captured by using polynomials defined by Fourier series, as the signal
would lend itself well to the cyclical nature of sinewaves. Spline fitting, on the other hand, is more flexible with
applicability in capturing both cyclic and acyclic motions. For asymmetrical motions, researchers can use polynomials
of different orders, or even piecewise continuous splines, to shape the curve according a moving average of the signal.
It can even be modified depending on the frequency content of subdivisions within the signal, a property which is
exploited by the proposed technique.

For the purposes of this study, a low bandpass frequency cutoff filters are compared to a standard polynomial fit
method, using Jackson’s cutoff technique[38] to automating the selection of cutoff frequencies or cubic spline
smoothness. Finally, a novel extension the cubic spline smoothing technique is proposed and compared to the
aforementioned methods in the time, space, and frequency domains. The results of these methods are used to illustrate
the effects of different filtering/smoothing techniques on an exemplar kinematic analysis, the vertical acceleration of
the ankle during impact.
29

2.2.5.1 Determining Signal Content
Since the trials being studied involve impact, the frequency content of each data set is hypothesized to range broadly
and may even overlap with the frequency of digital noise. The content of the digitized signal will thus need to be
explored in order to clarify the necessity of certain filtering techniques. Frequency content is determined by taking a
Fast Fourier Transform (FFT) of the data. This transformation treats a signal as a cyclical data set with a period equal
to the duration of the original signal of interest, Figure 18.

Figure 18. Analyzing the signal in terms of a repeating cycle results in discontinuities at the end of each period, as
illustrated by the vertical component of the ankle trajectory when it is repeated.

For this reason, signals which begin and end at different values develop artificial step function discontinuities between
the beginning and end of each cycle. Windowing functions is a signal scaling mask that serves to combat this problem
by attenuating signal values at their endpoints such that they begin and end at zero, Figure 19. While several different
windowing functions exist, the Hann function is commonly used for its low aliasing properties and is therefore applied
in this study, Figure 19.

30

Figure 19. Windows prevent the artificial discontinuities which arise as a result of looping the signal of interest. In
Fourier transform theory, a signal of interest is considered to be one period of an infinitely long repeating signal
composed of sinusoidal signals of various frequencies. Notice, the signal on the right can be repeated without
discontinuity as both its value and its derivative are zero at its beginning and end. To prevent a DC frequency offset
spike in the FFT at 0Hz, the signal is shifted vertically such that its average is zero.

The FFT decomposes the windowed signal into the set of sinewaves (with an associated magnitude) which, when
summed, reconstitute the windowed signal, Eq. 9. A power spectrum plot illustrates the associated magnitude of
oscillation for each of these sinewave subcomponents as a function of its frequency.

𝑥 ( 𝑡 )= ∑b
n
sin( 𝑛𝑡 + 𝜑 𝑛 )
∞
𝑛 =1
( 9 )

31

Figure 20. These power spectra indicate the amplitude at which particular frequency-associated sinewaves are
present in a time-variant signal (scaled by the window), as if it repeated cyclically. Note, the power spectrum
magnitude peaks at 0dB because it is normalized to the highest power amplitude. By normalizing the data to the
highest amplitude, one obtains some understanding of the relative importance of the frequencies in different types
of data, which may provide insight to the signal to noise ratio. For example, this power spectrum suggests that high
frequency noise plays a larger role in the ankle x-position data than the ankle y-position data. Similarly, the high
frequency content plateaus in the kinematic data, while declining more naturally in the force signals.

When an isolated spike in the power spectrum appears at a frequency much higher than the rest of the signal, one
could use this plot, Figure 20, to determine the frequency at which digital noise is introduced. At this point, choosing
an appropriate cutoff frequency for a low band-pass filter would simply be a matter of selecting any harmonic lower
than this which attenuates the signal at that frequency by -3dB, Figure 21. However, if no such spike is immediately
apparent, then the noise may be composed of frequencies similar to those of the true signal. This forces the investigator
to choose a more flexible filtering/smoothing technique. As expected, the frequency content of the data set being
studied ranges broadly because these trials each involve an impact [44]. Thus, merely filtering spatial data by a single
low band-pass filter may not provide sufficient smoothness across the trial.

32

(a)

(b)

Figure 21. (a) The FFTs show how a raw signal FFT (black) is affected by a low band pass filter (blue). Filtering at
a single frequency can provide sufficient smoothness during some intervals, but may fall short in others (b). Notice
how the smoother blue trajectory matches well at the bottom of the ankle trajectory, but it adheres too tightly to the
signal at the top of the ankle trajectory. Consequently, certain data series may require more sophisticated filtering
techniques depending on the signal being examined.

One way to improve the filtering approach is to acknowledge the time-variant nature of the frequency content,
sometimes referred to as time-frequency analysis. Creating a spectrogram from the data, one examines the frequency
content of a signal as a function of time. The data analyzed in this study, for example, involves a foot-first impact
which will induce higher frequency motion around that time. Therefore, instead of using a sampling window that
spans the entire signal, one can select a smaller window which spans a subdivision of the data such that the data passed
by the window will only contain frequencies present over this small sample of time, a short-time Fourier transform
(STFT)[45]. Moving the window along the original signal and taking an FFT at discrete time intervals results in a
power spectrum as a function of time, a spectrogram, which indicates how the frequency content of the signal varies
as a function of time, Figure 22. Note that variation in the spectral pattern moving along the time axis (left to right
along X), proves the hypothesis of time variant frequency content.

33

Figure 22. This spectrogram illustrates the frequency content of the force signals (Fx,Fy) as a function of time. As
expected, differences in force frequency content across the phases of landing justify kinetically defined phase
breakdown to isolate phases of kinematic frequency content by input to the dynamic system. The right hand side
of each plot shows how the force data during the impact phase (starting from zero, with its end denoted by the
vertical dotted line) contains significant high frequency signals. Because the force plates show minimal effects of
digital sampling and no derivatives are required of the force signal, this data is not filtered or smoothed. For clarity,
any signal frequency content with an associated magnitude of less than -70dB is omitted.

When delineating the force signal temporally according to the same kinetic markers mentioned in the Data Collection:
Force and Kinematics section, an increase in frequency content range coincides with the impact phase. This increase
in high frequency content matches expectations, as impact is effectively a bifurcation in the task dynamics which is
characterized by a sharp rise in force. Sharp changes in data require high frequency signal to fill out the corners,
similar to the Fourier series summation of a square wave.

2.2.5.2 Finding the Right Filter
Merely filtering the spatial data set by a single low band-pass filter may not provide adequate smoothness across the
entire trial, as the true signal is not the same frequency content over the entire trial. Instead, each trial is subdivided
into three phases (flight-phase, impact-phase, and post-impact) defined as the frames before contact with the force
plate, twice the time from contact to peak VGRF, and all frames thereafter, Figure 1. Kinetically bounded phases such
as these provide a standard mechanism for temporally dividing the associated kinematic data sets according to
expected frequency content, Figure 23.

34

Figure 23. These spectrograms illustrate how the frequency content of the ankle trajectory signals, and thus the cutoff
frequency of the filter should, change with time. The phases are delineated by dotted lines and thus, the spike in
frequency content between flight and impact phase is of particular interest. An overestimated cutoff frequency
filtering results in smoothed curves with the presence of significant noise in higher derivatives. Underestimated
cutoff frequency filtering results in overly smooth raw data which significantly deviates from raw signal. For clarity,
any signal frequency content with an associated magnitude of less than -70dB is omitted.

Given that a single cutoff frequency is insufficient, the next solution is to filter or spline-fit each phase of the data
according to its own cutoff frequency or polynomial order. Unfortunately, this approach does not guarantee signal
continuity, with the mismatch between phase endpoints producing step-like discontinuities, Figure 24. These
discontinuities cause mathematic artifacts in the signal derivatives, acting counter to the impetus for filtering in the
first place. To solve this problem, one proposes Phase-Weighted Cubic Splining (PWCS), a novel combination of the
cubic spline MATLAB algorithm (CSAPS) and Jackson’s cutoff method as a solution. This combination enables the
creation of a single continuous spline spanning the entire data set, formed using a time variant weighting function to
tighten data fit during phases of higher frequency content and loosen fit during other phases in favor of smoothness.

35

Figure 24. Filtering each phase of the data set at its own, distinct cutoff frequency (10Hz, 30Hz, 6Hz for flight
phase, impact phase, and post-impact phase, respectively) results in interphase discontinuities. While they are
difficult to see in the position signal, these phase-boundary steps cause spikes in the higher order derivatives
(examples shown at boundaries on the right) making continuous kinematic analysis difficult.

Using PWCS, position-time splines are generated for each dimension of the measured position data series. The cubic-
spline filter function CSAPS
1
, in MATLAB, uses a smoothing parameter, 𝑝 , to fit a data set, 𝑥 ( 𝑘 ) , of length n to a
spline function of class C
3
. According to the documentation, CSAPS finds the function, f(x), which minimizes the
smoothing cost function, S(x), using a discrete weighting function w(k) to vary the importance of any given sample
with respect to the fitted spline[46], Eq. 10. Note for this study, the roughness time series 𝜆 ( 𝑡 ) is left at the default
value of 1. As p decreases from a value of 1 to 0, the smoothed data set deviates from the cubic spline interpolation
where it evaluates to the original data set at every sample time until it reaches a least-squares linear fit.

𝑆 ( 𝑥 )= 𝑝 ∑𝑤 ( 𝑘 ) |𝑦 ( :,𝑘 )− 𝑓 ( 𝑥 ( 𝑘 ) ) |
2
+ ( 1− 𝑝 )∫𝜆 ( 𝑡 ) |𝑓 ̈( 𝑡 ) |
2
𝑑𝑡
𝑛 𝑘 =1
( 10 )

1
http://www.mathworks.com/help/curvefit/csaps.html
36

For each marker, the smoothness parameter cutoff value, p c, and associated spline smoothness is determined by
tracking the residual as a function of p. Residual is defined as the sum of the squared difference between a raw signal
and its smoothed equivalent (i.e. the first term in Eq. 11). Appropriate spline smoothness is chosen by applying a
threshold to the second derivative of the residual with respect to p (residual acceleration) similar to the cutoff
frequency work of Jackson[38], Eq. 12. The smoothness parameter cutoff, 𝑝 𝑐 , occurs when the residual acceleration
begins to increase exponentially as (1-p) increases, indicated by a knee in the data, as shown in Figure 26.. Thus, 𝑝 𝑐
is the lower bound on p below which the smoothed data series would quickly begin to deviate from the raw data series.

𝑅 ( 𝑝 )= |𝑦 ( :,𝑘 )− 𝑓 ( 𝑥 ( 𝑘 ) ,𝑝 ) |
2
( 11 )

𝑝 𝑐 = {max( 𝑝 )||
𝜕 2
𝑅 ( 𝑝 )
𝜕 2
𝑝 | ≤ 𝑇 ℎ𝑟𝑒 ℎ𝑜𝑙𝑑 } ( 12 )

According to MATLAB documentation, the interesting “close-fit” range for p occurs around n = 0 for:

𝑝 𝑛 =
1
(1+
(
1
fps
)
3
60/10
𝑛 )

( 13 )

where fps stands for frames per second and represents the sampling rate of the data set.

Figure 25. As the smoothing factor p decreases from a value of 1, the resulting piecewise cubic spline reduces its
adherence to the original raw data in favor of reducing the associated second derivative of the curve. Therefore, as
the smoothness increases (p decreases from 1) the residual increases, but at a decreasing rate because the smoothing
initially deviates from the noise then takes longer to deviate from the fundamental signal. Due to the CSAPS
formulation, the interesting range for p usually occurs very near to the value of 1[46].
37

The vigilant reader will notice that decreasing from perfect data adherence towards a linear fit is opposite to the
direction proposed by Jackson, Figure 25. However, one’s decision to cutoff in this direction is based on the
assumption that the curve fit should deviate from the raw data as minimally as possible while still reducing the second
derivative. It is speculated that this difference in Jackson’s approach was merely a limit of the available technology,
as there would be no way for them to know the starting point (i.e. perfect polynomial fit order) without proper
computational power. Approaching the cutoff from the perfect matching side is likely to result in a closer fit smoothed
function, overall.

Figure 26. Using a threshold on the second derivative of the first subject’s ankle marker residual during impact
phase with respect to p, the cutoff smoothness parameter, p c, occurs at the end of the initial exponential residual
growth (i.e. where the absolute value of the second derivative becomes small). The residual first increases as the
fit becomes smoother due to the way that the algorithm initially deviates from the noise of the signal. Then the rate
of deviation declines while further smoothing results in only minor deviations from the fundamental trajectory.
Between these two trends, the desired smoothness value occurs. Further smoothing does degrades the tightness of
fit, while less smoothing allows more noise to pass through.

38

Figure 27. Exemplary data set shows vertical marker acceleration of each segment during subject 1’s soft landing
condition. A good check for the quality of filtering is that the head acceleration is, at most, on the order of 3 times
gravity[47].

Selection of the residual acceleration threshold is informed by examining the residual acceleration curve and choosing
a value high enough that it occurs at the tail end of the exponential residual acceleration decline period (as p decreases
the rate of residual growth declines exponentially until it plateaus). Note, due to the vertical nature of the growth with
respect to p, the choice of this value is robust. Another way to choose this threshold is to look at the resulting segment
accelerations and decrease p until the resulting head accelerations are around 3 times gravity, as work by Zhang
suggests should be the case for a self-selected normal landing[47]. For this study, the residual acceleration threshold
was set at a value of 10
11
.

Table 1. The selected p-values for a set of markers during each trial illustrate how p might varies as function of both trial
condition type, as well as marker position on the body.
Trial / Marker Specific Smoothing Coefficient: p
Marker Softer than Normal Normal Harder than Normal
Toe (Tip of Foot) 0.9999999961602
Ankle (Talus) 0.9999999923387 0.9999999923387 0.9999999951660
Shank Cuff 1 (Top Right) 0.9999999939144
Thigh Cuff 1 (Top Right) 0.9999999923387

39

Since the smoothness of the each data series is highly phase dependent, a weighting factor is used to emphasize
position data during the impact phase (highest frequency content phase), effectively relaxing the spline-fit for
oscillations in the flight and post-impact phases (with lower frequency band data). Keeping a unit weight for data
points in the impact phase, the time dependent weighting function was piecewise constant with a unique value during
each of the three phases. The same process used to select p c from the impact phase residual was used to select the
weighting factors w FP and w PP from the flight phase and post-impact phase residual, respectively, Figure 28. Namely,
the trajectory residual was tracked while the phase specific weighting value was reduced from 1 to 0, with the cutoff
weight occurring as the phase-specific residual acceleration crosses the corresponding threshold.

Figure 28. The kinetic phases are used to separate the kinematic data sets into periods of distinct frequency content.
Due to these shifts in frequency content, it is necessary to modify the degree of smoothing for each phase. By
applying weighting values to each phase, a single piecewise cubic spline will allow the more roughness on phases
where higher frequency content is expected (impact: w IP = 1) and less on phases which should be smoother (flight
phase: w FP < 1, post-impact: w PP < 1). In the example shown here, the smoothing coefficient, p, is held constant
throughout the trial while the phase weighting coefficients serve to loosen the spline fit during the flight and post-
impact phases. Note, this case also suggests that the flight phase has a tighter fit than the post-impact phase.

The result is a data series represented with a single cubic-spline based function, providing twice-differentiable
polynomials for each marker trajectory, Figure 29. Given explicit polynomials, the resulting explicit time derivatives
no longer suffer from discretization induced noise. By this method, the largest deviation between filtered data and raw
data for an exemplar marker trajectory (ankle marker during the hard landing condition) is less than 7mm.

40

Figure 29. Smoothing each phase of the data set at its own phase weighting (0.0005568814, 1, 0.0000793410 for
flight phase, impact phase, and post-impact phase, respectively) using the piece-wise weighted cubic spline
technique results second order continuous trajectories with phase specific smoothness. While this is difficult to see
in the ankle marker position signals (left), these phase-boundary steps cause a significant reduction in the associated
higher order derivatives (right) making inverse dynamics analysis across dynamic boundaries plausible.

2.2.5.3 Technique Robustness
The data set being analyzed in this study is sampled at a rate of 11000Hz. Rather than being beneficial, this excessively
high sampling rate leaves the tracking software more susceptible to digital noise. Each marker spends a long period
of time in each pixel of the data resulting in jerky step motion each time the centroid crosses a pixel boundary. Despite
this shortcoming, the PWCS method performs well, reducing the calculated accelerations by 5 orders of magnitude.
For posterity, the raw data is down-sampled to determine the applicability of PWCS to data with more realistic
sampling rates. Because PWCS relies on cubic splining, the method is robust to changes in sampling rate showing
maximum trajectory position errors on the order of 5mm for sampling rates as low as 120Hz, Figure 30. To put this
in terms of a biomechanical indicator, the ankle vertical velocity at impact varies by 2.7[m/s] when the raw sampling
rate spans the 120Hz-1200Hz range. Importantly, it appears this down sampled ankle velocity converges to the wrong
value, as one intuitively knows the ankle should have negative velocity prior to contact. Applying PWCS, these values
remain more consistent, varying by less than 0.4[m/s] from -2.54[m/s] over the same frequency range, Figure 31.

41

Figure 30. PWCS robustness is illustrated by its ability to remain true to the original data set, despite significant
changes in the amount of data available for fitting. In this figure, the residual is shown as a function of the down-
sampling rate. At each frequency, the original data set is down-sampled, the PWCS method is applied, and finally
the resulting cubic spline is re-up-sampled to the original sampling rate for comparison to the raw signal. Results
at low sampling rates are likely be better than these, as the smoothing residual acceleration threshold was not
adjusted to the down sampled data sets.

Figure 31. The ankle velocity just before impact as calculated from taking the discrete time derivative varies much
more significantly across the 120Hz to 1200Hz sampling frequency range than the resulting smoothed trajectories
found by applying PWCS to the down sampled data sets. Importantly, the discrete velocity calculation immediately
prior to impact converges to a non-negative number, which one knows to be inaccurate to a vertical drop-landing
study.

42

Luckily, the incredibly high sampling rate for this collection offers additional insights into the necessity of sampling
rates. The down sampling of the kinematic data set is not only used to test the robustness of the PWCS filtering
method, but also to determine the frequency at which data loss begins to occur. Kinematic information loss is measured
in this study by the sum of the squared-difference between the original raw trajectory and those of a down-sampled
and re-interpolated data set. In essence, this study can illustrate how high the kinematic sampling rate must be in order
to faithfully capture motion during an impact. It is important that this study be applied to body segments directly
involved in impact, as these undergo the most severe rates of acceleration of any segment. Thus, the ankle marker is
selected, both for its ease of tracking and its proximity to the initial point of contact.

2.2.6 Kinematic Data Analysis

Biomechanical inverse dynamics requires raw kinematic information be converted into anatomically relevant
measures including linear and angular segment acceleration. Each segment parameter (length, mass, moment of
inertia, and longitudinal COM location) is determined using deLeva’s anthropometry data[36], for a 50
th
percentile
individual scaled to the subject.

In order to determine the angular kinematic information, reference frames are created for each segment from its rigid
cuff marker set two different ways, the triad method and the q-method (also known as the Wahba-problem)[48]. As
the q-method is the more robust[49], least-square fit solution, it was used to track segment orientations. By this
method, the quaternion describing the segment orientation at time step k is given by the K-matrix eigenvector whose
eigenvalue is largest[50], Eqs. 14-15.

𝒒 𝑘 = 𝒗
𝑘 .𝑡 . 𝜆
𝑘 > 𝜆 𝑗 𝑘 ∀ 𝜆 𝑗 𝑘 ≠ 𝜆
𝑘
( 14 )

where

[𝑽 𝒌 ,𝑫 𝒌 ,𝑼 𝒌 ] = 𝑣𝑑 ( 𝑲 𝒌 ) , 𝑽 𝑘 = [
𝒗 1
𝒗 2
… 𝒗 𝑛 ( 𝑛 +1) /2]
𝑘 , 𝑫 𝑘 = 𝑑𝑖𝑎𝑔 ( 𝜆 1
,𝜆 2
,…,𝜆 𝑛 ( 𝑛 +1) /2
)
𝑘 ( 15 )

Note, because the motion being studied is a 2D rotation, the resulting unit quaternion will have a unit vector
component, 𝒏̂, which always points perpendicular to the plane of motion. This assumption is used to help correct for
numerical errors, with the quaternion simplifying to a component in the z-direction and a scalar component, Eq. 16.

𝒒 𝑘 = [
𝒏̂ sin( 𝜃 𝑘 /2)
cos( 𝜃 𝑘 /2)
] = [0 0 sin( 𝜃 𝑘 /2) cos( 𝜃 𝑘 /2) ]
𝑇 ( 16 )

where θ k is the angle of rotation. Given a data set of n tracking marker positions, ( 𝒙 𝟏 ,𝒙 𝟐 ,…,𝒙 𝒏 ) , the K matrix is
created through a series of steps. First, vectors are calculated between every pair of positions on each segment for the
first frame, and these vectors are collected in a matrix, A 0, Eq. 17.
43

𝑨 0
= [( 𝒙 1
− 𝒙 2
)
0
( 𝒙 1
− 𝒙 3
)
0
… ( 𝒙 𝑛 −1
− 𝒙 𝑛 )
0
] ( 17 )

The same process is repeated each instant in time, k, and these are each collected into corresponding matrices, B k.

𝑩 𝑘 = [( 𝒙 1
− 𝒙 2
)
𝑘 ( 𝒙 1
− 𝒙 3
)
𝑘 … ( 𝒙 𝑛 −1
− 𝒙 𝑛 )
𝑘 ] ( 18 )

Next, these two matrices are multiplied together to form, W k.

𝑾 𝑘 = 𝑩 𝑘 𝑨 0
𝑇 ( 19 )

This matrix is used to form the K matrix components, with:

𝑺 𝑘 = 𝑾 𝑘 𝑇 + 𝑾 𝑘 ( 20 )

and

𝑍 𝑘 = [𝑾 𝑘 ( 2,3)− 𝑾 𝑘 ( 3,2) 𝑾 𝑘 ( 3,1)− 𝑾 𝑘 ( 1,3) 𝑾 𝑘 ( 1,2)− 𝑾 𝑘 ( 2,1) ]
𝑇 ( 21 )

𝜎 𝑘 = 𝑡𝑟𝑎𝑐𝑒 ( 𝑾 𝑘 ) ( 22 )

Finally, the K k matrix is formed:

𝑲 𝑘 = [
𝑺 𝑘 − 𝜎 𝑘 𝑰 𝒁 𝑘 𝒁 𝑘 𝑇 𝜎 𝑘 ] ( 23 )

The q-Method does suffer from a theoretical singularity point if the rotation being described is exactly 180⁰ away
from the reference orientation, 𝐴 0
. However, this can easily be handled by temporarily another frame as the reference
orientation, and multiplying the quaternions in series to determine the segment attitude during the desired frame.

2.2.6.1 Angular Velocity Vector
Given a quaternion-time history thus, calculating segment angular velocity is a simple case of using the relationship
between quaternions and angular velocity, Eq. 24.

𝜔 𝑘 = 2𝑞 ̇ 𝑘 𝑞 𝑘 ∗
( 24 )

where the dot accent indicates a time derivative and * indicates the complex conjugate. Similarly, the derivatives of
these segment angular velocities provides segment angular acceleration.

44

𝛼 𝑘 = 𝜔 ̇ 𝑘 = 2( 𝑞 ̈ 𝑘 𝑞 𝑘 ∗
− ( 𝑞 ̇ 𝑘 𝑞 𝑘 ∗
)
2
) ( 25 )

Since this motion is two dimensional the axis of rotation does not change direction, so one neglects the effects of
angular velocity directional changes. Note, this means one can also calculate these values simply through discrete time
derivatives of the segment angles, Eq. 25.

Figure 32. Illustration of potential segment shortening given motion which is not perfectly planar. As the
assumption of this study was planar motion, this shortening must be corrected by reconstruction of segment
endpoints.

Using the ankle marker as a base point[20], the kinematic chain was reconstructed using the segment orientation and
lengths to calculate the linear kinematic information via forward kinematics. Building up the subject kinematics using
this base segment method avoids apparent segment shortening which may be the result of motion out of plane of action
or digitization error, as shown in Figure 32. Assuming there is minimal relative motion between the cuff and the
segments during the trials, the segment endpoints and COMs are tracked by the associated cuff marker motion. These
calculated points provide COM trajectories and their derivatives, Eqs. 26-34.

𝑥 ℎ𝑒𝑒𝑙 ( 𝑡 )= 𝑥 𝑎𝑛𝑘𝑙𝑒 ( 𝑡 )+ 𝑞 𝑓𝑜𝑜𝑡 ( 𝑡 ) ( 𝑥 ℎ𝑒𝑒𝑙 0
− 𝑥 𝑎𝑛𝑘𝑙𝑒 0
) 𝑞 𝑓𝑜𝑜𝑡 ∗
( 𝑡 ) ( 26 )

𝑥 𝑡𝑜𝑒 ( 𝑡 )= 𝑥 ℎ𝑒𝑒𝑙 ( 𝑡 )+ 𝑞 𝑓𝑜𝑜𝑡 ( 𝑡 )(𝐿 𝑓𝑜𝑜𝑡 ( 𝑥 𝑡𝑜𝑒 0
− 𝑥 ℎ𝑒𝑒𝑙 0
)
|𝑥 𝑡 𝑜𝑒
0
− 𝑥 ℎ𝑒𝑒𝑙 0
|
2
)𝑞 𝑓𝑜𝑜𝑡 ∗
( 𝑡 ) ( 27 )

45

𝑥 𝑘𝑛𝑒𝑒 ( 𝑡 )= 𝑥 𝑎𝑛𝑘𝑙𝑒 ( 𝑡 )+ 𝑞 𝑠 ℎ𝑎𝑛𝑘 ( 𝑡 )(𝐿 𝑠 ℎ𝑎𝑛𝑘 ( 𝑥 𝑘𝑛𝑒𝑒 0
− 𝑥 𝑎𝑛𝑘𝑙𝑒 0
)
|𝑥 𝑘𝑛𝑒𝑒 0
− 𝑥 𝑎𝑛𝑘𝑙𝑒 0
|
2
)𝑞 𝑠 ℎ𝑎 𝑛 𝑘 ∗
( 𝑡 ) ( 28 )

𝑥 ℎ
( 𝑡 )= 𝑥 𝑘𝑛𝑒𝑒 ( 𝑡 )+ 𝑞 𝑡 ℎ 𝑔 ℎ
( 𝑡 )(𝐿 𝑡 ℎ 𝑔 ℎ
(𝑥 ℎ
0
− 𝑥 𝑘𝑛𝑒𝑒 0
)
|𝑥 ℎ
0
− 𝑥 𝑘𝑛𝑒𝑒 0
|
2
)𝑞 𝑡 ℎ 𝑔 ℎ
∗
( 𝑡 ) ( 29 )

𝑥 ℎ𝑒𝑎
( 𝑡 )= 𝑥 ℎ
( 𝑡 )+ 𝑞 𝑡𝑜𝑟𝑠𝑜 ( 𝑡 )(𝐿 𝑡𝑜𝑟𝑠𝑜 (𝑥 ℎ𝑒𝑎
0
− 𝑥 ℎ
0
)
|𝑥 ℎ𝑒𝑎
0
− 𝑥 ℎ
0
|
2
)𝑞 𝑡𝑜𝑟𝑠𝑜 ∗
( 𝑡 ) ( 30 )

Similarly, the centers of mass are found by:

𝑥 𝑓𝑜𝑜𝑡 ,𝐶𝑂𝑀 ( 𝑡 )= 𝑥 ℎ𝑒𝑒𝑙 ( 𝑡 )+ 𝐿 𝑓𝑜𝑜𝑡 ,𝐶𝑂𝑀 ( 𝑥 𝑡𝑜𝑒 ( 𝑡 )− 𝑥 ℎ𝑒𝑒𝑙 ( 𝑡 ) )
|𝑥 𝑡𝑜𝑒 ( 𝑡 )− 𝑥 ℎ𝑒𝑒𝑙 ( 𝑡 ) |
2

( 31 )

𝒙 𝑠 ℎ𝑎𝑛𝑘 ,𝐶𝑂𝑀 ( 𝑡 )= 𝒙 𝑎𝑛𝑘𝑙𝑒 ( 𝑡 )+ 𝐿 𝑠 ℎ𝑎𝑛𝑘 ,𝐶𝑂𝑀 ( 𝒙 𝑘𝑛𝑒𝑒 ( 𝑡 )− 𝒙 𝑎𝑛𝑘𝑙𝑒 ( 𝑡 ) )
|𝒙 𝑘𝑛𝑒𝑒 ( 𝑡 )− 𝒙 𝑎𝑛𝑘𝑙𝑒 ( 𝑡 ) |
2

( 32 )

𝒙 𝑡 ℎ 𝑔 ℎ,𝐶𝑂𝑀 ( 𝑡 )= 𝒙 𝑘𝑛𝑒𝑒 ( 𝑡 )+ 𝐿 𝑡 ℎ 𝑔 ℎ,𝐶𝑂𝑀 (𝒙 ℎ
( 𝑡 )− 𝒙 𝑘𝑛𝑒𝑒 ( 𝑡 ) )
|𝒙 ℎ
( 𝑡 )− 𝒙 𝑘𝑛𝑒𝑒 ( 𝑡 ) |
2

( 33 )

𝒙 𝑡𝑜𝑟𝑠𝑜 ,𝐶𝑂𝑀 ( 𝑡 )= 𝒙 ℎ
( 𝑡 )+ 𝐿 𝑡𝑜𝑟𝑠𝑜 ,𝐶𝑂𝑀 (𝒙 ℎ𝑒𝑎
( 𝑡 )− 𝒙 ℎ
( 𝑡 ) )
|𝒙 ℎ𝑒𝑎
( 𝑡 )− 𝒙 ℎ
( 𝑡 ) |
2

( 34 )

With the total body center of mass (COM) given by the weighted sum of the segment COM locations divided by the
total mass, Eq. 35.

𝒙 𝐶𝑂𝑀 ( 𝑡 )=
𝑚 𝑡𝑜𝑟𝑠𝑜 𝒙 𝑡𝑜𝑟𝑠𝑜 ,𝐶𝑂𝑀 ( 𝑡 )+ 𝑚 𝑡 ℎ 𝑔 ℎ
𝒙 𝑡 ℎ 𝑔 ℎ,𝐶𝑂𝑀 ( 𝑡 )+ 𝑚 𝑠 ℎ𝑎𝑛𝑘 𝒙 𝑠 ℎ𝑎𝑛𝑘 ,𝐶𝑂𝑀 ( 𝑡 )+ 𝑚 𝑓𝑜𝑜𝑡 𝒙 𝑓𝑜𝑜𝑡 ,𝐶𝑂𝑀 ( 𝑡 )
𝑚 𝑡𝑜𝑟𝑠𝑜 + 𝑚 𝑡 ℎ 𝑔 ℎ
+ 𝑚 𝑠 ℎ𝑎𝑛𝑘 + 𝑚 𝑓𝑜𝑜𝑡 ( 35 )

2.2.7 Parsed 4-Element Functional Joint Axis

Another way to characterize control is by describing the plane of motion as it relates to the mechano-functional axes
of the proximal segment, a task which becomes more difficult as the number of degrees of freedom increase. Thus,
one defines a novel means of characterizing relative segment motion from joints with more than one degree of
freedom, the Parsed 4-Element Functional Joint Axis.

46

Euler’s Rotation Theorem states that every rotational displacement of a rigid body in three dimensional space can be
described by a single rotation about a single axis. Thus, other researchers have explored the use of an attitude vector
as a means of representing relative segment orientation[51]. Using a single attitude vector parsed into three orthogonal
components, this representation avoids the mathematical artifacts associated with the rotation sequences altogether.
These attitude vectors are free of singularities and the rotational asymmetries of Euler angles. However, for non-hinge-
like joints (e.g. hip, ankle, shoulder, etc.) the orthogonal attitude vector component definition does not make a clear
distinction between internal/external rotation and the other three axes of rotation, leaving ambiguous crosstalk of
functional segment control in a kinetic context.

In order to gain insight into how a task is accomplished, one proposes the use of the relative segment angular velocity
vector (the instantaneous axis of rotation), parsed over 4 anatomical axes as a method of characterizing subject specific
segment control. Note, in order to avoid the aforementioned crosstalk of angular velocity signals, one includes a 4
th

anatomical axes of rotation: the longitudinal axis of the distal segment represents the first axis, internal/external
rotation, with the other three axes fixed to the proximal-segment reference frame. In the case of the hip joint, for
example, the internal-external axis points along the longitudinal axis of the thigh, the flexion/extension axis points
sagittally, the abduction/adduction axis points anteriorly, and the horizontal abduction/horizontal adduction axis points
superiorly.

Recall, due to the kinematic constraints of the human body (i.e. segments form a kinematic chain), the angular velocity
of each more distal segment is simply the sum of its proximal segment angular velocity and that which is added from
its proximally adjacent joint. Thus, one calculates the distal segment angular velocity relative to the proximal segment
(𝜔 ,
) as expressed in the inertial reference frame by simple vector subtraction, Eq. 36.

𝝎 ,
= 𝝎
− 𝝎
( 36 )

where 𝜔
represents the absolute distal segment angular velocity and 𝜔
represents the proximal segment angular
velocity. Internal/external angular velocity, 𝝎 𝐼𝐸
, can only be generated by rotation about the longitudinal axis of the
segment, so it is first parsed from the resultant angular velocity vector of the segment and treated as its own
independent vector, Eq. 37.

𝝎 𝐼𝐸
= |𝝎 ,
∙ 𝒚⃑ ⃑

|𝒚̂
𝒅 ( 37 )

where 𝒚⃑ ⃑

represents the longitudinal axis of the distal segment and
̂
indicates a unit vector in that same direction.
Notice, the internal-external angular velocity is a 3-element vector because its orientation points in the same direction
as the distal segment longitudinal axis, and it serves only to orient the segment rather than change where it is pointing.
By extracting the internal and external component of the angular velocity of the distal joint first, the remaining joint
angular velocity components can be compartmentalized into the standard orthogonal, proximal segment-fixed rotation
axes (e.g. relative to the proximal segment), Eqs. 38-42.
47

𝝎 3
= 𝝎 ,
− 𝝎 𝐼𝐸
( 38 )

𝜔 𝐴𝑏𝐴
= 𝝎 3
∙ 𝒙⃑ ⃑

( 39 )

𝜔 𝐻𝐴𝑏𝐻𝐴
= 𝝎 3
∙ 𝒚⃑ ⃑

( 40 )

𝜔 𝐹𝐸
= 𝝎 3
∙ 𝒛⃑

( 41 )

𝝎 4
= [
𝜔 𝐼𝐸
𝜔 𝐴𝑏𝐴
𝜔 𝐻𝐴𝑏𝐻𝐴
𝜔 𝐹𝐸
] ( 42 )

where 𝝎 3
represents the residual three element joint angular velocity that is parsed into the proximal segment reference
frame axes (𝒙⃑ ⃑

,𝒚⃑ ⃑

,𝒛⃑

). This residual joint angular velocity functions to point the distal segment, while the
internal/external component serves to orient the segment. The three residual components are the abduction/adduction
(𝜔 𝐴𝑏𝐴
), horizontal ab/adduction (𝜔 𝐻𝐴𝑏𝐻𝐴
) , and flexion/extension (𝜔 𝐹𝐸
) axes, respectively. Taking the magnitude
of the internal/external angular velocity vector (𝜔 𝐼𝐸
), one finally assembles the final 4-element vector, 𝝎 4
, Eq. 42.
Notice this parsed vector’s direction and magnitude reflect the rate of a segment’s anatomical rotation at each instant
in time, capturing precisely how the segment system moved from one relative position to another, also known as
technique. The 4-element parsing method is applied to the shoulder joint during a wheelchair push cycle to
demonstrate its utility in clarifying subject specific technique changes according to propulsion speed[52]. The study
was carried out in cooperation with a fellow PhD candidate in the USC Biomechanics Lab, Ian Russell[53].

2.2.8 Kinetic Data Analysis / Inverse Dynamics

The second half of the force balance equation is the environmental influence on the human body system. As such, one
must first determine proper placement of the effective ground reaction force vector. This means calculating the
effective center of pressure from the four piezo-electric signals at the corners of each force plate. Note, because the
magnitudes of twist, horizontal forces anterior-posterior (F x), and horizontal forces medial-lateral (F z) are insignificant
relative to the vertical ground reaction forces, the center of pressure equation is simplified to a single dimension force
balance, Eq. 43. In this case, the magnitude of the signal measured at each corner acts as a center of pressure (COP)
position weight to scale the x-coordinate position, and y must be at ground level.

𝒙 𝐹 ( 𝑡 )= [
𝐹 𝑦 1
( 𝑡 ) 𝑥 𝐹 1
+ 𝐹 𝑦 2
( 𝑡 ) 𝑥 𝐹 2
+ 𝐹 𝑦 3
( 𝑡 ) 𝑥 𝐹 3
+ 𝐹 𝑦 4
( 𝑡 ) 𝑥 𝐹 4
𝐹 𝑦 1
( 𝑡 )+ 𝐹 𝑦 2
( 𝑡 )+ 𝐹 𝑦 3
( 𝑡 )+ 𝐹 𝑦 4
( 𝑡 )
0
] ( 43 )

Defining the left-most edge of the plate as the inertial space origin, the anterior-posterior piezo-electric positions
(𝑥 𝐹 1
,𝑥 𝐹 2
,𝑥 𝐹 3
,𝑥 𝐹 4
) can be further simplified to (0, 0, L F, L F), where L F is the length of the force-plate. Thus, the center
48

of pressure equation becomes a ratio of force measured at the far end of the plate versus net vertical reaction force,
Eq. 44.

𝒙 𝐹 ( 𝑡 )= [
𝐿 𝐹 ∙
𝐹 𝑦 3
( 𝑡 )+ 𝐹 𝑦 4
( 𝑡 )
𝐹 𝑦 1
( 𝑡 )+ 𝐹 𝑦 2
( 𝑡 )+ 𝐹 𝑦 3
( 𝑡 )+ 𝐹 𝑦 4
( 𝑡 )
0
] ( 44 )

Pairing these kinetic measurements with their kinematic counterparts into the inverse dynamic equations completes
the dynamic force balance. The force time curves are synced with the kinematic data according the first frame of force
plate contact (F y greater than 10N). Since the force is sampled at a much lower rate, it has been up-sampled to match
that of the kinematic data using the cubic-spline filter function CSAPS. From the resulting data sets, the internal net
joint force (NJF) and net joint moment (NJM) time series for each of the lower extremity joints were calculated using
inverse dynamics in order to show the variation in technique and the resulting variety of VGRF-time curves.

The basic equations of motion are derived using Newton’s second law of motion, Eqs. 45 and 46.

∑𝑭
= 𝑚
𝒙 ̈
( 45 )

∑𝝉
= 𝐼
𝜽 ̈
( 46 )

These equations can be expressed more specifically in terms of the forces and torques at the proximal end of a segment
as a function of the forces and torques on the distal end of a segment as shown in Eqs. 47 and 48, with the variable
definitions as shown in Figure 33.

𝐹 𝑥 ,
= 𝑚
𝑥 ̈
− 𝐹 𝑥 ,
and 𝐹 𝑦 ,
= 𝑚
( 𝑦 ̈
+ 𝑔 )− 𝐹 𝑦 ,
( 47 )

𝜏 𝑧 ,
= 𝐼
𝜃 ̈ 𝑧 ,
− 𝐹 𝑥 ,
∆𝑦 𝑐𝑜𝑚 ,
𝑖 − 𝐹 𝑦 ,
∆𝑥 𝑐𝑜𝑚 ,
𝑖 − 𝐹 𝑥 ,
∆𝑦 𝑐𝑜𝑚 ,
𝑖 − 𝐹 𝑦 ,
∆𝑥 𝑐𝑜𝑚 ,
𝑖 − 𝜏 𝑧 ,
( 48 )

where g is gravity, F is force, m is mass, subscript d represents distal joint, subscript p represents proximal joint,
subscript i stands for the i
th
segment, 𝑥 ̈ is the COM acceleration in the horizontal direction, 𝑦 ̈ is the COM acceleration
in the vertical direction, 𝜃 ̈ 𝒛 is the angular acceleration along the joint axis, ∆𝑥 𝑐𝑜𝑚 ,__
represents the horizontal distance
between the COM and a point, ∆𝑦 𝑐𝑜𝑚 ,__
represents the vertical distance between the COM and a point, and 𝜏 is torque.
Using these equations, the internal joint forces and torques are calculated from the point of contact up the kinematic
chain until reaching the final free-ended segment.

49

Figure 33. Inverse dynamics free body diagrams for calculation of inter-segmental forces and torques, where the
terms being calculated are highlighted in red. Note, this diagram would be much more complex due to the
continuum of contact points. However, this can be simplified by summing the forces into an effective point reaction
force located at the time variant center of pressure which moves along the foot, one returns to the same simple
single end-point form.

The result of these calculations is a set of NJM-time series which show the joint control applied by the subject for
successful foot-first landing under different impact conditions, based the rigid body assumptions of inverse dynamics.
Collectively, they serve to demonstrate that the human body is an active system that can adapt its mechanical properties
to the requirements of task. In the following sections, one proposes a simple multilink rigid body model actuated by
nonlinear feedback control is sufficient to capture the essential elements of the human body, necessary to describe its
adaptive nature in successful land-and-stop tasks.

RESULTS

The first half of this section examines the results from investigations into the effects of data analysis techniques and
how they shape the biomechanical measures used as source material for investigations into human body control. First,
the underlying content of kinematic data involving impact is presented through frequency-time analysis. Next, the
relationship between marker placement and this underlying content is shown. Comparisons of different filtering
techniques show their effectiveness in capturing multiphase landing motion. Finally, the effects of sampling rate on
the biomechanical measures are presented to illustrate filtering technique robustness.

,
,
,
,
50

The second half of this section presents the implications of the filtered data in terms of human body control. First, the
initial conditions of each trial are compared to ensure that outcomes are the result of technique modification and not
simply unique starting conditions. Next, data is presented to demonstrate the human capacity for adapting control
techniques to task objective modifications. Next, kinematic indicators of control modification are presented along with
the associated kinetic calculations of joint control. These same mechanistic values serves as measures by which one
gauges the quality of simulated human body control.

2.3.1 Marker Filtering

After a data collection, the first and perhaps most important task, is proper processing of the data into biomechanical
measurements.

2.3.1.1 Finding the Right Filtering Technique
An exemplary marker trajectory, the ankle marker during a foot-first drop-landing task with impact demonstrates how
the signals require smoothing before differentiation for inverse dynamics. The noise present in the digitized signal
originates from automated marker centroid tracking digitization algorithms (DLTdv5/6, MATLAB). The process of
discrete differentiation, required to find the segment accelerations necessary for inverse dynamics analysis, amplifies
the small scale noise in this raw signal, Figure 34.

(a) (b) (c)

Figure 34. The horizontal, X, and vertical, Y, coordinate time-histories (a) depict the path of the subject’s right
ankle as digitized by the automatic marker centroid tracking algorithm the MATLAB sub-function DLTdv5 during
the soft landing condition. A significant amount of noise is present in the signal arising from the spatial
discretization, as the algorithm calculates the marker centroid location. These oscillations are amplified through the
process of taking discrete time-derivatives, resulting in unusable acceleration data.
51

The next logical step is to filter the raw signal, such that the derivatives are smoothed to values within a reasonable
range. A simple low bandpass 4
th
order Butterworth filter is designed with a cutoff frequency determined by Jackson’s
method[38], Figure 35.

Figure 35. The Jackson cutoff frequency method calculates the cumulative squared error, called residual, between
a trajectory and its smoothed counterpart. By tracking the rate at which this residual varies as a function of the
cutoff frequency, one gains a sense of the point at which further increases in the cutoff frequency do not
significantly improve the residual. Furthermore, this “knee” in the data can further be emphasized by taking the
second derivative of the residual with respect to the cutoff frequency. In this way, the desired cutoff frequency is
determined to occur immediately prior to the point where further improvement in the residual plateaus as the cutoff
frequency increases, identified by the red threshold line.

This single low-bandpass filter results in improved derivative scales, Figure 36. However, the single cutoff frequency
low band pass filtering technique proves insufficient for filtering this trajectory because the resulting ankle data still
exhibits large accelerations (on the order of 70m/s^2) in the post-impact phase, when the subject is standing still.

52

(a) (b) (c)

Figure 36. The low bandpass filtered horizontal, X, and vertical, Y, coordinate time-histories (a) depict the
smoothed path of the subject’s right ankle. A significant amount of noise has been removed by the filtering process.
However, a significant amount of noise is still present in the signal demanding a more complex filtering method.

Failing the produce adequate acceleration during flight and post-impact phases, the raw digitized signal is further
examined to determine how the filtering process could be improved. An FFT of the bandpass filtered signal is
compared to the original signal, Figure 37. As designed, the low band pass filtered signal shows attenuation of any
signal greater than the cutoff frequency. Thus, more complex techniques are required to extract the true trajectory
from the signals. The FFT does not show fundamental discrepancies with filter theory. Looking at the spatial trajectory
of the marker, however, one observes trajectory phases which are inadequately smoothed, Figure 38. Interestingly,
this inadequacy is not consistent, suggesting there may be other forces at play.

53

Figure 37. Comparing the filtered signal to that of the original raw data, one finds that the low bandpass is
functioning as designed, attenuating any frequencies greater than the cutoff frequency by -3dB or more. Since the
filtered accelerations are still inadequate during the other phases, one it led to believe a temporal dimension is
contributing to the problem.

Figure 38. Simply applying a low bandpass filter provides insufficient frequency filtering across multiple phases
of motion. While high frequency motion areas (like the bottom of the trajectory) match well, there are still points
which follow the raw signal too closely (like the top of the trajectory). Conversely, a lower cutoff frequency would
allow the top of the trajectory to match better, but the high frequency motion at the bottom of the trajectory would
begin to deviate significantly.
54

Because the ankle marker trajectory is insufficiently filtered by a single low bandpass filter, one looks to
environmental disturbances for the source of additional signal noise. For the purposes of this study, the environmental
interaction is measured through the ground reaction forces. Thus, one creates a spectrogram (also known as a moving
Short-time Fourier transform) of the vertical ground reaction forces to determine if high frequency tremors are present
during the post-impact phase, Figure 39. A spectrogram effectively creates an FFT of short periods of time,
sequentially, such that one is able to observe how signal frequency varies as a function of time. This type of analysis
is referred to as, time-frequency analysis.

Figure 39. The spectrogram clearly illustrates a bifurcation in the dynamic conditions to which the body is exposed.
During the flight phase, there are no environmental influences except the force of gravity, as illustrated by the pure
electrical noise in the force plate signal. This spectrogram suggests that the frequency content of signals during
flight and impact phase should be on the same order (i.e. minimal fluctuations). Thus, the ankle marker acceleration
spikes in the post-impact phase must be erroneous and improved filtering techniques are required. Note, frequency
content less than -90dB is omitted for clarity.

According to the spectrogram of the ground reaction force signals, the frequency content of the digitized marker is
expected to be minimal during the flight phase, spiking during the impact phase, and return to minimal content during
the flight and post-impact phases. Of course, exceptions to this may occur during the flight phase due to residuals
from the takeoff, Figure 40.
55

Figure 40. As expected, the ankle trajectory spectrogram mirrors the bifurcation illustrated by the GRF spectrogram,
particularly in the vertical force signal. These spectrograms have been normalized to their maximum magnitude
content, to illustrate the relative magnitude of the signal noise with respect to the true signal. Of course, this
normalization brings into sharp contrast the signal to noise ratio of the horizontal dimension. Focusing on the
vertical motion spectrogram on the right, this spectrogram suggests that the frequency content of signals during
flight and impact phase should be on the same order (i.e. minimal fluctuations) with a slight increase during the
impact phase. Thus, as observed, filtering to a single cutoff frequency is inadequate. For clarity, frequency content
less than -70dB is omitted.

The kinematic spectrogram suggests a need for time variant frequency filtering. Simply breaking the trajectory into
three pieces and implementing low bandpass filters with phase specific cutoff frequencies is likely to be insufficient,
because this method does not guarantee signal continuity at the phase boundaries, Figure 41. And indeed, due to these
discontinuities the resulting acceleration signal shows large (~2000m/s
2
) spikes in magnitude at the phase boundaries.

56

Figure 41. Filtering each phase of the landing is insufficient, as there is no guarantee of signal smoothness at the
boundaries between phases. While the discontinuities are not obvious in the position data, they manifest as large
spikes in the second derivative, acceleration data. Unfortunately, the crux of the analysis proposed in this work is
dependent on the second derivative for completion of the inverse dynamics equations. Thus, an alternative solution
must be developed.

Spline fitting is another common technique for smoothing position data, offering a continuous second derivative.
While this technique has, in the past, suffered from a need to define key data points to fit the smoothed functions to
the original data set, new techniques by MATLAB’s cubic smoothing spline algorithm, CSAPS allow for a much
greater degree of flexibility. However, even with this advanced spline fit method, one is faced with the same initial
difficulty of inconsistent signal to noise ratios. Holding the importance of each data point constant throughout the trial
results in a similar spread of inadequate smoothing regions, Figures 42-43.

57

Figure 42. Applying the CSAPS filtering method with constant weighting to the ankle trajectory data set results in
signals which suffer a constant degree of signal adherence throughout the entire trial, in much the same way as the
single low bandpass filtering technique. As shown in the top zoomed trajectory, it follows trajectory oscillations
too closely, despite an adequate degree of signal adherence during the impact phase of the trajectory in the bottom
zoomed trajectory.

Figure 43. With constant weighting function, the filtered trajectories produced by the cubic spline smoothing
method using Jackson’s cutoff technique are no better than those produced by a single low bandpass frequency
filter. Data during the impact phase appears to match well, but the acceleration spikes in the flight and post-impact
phases are still erroneous phenomena.

58

The novel piecewise weighted cubic spline (PWCS) method developed in this work provides an algorithm by which
one may adapt the CSAPS smoothing function to desired phase specific smoothnesses, while guaranteeing second
derivative signal continuity throughout the entire trial. Perhaps the most exciting part of this technique is that it
generates an explicit equation for the data which may be evaluated at any increased sampling rate and has at least two
explicit time derivatives, Figure 44.

Figure 44. Finally, the PWCS method provides a twice differentiable explicit equation which shows an adaptable
phase specific degree of smoothness. As expected, the result shows small accelerations during the flight phase, a
spike in marker acceleration during the impact phase, and minimal accelerations during the post-impact phase.

Implementing this technique requires applying the Jackson cutoff method three times. The first instance sets a
threshold on the second derivative of the impact phase residual taken with respect to the smoothing parameter, p, to
determine the necessary smoothness during the phase with the highest expected frequency content (determined by the
associated kinetics), p c. The weighting parameter of the impact phase, w IM, is now assigned a value of 1. The second
instance sets a threshold on the second derivative of the flight phase residual between the p c-smoothed trajectory and
the original signal as a function of the flight phase weighting parameter, w FP, as it decreases from a value of 1. The
cutoff phase weight for the flight phase occurs when the second derivative of the flight phase residual with respect to
w FP passes the user identified threshold. This process is repeated for the post-impact phase weight, w PP, resulting in a
piecewise continuous weighting scheme by phase, Figure 45.

59

Figure 45. The smoothing coefficient, p, is found using Jackson’s cutoff method on the impact phase residual as a
function of p, and it is applied to the entire data series. The weighting function changes as a step function, with
different data emphasis in each phase of the trial. A value of 1 is assigned to the impact phase, because the p value
already guarantees the proper level of matching, while the other two phase weights are significantly smaller to
increase the smoothness of the data during less tumultuous phases.

As a verification of smoothing quality, the adherence to the original signal is illustrated in a plot of the ankle trajectory,
with a 1 pixel error band applied to the smoothed curve such that the raw tracking curve stays within the band for all
time, Figure 46.

Figure 46. The trajectory of the raw digitized ankle marker (blue) vs. its cubic-spline smoothed trajectory (black)
during subject 1’s soft landing trial. The pink band around the black curve indicates a spatial error band of 1 pixel.
Because the raw trajectory curve stays within the error band, the PWCS smoothing technique is considered
sufficient for filtering the ankle trajectory across the mechanically unique phases of landing.

60

The differences in ankle marker frequency content before and after applying each type of filtering technique provides
a more intuitive look into the effects of these different filtering techniques, Figure 47. In summary, while there are
many different filtering techniques which may offer adequate trajectories and derivatives in more uniform conditions,
a more adaptable filtering technique is required for trials involving bifurcations in dynamics, especially impacts. The
piecewise weighted cubic spline smoothing technique provides this phase flexibility, while guaranteeing continuous
derivatives throughout the trial.

(a)

Figure 47. The frequency content of the ankle tracking marker varies with time, as shown by the raw signal
spectrograms. Thus, filtering techniques must be flexible to preserve high frequency content when present and
remove it otherwise. All techniques successfully retain the frequency spike during the impact phase. However, most
techniques also allow high frequency content through during other phases. Even the singe weight CSAPS method
fails to eliminate high frequency digital noise during the post-impact phase where there’s minimal ankle motion.
The PWCS method provides a means for preserving the impact phase high frequency content, while eliminating
high frequency noise during the flight and post-impact phases. A great interactive tool for understanding the
information presented in a spectrogram is found at: https://musiclab.chromeexperiments.com/Spectrogram
61

2.3.1.2 Filtering Differences between Markers
With the PWCS tool in hand, one is capable of filtering markers differently according to observably variant amplitudes
of high frequency content within a single trial. The frequency content of multiple marker raw trajectories are
characterized as a function of time to illustrate how marker placement position plays a role in frequency content,
Figure 48.

Figure 48. The distribution of frequency content illustrates how the impact waves traverse through the body.
Examining the vertical component in particular, one observes a slight time lag in the first frequency spike after
impact moving up the kinematic chain from foot to thigh. Because all of these peaks occur in the impact phase, the
PWCS method remains robust to these changes, and through the Jackson cutoff method, smoothing parameters and
phase weightings are adjusted automatically to fit the differences in frequency content.

Due to the differences in frequency content between markers, slightly different smoothing parameters and phase
weights are used for each marker, Figure 49.
62

Figure 49. The smoothing parameter and weighting values for each marker vary slightly depending on the frequency
content present in the signal. Of course the frequency content will be a function of their position on the body, with
markers closer to the point of contact showing greater magnitudes of high frequency content.
63

Applying this marker-specific adaptive filtering, one obtains smoothed trajectories for each marker whose
spectrograms illustrate the proper level of phase specific smoothing for each unique case, Figure 50.

Figure 50. After applying the PWCS method, the impact wave is more obvious, decreasing in frequency spectrum
going up the kinematic chain and lagging in time as expected[37]. Notice, in every case shown here, the frequency
content of the beginning and end phases has been reduced relative to the raw data set, while impact phase is smoothed
to a lesser extent.

By checking the vertical accelerations of key markers, one provides a final check that proper smoothing methods have
been applied. According to previous research[37], the typical head accelerations during a foot first impact are on the
order of three times gravity. Thus, plotting the vertical acceleration of the body segment centers of mass, one both
verifies the acceleration absorption by each segment in the kinematic chain which serves to protect the head, as well
as the resulting calculated acceleration of the head, Figure 51.
64

Figure 51. The vertical accelerations shown here are the result of the novel, PWCS method, which provides a means
of phase specific smoothing. A direct corollary of this is that the accelerations during the impact phase (bounded
by the vertical dashed lines) are significantly greater than those of the previous, flight phase, or the subsequent
post-impact phase. As expected, segment accelerations decrease as one moves up the kinematic chain from the foot
to the head. In addition, the maximal acceleration of the head is 25.90m/s
2
, which is around the benchmark of three
times gravity, indicating the filtering technique provides adequate trajectory and derivative results.

In summary, different markers undergo different levels of environmental influence and must therefore be filtered
according to their unique frequency content. The novel PWCS smoothing technique provides adequate results for all
markers under dynamic impact landing conditions as demonstrated by the preservation of high frequency content
during impact phase, and elimination of signal noise during phases where there should be less high frequency motion.
Finally, using the rule of thumb of head vertical acceleration during impact on the order of three times gravity provides
a good benchmark by which to compare filtered foot-first impact landing trajectory results.

2.3.2 Lower Sampling Rates

The PWCS method has proven successful in providing smoothed trajectories of kinematic data with an ultra-high
speed sampling rate (11000Hz), but the following section demonstrates that these results hold for data sets collected
at much lower sampling rates. In addition, these results demonstrate a lower limit for capturing biological impact
phenomena.
65

Downsampling the original data set generates a subset which simulates a kinematic signal taken at a lower sampling
rate. In order to make these downsampled signals comparable to the original data, they are re-upsampled through
linear interpolation. The PWCS fit applied to the ultra-high sampling rate acts as the gold-standard against which the
down-up sampled data sets are compared. Using the high sampling rate spline fit, rather than the original data set
minimizes the artificial fluctuations in data match quality which occur as a result of the specific choice of sample
subset, rather than the sampling rate alone. In this way, one gains an understanding of how much kinematic information
is lost as a function of the sampling rate, Figures 52-53.

Figure 52. The average trajectory error between the original PWCS fit and raw downsampled up-interpolated data
as a function of sampling rate indicates the undersampled ankle marker trajectory average error stays within 5mm
even for sampling rates as low as 30Hz. This measure means that the trajectory as a whole follows the benchmark,
high frequency PWCS fit, well for realistic sampling rates. The impact phase mirrors this trend, with the average
deviation from the benchmark crossing the 5mm threshold between 30-60Hz.

66

Figure 53. The maximum trajectory error between the original PWCS fit and raw downsampled up-interpolated
data as a function of sampling rate indicates the undersampled ankle marker trajectory does not deviate from the
benchmark by more than 5mm until the 90-120Hz range. The impact phase mirrors this trend, with the maximum
deviation from the benchmark crossing the 5mm threshold between 60-90Hz.

Plotting the instantaneous error between the downsampled/up-interpolated data set and the benchmark PWCS method
shows that the trajectory error is most problematic around the impact phase (immediately prior to and post) for
undersampled data, Figure 54.

67

Figure 54. Examining how the down-sampled, up-interpolated ankle marker trajectories compare with the
benchmark ultra-high frequency sampling rate PWCS trajectory gives one a sense of the problematic periods for
undersampled data. Clearly, a high sampling rate is necessary to fully capture the motion of the ankle during the
impact phase of a foot-first landing. Each horizontal dashed line depicts one pixel’s worth of error. Thus, even at
60Hz, the worst case scenario error, which occurs during the impact phase, is within 5pixels of the original data
set, which translates to error on the order of 8mm.

In an effort to demonstrate the robustness of the PWCS method, the same downsampling study is conducted on the
input signal to the PWCS method. By first downsampling the raw signal and then applying the PWCS method, the
resulting spline fit is only based on the downsampled set. This spline defines the trajectory fit that would be found if
only a subset of the original data were available (i.e. lower sampling rate). The downsampled data PWCS fit is
compared to the benchmark, high frequency PWCS fit to show the relative spline fitting technique quality for more
realistic sampling rates, Figures 55-56.

68

Figure 55. Similar to the downsampled up-interpolated trend, the average error of the PWCS fit ankle marker
trajectory remains low well beyond the anticipated 240Hz high speed sampling rate. In fact, the average trajectory
error does not pass the 1mm mark until the sampling rate is less than 120Hz. Note, this trend also holds for the
average error of the low sampling rate PWCS during the impact phase alone.

Figure 56. Interestingly the maximum error between the undersampled PWCS and the benchmark method shows
more variance than the downsampled up-interpolated equivalent. Similar to the downsampled up-interpolated trend,
the maximum error of the PWCS fit ankle marker trajectory remains low well beyond the anticipated 240Hz high
speed sampling rate. This means the worst case error between the undersampled PWCS and the benchmark PWCS
stays below 4mm until the sampling rate is less than 120Hz. The trend is slightly better for the maximum error of
the low sampling rate PWCS during the impact phase alone, crossing the 5mm threshold below 90Hz.
69

Plotting the instantaneous error between the downsampled data set PWCS trajectory and the benchmark PWCS
provides insight into the phases which are the most problematic for spline-fitting undersampled data, Figure 57.

Figure 57. Examining how the undersampled PWCS ankle marker trajectories compare with the benchmark ultra-
high frequency sampling rate PWCS trajectory gives one a sense of the problematic periods for low frequency
splined data. Interestingly, large errors occur immediately prior to, and after, the impact phase. This localized
deviation suggests that the step-like, phase weighting function transitions too quickly from high to low emphasis
at the phase boundaries. At sampling rates greater or equal to 120Hz, however, the worst case error never deviates
more than 2 pixels, or around 4.5mm from the benchmark PWCS curve. Again, each horizontal dashed line depicts
one pixel’s worth of error.

Illustrated in the spatial domain, one observes the trajectory deviations for undersampled PWCS trajectories as a
function of sampling first occur at times where sharp direction changes occur. Of course by 120Hz sampling rates, the
associated PWCS trajectories seem to follow the original digitized ultra-high frequency trajectory, even under these
conditions, Figure 58.

70

Figure 58. The resulting trajectories from the PWCS method applied to downsampled data continue to closely
match the original raw digitized ankle marker trajectory all the way down to 120Hz, even during the impact phase
(the end of which is denoted with a black circle). This shows the robustness of the method to changes in sampling
rate. In this case, the threshold on the residual acceleration was held constant for smoothing coefficient selection
across all sampling rates. Thus, these results are expected to improve if this threshold is also modified to the
sampling rate.

In summary, the downsampled up-interpolated trajectories adhere closely to the benchmark high frequency PWCS
trajectory for sampling rates as low as 120Hz. At this rate the maximum error during the entire trial is no more than
5mm. These downsampled up-interpolated trajectories have the poorest matching during the impact phase, while the
same measure for the undersampled PWCS trajectories match well during the impact phase, but not immediately prior
to or after the impact phase. Finally, the undersampled PWCS methods show the worst adherence during the sharpest
trajectory direction changes, but are within a 2 pixel limit for sampling rates greater than or equal to 120Hz.

2.3.3 Parsed 4-Element Functional Joint Axis Sample Study Results

As mentioned previously, this novel method of kinetically contextualized kinematic analysis provides an approach to
the analyzing complex multiplanar motion between two adjacent segments. In this study, the parsed 4-element angular
velocity technique has been applied to the shoulder joint, between the upper arm and the torso, during a wheelchair
user push cycle. Even in this relatively planar wheelchair push-cycle task, the method clarifies the changes in segment
control (i.e. coupling between segment angular velocity components). Because the axis of the internal/external
rotational motion tracks with the upper arm (distal segment), simply describing the angular velocity in terms of the 3
torso-fixed reference frame axes is insufficient for understanding the types of coordinations being utilized by the
subject. With the internal/external angular velocity contribution bleeding into each of the other three axes, coordination
trends are obscured. However, treating it as its own quantity quickly clarifies a shift in technique which occurs when
71

the wheelchair user is asked to push the wheelchair at two different rates, self-selected-free and self-selected-fast.
Notice, in this case, the graph in the bottom right corner shows that the subject has opted to increase the ratio of
horizontal ab/adduction to flexion/extension during their fast push cycles, as compared to their free push cycles.
Importantly, this shift is obscured when using only the 3 element functional axis definition (top right), Figure 59.

Figure 59. Comparing the angular velocity components from a wheelchair user’s normal-push cycles (blue) and
faster-then-normal push cycles (red), one observes how each of the angular velocity components evolve with
respect to the other. The top row of plots show ω parsed into the standard 3 torso-fixed axes. The bottom row shows
same three plots after internal-external rotation has been extracted illustrating the trends that become clarified in
the positioning angular velocity vector components when the crosstalk of internal-external rotation is parsed out.
Notice, the horizontal abduction/adduction component was particularly polluted by signals that would have hidden
the well-defined trends in the second row.

These types of technique patterns become particularly relevant when one begins to discuss task-techniques in terms
of controls which require definition of the functional (kinetic) context of an outcome. Recall, the task being studied in
this research is highly planar, so these trends are not as complex. However, one will see how joint coordination plays
a role in defining technique outcomes in Ch. 3 (Controls Based Modeling and Simulation), when the post-impact
phase joint coordinations are incorporated into the model.

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2.3.4 Initial Conditions

While each of these drop landing experiments demonstrates the effects of joint control as evidence for the inclusion
of feedback control in the model of the human body model, an important first step is verifying the uniformity of the
experimental initial conditions. The objective of this section is to illustrate that observed kinematic and kinetic
differences between drop landing conditions are the result of joint control modifications, not merely unique initial
conditions. Thus, key biomechanical measures are presented at standardized times with respect to impact. These key
measures include the center of mass vertical position at peak trajectory, joint angles, and body orientation.

The center of mass trajectory is consistent across the trials, Figure 60, with the each subject’s total body center of
mass reaching a repeatable peak height with respect to the surface.

Figure 60. A common point of comparison for the COM vertical velocity is difficult to define, with the relative
instant of impact changing according to trial specific reaching techniques. Thus, the COM vertical position at the
last instant that vertical velocity is greater than zero is shown for each condition. Any variation in this height
inevitably leads to variation in initial velocity at touchdown, though due to pull-up kinematics (explored in later
sections) this initial touchdown is unique to a condition as well. With a maximum difference of 5cm, these initial
conditions are assumed to be equivalent. Even if one assumes a difference in initial conditions at takeoff, the lack
of trend amongst the data points would suggest that this cannot be the cause of landing condition differences in the
impact phase.

Additional proof of initial condition commonality is shown in a film strip of digitized segment endpoints, Figure 61.
By overlaying the kinematics of each landing condition, time synced to impact, one observes both the relative
alignment of segments and their position in inertial space with respect to the digitized plate origin.

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Figure 61. This kinematic filmstrip of the segments of both subjects during all three landing conditions overlaid
for comparison, shows how the initial conditions during flight phase are nearly identical until shortly before
touchdown when the effects of subject joint control prepares the body for impact. All positions have been
calculated with respect to the digitized force plate origin, and the frames shown begin 0.25s prior to impact while
the subject is already in flight.

Given similar initial joint angles at take-off, differences in joint configuration at touchdown must be the result of flight
phase control. Thus, the joint angles for each subject are compared to the initial joint angles of the normal condition
at the peak point in the flight phase trajectory, Figure 62.

Figure 62. The joint angles at the peak of the flight phase trajectory serve as the check of initial conditions. The
human eye is typically capable of detecting angular differences on the order of 5-10deg. Since the angular
differences between initial joint angles for all three conditions are within this range, the joint angle initial conditions
are assumed to be equivalent.
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Finally, the total orientation of the body is difficult to define, so the initial body orientation will be measured by the
torso angle with respect to the inertial reference frame, Figure 63.

Figure 63. For all three landing conditions, the torso begins in a nearly vertical orientation. Both subjects show
initial torso angles around 88degrees, with variations of less than a degree between conditions. Thus, these initial
body orientations are considered equivalent for all tasks.

In summary, the vertical position of the center of mass at the peak of the flight phase trajectory is within 5cm for all
three conditions, in both subjects. The joint angles at this same instant are within +/-5 degrees of each other, for both
subjects across all three trial conditions. Finally, the total body orientation with respect to inertial begins nearly
vertical, as measured by the torso angle at the pinnacle of the flight phase trajectory. Overall, these initial experimental
conditions are considered to be uniform within each subject, so any experimental differences in kinematics or
dynamics across a subject’s trials after this apex trajectory point must be the result of the subject selected joint control
techniques.

2.3.5 Vertical Ground Reaction Force Variation

Given common initial conditions, one hypothesizes that a subject is capable of modulating the vertical ground reaction
forces according to intention. If a subject is capable of modifying their technique, and thus the vertical ground reaction
forces, then a responsive intentioned feedback element must be included in the model of the human body in order to
properly capture the causes behind the measureable differences in human dynamics.

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As hypothesized, there are measurable differences in ground reaction force between the trials as a function of landing
objective for both subjects. The maximum vertical ground reaction forces show a difference of 3.5 times body weight
in peak force (2200N), between the soft and hard trials respectively. Similarly, the impact phase duration increases as
impact peak reaction force decreases, Figure 64 and Table 2. These differences collectively modulate the impact phase
impulse, and consequentially, the effective vertical momentum reduction during that phase.

Figure 64. Both subjects show an ability to modulate their vertical ground reaction forces according to instructed
landing intensity conditions, self-selected normal, softer than normal, and harder than normal. Subject 1’s VGRF
peaks range from 2622 – 4815N for their soft to hard trials, respectively. Similarly, Subject 2’s VGRF peaks
range from 2636 – 7383N, for their soft to hard trials, respectively. Additionally, the duration of the impact phase
scales inversely to the VGRF peak, with both subjects impact phase decreasing by around 0.03s between the soft
and hard trials. Notice, the Y-axis are different for these charts due to the mass difference between subjects. The
force error bars, 1.3N, are too small to see.

Table 2: Comparing landing condition impulse properties for both subjects, the duration of impact decreases monotonically from
soft to hard, while peak reaction force increases monotonically from soft to hard.

SUBJECT 1 SUBJECT 2
SOFT NORMAL HARD SOFT NORMAL HARD
Peak Force: 2622.5 N 3060.4 N 4814.9 N 2636.0 N 3661.8 N 7382.8 N
ΔT Impact: 0.096 s 0.084 s 0.069 s 0.120 s 0.112 s 0.095 s
Impulse: 157.58 Ns 139.56 Ns 166.73 Ns 169.21 Ns 208.70 Ns 329.53 Ns
% Imp of Norm: 112.91 100 119.47 81.07 100 157.90

The impulse during the impact phase changes across the three landing objectives, with the duration of impact phase
increasing as peak ground reaction force decreases. For this reason, the hard and soft landing tasks show the highest
total impulse during the impact phase of landing. Additionally, the ground reaction force waveforms show a double
impact curve consistent with toe-heel contact described by Weyand[2], Figure 64.

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2.3.6 Observed Flight Phase Control: Kinematics

Despite initial kinematic equivalence, the three simple drop-landing experiments show a variety techniques for
successful drop landing. While the subjects exhibit an end-effector pull-up technique to reduce the absolute velocity
of the foot at impact, Figure 65, they used the technique to varying extents across their three different landing
conditions. In their soft landings, Subjects 1 and 2 reduce the vertical velocity of the ankle with respect to the ground
by as much as 0.44m/s and 0.52m/s, respectively. The hard landings, on the other hand only show vertical velocity
reduction by 0.1m/s and 0.25 m/s, respectively.

Figure 65. A first glance at the experimental kinematics demonstrates the each subject employs an active pull-up
technique in order to reduce the relative velocity of the foot with respect to the ground prior to impact. Perhaps
unsurprisingly, the soft landing trials show the greatest amount of pull-up for both subject 1 and 2, slowing the
ankle by 0.44m/s and 0.52m/s, respectively. Deviations from zero indicate active preparation for landing, with
differences in the positive direction showing segment pull-up, while negative values indicate segment reaching.

Each subject’s joints contributes to their overall pull-up technique to varying degrees. To understand the flight phase
technique, and thus how the pull-up is achieved, one examines the relative vertical velocity of each segment’s center
of mass with respect to its proximal joint. A positive difference between these two velocities shows the proximal joint
is dropping faster than the segment COM, which indicates that joint is contributing to the pull-up. The curves shown
in Figure 66 illustrate how the pull-up of each segment occurs in each trial. All three landings show the largest velocity
contribution from the hip, which contributes twice as much velocity reduction as the knee or ankle.

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Figure 66. Focusing on the moments leading to impact (the right edge), splines that end higher on the vertical axis
indicate a greater degree of segment specific pull-up in the trial. In general, the hard impact case shows the least
amount of pull-up, the normal impact shows medium pull-up, and the soft case shows the most pull-up, as shown
by the red > blue > green trend towards the end of the flight phase. Subject 2 shows some slight deviations in this
trend by reaching with the ankle in the soft landing, and pulling–up less with the hip during the normal landing. In
general, both subjects seem to favor hip-based pull-up techniques with either slight reaching or pull-up in the other
two segments.

In summary, both subjects exhibit a degree of pull-up during all the trials, with the soft trials showing the most pull-
up in general. While there are a variety of ways this pull up can occur, both subjects appear to favor a hip-based pull-
up technique.

2.3.7 Observed Flight Phase Control: Calculated Joint Moments

These kinematic differences must be the result of forces and torques, but the system body is not in contact with any
external elements during flight phase. Thus, the origin of this ability to change the ground reaction force time trace,
must derive from the application of internal net joint moments and forces. Flight phase control, as characterized by
these net joint moments during the flight phase, show significant differences in the coordination of the joints between
landing objectives.

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Figure 67. The column on the left shows the variation in control applied across the different landing conditions for
subject 1, while the column on the right shows the same for subject 2. The net joint moments shown are the product
of applying inverse dynamics to the observed kinematics. As expected, the hip shows the highest contribution to
the flight phase pull-up with NJMs on the order of 100s of Nm.

As the subjects are not in contact with the environment during the flight phase, these NJMs are applied solely in
preparation for landing, serving to orient and position segments in preparation for landing. In summary, hip torques
appear to be the main contributors to the pull-up techniques employed by both subjects.

2.3.8 Observed Impact Phase Control: Kinematics

With the landing gear positioning and orientation prepared during the flight phase, the body now must reduce the total
momentum of the system to a manageable level during the impact phase. Much like a vehicle, dissipating momentum
during an impact involves crumpling. The three simple drop-landing experiments show various levels of body collapse
in order to dissipate momentum over the impact phase, Figure 68. Both subjects show significantly less collapse in
the hard landing condition than the other two conditions. As this phase is about momentum reduction, the change in
vertical velocities of the three conditions are compared during the impact phase for both subjects, Figure 69.

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Figure 68. The degree of collapse during the impact phase is inversely proportional to the forces applied to resist
further collapse of the body. Thus, the relatively small collapse of the hard landing case indicates a large amount
of resistance to motion being generated within the subject. Alternatively, the other two cases show comparable
amounts of collapse, suggesting considerably less internal resistance to motion is supplied to the system.

Figure 69. During the impact phase, the downward vertical velocity of the center of mass is reduced by applying
different levels of resistive force to the ground. The net impulse from this resistance serves to reduce the overall
vertical momentum of the body. In this figure, the higher the curve finishes, the more momentum has been reduced.
Each subject reduces their momentum to the greatest degree during the hard landing condition followed by
comparable amounts in the normal and soft landing conditions. This evidence matches the ratios found in the
Vertical Ground Reaction Force Variation section, and serves as another check that both sides of the momentum
equation agree.
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In summary, the hard landing case shows the least amount of collapse and the greatest amount of momentum reduction
in both subjects. The soft and normal conditions show similar amounts of collapse in both subjects, but the soft case
reduces the momentum more in subject 1, while the normal case reduces more momentum for subject 2. Clearly,
momentum reduction is more complicated than simply the degree of collapse, with some internal mechanics at play.
The next section addresses the joint control techniques employed by these subjects to reduce vertical momentum
during the impact phase.

2.3.9 Observed Impact Phase Control: Calculated Joint Moments

The kinematic differences affect the measured impact phase ground reaction forces, but they do not provide the entire
explanation. In fact, ground reaction forces do not directly correlate to the degree to which a subject exhibits pull-
up/collapse, Figure 70. Thus, as is the case with the flight phase, the change the ground reaction force time trace must
also derive from the application of internal net joint moments and forces.

Figure 70. Evidence of effective subject stiffness during the impact phase showing how VGRF is more than a
function of collapse. With minimal difference in collapse between Subject 2’s normal and softer than normal
landings, VGRFs still show a decrease in maximum VGRF during impact. This suggests that the internal support
forces are reduced over the same degree of collapse, a phenomena which mirrors force generation of two springs
of different stiffness being compressed to the same extent. The stiffer spring will produce more force over the same
displacement than that of the softer spring. Thus, the human body control exhibited by this phase is comparable to
impedance control which manipulates a robot’s resistance to change in velocity according to a desired impedance.
The equivalent modification of human resistance manifests as effective stiffness and damping modulation,
appearing in the graph to the right as a change in the force-compression curve slope between the hard and soft
landings.

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Because impact phase is so short, and there is a large degree of high frequency acceleration, one should not over
interpret the net joint moments during the impact phase. However, in general, one sees a large increase the amount of
knee support across all three landing conditions. In addition, the hip exhibits significant oscillations in magnitude
throughout this phase, with increased oscillations occurring in the hard landing case, relative to the other two. The
main take away from this NJM information is a scaling effect occurs between the three landing conditions, with the
hard landing showing the most significant NJM magnitudes, with the ankle and knee showing the most directionally
consistent support moments. It is difficult to say exactly how each joint contributes to the momentum reduction process
as the contribution is also highly configuration dependent, Figure 71.

Figure 71. The impact phase NJMs are highly oscillatory and should not be over analyzed. Their net effect serves
to reduce the vertical momentum of the system to varying degrees, according to condition. Overall, both the knee
and ankle moments appear to consistently support of system during momentum reduction, and the magnitude of
the hard landing NJMs is of a greater magnitude than the other cases.

In summary, a greater amount of segment collapse does not provide the whole story for the reduction in vertical ground
reaction forces. The impact phase NJMs are noisy and highly oscillatory, so they should not be over-interpreted.
However, the knee and ankle show consistent support moments during the impact phase suggesting that they are
supporting the initial subset of mass, and momentum is reduced to a manageable amount for the post-impact phase.
In addition, the impact phase shows increased magnitude moments during the hard landing case, relative to the other
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two, suggesting that there is some level of NJM scaling taking place, though the dependency on segment configuration
may be obscuring the full picture.

2.3.10 Observed Post-Impact Phase Control: Kinematics

The objective of the post impact phase is final momentum reduction while maintaining overall system balance. Thus,
one first verifies that the velocity of the system is indeed brought to rest by plotting the vertical velocity of the system
during the post-impact phase, Figure 72.

Figure 72. In both cases, the system suspends its downward trajectory and begins to increase the center of mass
position to some desired equilibrium. While Subject 1 does not always reach the goal of rest before the trial period
is complete, Subject 2 continues the upward trajectory until reaching their equilibrium point. Note, achieving
equilibrium is delayed in conditions where the peak VGRF was lower, suggesting that missing the crucial initial
momentum window leaves one in a mechanically disadvantageous position which takes time to stabilize and
recover.

Next, one investigates the center of mass position with respect to the base of support to determine how the stabilization
process is achieved empirically, both in the horizontal and vertical directions, Figure 73.

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Figure 73. Horizontal equilibrium point appears to be ~1/3 the length of the foot as measured from the heel. Notice,
subject 1 stabilizes on the balls of their feet during the normal and soft landings, so this stability point shifts toward
the toes.

Similarly, the vertical COM position appears to gravitate to a different position depending on the individual. By
normalizing this measure to each subject’s height, one is able to compare the technique differences between subjects,
Figure 74.

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Figure 74. Vertical stability point is around 45% of subject 1’s total height, while subject 2 settles at 65% of their
height. This difference may be the result of gender differences in modelled mass distribution. As was the case with
the horizontal stability point, subject 1 does not return to their final stability point before the end of the data
collection period, though one can easily extrapolate the trends.

Interestingly, subjects not only gravitate to particular equilibrium points, but they also gravitate to particular joint
coordination ratios[54]. In broadening one’s understanding of the post-impact stabilization process, the joint
coordination patterns are identified suggesting that these manifolds should be a part of the control scheme used to
describe a subject’s technique, Figure 75

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Figure 75. Both subjects appear to operate on joint coordination manifolds specific to each subject. Subject 1 prefers
a hip-knee ratio around 1.3:1 (excluding the hard landing which does not span a broad enough angle range), with a
greater hip angle than knee angle. Conversely, subject 2 operates around a hip-knee ratio of 0.76:1, with a greater
change in knee angle than hip.

In summary, each subject brings their center of mass to rest at a desired equilibrium point over the base of support,
during the post-impact phase. The location of this stability point varies between subjects and may vary between trials,
though subject 1 does not reach equilibrium before the end of the soft or normal landing trials. Subject 2, on the other
hand, eventually returns to the same equilibrium point in all three trials. Throughout this phase, both subjects gravitate
to subject specific joint coordination manifolds, with subject 1 at a 1.3:1 hip-knee ratio while subject 2 employs a
0.76:1 ratio. Guiding the center of mass to a particular location with respect to the body fixed reference frame requires
multipurpose NJMs, which dissipate the residual system kinetic energy, counteract the effects of gravity and guide
the center of mass to the desired position. The empirical NJMs are discussed in the next section.

2.3.11 Observed Post-Impact Phase Control: Calculated Joint Moments

The post-impact phase kinetics must resolve the residual system momentum from the impact phase, and bring the
system to the final quasi-static rest condition. As such, one expects the resulting kinetics to show a variety of means
by which one reduces system momentum and recovers the system balance given differing initial conditions, Figure
76.

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Figure 76. NJMs during the post-impact phase peak at the tail end of the impact phase and gradually diminish
towards the end of the phase, settling into quasi-steady state conditions. In this steady state condition, hip and knee
joint angles are nearly maximal, and the ankle bears most of NJM cost.

These NJM time curves illustrate a variety of scaled NJM coordinations, but eventually all resolve towards a quasi-
static rest condition, characterized by plateauing minimal NJMs across all joints. While one easily observes these
stabilization trends in the NJMs, the mechanism by which stabilization is difficult to observe at the NJM level.

In summary, all NJMs peak at the beginning of the phase, when the subject must both reduce system momentum and
stabilize the COM, and they settle to a minimal constant as the system approaches quasi-static rest condition, when
the subject need only stabilize the COM. Subject 1’s knee and ankle NJMs do not approach zero because the subject
has not stabilized by the end of these two trials. Similarly, a sustained increase in knee moment magnitude for subject
2’s normal landing reflects a condition where their weight is still shifting horizontally and has not come to rest. In
general, this phase is characterized by large NJMs at the beginning and settling NJMs towards the end. Trends in the
COM point trajectory may indicate that the COM plays a more significant role in the task objective. Stabilization
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techniques adjust to handle the different levels of residual kinetic energy from the impact phase, bringing the COM to
rest over the base of support no matter these momentum differences.

DISCUSSION

In this section, the results of the drop landing experiments are discussed through the lens of human body control.

The novel PWCS data filtering method results represent a means of analyzing human control across dynamically
unique phases of a task. This ability to analyze kinematics across phases is important, because while the mechanical
objectives of each phase are unique, they must be understood in the common context of recovering stability of the
human body during a drop landing task. Thus, the phase specific results need to be tied together in order to fully
understand the reasons for mechanical phenomena.

The initial conditions of the experiment must be similar enough that their differences are not the primary explanation
for the observable differences in mechanical outcome.

The differences in vertical ground reaction forces, despite common initial conditions, supports the hypothesis that
human body control is an integral part of the mechanical system, and as such it must be included when modeling the
system.

The sources of these VGRF differences are multifaceted, with each phase of the task contributing to the dynamic
outcomes through different mechanisms.

1. During the flight phase, the subject adjusts their kinematics, such as segment pull-up, to provide different
initial conditions at the beginning of the impact phase.

2. During the impact phase, kinematic differences such as degree of body collapse contribute to the VGRF
differences. However, these do not provide the whole story, as multiple tasks demonstrate the same level of
collapse with differing VGRFs. Thus, internal mechanics, such as subject impedance must also be a
contributing factor.

3. During the post-impact phase, the mechanical objective involves bringing the body to rest, while maintaining
the COM over the base of support. Thus, differences in the post impact phase are characterized by the
kinematics which describe how the system is brought into alignment from the various initial conditions left
from the impact phase, as well as the kinetics which serve to push the system towards the common balanced
quasi-static mechanical objective despite these initial kinematic differences.
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Finally, many of these control based, mechanical characterizations are made possible by limiting the system to two
dimensions. In the context of a more complex multiplanar motion, characterization of control must begin through
analysis of kinetically contextualized kinematics. Thus, the parsed functional joint axis characterization provides a
means by which one can describe the complex range of shoulder motion in terms of the underlying mechanical
framework driving the motion.

The research questions presented in Ch. 1: Introduction / Background, at the beginning of this work, serve as a roadmap
by which the results from the previous section are addressed.

2.4.1 Frequency Content during Impact Landings

As expected, the frequency content of digitized kinematic signals varies significantly across the phases of foot-first
landings involving impact. Much like a hammer hitting an anvil, the ping of the collision begins at a high frequency
and quickly dissipates to low frequency motion. In the same way, one observes low frequency content during the flight
phase, followed by higher frequency content during the impact phase, and finally low frequency content again during
the post-impact phases. These frequency shifts are mirrored in spectrograms of the GRFs, providing independent
verification that these phase specific frequencies are authentic phenomena that should be preserved by the filter.

In addition to frequency shifts in kinematic data by phase, one also observes frequency shifts by marker placement on
the body. Similar to the results found in other studies[37], the markers towards the upper body experience far smaller
frequency shifts than those close to the sight of impact. This variation in content by placement suggests that a
sophisticated filtering technique is required to adequately preserve true high frequency content when it is present,
without the need to suffer erroneous high frequency noise when it isn’t. Thus, in the same way that filtering all phases
to the same smoothness is inadequate, filtering every marker trajectory signal in a phase to the same smoothness would
either preserve noise based oscillations in data which does not undergo large frequency changes, or it would smooth
out true oscillations in markers nearest to the site of impact.

2.4.2 PWCS Filter Comparison

The natural implication of this temporally shifting frequency content is that a multiphase task involving impact can
not be adequately filtered by previously existing methods. A single low frequency bandpass filter approach suffers the
problem of either filtering out significant high frequency motion during impact, or allowing errant high frequency
noise through during the flight and post-impact phases in an effort to preserve the impact phase. Similarly, filtering a
signal to various low bandpass frequencies throughout the trial results in trajectory discontinuities at phase boundaries.
While these discontinuities do not appear to be significant in the position data, the second time derivative, required
for inverse dynamics analysis, exhibits large spikes in magnitude from the discontinuous derivatives. Even traditional
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splining techniques, which do not rely on defining a cutoff frequency, suffer from similar phase specific trajectory
adherence issues. The smooth spline may follow a trajectory adequately during the impact phase, but fit too tightly
during the other two phases. Conversely, it can match the smooth low frequency motion of the flight and post-impact
phases, but smooth out fluctuations occurring during the high frequency impact phase. The answer to this problem is
the PWCS smoothing method, by which one may adaptively define the extent of smoothing temporally according to
kinetic indicators of frequency content. It can match the smooth low frequency data during the pre- and post-impact
phases, while decreasing its smoothness grade during the higher frequency impact phase.

2.4.3 PWCS Filtering Method

The novel filtering technique, PWCS, presented in this work demonstrates a methodology for designing adaptive
filtering methods such that different phases of a task may be smoothed according to their kinetically contextualized
frequency content. With the spectrogram of the GRF data serving as a phase content indicator, the PWCS method
produces a twice differentiable, smooth spline across three phases of motion without a common filtering frequency.
The differentiability of the spline guarantees that the acceleration calculations necessary for inverse dynamics analysis
are not plagued by the digital noise present in the original position measurements.

The novel PWCS data filtering method results represent a means of characterizing data across boundaries which, in
the past, would have meant conducting two separate studies. To be able to talk in specific terms about human body
control during a particular task, one needs to ensure that the initial conditions of the system at the beginning (or end)
of that task are equivalent so that there is a common point against which to measure technique differences. The same
necessity of commonality holds for phase specific analysis, otherwise one could not rule out initial condition
influences as being the root cause of phase specific differences. While this may be possible in some studies, it is not
a realistic requirement for controls based analysis of tasks such as the drop landing experiment. This is because some
reasons for the differences in mechanical outcomes of one phase are the ability to manipulate the initial conditions at
the beginning of the previous phase. For example, the measureable differences in VGRF during the impact phase are
partially explained by the kinematic initial conditions controlled during the flight phase. Thus, in the context of
controls, this parsing of tasks into different experiments is detrimental to the understanding of in situ controls, as the
dynamic context of one phase directly influences the next. Therefore, using the PWCS method allows one the luxury
of understanding the control mechanisms employed by the subject, in the context of the previous phase.

The PWCS method allows investigators to filter across dynamic condition bifurcations, opening the door to multi-task
analysis. An obvious extension of the ability to filter across phase boundaries is the ability to filter across task
boundaries. Because the dynamic context of some activities varies significantly, previous studies may have been
limited to dynamically common phases of the activity, like flight phases of the triple jump. However, given that this
technique as shown to produce useable kinematic results across task phases including impact, then by extension one
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is able to study activities which involve multiple impacts, like triple jump, in the context of the differing outcomes of
the previous phases.

2.4.4 Reduced Sampling Rates

Part of good data collection protocol is ensuring that one is sampling at rates adequate to capture the phenomena of
interest. For this research, this was not a problem, as the first experimental kinematic data set was captured with an
ultra-high speed camera at over 11000 frames per second. While this sampling rate is admittedly heavy-handed, it
provides an opportunity to determine the minimum rate required to capture the phenomena of foot first landings
involving impact. By downsampling the kinematic signals and re-upsampling them through interpolation, one
observes the frequency at which the maximum error between the raw trajectory and this artificial one begins to deviate
exponentially. For the ankle marker, the maximum error between the original data and the downsampled up-
interpolated trajectory remains 4-5mm until the sampling rate drops below 120Hz. Note, this is also the frequency
range where the impact phase becomes the one which contains the maximum error. This is not surprising as it is the
phase with the highest frequency content, and it will thus deviate from the sharp corners more significantly than the
relatively smooth flight and post-impact phases.

The PWCS method is only useful if this time-frequency based filtering technique is applicable at lower, more common
sampling rates. The results presented in the previous section demonstrate that the PWCS method does hold for reduced
sampling rates, with the maximum error of the PWCS method at lower frequencies remaining below 4mm with respect
to the high sampling rate spline, all the way until the sampling rate drops below 120Hz. Undersampled PWCS data
continues to match the high sampling rate PWCS trajectory well during the impact phase over the same frequency
range. This robustness even in the most demanding phase is likely due to the emphasis assigned to these data points
during the PWCS method. Recall, the smoothing factor is chosen on the residual fit to these points alone, and the other
two phases are fit by reducing the phase specific weights from that point. Because these data sets match the original
trajectory at sampling rates below 120Hz, the second drop landing experiment, collected at a 120Hz sampling rate can
use the same filtering techniques to generate second order smooth, acceleration time histories for the associated
digitized position data. Due to limits in data processing power, this suggests researchers can use data collection
techniques with lower temporal sampling rates, in favor of higher spatial resolution.

2.4.5 Parsed 4-Element Kinetically Contextualized Functional Joint Axis

While not used directly in the primary vein of this research project due to its 2D nature, the 3D 4-element parsing
technique serves to clarify the observed kinematics in terms of their kinetic influences. The results shown in the
previous section illustrate how the internal/external angular velocity crosstalk can serve to obfuscate the motion
coordination trends. Interestingly, the figure presented demonstrates a subject specific shift in technique as they switch
from normal push-cycles to faster-than-normal push-cycles. This shift is most evident in the horizontal
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abduction/adduction angular velocity plotted against the flexion extension axis. In this graph, the ratio of components
shifts to a new slope as a result of the task required change in propulsion speed. This shift alone indicates that task
time constraints may play a role in defining subject specific coordination strategies.

A strength of this type of analysis is the wide ranging applicability. The method does not rely on two adjacent joints
meeting any functional criteria, only that they demonstrate relative motion which can be compartmentalized. Thus,
while the example presented focuses on shoulder joint motion descriptions, this method applies to any joint in the
human body. Particularly useful for looking at ball and socket type joints, this technique may also present evidence of
internal/external rotation strains on knee joints during athletic activities.

Future work with this analysis tool will apply this approach to other joints and other tasks to extract the underlying
control coordination patterns. In fact, these control based kinematic motion studies inevitably lead one to investigate
the connection between the parsed angular velocities and the similarly parsed NJMs which drive them. Taking this
one step further, one may even evaluate the condition of these NJMs in terms of the power they supply to the joint,
describing moments in terms of stabilizing or actuating efforts, though these studies are left to future work. A parallel,
though different type of coordination trend is exploited in the post-impact phase modeling stage to capture subject
specific preferences for joint angle coordinations, a preference which tends to hold across tasks[54].

2.4.6 Initial Conditions

Another important check of experiment quality is ensuring only the phenomena being studied are allowed to vary
from trials to trial. This practice prevents unintentional, external factors from providing alternative explanations for
empirical observations. With this in mind, the initial conditions for each drop landing task are effectively equivalent,
with variations between peak trajectory height on the order of 5cm. Similarly, the both subjects’ joint angles at this
same instant in time vary by less than 5 degrees across the trials. Finally, the total body orientation, here measured as
the torso angle with respect to the inertial reference frame varies by less than 1 degree at this same instant in time. The
torso is effectively vertical for both subjects in all three trials. Verification of the commonality of initial conditions
provides assurance that the empirical differences in system dynamics are the result of changes in control technique,
not merely the result of different conditions.

2.4.7 Vertical Ground Reaction Force Variation

Neither the initial conditions, nor the physical system change between the three drop landing trials for each subject,
so any variation in VGRF waveforms derives from the inherent human ability to modify their system dynamics through
coordinated NJM control. As expected, subjects showed an ability to modulate the ground reaction forces during
impact according to investigator instruction, with peak ground reaction forces ranging from 2600N to 4800N for
subject 1 and 2600 to 7400N for subject 2. This ability to modify technique as a function of mechanical objective
92

provides strong evidence for the key claim of this research. Namely, mechanical models of the human body are
incomplete without the control algorithms which drive the dynamics of the system. Thus, the following sections detail
the characterization of observable biomechanical technique indicators such that they can be integrated into the human
body model via mechanically equivalent control laws. Accordingly, these techniques are broken down according to
the mechanical objectives of the phase in which they occur.

2.4.8 Observed Flight Phase Control

As mentioned, a key objective of this research is understanding the control schemes behind technique modification in
terms of a mechanical objective. During the flight phase, the velocity of the center of mass with respect to the ankle
prior to contact is a measureable technique modification. Using the total body COM as the system reference frame,
differences in this quantity suggest ankle velocity reduction prior to contact is a mechanism by which VGRF during
the impact phase is modified. Reducing the relative difference in speed between the ankle and ground by as much as
0.3m/s between hard and soft, both subjects employ this technique to achieve smaller VGRFs during impact.

2.4.9 Observed Impact Phase Control

Similar kinematic phenomena, observed in the impact phase, contribute to the reduction in VGRFs from hard to soft.
Much like the flight phase, the impact phase is characterized by the degree of body collapse. Namely, trials which
attempt to reduce VGRFs show a greater degree of collapse. Subject 1 collapses by an additional 8cm more than that
of the hard landing condition by the end of the impact phase during the soft landing. Likewise, subject 2 collapses by
an additional, 14cm in the soft and normal landings, than they do in the hard landing. However, both the soft landing
and the normal landing for subject 2 show the same amount of collapse during the impact phase, yet different VGRFs.

Another source of VGRF modulation must be the kinetics internal to the human system. These kinetics suggest that
one can generate different levels of force over the same span of collapse. Mechanically, this correlates to comparing
two springs of different stiffness. Compressing these two springs to the same extent produces different amounts of
spring resistance forces. Thus, the second mechanism by which impact phase force is reduced mirrors the modification
of spring stiffness according to a desired stiffness, a type of control known as impedance control.

A final note about the impact phase control is that oscillations in the impact phase NJMs may not be trustworthy due
to the rigid body assumptions applied through inverse dynamics, which assume immediate transfer of contact forces
through every segment. Contrarily, one directly observes the degree to which marker accelerations differ up the
kinematic chain, as mentioned in the previous section. Eventually, these acceleration fluctuations are mitigated over
the short time span of the drop-landing task. But with this caveat in mind, one should not over analyze the exact NJM
for comparison to simulated results.

93

2.4.10 Observed Post-Impact Phase Control

During the final phase of impact, the subjects reduce their residual kinetic energy, while bringing their total body
center of mass over their base of support, the foot. In this phase, the system is not characterized by what is different
so much as it is defined by how one achieves the same ends, from vastly different beginnings. Objectively, the subject
brings their center of mass over the base of support, but in doing so they operate on self-selected solution manifolds
defined by a joint coordination tendency of one joint angle to another. Both subjects show a strong tendency to
coordinate their joint angles, with subject 1 preferring a hip-knee ratio of 1.3:1. Subject 2, on the other hand, operates
with a 0.7:1 hip-knee ratio. As both subjects illustrate a tendency for joint coordination, this phenomena should be
captured by a model of the human body during the post-impact phase.

Again, the NJM at the beginning of this phase should not be over interpreted, as they are still oscillatory, however,
one can take away the general trend in the post-impact phase NJM waveforms. The NJMs for both subjects, at the
beginning of the post-impact phase, are of a large magnitude. This NJM magnitude reduces with time and eventually
plateaus at a NJM minimum. At this point, the NJMs are expected to demonstrate minimal levels of extensor torques,
with an exception being subject 2 normal landing, in which the subject’s weight shifts forward momentarily at the end
of the task.

CHAPTER SUMMARY

In this chapter, drop landing experiments were presented and analyzed in the context of human body control and its
importance to the dynamics of the mechanical system. First the task and collection setup is described. Next, the
kinematic data is processed using a novel filtering method resulting in a continuous description of human kinematics
across multiple mechanically-unique phases of a task. These measures allow for a cohesive characterization of
mechanically-oriented, phase-specific control techniques, demonstrating both the human capacity for control as well
as the interphase influence of each control architecture.

The initial conditions of each landing condition are found to be similar enough that their differences are not the primary
explanation for the observable differences in mechanical outcome. Despite common initial conditions, differences in
vertical ground reaction forces are observed between landing conditions, supporting the hypothesis that human body
control is an integral part of the mechanical system, and as such it must be included when modeling the system. The
sources of these VGRF variances are multifaceted, with human control contributing to the kinetic outcomes differently
during each phase. During the flight phase, changes in the subject’s kinematics, such as segment pull-up, provide
different initial conditions at the beginning of the impact phase. During the impact phase, kinematic differences such
as degree of body collapse contribute to the VGRF differences. However, these do not provide the whole story, as
multiple tasks demonstrate the same level of collapse with differing VGRFs. Thus, internal mechanics, such as subject
94

impedance must also be a contributing factor. During the post impact phase, the mechanical objective involves
bringing the body to rest, while maintaining the COM over the base of support. Thus, differences in the post impact
phase are characterized by the kinematics which describe how the system is brought into alignment from the various
initial conditions left from the impact phase, as well as the kinetics which serve to push the system towards the
balanced quasi-static mechanical objective despite these initial kinematic differences.

Finally, many of these control based, mechanical characterizations are made possible by limiting the system to two
dimensions. In the context of a more complex, multiplanar motion characterization of control must begin through
analysis of kinetically contextualized kinematics. Thus, the parsed functional joint axis characterization provides a
means by which one can describe the complex range of shoulder motion in terms of the underlying mechanical
framework driving the kinematics.

In summary, this chapter covered the collection and analysis of kinetic and kinematic data drop landing experiments
for two subjects with three different landing condition objectives. Tools and techniques used to analyze this data were
presented, including a novel method for smoothing kinematic data. The outcomes from these techniques are used with
inverse dynamics to illustrate the importance of human body control. The body control is characterized in the context
of drop-landing tasks, for the purposes of developing a sufficiently complex model of the human body to replicate
these phenomena. The next chapter proposes that a simple, rigid segment model that has been trained on an
experimental data set is capable of this capturing these phenomena during a drop landing task when driven by driven
nonlinear phase specific control.

95

CHAPTER 3

CONTROLS BASED MODELING AND SIMULATION

CHAPTER OVERVIEW

The following chapter details the methods and techniques used during each stage of this experimentally validated
modeling process. The model is first defined in terms of plant, anatomical components scaled to the subjects they
represent. Next, the simulation environment is defined providing a mechanical landscape for the human body model.
Then, the main impetus of this research, phase specific nonlinear control theory correlates to empirical human motor
control are defined for each phase of landing according to the phase-specific mechanical objectives. The model
validation process is explained with phase specific checks designed to verify that the model is faithful to
experimentally measured phenomena. The model is applied with modified conditions and control parameters to
human motor control based questions. Finally, one describes the results and their implications for the sufficiency of
nonlinear feedback control based human modeling.

MODEL DESIGN

The human body is composed of around 650 skeletal muscles which act in a coordinated effort to achieve a defined
kinematic and/or mechanical objective. Thus, the goal of this study is to develop a simple, though sufficiently complex,
experimental validated, forward dynamics 2D model of the human body driven by NJM control to capture the
dynamics of symmetric, foot-first drop landings and assesses it for accuracy by comparing the vertical ground reaction
forces found in simulation to those collected from its experimental counterpart. The dynamic model the human body
presented in this work is comprised of two fundamental elements: the physical plant and the control algorithms which
drive its behavior.

3.2.1 Mechanical Model Anatomy

Gruber et al. represented the human body as a 2D kinematic chain of three rigid links (a combined head, torso, and
arms segment, a thigh segment, and a shank segment) connected by ideal hinge joints with wobbling masses
attached[19]. Building upon this work, this study proposes a 2D model comprised of 4-rigid links connected by hinge
joints, reducing the number of degrees of freedom. A rigid foot segment is added to the end of the shank by another
hinge joint, similar to the model proposed in jumping simulation study by Bobbert & vanSoest[55]. Rather than using
the specific muscle activations for actuation, uniaxial net joint moment actuators were implemented to simplify
96

analysis. Each link is the summation of the symmetrical body segments, such that all of the mass lies in the sagittal
plane, and the single hinge joints represent both hip, knee, and ankle joints respectively.

3.2.1.1 Segment Parameters
Each model segment parameter (length, mass, moment of inertia, and COM location) is determined using deLeva’s
anthropometry data[36]. According to deLeva, one first defines average male/female body dimensions and mass
distributions normalized to their height and mass, respectively. Then, the length of each segment is determined by
multiplying the subject’s height by segment specific scaling ratios for the 50
th
percentile individual, which relates an
average person’s segment lengths to their height. Similarly, both the total body mass and the associated moment of
inertia are distributed across each body segment according to the anthropometry data, Tables 3-4. Employing the same
mass distributions and dimensions as used for inverse dynamics calculations, body segments lengths and masses, and
COM locations, and moments of inertia of symmetrical segments are combined to produce the 4-rigid links: a
combined head, torso, and arms segment, a thighs segment, a shanks segment, and a feet segment. The result is a
scaled 2D model with 4-rigid segments, connected at their endpoints by hinges, with lengths, masses, and moments
of inertia scaled to a 50
th
percentile subject of similar height and mass, analogous to that which is described in work
by Bobbert & vanSoest[55], Figure 77.

Table 3. Mass distribution parameters for model segments scaled to drop-landing subject 1 (1.575m, 64.5kg).
Segment
Length
[m]
Mass
[kg]
Longitudinal CoM L
(from proximal) [m]
Moment of Inertia
[kg×m
2
]

Torso 0.801 38.88 0.496 1.760
Thigh 0.334 18.27 0.137 0.278
Shank 0.392 5.58 0.175 0.063
Foot 0.207 1.77 0.091 0.007

Table 4. Mass distribution parameters for model segments scaled to drop-landing subject 2 (1.829m, 84.3kg).
Segment
Length
[m]
Mass
[kg]
Longitudinal CoM L
(from proximal) [m]
Moment of Inertia
[kg×m
2
]

Torso 0.930 50.83 0.577 3.298
Thigh 0.443 23.88 0.182 0.508
Shank 0.456 7.30 0.203 0.099
Foot 0.271 2.31 0.120 0.011

Upper
Body
Thigh
Shank
Foot
Upper
Body
Thigh
Shank
Foot
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Figure 77. Using SimMechanics, MATLAB simulation environment, the system plant is defined through deLeva
mass and segment length distributions[36]. Before this model can be integrated, however, each of the terms under
initial states must be quantified in order to fully define this simple 4 rigid segment model and its initial conditions
for simulation.

3.2.1.2 Torsional Hinge Joint Actuators
The standard 2D rigid-body model has also been modified to have ideal torsional actuators at each hinge joint. Rather
than a redistribution of all the model parameters (i.e. wobbling mass, spring constants, etc.) for each new landing
attempt, this model simulates different landing strategies by using control theory to manipulate the NJM inputs. This
manipulation more faithfully reflects the forces at work in the human body, which controls the system dynamics not
by body redefinition, but by alternate actuator coordination. Physically, actuation is caused by the intricate interplay
of muscles of different capacity with various origins and insertions acting in a coordinated effort to achieve specific
segment motion as well as desired environmental interaction. In an effort to maintain model simplicity, this study
attempts to replicate the net effects of these muscles through nonlinear phase specific control of net joint moments,
ideally similarly affecting the rigid body-environment interaction. Thus, torque actuators are used to produce the
resulting NJM at each lower extremity joint of the body.

System Plant Definition
• Mass Distribution: Masses and Moments of Inertia
– 𝑚 𝑡𝑜𝑟𝑠𝑜 , 𝑚 𝑡ℎ 𝑔ℎ
, 𝑚 𝑠ℎ𝑎𝑛𝑘 , 𝑚 𝑓𝑜𝑜𝑡
• Note: 𝑚 𝑡𝑜𝑟𝑠𝑜 = 𝑚 𝑡𝑜𝑟𝑠𝑜 + 𝑚 ℎ𝑒𝑎
+ 𝑚 𝑛𝑒𝑐𝑘 + 𝑚 𝑎𝑟𝑚𝑠 – 𝐼 𝑡𝑜𝑟𝑠𝑜 , 𝐼 𝑡ℎ 𝑔ℎ
, 𝐼 𝑠ℎ𝑎𝑛𝑘 , 𝐼 𝑓𝑜𝑜𝑡
• See next slide for 𝐼 𝑡𝑜𝑟𝑠𝑜 details
• Segment lengths
– 𝐿 𝑡𝑜𝑟𝑠𝑜 , 𝐿 𝑡ℎ 𝑔ℎ
, 𝐿 𝑠ℎ𝑎𝑛𝑘 , 𝐿 𝑓𝑜𝑜𝑡
– Heel to Ankle dimensions: 𝐿 ℎ𝑒𝑒𝑙
, 𝐿 ℎ𝑒𝑒𝑙
– COM Longitudinal Positions (deLeva):
𝐶
𝑡𝑜𝑟𝑠𝑜 , 𝐶
𝑡ℎ 𝑔ℎ
, 𝐶
𝑠ℎ𝑎𝑛𝑘 , 𝐶
𝑓𝑜𝑜𝑡
• See next slide for 𝐶
𝑡𝑜𝑟𝑠𝑜 details
• Initial States
– Joint angles: 𝜃 ℎ
, 𝜃 𝑘𝑛𝑒𝑒 , 𝜃 𝑎𝑛𝑘𝑙𝑒 – Joint angular velocities: 𝜔 ℎ
, 𝜔 𝑘𝑛𝑒𝑒 , 𝜔 𝑎𝑛𝑘𝑙𝑒 – Base Segment (Torso) COM position: 𝑥 𝑡𝑜𝑟𝑠𝑜 , 𝑦 𝑡𝑜𝑟𝑠𝑜 – Base Segment (Torso) COM velocity: 𝑣 𝑥 ,𝑡𝑜𝑟𝑠𝑜 , 𝑣 𝑦 ,𝑡𝑜𝑟𝑠𝑜 – Base Segment (Torso) angle: 𝜃 𝑡𝑜𝑟𝑠𝑜 – Base Segment (Torso) angular velocity: 𝜔 𝑡𝑜𝑟𝑠𝑜 𝜃 𝑓𝑜𝑜𝑡 𝜃 𝑎𝑛𝑘𝑙𝑒 𝜃 𝑘𝑛𝑒𝑒 𝜃 ℎ
Y
X
98

3.2.2 Ground-Foot Interaction

Like Gruber[23] and Pain[22], the ground reaction force in the vertical (Z) direction is modelled as a nonlinear spring
and damper, while the horizontal (X) ground contact force is modeled as Coulomb friction, Eq. 49. The model
presented in this study uses a modified version of these ground reaction force equations, as originally described by
Munaretto[56]:

[
𝐹 𝑥 ,
𝐹 𝑦 ,
] =
{

[
𝜇 𝐹 𝑦 ,
tanh( 100𝑥 ̇
)
𝐶 𝑦 |𝑦
|
𝐶 𝑒𝑥𝑝
− 𝐷 𝑦 |𝑦
|𝑦 ̇
], 𝑦
< 0
[
0
0
], 𝑦
≥ 0
( 49 )

where 𝐶 𝑦 , 𝐶 𝑒𝑥
, and 𝐷 𝑦 are the optimizable VGRF-coefficients representing floor-stiffness, stiffness-hardening, and
floor-damping respectively. Similarly, the coefficient of friction, 𝜇 , has a value of 1 to prevent horizontal slipping
during impact. The subscript, i, represents the point at which the force is applied (heel, toe), the dot accent represents
a time derivative, and (x, y) are the horizontal and vertical position coordinates, respectively. However, the equations
of the VGRF, as written, do not prevent a net VGRF in the -Y direction (i.e. suction), should the damping force
component magnitude be larger and more negative than the stiffness component. This scenario occurs when the
contacting body point rebounds upwards at a high enough rate. For this reason, the definition is modified such that the
magnitude of the damping force pointed in the -Y direction is upper bounded by the positively oriented stiffness force,
Eq. 50.

[
𝐹 𝑥 ,
𝐹 𝑦 ,
] =
{

[
𝜇 𝐹 𝑦 ,
tanh( 100𝑥 ̇
)
𝐶 𝑦 |𝑦
|
𝐶 𝑒𝑥𝑝
− 𝐷 𝑦 |𝑦
|𝑦 ̇
], 𝑦
≤ 0 .𝑡 . 𝐹 𝑦 ,
≥ 0
[
0
0
], 𝑦
> 0
( 50 )

As discussed in the Open Loop Control: Kinematics and Kinetics Section, initial open loop control simulations using
Gruber’s reaction force coefficients[23] result in VGRF-time curves which fall outside the range of those measured
experimentally, Figure 78. One contributing factor to this mismatch is that the simulation environment ground reaction
force equations are kinematics dependent as demonstrated by Eq. 50. Evaluating this forward dynamic simulation
shows that simulated kinetics and kinematics have a circular relationship, the kinetics drive the kinematics (via F =
ma) and the kinematics define the kinetics (via the equations above). Therefore, in order to properly tune the simulated
ground reaction force equations to reflect those measured experimentally, these kinetic, ground reaction force
coefficients must be defined during the same optimization which finds the control coefficients that govern the human
kinematics.

99

Figure 78. Running the model in open loop control, as expected, does not produce the desired VGRF results. The
mere fact that this model simplifies the real system means there are differences between the NJMs according to
inverse dynamics and those required to reproduce the VGRFs in simulation. While initial frames of the simulation
show the open loop NJMs helping the model achieve some flight phase landing preparation, the slight
simplifications of the model, even in this simple phase, lead to compounding differences between VGRF result
sets. This means control and ground coefficients require further optimization as well as feedback control to drive
real time compensation for the effects of plant simplification. The exemplar example shown here is the VGRF
result from driving the model of subject 1 with empirical NJMs from subject 1’s normal landing condition.

Ground reaction force coefficient optimization is initialized with a linear fit calculation, using the experimental
kinematic and kinetic data in a least squares formulation, Eq. 51. Cropping each experimental data set to frames in
which the subject is in contact with the ground, all three drop landing trials (self-selected softer than normal, normal,
and harder than normal) are appended one after another such that the resulting least squares fit is applicable to every
trial. Note, kinematic contact is defined as times when the segment contact point in question is lower than or equal to
its position at the end of the trial (flat footed), measured in the vertical direction. The duration of contact varies for
each point of contact, so the corresponding vector entries of any unpaired samples are backfilled with zeros to ensure
common vector dimensions.

𝑨𝒙 = 𝒃 ( 51 )

100

where the matrix 𝑨 represents the measured kinematic data, vector 𝒃 represents the measured kinetic data, Eq. 54, and
vector 𝒙 contains the coefficients of interest, Eq. 53. The measured kinematic data matrix, 𝑨 , is modified such that
each of its columns contain on of the vertical ground reaction force (VGRF) terms from the Gruber equations, Eq. 52.

𝑨 𝒊 = [|𝒚 𝑡𝑜𝑒 |
𝐶 𝑒𝑥𝑝 ,𝑖 + |𝒚 ℎ𝑒𝑒𝑙
|
𝐶 𝑒𝑥𝑝 ,𝑖 −|𝒚 𝑡𝑜𝑒 |𝒚 ̇ 𝑡𝑜𝑒 − |𝒚 ℎ𝑒𝑒𝑙
|𝒚 ̇ ℎ𝑒𝑒𝑙
] ( 52 )

𝒙 = [
𝐶 𝑦 𝐷 𝑦 ] ( 53 )

𝒃 = 𝑭 𝑦 ( 54 )

where 𝐶 𝑒𝑥 ,
indicates the nonlinear spring stiffening power. Because this term is nonlinear, the linear fit equation is
solved for a range of different 𝐶 𝑒𝑥
values in a brute force search approach to determine which stiffness power provides
the best fit, resulting in the values shown in Table 5. The corresponding fit shows how closely a modelled VGRF
would match the actual VGRF, using only these experimental kinematic data as inputs, Figure 79.

(a) (b)

Figure 79. A simple linear fit technique produces a simulated VGRF curve as a function of the experimentally
measured contact points (Toe, Heel) kinematics. (a) When varying the spring stiffening power, one finds a local
minimum in the overall fit residual at C exp = 3.77. (b) The resulting signal is a poor fit to the VGRFs found
experimentally, as the fitting process should actually be subject to bidirectional influence (i.e. kinematics define
kinetics, kinetics influence kinematics). Because this fit only provides a unidirectional fit (kinematics to kinetics),
the ground coefficients are further optimized in the upcoming Impact Phase: Impedance Control section.

Table 5. List of initial VGRF coefficients determined through linear fit optimization using experimentally measured
kinematic data of the two contact points (Heel and Toe) and experimental VGRF measurements as the source data.
Contact Points Stiffness (𝑪 𝒚𝟎
) [
𝐍 𝐦 ] Stiffening (𝑪 𝒆𝒙𝒑𝟎 ) Damping (𝑫 𝒚𝟎
) [𝐍 𝐬 𝐦 ]
Heel, Toe 229305977 3.77 4149

These ground reaction force coefficients are further optimized during subject 1’s normal landing impedance control
optimization due to the interdependent nature of environmental admittance and subject impedance. This trial is
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arbitrarily selected to serve as the environment training trial. After the reaction force coefficients are defined from this
normal landing case, they are not modified again. Holding these floor coefficients constant and successfully recreating
empirical VGRFs in all simulations serves two functions. First, because the floor does not need to change for any
other trial in order to find an acceptable control coefficient set for recreating the experimental VGRFs, it suggests that
there is no loss of generality from the this arbitrary selection of environment training trial. Second, any subsequent
modification of VGRF-time curves across trials must be the result of feedback control modification, as the floor is
held constant and the initial conditions are effectively equivalent. This process is expanded upon in the Calibration:
Control Optimization Algorithm Section.

NONLINEAR CONTROL THEORY APPLIED TO HUMAN BODY CONTROL

Without motor control, the human body is merely a kinematic chain of masses connected by synovial joints, a system
which would collapse in a heap under its own weight. Recognizing that conscious humans do not move in this way
suggests there is a continuous series of commands being sent to the system actuators, the muscles, to prevent this
natural collapse. Similarly, without feedback from the many sensory organs, these signals applied to the actuators
would cause blind, uncoordinated motion unless they were perfectly tuned. Even in the case of perfectly tuned signals,
the system would deviate from its goal when presented with any degree of disturbance or noise. As humans are
regularly able to achieve their task objectives, one can infer the presence of feedback, by which the body is able to
correct for differences between the intended motion and those actualized by the body.

In this section, one addresses the main focus of this dissertation, the addition of phase specific nonlinear feedback
control to the simple, rigid-segment plant detailed in the previous sections, as a means of capturing the adaptive
dynamics of the human body observed during drop landing experiments. First, as a proof of concept, open-loop inverse
dynamics derived NJMs are applied to the system in an open-loop control architecture. Then, the three different types
of joint torque control are derived and applied to this drop landing model: joint angle-based following control, force-
based impedance control, and position-based energy-shaping control. These three algorithms are applied to the flight
phase, impact phase, and post-impact phase of landing, respectfully.

3.3.1 Open Loop Control: Kinematics and Kinetics

In a world of perfect models, one would be able to apply the net joint moments calculated from experimental inverse
dynamics directly to each modelled joint actuator, and the resulting simulated kinematics and kinetics would match
those found experimentally, Figure 80. However, models are, by definition, simplifications of complex physical
properties that provide insight into the underlying mechanisms which can explain observations by boiling them down
to explicit mathematical relationships. These simplifications inherently introduce disagreement between practical and
theoretical human dynamics due to digitization, linearization, and other assumptions. Even ones calculations with
102

inverse dynamics assume rigid body motion, neglect soft tissue artifacts. Due to these limitations, it is unlikely that
empirical NJMs drive the modelled system in the same way as its experimental counterpart.

Figure 80. An open loop control configuration (a) offers no means of compensating for disturbances, d(s), or errors
between an intended reference trajectory, q r(s), and the actualized trajectory, q’(s). In human terms, this would
mean a total lack of knowing whether or not the supplied NJM time-series is following their intended set of joint
angle trajectories as a function of time. From a modeling perspective, this is implemented by calculating the
experimental NJMs via inverse dynamics, and applying these directly to the model as a function of time.

When applied in the context of a forward dynamic simulation, the empirical NJMs derived from experimental data
through inverse dynamics are generally insufficient for reproducing the experimental trial results. This is disconnect
with reality is particularly evident when the modelled subject comes into contact with modelled environments,
compounding the number of simplifications and assumptions. In the laboratory conditions, the environment is defined,
so the human subject need only adapt their technique to that environment to successfully execute a drop landing task.
By contrast, the simulation environment must be properly defined and tuned to the associated experimental setup, and
then the model must adapt to these environmental approximations in order to reproduce the desired kinetic and
kinematic measures for which it is designed. Recall from the previous section, the modelled ground reaction forces,
in a necessary departure from reality, rely on kinematic penetration of the floor to quantify the degree of environmental
influence. This definition alone guarantees the experimental NJMs are insufficient, because the subject does not
penetrate the floor during the experiment.

In a baseline proof of concept study, one simply applies the inverse dynamic moment time curves to the model in an
open loop control architecture to demonstrate the need for closed loop control, Figure 81. One hypothesizes, open
loop application of these torques does not produce the desired kinematic outcomes due to imperfections between the
model and the subject (simplifications, digital, mathematical noise, etc.). The reader may once again refer to Figure
78 of the previous section for evidence to this end, showing how the VGRF waveform generated through blind torque
application does not reflect the experimental outcomes. The fundamental differences between the physical and the
System Plant Definition
• Mass Distribution: Masses and Moments of Inertia
– 𝑚 𝑡𝑜𝑟𝑠𝑜 , 𝑚 𝑡ℎ 𝑔ℎ , 𝑚 𝑠ℎ𝑎𝑛𝑘 , 𝑚 𝑓𝑜𝑜𝑡
• Note: 𝑚 𝑡𝑜𝑟𝑠𝑜 = 𝑚 𝑡𝑜𝑟𝑠𝑜 + 𝑚 ℎ𝑒𝑎 + 𝑚 𝑛𝑒𝑐𝑘 + 𝑚 𝑎𝑟𝑚𝑠 – 𝐼 𝑡𝑜𝑟𝑠𝑜 , 𝐼 𝑡ℎ 𝑔ℎ , 𝐼 𝑠ℎ𝑎𝑛𝑘 , 𝐼 𝑓𝑜𝑜𝑡
• See next slide for 𝐼 𝑡𝑜𝑟𝑠𝑜 details
• Segment lengths
– 𝐿 𝑡𝑜𝑟𝑠𝑜 , 𝐿 𝑡ℎ 𝑔ℎ , 𝐿 𝑠ℎ𝑎𝑛𝑘 , 𝐿 𝑓𝑜𝑜𝑡
– Heel to Ankle dimensions: 𝐿 ℎ𝑒𝑒𝑙 , 𝐿 ℎ𝑒𝑒𝑙
– COM Longitudinal Positions (deLeva):
𝐶 𝑡𝑜𝑟𝑠𝑜 , 𝐶 𝑡ℎ 𝑔ℎ , 𝐶 𝑠ℎ𝑎𝑛𝑘 , 𝐶 𝑓𝑜𝑜𝑡
• See next slide for 𝐶 𝑡𝑜𝑟𝑠𝑜 details
• Initial States
– Joint angles: 𝜃 ℎ , 𝜃 𝑘𝑛𝑒𝑒 , 𝜃 𝑎𝑛𝑘𝑙𝑒 – Joint angular velocities: 𝜔 ℎ , 𝜔 𝑘𝑛𝑒𝑒 , 𝜔 𝑎𝑛𝑘𝑙𝑒 – Base Segment (Torso) COM position: 𝑥 𝑡𝑜𝑟𝑠𝑜 , 𝑦 𝑡𝑜𝑟𝑠𝑜 – Base Segment (Torso) COM velocity: 𝑣 𝑥 ,𝑡𝑜𝑟𝑠𝑜 , 𝑣 𝑦 ,𝑡𝑜𝑟𝑠𝑜 – Base Segment (Torso) angle: 𝜃 𝑡𝑜𝑟𝑠𝑜 – Base Segment (Torso) angular velocity: 𝜔 𝑡𝑜𝑟𝑠𝑜 𝜃 𝑓𝑜𝑜𝑡 𝜃 𝑎𝑛𝑘𝑙𝑒 𝜃 𝑘𝑛𝑒𝑒 𝜃 ℎ
Y
X
Simulated
VGRF
Simulated
Joint Angles
Simulated
NJMs
Experimental
NJMs
Subject VGRF
Intentions
Inverse
Dynamics
Experimental
VGRFs
103

virtual task are not limited to ground reaction force contact. The joint angles for two systems with different control
schemes (open-loop vs closed loop) are compared during flight phase to illustrate the fundamental difference between
the two architectures. In effect, the open loop control case deviates and continues to deviate from the empirical joint
angles, while the closed loop system detects joint angle errors and corrects the NJMS accordingly, following the joint
angles within 7 degrees in all three cases, Figure 81.

(a) Open loop control joint angle comparison (b) PD Following control joint angle comparison

Figure 81. As an initial proof of the necessity of closed loop control, these simulated joint angle time curves (dashed
blue) illustrate how model simplifications result in kinematics mismatch with the equivalent experimental measures
(black) when directly applying the torque time curves which result from inverse dynamics in an open-loop control
scheme. Closed loop joint angle feedback control (b), on the other hand, results in tighter joint angle following, as
described in the Flight Phase: Following Control section.

3.3.2 Flight Phase: Following Control

While it is difficult to know a subject’s intended joint trajectories (sometimes referred to as “efferent copy”[57]) with
any certainty, one can use closed-loop feedback control architecture to describe the inferable trajectory tracking that
voluntary configuration and positioning would entail. Much like a robot, when a subject desires to strike a certain
time-variant pose in flight, and they may apply joint torques scaled to the difference between their current joint angles
and those of their goal, until their segments and joint angles match this configuration. The closeness by which the
body actualizes these poses correlates to the effective control-driven joint stiffnesses, as closer following / stronger
disturbance rejection would be the result of stiffer joints. In a similar way, the proposed model uses joint-angle
following control to track the simulation joint angles to the empirical kinematics during the flight phase, Figure 82.

104

Figure 82. A block diagram of the flight-phase following control feedback, shows the use of experimental joint
angles and associated derivatives as the reference trajectory inputs to the system. While these reference trajectories
would ideally be functions of the objective, this is a complex problem left to future work in this field.

As one has no direct knowledge of the subject’s configuration and positioning intentions, having access to observable
flight phase control kinematic outcomes only, one assumes the experiment subjects exhibit ideal control in flight to
achieve their intentions exactly. Thus, during the flight phase, the model uses these empirical joint angle time-histories
as functional reference joint trajectories. Subsequently, the modelled joint moment actuators operate on the scaled
joint angle errors through PD-following control to compensate differences between the simulated model kinematics
and these empirical joint angle time histories.

The objective of this flight phase control is proper segment and joint moment preparation for the impact phase[58].
Thus, the associated flight phase control gains are assigned values high enough to ensure that the segment
configurations (and their derivatives) tightly match those of the experiment at touch down. As a result, the simulated
initial conditions for impact are true to the experiment, Figure 83.

NJM PD Tracking Control:
𝑇 = 𝐾
𝜃 𝑒𝑥
− 𝜃 𝑠 𝑚

+ 𝐾
𝜃 ̇ 𝑒𝑥
− 𝜃 ̇ 𝑠 𝑚
Joint
Angle
Error
Experimental Joint Angles
𝜃 ̇ 𝜃 𝜃 ̇ 𝜃 Simulated VGRF
Simulated NJMs
Sim Joint Angles
-
+
System Plant Definition
• Mass Distribution: Masses and Moments of Inertia
– 𝑚 𝑡𝑜𝑟𝑠𝑜 , 𝑚 𝑡ℎ 𝑔ℎ , 𝑚 𝑠ℎ𝑎𝑛𝑘 , 𝑚 𝑓𝑜𝑜𝑡
• Note: 𝑚 𝑡𝑜𝑟𝑠𝑜 = 𝑚 𝑡𝑜𝑟𝑠𝑜 + 𝑚 ℎ𝑒𝑎 + 𝑚 𝑛𝑒𝑐𝑘 + 𝑚 𝑎𝑟𝑚𝑠 – 𝐼 𝑡𝑜𝑟𝑠𝑜 , 𝐼 𝑡ℎ 𝑔ℎ , 𝐼 𝑠ℎ𝑎𝑛𝑘 , 𝐼 𝑓𝑜𝑜𝑡
• See next slide for 𝐼 𝑡𝑜𝑟𝑠𝑜 details
• Segment lengths
– 𝐿 𝑡𝑜𝑟𝑠𝑜 , 𝐿 𝑡ℎ 𝑔ℎ , 𝐿 𝑠ℎ𝑎𝑛𝑘 , 𝐿 𝑓𝑜𝑜𝑡
– Heel to Ankle dimensions: 𝐿 ℎ𝑒𝑒𝑙 , 𝐿 ℎ𝑒𝑒𝑙
– COM Longitudinal Positions (deLeva):
𝐶 𝑡𝑜𝑟𝑠𝑜 , 𝐶 𝑡ℎ 𝑔ℎ , 𝐶 𝑠ℎ𝑎𝑛𝑘 , 𝐶 𝑓𝑜𝑜𝑡
• See next slide for 𝐶 𝑡𝑜𝑟𝑠𝑜 details
• Initial States
– Joint angles: 𝜃 ℎ , 𝜃 𝑘𝑛𝑒𝑒 , 𝜃 𝑎𝑛𝑘𝑙𝑒 – Joint angular velocities: 𝜔 ℎ , 𝜔 𝑘𝑛𝑒𝑒 , 𝜔 𝑎𝑛𝑘𝑙𝑒 – Base Segment (Torso) COM position: 𝑥 𝑡𝑜𝑟𝑠𝑜 , 𝑦 𝑡𝑜𝑟𝑠𝑜 – Base Segment (Torso) COM velocity: 𝑣 𝑥 ,𝑡𝑜𝑟𝑠𝑜 , 𝑣 𝑦 ,𝑡𝑜𝑟𝑠𝑜 – Base Segment (Torso) angle: 𝜃 𝑡𝑜𝑟𝑠𝑜 – Base Segment (Torso) angular velocity: 𝜔 𝑡𝑜𝑟𝑠𝑜 𝜃 𝑓𝑜𝑜𝑡 𝜃 𝑎𝑛𝑘𝑙𝑒 𝜃 𝑘𝑛𝑒𝑒 𝜃 ℎ
Y
X
Simulated
NJMs
105

Figure 83. The simulated kinematics overlaid on the experimental kinematics illustrate the close match of the
simulated flight phase joint angles. This suggests that the model accurately captures the human capacity for
preparation for impact, mirroring the empirical pull-up techniques demonstrated by the subjects.

3.3.3 Impact Phase: Impedance Control

In the field of mechanical engineering, the collective resistive effect of a system’s mass, stiffness, and damping to the
rate of external disturbances is referred to as its impedance. For impedance control, a system uses directly measured
or indirectly calculated (via environment modeling) reaction force feedback to regulate reference trajectory inputs
such that the plant’s local environment (point of interaction) experiences forces equivalent to a desired effective
impedance[59]. Using both kinetic and kinematic feedback to shape the plant-environment interaction stiffness mirrors
the subject-ground stiffness formalisms of Farley and Ferris[9,10], formalisms which describe a runner’s dynamics in
terms of effective leg stiffness.

The proposed study models the human body using the kinematic chain enhanced with impedance control to express
this active variation of subject stiffness. Because the experimental subjects performed each landing trial with different
mechanical objectives (i.e. increasing or decreasing vertical ground reaction force with respect to a self-selected
normal landing), each trial exemplifies a unique subject-specific impedance set. Therefore, the impedance control
coefficients alone were optimized across the trials and the resulting variation in values illustrates this adaptable
intended stiffness.

One would be remiss if discussion of the other half of the plant-environment interaction was neglected. The definition
of ground reaction forces presents significant difficulties to the impedance optimization process because it requires
the extent of environmental interaction to be defined kinematically, in terms of plant motion with respect to an
environmental boundary (floor). As mentioned previously, this means defining a nonlinear spring-damper to act as a
106

floor when the segment-fixed contact points exceed the boundary[23], [56]. Notice, there are two major drawbacks to
this formulation:

1. During contact, the colliding body must unrealistically penetrate the boundary to generate a reaction force,
which means that simply following the empirical subject motion could not result in the empirical ground
reaction forces, as shown in Section Ground-Foot Interaction.

2. Defining kinetics kinematically is an exercise in circular logic. In forward dynamic simulations, reaction
forces are unknowable until the kinematics are observed, and these very same kinematics are influenced by
those aforementioned forces. Thus, there is no way to predict the kinematic or kinetic outcomes of this
interaction short of evaluating the simulation.

For these reasons, the ground reaction force spring-damper coefficients are included in the impedance optimization
process for the (subject 1), normal landing trial, as discussed in Section Ground-Foot Interaction. By limiting their
inclusion to one trial, any differences in simulated VGRFs across trials are the result of impedance control
manipulation, not changes to the environment. The result of this integrated impedance control, ground reaction force
optimization is a set of VGRF coefficients which produces VGRFs similar to those measured experimentally, when
exposed to the corresponding simplified model landing kinematics, Table 6.

Table 6. Finalized VGRF coefficients found via simultaneous nonlinear VGRF and impedance coefficient optimization trained on
subject 1’s normal landing trial, minimizing the difference between simulated VGRF and experimental VGRF measures.
Contact Points Stiffness (𝑪 𝒚 ) [𝐍 /𝐦 ] Stiffening (𝑪 𝒆𝒙𝒑 ) Damping (𝑫 𝒚 ) [𝐍𝐬 /𝐦 ]
Heel, Toe 4.5632e+08 3.4037 11926

3.3.3.1 Impedance Control Structure
Weyand’s work[2] suggests each foot-first impulse is the summation of two smaller impulses which describe the
momentum reduction of two subsystems, a smaller and a larger mass. Accordingly, the body is divided into two
subsystems, the lower extremity system and the rest of the body. Each subsystem must therefore possess its own
effective impedance during an impact, resulting in two distinct momentum reductions which sum to the measured
experimental VGRFs for each landing. Two lines of action are proposed for impedance regulation, total body COM
to ankle and femur condyle to calcaneus, Figure 84.

107

Figure 84. Impedance control uses force and position feedback to coordinate the modulation of joint torques such
that the interaction between a system and its environment forces matches the dynamics of a desired impedance. In
this case, the distance and relative velocity between two endpoint pairs are each compared to a corresponding
desired length and rate time series to generate spring-damper-like behavior between the pairs of points. Each line
of action shows the axis along which the associated spring-damper forces are applied.

Beyond simply controlling one’s stiffness, a subject can also modify their reference trajectory (i.e. the path which they
desire an end-effector to follow), which St-Onge and Feldmen[60] refer to as the “referent configuration”. To
understand this, imagine a set of springs along the aforementioned lines of action. The unloaded length of each spring
is a time dependent virtual trajectory, such that it produces force when impedance endpoints deviate from this length.
In a similar way, the damping component of the impedance also generates force if the rate reference trajectory length
change is different than that of the impedance endpoints. These internal impedance forces serve to both drive segment
motion and/or oppose external forces. With two sets of these active spring-dampers, the model is able to independently
modulate the momentum of both subsystems, accommodating the bimodal impulse theory mentioned previously.

3.3.3.2 Forward Kinematics
The impedance forces described previously are applied to the model through a transformation matrix known as the
pseudo-inverse Jacobian, which uses the relationship between task-space and joint-space to convert the task-space
control forces into joint space torques. In order to create the Jacobian, one must first define the task-space location of
an end effector in terms of joint coordinates. This process, known as “forward kinematics” allows one to describe the
Cartesian location of any point on the body with respect to an inertial space origin, simply by knowing the lengths of
each segment and the angles of each joint. Using the Denavit-Hartenberg (DH) convention, process is made simple
through ordered matrix multiplication, with each transformation matrix describing the translation and rotation required
to move from one segment origin to the next as shown in, Eqs. 55-58. Deriving the forward dynamics, the torso is
used as the task space reference frame relative to which key segment point locations are expressed. For simplicity, the
superior end of the torso segment (referred to here as the head) serves as the torso reference frame origin. At the end
of this mathematical process, a matrix transformation is applied such that the results are expressed with respect to the
foot (heel) reference frame in order to make use of the assumption that the foot is flat on the ground. Recall the task
is assumed to be planar and thus all kinematics are limited to the XY-plane.

108

3.3.3.2.1 Kinematic Notation
The shorthand notation used for each element of the forward kinematic process is described in this section.
L ix : the length of the i
th
segment
L COMix : the longitudinal position of the i
th
segment COM, as measured from the proximal end
L COM3y : the superior-inferior position of the foot COM described with respect to the ankle
L 3Hx : the longitudinal position of the heel described with respect to the ankle
L 3Hy : the superior-inferior position of the heel with respect to the ankle
𝑹
𝑗
( 𝑞
) : the attitude of the i
th
frame with respect to the j
th
frame, as expressed in j
th
reference frame coordinates
𝑹 𝑗
( 𝑞
)= [
cos𝑞
−sin𝑞
sin𝑞
cos𝑞
]
Note, this describes a rotation by 𝑞
about the z-axis (i.e. perpendicular to the plane of motion)
𝒅
𝑗
: the position of the i
th
frame origin relative to the j
th
frame origin, as expressed in j
th
reference frame
coordinates
𝒅 𝑗
= [ 𝑥 𝑗
𝑦 𝑗
1]
𝑇
Note, the final row of this vector does not represent position in the z-direction, but rather acts as a place
holder tying the rotation and translation calculations together when following the DH multiplication
convention. As such, it is excluded when the vectors are not being used in the DH context.
𝑻
𝑗
: the DH transformation matrix describes a position in the i
th
frame as viewed from the j
th
reference
frame
𝑻 𝑗
= [
𝑹 𝑗
( 𝑞
)
0 0
𝒅 𝑗
]
𝒑
𝑗 ,𝑘 : the position of the k
th
point with respect to the i
th
reference frame origin, as expressed in j
th
reference
frame coordinates
𝒑 𝑗 ,𝑘 = [ 𝑥 𝑗 ,𝑘 𝑦 𝑗 ,𝑘 1]
𝑇

3.3.3.2.2 Denavit-Hartenberg Forward Kinematics Calculations
The components of the forward kinematics calculations are defined in advance of their ordered assembly to reduce
the need for side derivations during the forward kinematic chain descriptions. First, the orientation of a link (child)
reference frame is defined with respect to the orientation of the previous link (parent) in the kinematic chain. Next,
using parent link coordinates, the displacement of the child’s reference frame origin is described with respect to the
origin of the parent’s link. Finally, each key point of the child link is defined in child reference frame coordinates.

Torso:
R: Orientation of the torso with respect to the inertial reference frame: 𝑹 0
0
( 0)= 𝑰
d: Displacement of Torso Origin (Head) w.r.t. Torso Reference Frame: 𝒅 0
0
= [0 0 1]
𝑇
T: DH Transformation from Torso to Head Reference Frame:
109

𝑻 0
0
= [
𝑹 0
0
( 0)
0 0
𝒅 0
0
] = [
1 0 0
0 1 0
0 0 1
] ( 55 )
Position of Head in Torso Reference Frame:
𝒑 0
0,1
= [0 0 1]
𝑇
Position of Torso COM in Torso Reference Frame:
𝒑 0
0,2
= [𝐿 𝐶𝑂𝑀 0𝑥 0 1]
𝑇
Position of Hip in Torso Reference Frame:
𝒑 0
0,3
= [𝐿 0𝑥 0 1]
𝑇

Thigh:
R: Orientation of Thigh w.r.t. Torso Reference Frame: 𝑹 0
1
( −𝑞 1
)
- Here 𝑞 1
, represents the hip angle as the outer angle swept from knee-to-hip vector to the hip-to-head vector.
d: Displacement of Thigh Origin (Hip) w.r.t. Torso Reference Frame: 𝒅 0
1
= 𝒑 0
0,3

T: DH Transformation from Thigh to Torso Reference Frame:
𝑻 0
1
= [
𝑹 0
1
( −𝑞 1
)
0 0
𝒅 0
1
] ( 56 )
Position of Hip in Thigh Reference Frame:
𝒑 1
1,1
= [0 0 1]
𝑇
Position of Thigh COM in Thigh Reference Frame:
𝒑 1
1,2
= [𝐿 𝐶𝑂𝑀 1𝑥 0 1]
𝑇
Position of Knee in Thigh Reference Frame:
𝒑 1
1,3
= [𝐿 1𝑥 0 1]
𝑇
Approximate position of Femoral Epicondyle (FE) in Thigh reference frame: (discussed later)
𝒑 1
1,4
= [
19
20
𝐿 1𝑥 0 1]
𝑇

Shank:
R: Orientation of Shank w.r.t. Thigh Reference Frame: 𝑹 1
2
( −𝑞 2
)
- Here 𝑞 2
, represents the angle of the knee.
d: Displacement of Shank Origin (Knee) w.r.t. Thigh Reference Frame: 𝒅 1
2
= 𝒑 1
1,3

T: DH Transformation from Shank to Thigh Reference Frame:

𝑻 1
2
= [
𝑹 1
2
( −𝑞 2
)
0 0
𝒅 1
2
]
( 57 )
Position of Knee in Shank Reference Frame:
𝒑 2
2,1
= [0 0 1]
𝑇
Position of Shank COM in Shank Reference Frame:
𝒑 2
2,2
= [𝐿 𝐶𝑂𝑀 2𝑥 0 1]
𝑇
Position of Ankle in Shank Reference Frame:
110

𝒑 2
2,3
= [𝐿 2𝑥 0 1]
𝑇

Foot:
R: Orientation of Foot w.r.t. Shank Reference Frame: 𝑹 2
3
( 𝜋 − 𝑞 3
)
- Note: Here 𝑞 3
, represents the angle of the ankle.
d: Displacement of Foot Origin (Ankle) w.r.t. Shank Reference Frame: 𝒅 2
3
= 𝒑 2
2,3

T: DH Transformation from Foot to Shank Reference Frame:
𝑻 2
3
= [
𝑹 2
3
( 𝜋 − 𝑞 3
)
0 0
𝒅 2
3
] ( 58 )
Position of Ankle in Foot Reference Frame:
𝒑 3
3,1
= [0 0 1]
𝑇
Position of Foot COM in Foot Reference Frame:
𝒑 3
3,2
= [𝐿 𝐶𝑂𝑀 3𝑥 − 𝐿 3𝐻𝑥
−𝐿 𝐶𝑂𝑀 3𝑦 1]
𝑇
Position of Toe in Foot Reference Frame:
𝒑 3
3,3
= [𝐿 3𝑥 − 𝐿 3𝐻𝑥
−𝐿 3𝐻𝑦
1]
𝑇
Position of Heel in Foot Reference Frame:
𝒑 3
3,4
= [−𝐿 3𝐻𝑥
−𝐿 3𝐻𝑦
1]
𝑇

Using the aforementioned notation, expressing each point in the torso reference frame is merely a process of defining
the transformation matrices describing the effect of each joint angle and segment length, Eqs. 55-58.
Head: 𝒑 0
0,1
= [0 0 1]
𝑇
Torso COM: 𝒑 0
0,2
= [𝐿 𝐶𝑂𝑀 0𝑥 0 1]
𝑇
Hip: 𝒑 0
0,3
= [𝐿 0𝑥 0 1]
𝑇
Thigh COM: 𝒑 0
1,2
= 𝑻 0
1
𝒑 1
1,2
= [
𝑹 0
1
( −𝑞 1
)
0 0
𝒅 0
1
][𝐿 𝐶𝑂𝑀 1𝑥 0 1]
𝑇
= [
cos𝑞
−sin𝑞
𝐿 0𝑥 sin𝑞
cos𝑞
0
0 0 1
][
𝐿 𝐶𝑂𝑀 1𝑥 0
1
]
= [
𝐿 𝐶𝑂𝑀 1𝑥 cos( −𝑞 1
)+ 𝐿 0𝑥 𝐿 𝐶𝑂𝑀 1𝑥 sin( −𝑞 1
)
1
]
Knee: 𝒑 0
1,3
= 𝑻 0
1
𝒑 1
1,3
= [
𝑹 0
1
( −𝑞 1
)
0 0
𝒅 0
1
][𝐿 1𝑥 0 1]
𝑇
= [
𝐿 1𝑥 cos( −𝑞 1
)+ 𝐿 0𝑥 𝐿 1𝑥 sin( −𝑞 1
)
1
]
Thigh FE: 𝒑 0
1,4
= 𝑻 0
1
𝒑 1
1,4
= [
𝑹 0
1
( −𝑞 1
)
0 0
𝒅 0
1
][
19
20
𝐿 1𝑥 0 1]
𝑇
111

= [
19
20
𝐿 1𝑥 cos( −𝑞 1
)+ 𝐿 0𝑥 19
20
𝐿 1𝑥 sin( −𝑞 1
)
1
]
Shank COM: 𝒑 0
2,2
= 𝑻 0
1
𝑻 1
2
𝒑 2
2,2
= [
𝑹 0
1
( −𝑞 1
)
0 0
𝒅 0
1
][
𝑹 1
2
( −𝑞 2
)
0 0
𝒅 1
2
][𝐿 𝐶𝑂𝑀 2𝑥 0 1]
𝑇
= [
𝑹 0
1
( −𝑞 1
)
0 0
𝒅 0
1
][
𝐿 𝐶𝑂𝑀 2𝑥 cos( −𝑞 2
)+ 𝐿 1𝑥 𝐿 𝐶𝑂𝑀 2𝑥 sin( −𝑞 2
)
1
]
= [
( 𝐿 𝐶𝑂𝑀 2𝑥 cos( −𝑞 2
)+ 𝐿 1𝑥 )cos( −𝑞 1
)− ( 𝐿 𝐶𝑂𝑀 2𝑥 sin( −𝑞 2
) )sin( −𝑞 1
)+ 𝐿 0𝑥 ( 𝐿 𝐶𝑂𝑀 2𝑥 cos( −𝑞 2
)+ 𝐿 1𝑥 )sin( −𝑞 1
)+ ( 𝐿 𝐶𝑂𝑀 2𝑥 sin( −𝑞 2
) )cos( −𝑞 1
)
1
]
= [
𝐿 𝐶𝑂𝑀 2𝑥 cos( −𝑞 1
− 𝑞 2
)+ 𝐿 1𝑥 cos( −𝑞 1
)+ 𝐿 0𝑥 𝐿 𝐶𝑂𝑀 2𝑥 sin( −𝑞 1
− 𝑞 2
)+ 𝐿 1𝑥 sin( −𝑞 1
)
1
]
Ankle: 𝒑 0
2,3
= 𝑻 0
1
𝑻 1
2
𝒑 2
2,3
= [
𝑹 0
1
( −𝑞 1
)
0 0
𝒅 0
1
][
𝑹 1
2
( −𝑞 2
)
0 0
𝒅 1
2
][𝐿 2𝑥 0 1]
𝑇
= [
𝑹 0
1
( −𝑞 1
)
0 0
𝒅 0
1
][
𝐿 2𝑥 cos( −𝑞 2
)+ 𝐿 1𝑥 𝐿 2𝑥 sin( −𝑞 2
)
1
]
= [
𝐿 2𝑥 cos( −𝑞 1
− 𝑞 2
)+ 𝐿 1𝑥 cos( −𝑞 1
)+ 𝐿 0𝑥 𝐿 2𝑥 sin( −𝑞 1
− 𝑞 2
)+ 𝐿 1𝑥 sin( −𝑞 1
)
1
]

Foot COM: 𝒑 0
3,2
= 𝑻 0
1
𝑻 1
2
𝑻 2
3
𝒑 3
3,2

= [
𝑹 0
1
( −𝑞 1
)
0 0
𝒅 0
1
][
𝑹 1
2
( −𝑞 2
)
0 0
𝒅 1
2
][
𝑹 2
3
( 𝜋 − 𝑞 3
)
0 0
𝒅 2
3
][𝐿 𝐶𝑂𝑀 3𝑥 − 𝐿 3𝐻𝑥
−𝐿 𝐶𝑂𝑀 3𝑦 1]
𝑇
= [
𝑹 0
1
( −𝑞 1
)
0 0
𝒅 0
1
][
𝑹 1
2
( −𝑞 2
)
0 0
𝒅 1
2
][
( 𝐿 𝐶𝑂𝑀 3𝑥 − 𝐿 3𝐻𝑥
)cos( 𝜋 − 𝑞 3
)− ( −𝐿 𝐶𝑂𝑀 3𝑦 )sin( 𝜋 − 𝑞 3
)+ 𝐿 2𝑥 ( 𝐿 𝐶𝑂𝑀 3𝑥 − 𝐿 3𝐻𝑥
)sin( 𝜋 − 𝑞 3
)+ ( −𝐿 𝐶𝑂𝑀 3𝑦 )cos( 𝜋 − 𝑞 3
)
1
]
= [
𝑹 0
1
( −𝑞 1
)
0 0
𝒅 0
1
][
( 𝐿 𝐶𝑂𝑀 3𝑥 − 𝐿 3𝐻𝑥
)cos( 𝜋 − 𝑞 3
− 𝑞 2
)+ 𝐿 𝐶𝑂𝑀 3𝑦 sin( 𝜋 − 𝑞 3
− 𝑞 2
)+ 𝐿 2𝑥 cos( −𝑞 2
)+ 𝐿 1𝑥 ( 𝐿 𝐶𝑂𝑀 3𝑥 − 𝐿 3𝐻𝑥
)sin( 𝜋 − 𝑞 3
− 𝑞 2
)− 𝐿 𝐶𝑂𝑀 3𝑦 cos( 𝜋 − 𝑞 3
− 𝑞 2
)+ 𝐿 2𝑥 sin( −𝑞 2
)
1
]
= [
( 𝐿 𝐶𝑂𝑀 3𝑥 − 𝐿 3𝐻𝑥
)cos( 𝜑 )+ 𝐿 𝐶𝑂𝑀 3𝑦 sin( 𝜑 )+ 𝐿 2𝑥 cos( −𝑞 2
− 𝑞 1
)+ 𝐿 1𝑥 cos( −𝑞 1
)+ 𝐿 0𝑥 ( 𝐿 𝐶 𝑂 𝑀 3𝑥 − 𝐿 3𝐻𝑥
)sin( 𝜑 )− 𝐿 𝐶𝑂𝑀 3𝑦 cos( 𝜑 )+ 𝐿 2𝑥 sin( −𝑞 2
− 𝑞 1
)+ 𝐿 1𝑥 sin( −𝑞 1
)
1
]
where 𝜑 = 𝜋 − 𝑞 3
− 𝑞 2
− 𝑞 1

Toe: 𝒑 0
3,3
= 𝑻 0
1
𝑻 1
2
𝑻 2
3
𝒑 3
3,3

= [
𝑹 0
1
( −𝑞 1
)
0 0
𝒅 0
1
][
𝑹 1
2
( −𝑞 2
)
0 0
𝒅 1
2
][
𝑹 2
3
( 𝜋 − 𝑞 3
)
0 0
𝒅 2
3
][𝐿 3𝑥 − 𝐿 3𝐻𝑥
−𝐿 3𝐻𝑦
1]
𝑇
= [
𝑹 0
1
( −𝑞 1
)
0 0
𝒅 0
1
][
𝑹 1
2
( −𝑞 2
)
0 0
𝒅 1
2
][
( 𝐿 3𝑥 − 𝐿 3𝐻𝑥
)cos( 𝜋 − 𝑞 3
)− ( −𝐿 3𝐻𝑦
)sin( 𝜋 − 𝑞 3
)+ 𝐿 2𝑥 ( 𝐿 3𝑥 − 𝐿 3𝐻𝑥
)sin( 𝜋 − 𝑞 3
)+ ( −𝐿 3𝐻𝑦
)cos( 𝜋 − 𝑞 3
)
1
]
112

= [
𝑹 0
1
( −𝑞 1
)
0 0
𝒅 0
1
][
( 𝐿 3𝑥 − 𝐿 3𝐻𝑥
)cos( 𝜋 − 𝑞 3
− 𝑞 2
)+ 𝐿 3𝐻𝑦
sin( 𝜋 − 𝑞 3
− 𝑞 2
)+ 𝐿 2𝑥 cos( −𝑞 2
)+ 𝐿 1𝑥 ( 𝐿 3𝑥 − 𝐿 3𝐻𝑥
)sin( 𝜋 − 𝑞 3
− 𝑞 2
)− 𝐿 3𝐻𝑦
cos( 𝜋 − 𝑞 3
− 𝑞 2
)+ 𝐿 2𝑥 sin( −𝑞 2
)
1
]
= [
( 𝐿 3𝑥 − 𝐿 3𝐻𝑥
)cos( 𝜑 )+ 𝐿 3𝐻𝑦
sin( 𝜑 )+ 𝐿 2𝑥 cos( −𝑞 2
− 𝑞 1
)+ 𝐿 1𝑥 cos( −𝑞 1
)+ 𝐿 0𝑥 ( 𝐿 3𝑥 − 𝐿 3𝐻𝑥
)sin( 𝜑 )− 𝐿 3𝐻𝑦
cos( 𝜑 )+ 𝐿 2𝑥 sin( −𝑞 2
− 𝑞 1
)+ 𝐿 1𝑥 sin( −𝑞 1
)
1
]

Heel w.r.t. Thigh: 𝒑 1
3,4
= 𝑻 1
2
𝑻 2
3
𝒑 3
3,4

= [
𝑹 1
2
( −𝑞 2
)
0 0
𝒅 1
2
][
𝑹 2
3
( 𝜋 − 𝑞 3
)
0 0
𝒅 2
3
][−𝐿 3𝐻𝑥
−𝐿 3𝐻𝑦
1]
𝑇
= [
𝑹 1
2
( −𝑞 2
)
0 0
𝒅 1
2
][
−𝐿 3𝐻𝑥
cos( 𝜋 − 𝑞 3
)− ( −𝐿 3𝐻𝑦
)sin( 𝜋 − 𝑞 3
)+ 𝐿 2𝑥 −𝐿 3𝐻𝑥
sin( 𝜋 − 𝑞 3
)+ ( −𝐿 3𝐻𝑦
)cos( 𝜋 − 𝑞 3
)
1
]
= [
−𝐿 3𝐻𝑥
cos( 𝜋 − 𝑞 3
− 𝑞 2
)+ 𝐿 3𝐻𝑦
sin( 𝜋 − 𝑞 3
− 𝑞 2
)+ 𝐿 2𝑥 cos( −𝑞 2
)+ 𝐿 1𝑥 −𝐿 3𝐻𝑥
sin( 𝜋 − 𝑞 3
− 𝑞 2
)− 𝐿 3𝐻𝑦
cos( 𝜋 − 𝑞 3
− 𝑞 2
)+ 𝐿 2𝑥 sin( −𝑞 2
)
1
]

Heel w.r.t. Torso: 𝒑 0
3,4
= 𝑻 0
1
𝒑 1
3,4

= [
𝑹 0
1
( −𝑞 1
)
0 0
𝒅 0
1
][
−𝐿 3𝐻𝑥
cos( 𝜋 − 𝑞 3
− 𝑞 2
)+ 𝐿 3𝐻𝑦
sin( 𝜋 − 𝑞 3
− 𝑞 2
)+ 𝐿 2𝑥 cos( −𝑞 2
)+ 𝐿 1𝑥 −𝐿 3𝐻𝑥
sin( 𝜋 − 𝑞 3
− 𝑞 2
)− 𝐿 3𝐻𝑦
cos( 𝜋 − 𝑞 3
− 𝑞 2
)+ 𝐿 2𝑥 sin( −𝑞 2
)
1
]
= [
−𝐿 3𝐻𝑥
cos( 𝜑 )+ 𝐿 3𝐻𝑦
sin( 𝜑 )+ 𝐿 2𝑥 cos( −𝑞 2
− 𝑞 1
)+ 𝐿 1𝑥 cos( −𝑞 1
)+ 𝐿 0𝑥 −𝐿 3𝐻𝑥
sin( 𝜑 )− 𝐿 3𝐻𝑦
cos( 𝜑 )+ 𝐿 2𝑥 sin( −𝑞 2
− 𝑞 1
)+ 𝐿 1𝑥 sin( −𝑞 1
)
1
]

Calculating the total body COM location, for example, entails defining it with respect to the body origin, Eq. 46.

𝒑
0
𝐶𝑂𝑀 =
( 𝒑 0
0,2
𝑚 𝑡𝑜𝑟𝑠𝑜 + 𝒑 0
1,2
𝑚 𝑡 ℎ 𝑔 ℎ
+ 𝒑 0
2,2
𝑚 𝑠 ℎ𝑎𝑛𝑘 + 𝒑 0
3,2
𝑚 𝑓𝑜𝑜𝑡 )
𝑚 𝑡𝑜𝑟𝑠𝑜 + 𝑚 𝑡 ℎ 𝑔 ℎ
+ 𝑚 𝑠 ℎ𝑎𝑛𝑘 + 𝑚 𝑓𝑜𝑜𝑡 ( 59 )

3.3.3.3 Feedback Definition
The two aforementioned lines of impedance are described in relative terms, expressed in the Torso reference frame.
As mentioned previously, the 3
rd
row of these vectors is not used beyond this point, as its use as a placeholder is
limited to that of the DH transformations.

First, one defines the line of impedance between the ankle and the total body center of mass, LOI1 (simplified in
Appendix A).

𝒑
3
𝑎𝑛𝑘 −𝐶𝑂𝑀 = 𝑳 𝑒𝑒 1
( 𝒒 )
= 𝒑
3
2,3
− 𝒑
3
𝐶𝑂𝑀
( 60 )

113

The second line of impedance, LOI2, is defined as the vector between the heel and the approximate femoral epicondyle
positions (simplified in Appendix A).

𝒑
3
ℎ𝑒𝑒𝑙 −𝐹𝐸
= 𝑳 𝑒𝑒 2
( 𝒒 )
= 𝒑
3
3,4
− 𝒑
3
1,4

( 61 )

The task-space impedance control forces are converted to the joint-space control torques through pre-multiplication
by the pseudo-inverse Jacobian. This conversion expresses the desired, effective task space control forces in terms of
torques that can be generated by the ideal hinge joint actuators available in the modelled human system. Thus, a
Jacobian is calculated for each line of impedance (LOI1, LOI2) by describing the task-space velocity of each end
effector (i.e. distal end of each LOI) as a function of the joint angles and angular velocities preceding them in the
kinematic chain, Eqs. 63 and 65.

𝒑 ̇
3
𝑎𝑛𝑘 −𝐶𝑂𝑀 =
𝑑 𝑳 1
( 𝒒 )
𝑑𝑡
=
𝑑 𝑳 1
( 𝒒 )
𝑑 𝒒 ∙
𝑑 ( 𝒒 )
𝑑𝑡
= 𝑱 1
( 𝒒 ) 𝒒 ̇ ( 62 )

𝑱 𝑒𝑒 1
( 𝒒 )=
𝜕 𝑳 1
( 𝒒 )
𝜕 𝒒 = [
𝜕 𝑳 1
( 𝒒 )
𝜕 𝑞 1
𝜕 𝑳 1
( 𝒒 )
𝜕 𝑞 2
𝜕 𝑳 1
( 𝒒 )
𝜕 𝑞 3
] ( 63 )

𝒑 ̇
3
ℎ𝑒𝑒𝑙 −𝐹𝐸
=
𝑑 𝑳 2
′( 𝒒 )
𝑑𝑡
=
𝑑 𝑳 2
′( 𝒒 )
𝑑 𝒒 ∙
𝑑 ( 𝒒 )
𝑑𝑡
= 𝑱 2
( 𝒒 ) 𝒒 ̇ ( 64 )

𝑱 𝑒𝑒 2
( 𝒒 )=
𝜕 𝑳 2
( 𝒒 )
𝜕 𝒒 = [
𝜕 𝑳 2
( 𝒒 )
𝜕 𝑞 1
𝜕 𝑳 2
( 𝒒 )
𝜕 𝑞 2
𝜕 𝑳 2
( 𝒒 )
𝜕 𝑞 3
] ( 65 )

Combining these transformations, one forms the 4x3 matrix end-effector transform, J ee, in which each LOI velocity
is related to the joint velocities, Eq. 66. Note, for clarity the Jacobian dependency on the joint states is assumed from
now on, so it will not be included in the equations.

[
𝒑 ̇
3
𝑎𝑛𝑘 −𝐶𝑂𝑀 𝒑 ̇
3
ℎ𝑒 𝑒𝑙 −𝑘𝑛𝑒𝑒 ] = [
𝑱 𝑒𝑒 1
𝑱 𝑒𝑒 2
][
𝑞 ̇ 1
𝑞 ̇ 2
𝑞 ̇ 3
] ( 66 )

The Jacobian transformation is henceforth used to relate task-space states to their equivalent joint space states. Note,
despite its original definition being expressed in terms of the velocity states, position states can also be related via the
Jacobian. Angular displacement is approximately equivalent to rectilinear displacement if the angle being traversed is
small enough (i.e. small angle theorem). Thus, over a small angle (or with a high enough sampling rate) the angular
position change approximates to this linear relationship. This flexibility is exploited the in definition of control torques
with respect to task-space control forces, using the pseudo-inverse Jacobian to describe angular efforts as a function
of task-space, rectilinear control forces.

114

3.3.3.4 Task-Space Impedance Forces
The end effector positions presented in the previous section do not fully describe the end effector position with respect
to inertial space. Because the body isn’t rigidly fixed in space, a number of unactuated degrees of freedom exist
between the body and its environment. In order to fully describe the state of a body fixed end effector in inertial space,
additional terms defining the torso’s absolute position and absolute orientation are needed. As a result, the absolute
Jacobian has columns of zeros which highlight the underactuated system’s ineffectual joints with respect to inertial
space. Since these joints are impossible to actuate, one must define desired control forces in a local, body-fixed task
space, similar to the way Schall and Mistry[61] define their “floating base reference frame”. These impedance forces
need only describe the relative, internal mechanics of the system with respect to the body-fixed base reference frame,
and the environment is treated as an external disturbance to this isolated system.

Defining impedance based control forces internally means characterizing them as a function of both the body fixed
LOI vector length and its rate of change to realize spring-damper-like behavior along the LOI vector, acting its
endpoints. These impedance control forces are defined as PD controllers acting on the error and error rate between the
LOI vector and its respective optimized LOI reference trajectory, 𝒑
, Eqs. 67-68.

𝑭
3
𝑐 1
= (𝐾 1
( 𝑝 1
− | 𝒑
3
𝑎𝑛𝑘 −𝐶𝑂𝑀 |)+ K
d1
(𝑝 ̇ 1
−
𝑑 ( | 𝒑
3
𝑎𝑛𝑘 −𝐶𝑂𝑀 |)
𝑑𝑡
))∙ 𝒑̂

3
𝑎𝑛𝑘 −𝐶𝑂𝑀 ( 67 )

𝑭
3
𝑐 2
= (𝐾 2
( 𝑝 2
− | 𝒑
3
ℎ𝑒𝑒𝑙 −𝐹𝐸
|)+ K
d2
(𝑝 ̇ 2
−
𝑑 ( | 𝒑
3
ℎ𝑒𝑒𝑙 −𝐹𝐸
|)
𝑑𝑡
))∙ 𝒑̂

3
ℎ𝑒𝑒𝑙 −𝐹𝐸
( 68 )

In this form, one need only know the relative positions of LOI endpoints (and their rate of change), and the desired
control force can be calculated to provide the desired effective impedance. Note, these equations also illustrate the
desired rate trajectory of change, 𝒑 ̇
, is targeted with the PD controller seeking to match both length and rate of length
change.

3.3.3.5 Joint-Space Impedance Torques
Unlike external environmental forces which are converted to the joint space through the Jacobian transpose, task-
space control forces can be converted to the joint space by pre-multiplication by the pseudo-inverse Jacobian. The
conversion is made possible because the impedance forces are defined in terms of task-space coordinates, meaning
that they relate to the joint space in a least squares sense[62]. The optimal joint velocity set for this application, which
minimizes the energy cost to the system (squared magnitude of the joint angular velocity vector), can be derived with
the use of Lagrangian multipliers and the definition of the Jacobian in the form of a constraint, Eq. 69.

𝐶 ( 𝒒 ̇,𝝀 )= 𝒒 ̇ 𝑇 𝒒 ̇ − 𝝀 𝑇 ( 𝑱 𝒒 ̇ − 𝒑 ̇) ( 69 )

115

Taking the partial gradient of this cost function and setting it equal to zero provides the optimal solution set, ( 𝒒 ̇ ′
,𝝀 ′) ,
which occurs at the minimized extremum, Eqs. 70-71. One knows this extremum to be minimum, as there is no
theoretical upper limit on the joint angular velocity magnitude, thus it cannot be a maximum.

𝜕𝐶 ( 𝒒 ̇,𝝀 )
𝜕 𝒒 ̇ = 2𝒒 ̇ − 𝑱 𝑇 𝝀 = 𝟎 ( 70 )

𝜕𝐶 ( 𝒒 ̇,𝝀 )
𝜕 𝝀 = 𝑱 𝒒 ̇ − 𝒑 ̇ = 𝟎
( 71 )

Solving Eq. 70 for the angular velocity:

𝒒 ̇ =
1
2
𝑱 𝑇 𝝀
( 72 )

Substituting this expression into Eq. 71 gives the Lagrangian as a function of the task space velocity:

𝒑 ̇ =
1
2
𝑱 𝑱 𝑇 𝝀
( 73 )

Finally, using the invertibility of 𝑱 𝑱 𝑇 to solve for the Lagrangian and substituting it back into Eq. 72, the conversion
from task-space to joint space is given by Eq. 74.

𝒒 ′
̇ = 𝑱 𝑇 ( 𝑱 𝑱 𝑇 )
−𝟏 𝒑 ̇
( 74 )

This transformation matrix, 𝑱 𝑇 ( 𝑱 𝑱 𝑇 )
−𝟏 , is commonly referred to as the pseudo-inverse of the Jacobian, and it is denoted
𝑱 #
. Not only does the pseudo-inverse convert task space velocity to joint space velocity, but it can also be used to
convert each task-space impedance force into their corresponding joint space impedance torques.

𝝉 𝑐 ,
= 𝑱
#
𝑭 𝑐 ,
( 75 )

Substituting the previous equations for control force into Eq. 75, and making use of Eq. 66, one finds this
transformation simply handles the conversion from task-space PD-control to joint-space PD control, Eq. 76.

𝑱
#
(𝐾
( 𝒑
− | 𝒑 3

|)+ K
di
(𝒑 ̇
−
𝑑 ( | 𝒑
3

|)
𝑑𝑡
)) 𝒑̂

3

= 𝐾
( 𝒒
− | 𝒒
3

|)+ K
di
(𝒒 ̇
−
𝑑 ( | 𝒒
3

|)
𝑑𝑡
) ( 76 )

Each set of task space forces is converted to a set of joint moments by the associated LOI’s pseudo inverse Jacobian.
Finally, the summation of these joint torque sets form the net joint moments applied at each joint as a function of time,
Eq. 77.

116

𝑻 𝑐 = 𝑱 1
#
𝑭 𝑐 1
+ 𝑱 2
#
𝑭 𝑐 2
( 77 )

In this form, however, there is no guarantee of orthogonality between these two impedance torque sets. In other words,
the torque from one task space impedance control may influence or even cancel out efforts of the other (i.e. 𝑱 1

is not
orthogonal to 𝑱 2

). Thus, the impedance forces are parsed such that they do not interfere with one another, absorbing
the components of 𝑭 𝑐 2
parallel to 𝑭 𝑐 1
, 𝑭 𝑐 2||1
, into the 𝑭 𝑐 1
signal for use with LOI1. 𝑭 𝑐 2⊥1
, the perpendicular residual
of 𝑭 𝑐 2
is returned to its original assignment for use with LOI2. The component of 𝑭 𝑐 2
parallel to 𝑭 𝑐 1
is defined as
𝑭 𝑐 2||1
, Eq. 78.

𝑭 𝑐 2||1
= 𝑭 𝑐 2
𝑇 𝑭̂
𝑐 1
∙ 𝑭̂
𝑐 1
( 78 )

where 𝑭̂
𝑐 1
, is a unit vector pointing in the 𝑭 𝑐 1
direction. The remainder of 𝑭 𝑐 2
is now guaranteed perpendicular to 𝑭 𝑐 1

and is separately applied along LOI2, Eq. 79.

𝑭 𝑐 2⊥1
= 𝑭 𝑐 2
− 𝑭 𝑐 2||1
( 79 )

The final orthogonalized form of the impedance control torque distribution is given by Eq. 80.

𝑻 𝑐 = 𝑱 1
#
( 𝑭 𝑐 1
+ 𝑭 𝑐 2||1
)+ 𝑱 2
#
𝑭 𝑐 2⊥1
( 80 )

This input signal orthogonality guarantees that the converted joint torque expressions will not interfere with each
other. The proof of this orthogonality is simply a matter of taking the dot product of two impedance torque
components, Eq. 81.

𝑱 1
#
( 𝑭 𝑐 1
+ 𝑭 𝑐 2||1
) 𝑭 𝑐 2⊥1
𝑇 𝑱 2
#𝑇 = 𝑱 1
#
0𝑱 2
#𝑇 = 0 ( 81 )

As mentioned, this pseudo-inverse application of the Jacobian must, by definition, result in a minimized joint angular
velocity solution set known as the least norm solution, ( 𝒒 ̇′)
𝑇 𝒒 ̇′. However, because this solution is one obtained in a
least squares sense, the pseudo-inverse Jacobian is not the only way to distribute control effort. One can also weight
the Jacobian according to joint capacity or more complex algorithms. This intentional design of joint weight
distribution will be both discussed at length and implemented in the section on Post-Impact Phase: Passivity Based
Energy Shaping Control.

3.3.3.6 Off-line Impedance Feedback
Humans learn basic skills through the process of trial and error[63], the biological equivalent of a nonlinear
optimization process. In fact, studies have even modeled human acquisition of motor skills through an optimization
of task-specific weighting coefficients used to combine simple hardwired base functions[64]. Similarly, when
implementing the impedance control algorithm, real-time force feedback was not used in the strictest sense. True
117

impedance control, which uses real-time force feedback to inform the adjustment of endpoint reference trajectories,
was incorporated indirectly by optimizing the reference impedance trajectories off-line during training of the model
to experimental data sets. With the difference between a simulated force-time series and its associated experimental
data set as a measure of reference trajectory quality, the optimization process adjusted the reference impedance
trajectories for the simulated task motions until the resulting simulated force-time curves were comparable to the
experimental results.

The experimental data from the self-selected normal landing of subject 1 was used to optimize both the ground reaction
force coefficients and these impedance coefficients. With adequate VGRF time-series matching achieved for this trial,
all other experimental data sets are matched through optimization of the impedance coefficients alone. In this way,
the model captures the human capacity for impedance manipulation (both gains and reference trajectory modification)
as the mechanism by which one changes their measured VGRF.

Figure 85. Block diagram of impact phase impedance control showing the use of ground reaction force feedback
during the offline nonlinear impedance trajectory and stiffness/damping coefficient optimization process (gold).
Notice, the force and kinematic feedback to the impedance block are dashed because, unlike typical day to day
activities, an impact does not provide enough time for the human body to take these measures into account. Humans
are extremely good at pattern recognition, however, and they learn to adjust their actions according expected
forces[65]–[67]. In this way, impedance is still a part of human impact control through optimization. Thus, the
impedance block is included here as being representative of these learned impedance correction behaviors, which
serve to modify one’s intended equilibrium position (aka the efferent copy) to account for expected ground reaction
forces. Finally, because the control forces are defined in the Cartesian task space, they are converted to joint torques
through the mechanical relationship of the Jacobian transpose.

3.3.3.7 Calibration: Control Optimization Algorithm
The mechanical objective of the impact phase is to reduce the body’s total momentum to an amount that can be
controlled during the post-impact phase[68]. Impedance coefficient set quality is measured by the degree to which
simulated, impact phase VGRF-time curves match that of corresponding experimental landings, as measured by the
PD Stiffness Control
𝐹
= 𝐾 ,
𝑥 𝑟
,
− 𝑥 𝑠 𝑚 ,
+𝐾 ,
𝑥 ̇ 𝑟
,
− 𝑥 ̇ 𝑠 𝑚 ,
FSL
Error
𝒙 ,𝟐 ,𝒙 ̇ ,𝟐
𝒙 ,𝟏 ,𝒙 ̇ ,𝟏
Simulated VGRF
Simulated NJMs
Sim Joint Angles
𝑥 1,2
𝑞 𝑥 ̇ 1,2
𝑞 ,𝑞 ̇ Experimental NJMs
“Force Tracking Control”
Learned Impedance Trajectory
𝐾 1
,𝐾 1
,𝐾 2
,𝐾 2
,𝐶 𝑦 ,𝐶
,𝐷 𝑦 Experimental VGRFs
Estimated LOI
Trajectories:
t
x1
, x
01
, x
11
, x
21
t
x2
, x
02
, x
12
, x
22
x
r,1
(t) = x
01
+ x
11
t + u(t-t
x1
)(-x
11
t + x
21
(t-t
x1
))
x
r,2
(t) = x
02
+ x
12
t + u(t-t
x2
)(-x
12
t + x
22
(t-t
x2
))
System Plant Definition
• Mass Distribution: Masses and Moments of Inertia
– 𝑚 𝑡𝑜𝑟𝑠𝑜 , 𝑚 𝑡ℎ 𝑔ℎ , 𝑚 𝑠ℎ𝑎𝑛𝑘 , 𝑚 𝑓𝑜𝑜𝑡
• Note: 𝑚 𝑡𝑜𝑟𝑠𝑜 = 𝑚 𝑡𝑜𝑟𝑠𝑜 + 𝑚 ℎ𝑒𝑎 + 𝑚 𝑛𝑒𝑐𝑘 + 𝑚 𝑎𝑟𝑚𝑠 – 𝐼 𝑡𝑜𝑟𝑠𝑜 , 𝐼 𝑡ℎ 𝑔ℎ , 𝐼 𝑠ℎ𝑎𝑛𝑘 , 𝐼 𝑓𝑜𝑜𝑡
• See next slide for 𝐼 𝑡𝑜𝑟𝑠𝑜 details
• Segment lengths
– 𝐿 𝑡𝑜𝑟𝑠𝑜 , 𝐿 𝑡ℎ 𝑔ℎ , 𝐿 𝑠ℎ𝑎𝑛𝑘 , 𝐿 𝑓𝑜𝑜𝑡
– Heel to Ankle dimensions: 𝐿 ℎ𝑒𝑒𝑙 , 𝐿 ℎ𝑒𝑒𝑙
– COM Longitudinal Positions (deLeva):
𝐶 𝑡𝑜𝑟𝑠𝑜 , 𝐶 𝑡ℎ 𝑔ℎ , 𝐶 𝑠ℎ𝑎𝑛𝑘 , 𝐶 𝑓𝑜𝑜𝑡
• See next slide for 𝐶 𝑡𝑜𝑟𝑠𝑜 details
• Initial States
– Joint angles: 𝜃 ℎ , 𝜃 𝑘𝑛𝑒𝑒 , 𝜃 𝑎𝑛𝑘𝑙𝑒 – Joint angular velocities: 𝜔 ℎ , 𝜔 𝑘𝑛𝑒𝑒 , 𝜔 𝑎𝑛𝑘𝑙𝑒 – Base Segment (Torso) COM position: 𝑥 𝑡𝑜𝑟𝑠𝑜 , 𝑦 𝑡𝑜𝑟𝑠𝑜 – Base Segment (Torso) COM velocity: 𝑣 𝑥 ,𝑡𝑜𝑟𝑠𝑜 , 𝑣 𝑦 ,𝑡𝑜𝑟𝑠𝑜 – Base Segment (Torso) angle: 𝜃 𝑡𝑜𝑟𝑠𝑜 – Base Segment (Torso) angular velocity: 𝜔 𝑡𝑜𝑟𝑠𝑜 𝜃 𝑓𝑜𝑜𝑡 𝜃 𝑎𝑛𝑘𝑙𝑒 𝜃 𝑘𝑛𝑒𝑒 𝜃 ℎ
Y
X
Softer:𝑥 𝐹 ,𝑖 𝑥 𝑖 ,𝑥 ̇ 𝑖 ,𝐹 ,𝐹 ̇ Normal:𝑥 𝐹 ,𝑖 𝑥 𝑖 ,𝑥 ̇ 𝑖 ,𝐹 ,𝐹 ̇ Harder:𝑥 𝐹 ,𝑖 𝑥 𝑖 ,𝑥 ̇ 𝑖 ,𝐹 ,𝐹 ̇ Sim
NJMs

𝑇 𝑭 𝟏 ,𝑭 𝟐 Stiffness Estimations:
Subject: K
p1
, K
v1
, K
p2
, K
v2
Floor: C
y
, C
exp
, D
y
+
-
𝒙

,𝟐 ,𝒙 ̇

,𝟐
𝒙

,𝟏 ,𝒙 ̇

,𝟏
+
-
118

residual between these two curves. The residual, Eq. 82, is calculated as the summed square difference between these
two curves.

𝑅
= ∑|𝐹 𝑦 ( 𝑡 )− 𝐹 𝑦 ,𝑠 𝑚 𝑖 ( 𝑥 0
𝑖 ,𝑥 1
𝑖 ,𝑥 2
𝑖 ,𝑡 𝑥 𝑖 ,𝐾
𝑖 ,𝐾
𝑖 ,𝑡 ) |
2
( 82 )

where the first four parameters (𝑥 0
𝑖 ,𝑥 1
𝑖 ,𝑥 2
𝑖 ,𝑡 𝑥 𝑖 ) describe two sets of piecewise-linear impedance reference trajectories,
Eq. 83.

𝑝
( 𝑡 )= 𝑥 0
𝑖 + 𝑥 1
𝑖 𝑡 + ( 𝑥 2
𝑖 − 𝑥 1
𝑖 ) ( 𝑡 − 𝑡 𝑥
) 𝐻 ( 𝑡 − 𝑡 𝑥
) ( 83 )

where 𝑡 𝑥
describes the each reference trajectory slope breakpoint, H(t) is the Heaviside function, x i’s are the
impedance trajectory coefficients, and (𝐾
𝑖 ,𝐾
𝑖 ) describe the stiffness, stiffness-rate at which the i
th
reference
trajectory is followed, respectively. The resulting impedance reference trajectories for Subject 1’s soft, normal, and
hard landings are shown in Figure 86. Since simulations are run using a variable integration step size algorithm which
automatically reduces the step size according to the rate of state change[69], each simulation output is resampled at a
constant step size to ensure sample size does not affect the effective residual calculation.

119

(a)

(b)

Figure 86. The effective support forces generated internal to the body model are described by two pairs of springs
and dampers, each of which follow an associated free-spring length trajectory. These trajectories (a), when used in
combination with their spring and damper coefficients (b), serve to describe the nonlinear lower-extremity support
force behavior during impact.

The optimization process is a multivariable combination of Newtonian gradient of descent and discrete Monte-Carlo
iterations, adaptive to the degree of residual improvement with respect to the previous iteration. During the initial
120

Monte-Carlo iteration, each coefficient is varied ±5% from its previous value, and simulations are evaluated with
combinations of these modified coefficients. To reduce the optimization run time and computational expense, at most
two coefficients were varied from their previous value for any single simulation. At the end of a stage, where every
combination of coefficients has been used in simulation, the coefficient set associated with the smallest residual
determines the direction of the Newtonian gradient of descent step. The next iteration of coefficients is initialized by
adding a weighted difference to the residual-minimizing coefficient set. The difference between the current and
previous residual minimizing coefficient set is calculated, and it is weighted by the fractional improvement in residual
value between sets, Eq. 84.

𝒙 𝒊 +𝟏 = 𝒙 𝒊 + ( 𝒙 𝒊 − 𝒙 𝒊 −𝟏 )(
𝟐 𝑹 𝒊 −𝟏 − 𝑹 𝒊 𝑹 𝒊 −𝟏 ) ( 84 )

where i, is the optimization iteration count, x i is the associated vector of impedance coefficients, and 𝑹 𝒊 is the residual
for that coefficient set. Finally, in order to ensure that the optimization does not get stuck in a local minimum, the
Monte-Carlo span percentage increases when the iteration to iteration residual improvement drops below 5%. If the
residual does not improve by more than 5% after 4 consecutive iterations of span percentage modification, the Monte-
Carlo span is randomized to prevent incidental systemic solution space limits. The optimization finally terminates
after either a user defined number of iterations are evaluated or the matching criteria has been met. If the matching
criteria are unmet, the process is restarted using the coefficient set from the stopping condition.

3.3.4 Post-Impact Phase: Passivity Based Energy Shaping Control

During the final phase of a landing, the subject attempts to remain balanced while bringing the system to rest. In terms
of task objectives, this means they attempt to keep their COM over their base of support while reducing their
momentum, or kinetic energy, to zero[68]. Formulating the task objectives in this way highlights their similarity to
the passivity based energy shaping control structure. Passivity based control is designed in the system energy domain,
using the natural effects of potential energy as a mechanism for shaping the system equilibrium point via controlled
actuators. By introducing virtual potential elements (springs, magnets, etc.) in the form of control feedback, one
manipulates the shape of the natural potential energy field to user-defined stability points.

As explained by Koditschek in his review of energy based control[33], this type of control allows one to define an N-
dimensional potential function with energy dissipation. When coupled with the natural dynamics of a system, this
control guides the point of interest to a designed equilibrium point asymptotically (assuming its trajectory does not
pass through any zero-velocity states). Arimoto shows the application of this type of control to nonlinear robots by
way of passivity in several papers[34]. Essentially, if the nonlinear system in question is passive, storing no more
energy than is input to it, then the system will also be stable when coupled with a passive controller. Thus, using an
N-dimensional positive definite potential function with an equilibrium point over the feet combined with an energy
dissipating function as inputs, the control law asymptotically drives the total body COM to a point the base of support,
121

dissipating energy until the system comes to rest. Note, a weakness of this methodology comes from the need to define
a suitable potential function. However, the passivity of the natural system and its external interaction forces help with
this. This type of control has been explained in other fields, such as when Gu et al. describes control of humanoid
robotics as moving between a series of equilibrium points[70].

3.3.5.1 Potential Energy
In this study, passivity-based energy shaping control is used to guide the subject’s COM to a point above the base of
support. The system potential energy is shaped by adding control effort in the form of a virtual spring to attract the
COM and a gravity compensation term, Eq. 85.

𝐹 𝐶 = 𝐾
( ( 𝑝 ℎ𝑒𝑒𝑙 + 𝑝
)− 𝑝 𝐶𝑂𝑀 )+ [
0
𝑚 𝑡𝑜𝑡 𝑔 ] ( 85 )

where 𝒑 𝒅 is the desired position of the end-effector defined in the Cartesian task space relative to the heel (i.e. w.r.t.
the base of support), 𝑚 𝑡𝑜𝑡 is the total mass of the body, and 𝑚 𝑡𝑜𝑡 𝑔 compensates for the force of gravity. By attracting
the total center of mass to a position located horizontally between the heel and the toe and vertically greater than the
subject’s lowest standing squat point, one creates an equilibrium point that guides the subject to a supported stance
configuration. Allowing the system to evolve in this designed potential field, the body reaches a dynamic equilibrium
while its COM is above the feet, Figure 87. Note, the stability of this dynamic system depends on the whether or not
the amplitude of COM horizontal oscillation extends outside the base of support long enough for the moment from
the reaction force to tip the system beyond recovery.

122

(a)

(b)

Figure 87. The natural potential energy field shows that an uncontrolled human body reaches a minimum potential
energy when lying prone on the ground (a). After applying gravity potential compensation, the potential energy
space is neutralized. Finally, the addition of a virtual spring creates a new equilibrium point in the task-space at the
desired location relative to the heel fixed kinematic origin (b).

3.3.5.2 Energy Dissipation
Without a dissipative force, the COM end-effector would continuously oscillate back and forth, failing to come to
rest. Ensuring the body decreases this kinetic energy means adding an energy dissipation element, or damper, to the
control. The final form of the energy shaping control effort is given by Eq. 87, where the equilibrium position, 𝒑 𝒅 is
defined relative to the heel position.

𝒑
′
= 𝒑 ℎ𝑒𝑒𝑙
+ 𝒑
( 86 )

𝑭 𝐶 = 𝐾
( 𝒑
′
− 𝒑 𝐶𝑂𝑀 )− 𝐾
( 𝟎 − 𝒑 ̇ 𝐶𝑂𝑀 )+ [
0
𝑚 𝑡𝑜𝑡 𝑔 ] ( 87 )

The control defined here simultaneously guides the total body COM towards the desired equilibrium, 𝒑 𝒅 ′
, dissipates
energy, and compensates for the force of gravity. Recall from the chapter on Impact Phase: Impedance Control, the
pseudo-inverse Jacobian can be used to convert task-space impedance forces into the corresponding joint space
impedance torques, Eq. 88. In this case, the forward kinematic transformation, L(q), and associated Jacobian, J(q),
describe the total body center of mass as an end-effector position and velocity with respect to the foot reference frame
located at the heel, respectively.

Y
X
Natural Potential Energy
Shaped Potential Energy
X
Y
123

𝝉 𝑐 = 𝑱 #
𝑭 𝑐 ( 88 )

Substituting the previous equations for control force into this equation and making use of Eq. 66 one finds this
transformation simply handles the conversion from task-space PD-control to joint-space PD control, Eq. 89.

𝝉 𝑐 = 𝑱 #
(𝐾 𝒑 ( 𝒑 𝒅 ′
− 𝒑 𝑪𝑶𝑴 )+ 𝐾 𝒅 ( 𝟎 − 𝒑 ̇ 𝑪𝑶𝑴 )+ [
0
𝑚 𝑡𝑜𝑡 𝑔 ])
= 𝐾
( 𝒒
′
− 𝒒 𝐶𝑂𝑀 )+ 𝐾
( 𝟎 − 𝒒 ̇ 𝐶𝑂𝑀 )+ 𝑱 #
[
0
𝑚 𝑡𝑜𝑡 𝑔 ]
( 89 )

The final term in this equation presents a problem, as the force of gravity is not a function of the task space and
therefore is not converted to the joint space by the pseudo-inverse Jacobian. Instead, one leverages the natural
mechanical relationship between force and torque, the Jacobian transpose, 𝑱 𝑇 , to convert any force from the Cartesian
space into the joint space, Eq. 90.

𝝉 = 𝑱 𝑇 𝑭 ( 90 )

3.3.5.3 Conversion to the Jacobian Transpose
Khatib’s explains[71] without loss of generality, one can determine the natural relationship between torque and force
vectors through analyzing a simple system with a single rigid link (described here by the vector, r) with a 1DoF hinge
joint connected to the inertial reference frame at one end and the end effector at the other. The linear velocity of this
end-effector as a function of the segment angular velocity is the well documented, Eq. 91.

𝒗 = 𝝎 × ( 91 )

Rewriting this using the skew-symmetric cross product tensor, this equation is instead defined by a simple linear
algebraic statement, Eq. 92.

𝒗 = ⟦𝝎 ⟧ = ⟦− ⟧𝝎 ( 92 )

where the cross-product tensor is the matrix equivalent of the cross-product operation, Eq. 93.

⟦ ⟧ = [
0 −𝑟 3
𝑟 2
𝑟 3
0 −𝑟 1
−𝑟 2
𝑟 1
0
] ( 93 )

Because the motion being studied is assumed planar (X-Y), the angular velocity axis cannot change direction from
perpendicular to this plane (Z), only its magnitude. Thus, Eq. 92 can be rewritten in terms of task and joint space time
derivatives, Eq. 94. This formulation is now directly related to the Jacobian transformation from an angular motion
description (joint space) to linear descriptors (task space) for the single degree of freedom system.

124

𝒙 ̇ = ⟦− ⟧𝒒 ̇ = 𝑱 𝒒 ̇ ( 94 )

In a similar way, one can use screw theory as applied to rigid body dynamics to translate force applied at an end-
effector positioned at point, r, relative to a hinge joint into its corresponding torque applied at the joint, Eq. 95. Notice,
this too can be written in terms of the cross-product tensor, Eq. 96.

𝝉 = × 𝑭 ( 95 )

𝝉 = ⟦ ⟧𝑭 ( 96 )

First, one recognizes this cross-product tensor is a skew symmetric matrix, and as such it is the negative of its own
transpose. Employing this fundamental property, the equation is rewritten in terms of the negative cross-product
tensor, Eq. 97.

𝝉 = ⟦− ⟧
𝑻 𝑭 ( 97 )

Eq. 90 demonstrates 𝑱 is functionally equivalent to the negative cross-product tensor of the end-point offset vector, r.
Replacing that term with the Jacobian, one recovers the relationship illustrating how end-effector inertial space forces
are equivalently generated by joint space torques with the effort distribution described by the transpose of the
associated end-effector Jacobian, Eq. 98.

𝝉 = 𝑱 𝑇 𝑭 ( 98 )

With the all of the forces, F, described in the Cartesian space, one can directly describe the joint space torques which
account for any internally defined or externally experienced forces, 𝝉 . Thus, applying this relationship to the task-
space control and gravity compensation forces, one arrives at the complete description of NJM control torques during
the post-impact phase, Eq. 99.

𝝉 = 𝝉 𝑐 + 𝝉 𝑒 = 𝑱 𝑇 (𝐾 𝒑 ( 𝒑 𝒅 ′
− 𝒑 𝑪𝑶𝑴 )+ 𝐾 𝒅 ( 𝟎 − 𝒑 ̇ 𝑪𝑶𝑴 )+ [
0
𝑚 𝑡𝑜𝑡 𝑔 ]) ( 99 )

The control description up to this point only operates in two Cartesian degrees of freedom (DoF), while the joints
consist of 3 DoF total. When the number of actuated DoF exceeds the DoF of the mechanical objective, the system
can achieve the objective in a number of different ways by modifying the extra degrees of freedom, known as the null
space control. In much the same way as the least squares solution from the Joint-Space Impedance Torques section,
constraints can be applied to the solution space to promote solutions that meet designed criteria.

3.3.5.4 Constraining the Motion (UWLN)
Healthy human motion is constrained in several ways. The first and perhaps most obvious set of limits is described by
each joint’s range of motion in either direction, joint limits. Motion continued beyond these limits would cause damage
125

to the surrounding tissue and could permanently inhibit the functionality of the system. A slightly more subtle limit is
that which is driven by subject flexibility. While the skeletal system creates hard stops to limit one’s range of motion,
some individuals are unable to reach these limits at all due to increases in muscle tension prior to that point. These
may be considered softer limits which could be modified with time. The most implicit source of kinematic limitation
addressed in this study is the subject specific coordination of joint angles which arises as a result of one’s
preferred/learned technique[54]. Therefore, during simulation a model should only evolve in joint angle ranges which
meet these criteria. These constraints are implemented through carefully designed nonlinear weighting of the Jacobian.

In addition to the primary mechanical task objectives, studies have shown that subjects secondary objectives like joint
limits[26]. In this way, control objectives are multi-tiered, with self-preservation objectives dominant over other task
objectives, like stability[61]. Extensive work in the field of robotics[61], [72]–[74] has demonstrated different
frameworks for defining a hierarchical set of constraints in the form of control objectives such that a system can satisfy
both primary and secondary objectives, as long as they are mathematically orthogonal to each other[75]. In this
formulation, the number of enforceable constraints is limited to the number of residual degrees of freedom beyond
those used by the primary constraint, also known as the system’s nullity (dimension of the null space). Null space
control is less than ideal for this application however, as the body has more kinematic constraints to consider than
available additional degrees of freedom.

Recent research by Chen et al[74], describes a method of applying an unlimited number of competing constraints in a
quadratic form called the Unified Weighted Least Norm (UWLN) method. In this method, Chen shows a state-
dependent weighting matrix that adapts its elements according to their distance from the constraint boundary. When
the system approaches a constraint boundary condition, this formulation increases the corresponding constraint weight
to assign the most cost to further evolution along that trajectory. Note, this quadratic criterion scheme always has a
solution due to its positive definite definition. Additionally, it reduces to the least-norm solution in the task space
where constraints are not being contested.

Interestingly, the resulting solution minimizes a pseudo-angular-kinetic energy of the system, here referred to as
“pseudo” because it is not weighted by the inertias of each segment[73]. Each UWLN weight corresponds to a specific
user defined constraint which may be complex and multivariate. Furthermore, each weight is state dependent such
that it smoothly transitions between active and inactive near its boundaries as shown in Eq. 100. The limit weighting
matrix, 𝑾 𝐻 , is formed with the square of these weights along its diagonal, ensuring positive semi-definiteness, Eq.
101.

𝑤
( 𝒒 ,𝒒 ̇)= {
𝛼 𝑐𝑜 (
𝜋 𝐻
( 𝒒 )
2𝜁
)
1
𝐻
( 𝒒 )
𝑖𝑓 𝐻 ̇
( 𝒒 ,𝒒 ̇)< 0 𝑎𝑛𝑑 𝐻
( 𝒒 )≤ 𝜀
0 𝑒𝑙 𝑒 ( 100 )

126

𝑾 𝐻 ( 𝒒 ,𝒒 ̇)=
[

𝑤 1
2
0 ⋯ 0
0 𝑤 2
2
⋯ 0
⋮ ⋮ ⋱ 0
0 0 0 𝑤 𝑛 2
]

( 101 )

3.3.5.5 Constraint Vector, H(q)
Each constraint, 𝐻
( 𝒒 ) is written as an inequality with respect to zero. In their simplest form, these constraints have a
linear form relating a joint to its end-of-range limits. The first six constraints of this study are defined in this format,
identifying the upper and lower bounds of the three joints, respectively Eqs. 102-107.

H
1
( 𝐪 )= 100
q
1hi
− q
1
q
3hi
− q
3lo
− 𝜀 ≥ 0 ( 102 )

H
2
( 𝐪 )= 100
q
1
− q
1lo
q
3hi
− q
3lo
− 𝜀 ≥ 0
( 103 )

H
3
( 𝐪 )= 100
q
2hi
− q
2
q
2hi
− q
2lo
− 𝜀 ≥ 0
( 104 )

H
4
( 𝐪 )= 100
q
2
− q
2lo
q
2hi
− q
2lo
− 𝜀 ≥ 0
( 105 )

H
5
( 𝐪 )= 100
q
3hi
− q
3
q
1hi
− q
1lo
− 𝜀 ≥ 0 ( 106 )

H
6
( 𝐪 )= 100
q
3
− q
3lo
q
1hi
− q
1lo
− 𝜀 ≥ 0
( 107 )

In order to prevent any one joint from unintentionally dominating the others due to magnitude of range alone, each
joint range was normalized and bounds were placed on the percentages rather than the absolute values themselves.
In addition to range normalization, an activation margin, 𝜀 , initiates weight increase before the boundary condition is
exceeded. In this study, activation margin is arbitrarily set to 5 percent of the full range. The values for these bounds
are taken from the Acta Orthopaedica Scandinavica [76] are shown in Table 7.

127

Table 7. List of flexion/extension joint end-of-range constraints used to determine when a joint associated weight should increase
to prevent continued propagation along the associated joint trajectory. *The Orthopaedica Scandinavica[76] measured a maximum
dorsiflexion angle of 75 degrees, however, because the model does not have a flexible foot this number was reduced to 25 degrees.
This range increase is meant to reflect one’s ability to continue joint collapse beyond maximum dorsiflexion by riding over the ball
of the foot.
Joint Notation
Lower Bound
𝑞 lo
[rad] (deg)
Upper Bound
𝑞 hi
[rad] (deg)
Hip 𝑞 1

−
2𝜋 3
( -120 )
𝜋 18
( 10 )
Knee 𝑞 2

𝜋 90
( 2 )
12𝜋 15
( 144 )
Ankle 𝑞 3

5𝜋 36
( 25 )*
13𝜋 18
( 130 )

The constraint vector, thus far, simply contains joint range constraint descriptions. However, the flexibility of the
UWLN method means one to define an unlimited number of constraints, with complexity limited only by the user’s
ability to define it. The true value of this method to the field of human modelling is illustrated by its ability to include
a constraint which captures a specific subject’s natural inclination to particular solution space regimes relating multiple
joint states. Therefore, one proposes a lower extremity joint coordination constraint, Eq. 108, to capture hip-knee
synergies that have been observed across lower extremity tasks for a subject[54].

𝐻 8
( 𝒒 )= 100
𝑞 21ℎ 𝑙𝑜 −
1
2
( 𝜆 21
( 𝜋 − 𝑞 2
)− ( 𝜋 + 𝑞 1
) )
2
20𝑞 21ℎ 𝑙𝑜 − 𝜀 ≥ 0
( 108 )

This constraint equation sets limits on the difference between an empirical joint ratio, 𝜆 21
, of the acute knee and hip
angles (angle which decrease during flexion). In addition to the joint ratio, a coordination range, 𝑞 21ℎ 𝑙𝑜 , is defined to
increase or soften domination of this constraint by the others. As before, the activation margin 𝜀 is set to 5 percent of
the range such that the constraint is activated prior to violation of the constraint limit. Notice, when the 𝑞 21ℎ 𝑙𝑜 range
is set to 5 degrees, as is the case with this application, the coordination constraint is always active. Evidence of this
cooperation is present in the kinematic data from the experiments conducted for this research project, Figure 88. As
shown in this state space diagram, the values of 𝜆 21
are directly observable as the slope between measured joint angles,
and it is subsequently implemented at 1.3 and 0.7 for Subject 1 and Subject 2, respectively.

128

(a)

(b)

Figure 88. Evidence of subject specific joint coordination techniques is found to hold across multiple trials. Subject
1 (a) shows a tendency to a 1.3:1 hip-knee joint angle ratio, while Subject 2 (b) has a more tightly controlled
coordination tendency of 0.7:1. In terms of its implementation, this control coordination rigidity can be reflected in
the coordination weight applied to that constraint, with a higher weight indicating a stronger joint ratio adherence.

Finally, the constraint vector, 𝑯 ( 𝒒 ) , is formed by placing each constraint on its own row. Note, the order of these
constraints does not matter as the constraint matrix handles proper distribution to associated joint weights, a feature
that is illustrated in the Constraint Matrix, C( q) section. By the UWLN method, these constraints are converted into
weights which increase the cost on a joint velocity when the associated constraint boundary is in danger of being
violated.

3.3.5.6 Constraint Matrix, C(q)
The gradient of the constraint vector with respect to the joints, Eq. 109, describes the way in which these constraints
are distributed to the joints, and it is referred to here as the constraint matrix, 𝑪 ( 𝒒 ) . As the gradient of a vector, this
matrix is a local linearization showing how the constraint values change as a function of each joint angle. For most of
the constraints described in the previous section, this is simply a matter of a signed scalar. For the last constraint,
however, the constraint matrix shows how the evolution of two joints can be tied to each other by constraint definition.

129

𝑪 ( 𝒒 )=
𝜕 𝑯 ( 𝒒 )
𝜕 𝒒 ( 109 )

Evaluating this partial derivative, provides the matrix relating weights to the appropriate joints, Eq. 110.

𝑪 ( 𝒒 )=
[

−100
q
3hi
− q
3lo
0 0
100
q
3hi
− q
3lo
0 0
0
−100
q
2hi
− q
2lo
0
0
100
q
2hi
− q
2lo
0
0 0
−100
q
1hi
− q
1lo
0 0
100
q
1hi
− q
1lo
5
𝑞 21ℎ 𝑙𝑜 ( 𝜆 21
( 𝜋 − 𝑞 2
)− ( 𝜋 + 𝑞 1
) )
5𝜆 21
𝑞 21ℎ 𝑙𝑜 ( 𝜆 21
( 𝜋 − 𝑞 2
)− ( 𝜋 + 𝑞 1
) ) 0
]

( 110 )

Not only does this matrix help illustrate how each constraint relates to the joints, but Chen has cleverly leveraged its
associative nature in the UWLN method in order to mathematically tie each constraint weight to its associated joint
by forming the positive definite constraint weighting matrix, 𝑸̅
, Eq. 111.

𝑸̅
( 𝒒 ,𝒒 ̇)= 𝑰 + 𝑪 𝑻 ( 𝒒 ) 𝑾 𝑯 ( 𝒒 ,𝒒 ̇) 𝑪 ( 𝒒 ) ( 111 )

Without loss of generality, one demonstrates the associative capacity of the constraint matrix by assuming a single
constraint condition using only the aforementioned coordination constraint. Given the single constraint condition, the
coordination weight is the only entry of the limit weighting matrix, Eq. 112, and assuming unit ratios and gains for
the constraint, the corresponding 𝑸̅
1
matrix relates this weight to the two joints involved, Eq. 113.

𝑊 1
= [𝑤 𝑐𝑜𝑜𝑟
2
] ( 112 )

𝑸̅
1
( 𝑞 )= 𝑰 + [
−( 𝑞 2
+ 𝑞 1
)
−( 𝑞 2
+ 𝑞 1
)
0
]𝑊 1
[−( 𝑞 2
+ 𝑞 1
) −( 𝑞 2
+ 𝑞 1
) 0] ( 113 )

Simplification of this matrix demonstrates the ankle joint velocity weight is unaffected by the constraint, as expected,
Eq. 114.

𝑸̅
1
( 𝑞 )= [
1+ 𝑊 1
( 𝑞 2
+ 𝑞 1
)
2
𝑊 1
( 𝑞 2
+ 𝑞 1
)
2
0
𝑊 1
( 𝑞 2
+ 𝑞 1
)
2
1+ 𝑊 1
( 𝑞 2
+ 𝑞 1
)
2
0
0 0 1
] ( 114 )

As predicted, should this matrix be applied in a joint velocity vector weighting capacity, the hip and knee joint angular
velocities receive the weighting associated with this constraint, while the ankle weight is left unmodified.

130

3.3.5.7 Joint Specific Weighting
In addition to the constraint based weighting, one may wish to include constant joint-specific weights in the form of
a diagonal matrix, 𝑾 𝑞 , to capture the mobility or strength of the corresponding joint. A higher weight means the
weighted least norm assigns a higher cost to angular velocity about the associated joint, such that the angular velocity
about that joint is reduced relative to the others. As defined, 𝑸̅
is not always diagonal which means that it doesn’t
necessarily commute with 𝑾 𝑞 . Since the joint weighting matrix serves to scale each joint directly, 𝑾 𝑞 should be pre-
multiplied by 𝑸̅
, Eq. 115. In this way, 𝑾 𝑞 scales each joint angular velocity before their weighted forms are adjusted
relative to one another by 𝑸̅
.

𝑸̅
𝑊 ( 𝒒 ,𝒒 ̇)= 𝑸̅
( 𝒒 ,𝒒 ̇) 𝑾 𝑞 ( 115 )

As before, Slotine’s work[62] illustrates that application of this weighted Jacobian produces an angular velocity
solution which minimizes both the task-space velocity error as well as the weighted angular velocity norm,
1
2
𝒒 ̇ 𝑇 𝑸̅
𝑊
𝒒 ̇ .
Proof of this claim mirrors the derivation in the Joint-Space Impedance Torques section, with the addition of the
weighting matrix, 𝑸̅
𝑊
. Starting from Eq. 69, one replaces the quadratic joint velocity term with its weighted
equivalent, Eq. 116.

𝐶 ( 𝒒 ̇,𝝀 )= 𝒒 ̇ 𝑇 𝑸̅
𝑊
𝒒 ̇ − 𝝀 𝑇 ( 𝑱 𝒒 ̇ − 𝒑 ̇) ( 116 )

This serves to modify the partial derivative of 𝐶 ( 𝒒 ̇,𝝀 ) with respect to 𝒒 ̇ by including the weighting matrix, Eq. 117.

𝜕𝐶 ( 𝒒 ̇,𝝀 )
𝜕 𝒒 ̇ = 2𝑸̅
𝑊
𝒒 ̇ − 𝑱 𝑇 𝝀 = 𝟎 ( 117 )

Continuing with this derivation, one arrives at the weighted pseudo-inverse matrix describing the transformation
between task-space and the minimized joint-space, 𝒒 ′
̇ , subject to the relative cost of each joint, Eq. 118.

𝒒 ′
̇ = 𝑸̅
𝑊 −1
𝑱 𝑇 ( 𝑱 𝑸̅
𝑊 −1
𝑱 𝑇 )
−𝟏 𝒑 ̇
( 118 )

These weighted constraints are combined into the weighted pseudo-inverse Jacobian matrix, Eq. 119, which serves to
convert desired task-space velocities into a set of minimal joint-space velocities while observing the relative cost of
each joint’s motion as indicated by the UWLN matrix.

131

𝑱 𝑄 #
= 𝑸̅
𝑊 −1
𝑱 𝑻 ( 𝑱 𝑸̅
𝑊 −1
𝑱 𝑻 )
−1
( 119 )

Notice, in its simplified form, this equation is not strictly a pseudo-inverse as it is pre-multiplied by another matrix,
Eq. 120. However, it does share 3 out of the 4 pseudo-inverse properties.

𝑱 𝑄 #
= 𝑸̅
𝑊 −1/2
( 𝑱 𝑸̅
𝑊 −1/2
)
#

( 120 )

Recall from the Joint-Space Impedance Torques Section, the angular velocity solution sets that result from use of this
pseudo-inverse Jacobian, by definition, satisfy minimization requirements of both the weighted pseudo-kinetic energy
measure task space errors, ( 𝒒 ̇′)
𝑇 𝑸̅
𝑊
𝒒 ̇′, as well as the minimizing the task-space error constraint, |𝑱 𝒒 ̇′− 𝒑 ̇|. Applying
this new pseudo-inverse to the control forces defined earlier in this chapter (without gravity compensation), the post-
impact phase controller uses energy shaping to control the position of the total body COM and UWLN constraint
based weighting to ensure proper motion distribution across the joints, Eq. 121.

𝝉 = 𝑱 𝑄 #
(𝑲
( 𝒑
′
− 𝒑 𝐶𝑂𝑀 )+ 𝑲 𝒅 ( 0 − 𝒑 ̇ 𝐶𝑂𝑀 ) ) ( 121 )

In this form one directly observes how the UWLN method works, with weights increasing as a joint evolves toward a
joint limit causing the corresponding joint angular velocity magnitude to decrease, minimizing the weighted kinetic
energy measure ( 𝒒 ̇′)
𝑇 𝑸̅
𝑊
𝒒 ̇′.

Recall, however, this control definition is not flexible enough to be applied to all control forces, as it serves only to
translate forces explicitly defined in local Cartesian task space into joint space torques. As a result, this weighted
Jacobian pseudo-inverse can not be applied to the gravity compensation term. Instead, one proposes the use of the
weighted Jacobian transpose, such that the same weighting scheme serves to distribute any forces across the torques
directly, rather than defining the torque effects through the conversion of task space velocities to angular velocities.
Of course, this means redefining the control equations one last time.

3.3.5.8 The Modified UWLN Method
As stated in the previous section, one proposes the use of the weighted Jacobian transpose, such that the UWLN
weighting scheme can distribute any forces across the torques directly. The goal of this definition is to implement the
Jacobian transpose in its weighted form, such that any force may be included and scaled in the weighted torque control
distribution process, Eq. 122.

𝜏 𝑐 =
𝑊 𝑇 𝐹 ( 122 )

Starting from the pseudo-inverse weighted Jacobian which captures the desired constraints, defined in Eq. 120, one
need only to invert the pseudo-inverse Jacobian to form a weighted Jacobian. At this point, the reader will recall from
Eq. 120 that 𝑱 𝑄 #
is not a pure pseudo-inverse in the strict mathematical sense because the true pseudo-inverse
132

component is premultiplied by 𝑸̅
𝑊 −1/2
. Consequently, one may not simply distribute the desired inversion across the
pseudo-inverse and its pre-multiplicative weighting matrix. A benefit of this fact is that inverting this pseudo-pseudo-
inverse does not cause the weighting matrices to cancel out. Instead, one is left with a rather complicated pseudo-
inverse of the weighted pseudo-inverse which cannot be further simplified, Eq. 123.

( 𝑱 𝑄 #
)
#
= (𝑸̅
𝑊 −1/2
( 𝑱 𝑸̅
𝑊 −1/2
)
#
)
#
= (𝑸̅
𝑊 −1/2
( 𝑱 𝑸̅
𝑊 −1/2
)
#
)
𝑇 (𝑸̅
𝑊 −1/2
( 𝑱 𝑸̅
𝑊 −1/2
)
#
(𝑸̅
𝑊 −1/2
( 𝑱 𝑸̅
𝑊 −1/2
)
#
)
𝑇 )
−1

= ( 𝑸̅
𝑊 −1/2
𝑱 𝑇 )
#
𝑸̅
𝑊 −1/2
(𝑸̅
𝑊 −1/2
( 𝑱 𝑸̅
𝑊 −1/2
)
#
( 𝑸̅
𝑊 −1/2
𝑱 𝑇 )
#
𝑸̅
𝑊 −1/2
)
−1

( 123 )

Due to this additional inversion, this pseudo-pseudo-inverse-inverse weighted Jacobian now implicitly contains
weighting matrices to positive powers. As these would serve to reverse the intended distribution effects of the original
weighting matrices, one must also invert the weighting matrix supplied to this new matrix, Eq. 124, which is
henceforth referred to as the weighted Jacobian, 𝑱 𝑊
. Here the subscript 𝑾 serves to remind the reader that this
weighting scheme is now the inverse of Chens definition, 𝑾 = 𝑸 −1
.

𝑱 𝑊
= ( 𝑸̅
𝑊 1/2
𝑱 𝑇 )
#
𝑸̅
𝑊 1/2
(𝑸̅
𝑊 1/2
( 𝑱 𝑸̅
𝑊 1/2
)
#
( 𝑸̅
𝑊 1/2
𝑱 𝑇 )
#
𝑸̅
𝑊 1/2
)
−1

( 124 )

In the uninverted form, this weighted Jacobian is applicable in the standard conversion from force to torque, Eq. 125.

𝝉 𝒄 = 𝑱 𝑊 𝑇 𝑭 𝒄 ( 125 )

An immediate value of this new definition is the flexibility to incorporate both inertial and local force definitions into
the control force definition directly. The gravity compensation component may now be included in the transformation
along with those forces defined through forward dynamics, Eq. 126.

𝝉 𝑐 = 𝑱 𝑊 𝑇 (𝐾
( 𝒑
′
− 𝒑 𝐶𝑂𝑀 )+ 𝐾 𝒅 ( 𝟎 − 𝒑 ̇ 𝐶𝑂𝑀 )+ [
0
𝑚 𝑡𝑜𝑡 𝑔 ]) ( 126 )

As will be illustrated in the Post-Impact Phase Joint Coordination and System Stability section, an outcome of this
increased control force flexibility is stabilization robustness. In effect, the weighted Jacobian transpose definition
allows for the inclusion of inertial space damping which prevents global system deviations which otherwise occur
when the control is only focused on local measures, internal to the system. Thus, the final form of the control includes
three elements: local task space energy-shaping in the form of PD control, global inertial space energy-shaping in the
form of PD control, and finally gravity compensation, Eq. 127.

𝝉 𝒄 = 𝑱 𝑊 𝑇 (𝐾 𝒑 ( 𝒑 𝒅 ′
− 𝒑 𝑪𝑶𝑴 )+ 𝐾 𝒅 ( 𝟎 − 𝒑 ̇ 𝑪𝑶𝑴 )+ 𝐾 𝒑 ( 𝒙 𝒅 ′
− 𝒙 𝑪𝑶𝑴 )+ 𝐾 𝒅 ( 𝟎 − 𝒙 ̇ 𝑪𝑶𝑴 )+ [
0
𝑚 𝑡𝑜𝑡 𝑔 ]) ( 127 )

133

where 𝒑 represents positions in local Cartesian task-space, 𝒙 represents positions in global Cartesian inertial-space,
and 𝑚 𝑡𝑜𝑡 𝑔 represents the gravity compensation control. While the Jacobian does not explicitly translate forces from
global space into joint space torques, one’s assumption of a flat foot means that the global inertial-space and the local
task-space are aligned. Thus, by defining the desired global position, 𝒙 𝒅 ′
, relative to the local origin position in inertial
space, these two control definitions operate in nearly identical spaces. The primary advantage of this addendum is that
the inertial forces are not subject to the tipping which occurs from the flexible ground, so its damping function is
relative to absolute rest.

3.3.5.9 Robust UWLN Weight Definitions
In addition to inverting the weighting matrix 𝑸̅
𝑊
to fit the weighted Jacobian transpose definition, there are other
features of Chen’s weighting scheme[74], restated here for convenience, that may prove problematic for simulations
using variable step size algorithms, Eq. 128.

𝑤
( 𝒒 ,𝒒 ̇)= {
𝛼 𝑐𝑜 (
𝜋 𝐻
( 𝒒 )
2𝜁
)
1
𝐻
( 𝒒 )
𝑖𝑓 𝐻 ̇
( 𝒒 ,𝒒 ̇)< 0 𝑎𝑛𝑑 𝐻
( 𝒒 )≤ 𝜀
0 𝑒𝑙 𝑒 ( 128 )

The following section discusses how Chen’s weighting matrix is modified before it is eventually inverted for
implementation in the weighted Jacobian transpose. Notice in the equation, there are two main features of the
weighting system, such that the joints are constrained to the controller defined ranges:

1) When the i
th
kinematic constraint is within the boundary safety margin, 𝜀 , and continues to approach its
boundary (i.e. 𝐻
( 𝒒 )→ 0), the associated joint weights increase (𝑤
→ ∞) such that the system stops moving
in the i
th
direction due to infinitely high damping
2) When the i
th
kinematic constraint is not within the boundary safety margin (𝐻
( 𝑞 )> 𝜀
), the system moves
along the least norm solution should not be influenced by the constraint (𝑤
= 0).

The first problem inherent to this if-statement based weighting scheme is that a simulation, using a sufficiently small
step size, will invariably chatter between decreasing along the 𝐻
( 𝒒 ) trajectory and a damping induced rest condition
(i.e. between 𝐻 ̇
( 𝒒 )< 0 and 𝐻 ̇
( 𝒒 )≥ 0). By Chen’s weight definitions, this transition may occur within the activation
margin of the constraint condition, 𝜀 , leading to infinite jumps in the weighting values and subsequently large control
discontinuities. In some cases, this conditional discontinuity leads to simulation failure due to an inability to converge
to continuous derivatives.

The second problem with this weighting scheme is due to its inclusion of the constraint inverse, 1/𝐻
( 𝒒 ) . Using a
sufficiently large integration step-size allows the integrator to pass over the increasing weight condition entirely,
leading to a negative weight of a small magnitude, when what is desired instead is a large damping weight on the i
th

constraint associated joint set. In this case, the constraint condition is actively destabilized.
134

Both of these shortcomings are addressed in the weighting definitions described in this work. First, rather than
formulating the weighting factor as a simple on/off condition from the constraint velocity (𝐻 ̇
( 𝑞 ) ) perspective, the
constraint velocity is multiplied into the constraint weight, Eq. 129. This has the effect of scaling the damping
according to both distance from constraint boundary and the magnitude of the undesired velocity. Second, the weight
increase due to the distance from constraint boundary (i.e. 𝐻
( 𝒒 )= 𝜀
) has also been modified to increase
monotonically for 𝐻
( 𝒒 ) values less than 𝜀
, as shown in the state-space comparison to Chen’s definition, Figure 89.
As a result, even simulations using large integration step sizes are not destabilized by inadvertently jumping the
constraint boundary.

𝑤
= {
𝛼
𝐶
𝒒 ̇( 𝐻
( 𝒒 )− 𝜀
)
3
𝑖𝑓 𝐻 ̇
( 𝒒 )< 0 𝑎𝑛𝑑 𝐻
( 𝒒 )< 𝜀
0 𝑒𝑙 𝑒 ( 129 )

Figure 89. Comparison of Chen’s constraint weight definitions (left) for UWLN[74], against those used in this
work (right). The discontinuity in the Chen definition when moving from positive 𝐻
( 𝒒 ) to negative can cause
simulation failure if the integration step-size is too large, stepping over the discontinuity (red). Similarly, the
discontinuity when moving from negative 𝐻 ̇
( 𝒒 ) to positive can cause simulation failure through weight chatter if
𝐻
( 𝒒 ) does not become sufficiently positive before 𝐻 ̇
( 𝒒 ) again becomes negative (black). Both of these
discontinuities have been resolved, as shown by the smoothness of the surface on the right.

Recall, the work by Chen defines a system of weighting which successfully incorporates obstacle avoidance
constraints for a multilink system with kinematic redundancies operating in the horizontal plan[74], and thus it does
not need to account for any external forces. Chen’s control system achieves its means by increasing the effective
weight on any joint whose continued contribution would lead the end effector to collide with an obstacle. In the work
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presented in this research, however, one implements this same weighting scheme in a vertical dynamics context,
scaling the Jacobian transpose such that torque supplied by a joint increases to avoid constraint boundaries. Therefore,
the weighting matrix is implemented in an inverted form with respect to Chen’s definition.

In summary, this section describes the means by which one models multiphase human body landing dynamics using
a simplistic model driven by phase-specific nonlinear control. The flight phase uses joint angle following control to
position the segments for impact. The impact phase uses impedance control to dissipate system momentum. Finally,
the post-impact phase uses weighted energy shaping control to guide the system’s total body COM to a positon above
the base of support while further dissipating the remaining kinetic energy. In the next section, one discusses the various
experimentally based criteria by which the validity of the model is quantifiably tested.

MODEL: VALIDATION

A mechanical model is not useful unless it predicts the dynamic behavior of a system within a desired degree of
accuracy. Thus, important final steps in model the definition process are validation and sensitivity analysis. By the
ASME definition, validation is “the process of determining the degree to which a model is an accurate representation
of the real world from the perspective of the intended uses of the model”[4]. The model proposed in this work is
validated against the experimental data from the collections described in Ch 2: Drop Landing Experiments.

3.4.1 Open Loop Control

Running the model in an open loop control architecture illustrates the need for feedback in the modeling context.
Appropriate model complexity and control is assessed by comparing three simulated drop-landing kinematics and
VGRFs to those of the drop-landing experiments. A first look at the model may suggest that one could simply drive
the simulation kinematics by those of the experiment to arrive at the same results. However, one expects differences
in model definition from simplifications, as well as mass distribution assumptions, to contribute to simulation errors
to the extent which requires closed-loop feedback control to make the proper corrections.

3.4.2 Phase Criteria

For a controls based model of the human body interested in lower extremity dynamics, model sufficiency is defined
in terms of its ability to replicate empirical kinetic and kinematic phenomena observed during drop-landing
experiments. As the phenomena of interest for each phase are functions of time-dependent mechanical objectives, the
rubric by which one measures the sufficiency of the model similarly varies according to phase.

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3.4.2.1 Flight Phase
Based upon the previous studies, several flight phase features are influential in determining the shape and magnitude
of the GRF-waveform during landings:
 absolute base segment orientation (land flat footed vs. heel vs. toes)[77]
 joint configurations / joint angles[26]
 segment velocities[39], particularly heel/ankle velocity[24]
The flight phase control is, therefore, based on manipulating the orientation and velocity of the end effector (foot)
relative to the contact surface. During flight phase, the model’s joint angles are tracked to those of the subject’s relative
segment motion because kinematic landing preparation is the key objective during flight phase. Thus, measures of
model success in this phase are manipulation of segment orientation (±10 ⁰), joint angles (±10%), and relative
velocities (±10%) prior to touchdown. With the segments properly positioned and end-effector slowed relative to the
point of contact, the body must next prepare to steady the COM above the feet by first significantly reducing its
momentum.

3.4.2.2 Impact phase
During impact phase, the degree to which the body collapses correlates inversely to the rate of momentum dissipation.
Evidence of this phenomena is observable when comparing the body segment configuration from the three different
land-and-stop experimental tasks, Figure 90. Softer landings show more body collapse, reducing the rate at which
momentum is dissipated, also measured as a decrease in peak VGRF magnitude. One may conceptualize this body
segment collapse as a softer effective stiffness between the COM and the point of contact. In fact, this effective
stiffness has even been observed in different tasks which include human body impact, such as running gait[13]. Thus,
during the impact phase, the model is assigned an effective impedance in an effort to mirror how the body dissipates
the momentum of the system.

The impact phase is characterized by the degree to which vertical momentum is reduced, so model success in this
phase is measured by the agreement between simulated and experimental impact phase impulse magnitude (±10%).
Some studies suggest that peak reaction force magnitude is a potential contributing factor in injury during foot first
landings involving impact[78]. Therefore, one also uses the accuracy of predicted peak VGRF as a secondary measure
of model success (±10%). Similarly a continued kinematic measure from the flight phase, the degree of segment pull-
up, or in the case of ground contact the degree of body collapse, is provides kinematic measure of technique
modification. Thus, capturing both the kinetic and kinematic markers of the experimental impact phase, the model
predicted vertical change in COM position is also compared to its experimental counterpart. Finally, because humans
are capable of generating a variety of GRF waveform patterns[2], one qualitatively verifies the presence of a similarly
scaled characteristic dual-peak. Proper distribution of momentum reduction across the toe-heel foot strike is a result
which speaks to the importance of including a foot in the model, and so it serves as a measure of foot contact technique
equivalence.

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Figure 90. Relative collapse of body segments (as measured by the bottom left thigh cuff marker) across three
different landing techniques, showing the empirical relationship between peak GRF and degree of collapse at lowest
point during trial.

3.4.2.3 Post-impact Phase
The post-impact phase is characterized by a settling of the mechanical system to a stable equilibrium according to
subject-specific joint coordinations. Within this scope, the human body must accomplish the stabilization objective
when presented with a variety of initial momentum conditions (residuals from the impact phase) as well as a variety
of kinematic states owing to the extent of body collapse during impact. Thus, one measures the post-impact phase
model quality by its ability to produce similarly stable, static equilibrium outcomes. For simplicity, one defines the
equilibrium objective position with a vertical position equal to the simulated impact phase COM end position to reflect
the increased degree of collapse in trials with reduced VGRFs. The horizontal position is, on the other hand, set at a
standard position along the foot length, reflective of the apparent gravitation to a common relative balance point in
experimental trials. Finally, studies have shown subject specific tendencies towards joint coordination manifolds[54].
Thus, one uses the adherence to these ratios of joint angle evolution as a measure of how the model captures subject
specific techniques.

During the post-impact phase, the model is assigned an equilibrium point via energy shaping control. This phase is
characterized by the model’s ability to capture the system stabilization process, bringing the COM to rest over the
base of support. The model should also exhibit the subject specific joint coordination found in the experiment. The
corresponding measure of model success in this phase is how well the simulated joint coordination matches that of
the empirical subject specific solution space (±10%), while bringing the system to rest over the base of support. A last
binary measure of model accuracy is its success in stabilizing the system over the base of support.

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Upon completion of these phase based validation studies, one must test model robustness to variations in coefficient
definitions to qualify the strength of these predictions. If a model’s outcomes are highly sensitive to a particular
parameter, the investigators must have strong confidence in the parameter value to guarantee the substance of model
predictions. To this end, the next section addresses the model parameters which are most uncertain and their influence
over the simulation results. Significant parameter influence over the model outcomes suggests investigators must
develop robust testing schemes to guarantee the value of the parameters to trust model results.

MODEL: SENSITIVITY

In a general review of biomechanical model quality, Hicks states that a simulation is “most credible when the outputs
of interest are insensitive to variables with high uncertainty”[4]. During the impact phase, the modelled components
with greatest uncertainty are those which serve to define the environment. The control coefficients which define the
dynamic interaction with this environment must, by association, also be placed under scrutiny as they evolve from the
same optimization process. These coefficients (the ground reaction force coefficients and each impedance coefficient)
are individually adjusted ±10% to examine their influence on the VGRF-time curves (impulse and peak), as well as
the extent of body collapse during impact. During the post-impact phase, the weight of each kinematic constraint is
somewhat arbitrary, owing to the abstract nature of the concepts. Thus, each post-impact phase constraint weight is
also modulated by 10% to show their influence over the stable equilibrium objective. These outcomes are measured
in a binary landing success measure, as well as a scalar measure of COM positioning error with respect to the
commanded stability point.

In summary, a model should not be sensitive to values which are highly uncertain. The sensitivity analysis modulates
each parameter by ±10% of its original value to determine its degree of influence in deciding the model predictions.
The next section suggests applications for this model, providing insight into otherwise untestable experimental
phenomena.

MODEL: APPLICATIONS

The final phase of research inquiry involves the application of this model’s predictive capabilities. First, the impact
phase control technique is held constant while the floor definition is modified in order to investigate the dynamics
implications of miscued body control objectives. Subsequently, the model is evaluated under modulated joint effort
distributions to demonstrate how impact phase mechanical control outcomes might be influenced by joint specific
training regiments. Finally, the post-impact phase joint limit ranges are increased and decreased to examine the
importance of joint flexibility ranges to the solution space for system stabilization.

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3.6.1 Miscued Control Objectives

In an effort to examine how disparities between a subject’s expectations for environmental interaction dynamics and
those presented in execution affect subject kinetics and kinematics, the model is simulated under different ground
reaction force coefficient conditions than those for which it is optimized. These miscues occur in day to day activities
when one misses an intended point of contact. More extreme examples of these circumstances have been examined
by investigators looking into injuries to circus acrobats performing on raised platforms with spatially inconsistent
stiffness conditions[40]. Thus, while these experiments cannot be conducted without risk of injury to the subjects
under investigation, simulations can provide invaluable insight into potential technique inclusive dynamics outcomes.

3.6.2 Modifying Relative Joint Weight Distributions

The field of sports based biomechanics often attempts to offer dynamics based technique feedback to improve athlete
performance. One of the shortcomings of this approach is an inability to predict the technique based changes which
occur due to plant enhancement through prescribed training programs. Thus, this model’s ability to instantaneously
redistribute control efforts according to different relative joint capacities may provide insights into how techniques
are initially influenced by this reallocation of strength. With a greater understanding of the influence of changes in
relative capacity, one may better be able to design the training regimen to the intended outcomes. The study of these
changes does not require modification of the impedance coefficient sets, rather, one may simply change the associated
joint weights to explore effort redistribution. Similar to a sensitivity study, each impact phase joint weight is
individually modified by ±10% of its standard weighting value of 1 (i.e. 0.9, 1, and 1.1).

3.6.3 Joint Flexibility

The joint flexibility ranges play a role in defining where control effort is distributed during the post-impact phase due
to the rigorous momentum reduction requirements competing simultaneously with stabilization requirements. These
sometimes conflicting objectives may serve to drive joints to their range limits, particularly the ankle in dorsiflexion.
As a result, the use of standard joint limits may not capture the subject specific outcomes of subjects exhibiting subpar
or above average joint flexibility. Similarly, one may also be interested in the robustness benefits to be gained through
an increased range of flexibility. Thus, the model is simulated for modulations on each joint range, increasing and
decreasing them by 10 ⁰ each (±5 ⁰ at both ends of the joint range). The success of these landings is measured in a
binary context to determine how (in)flexible joints may support (or detract from) achieving system stability.

3.6.4 Characteristic Stabilization Force Stiffness Study

Different modes of stabilization occur for different degrees of initial kinetic energy. When the kinetic energy present
in the system is too large relative to the subject control capacity, there is a bifurcation of stabilization behavior which
occurs. This change in evident in cases where the impact phase does not sufficiently reduce the vertical momentum
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of the system, leaving the post impact phase to handle additional energy whilst maintaining balance and bringing the
system to a stable equilibrium point. To understand how increased initial kinetic energy might affect the stabilization
process, the energy-shaping control gains are decreased simulating a decreased capacity relative to the system
momentum. In effect, this modified ratio of momentum reduction rate to initial momentum conditions, offering insight
into how the body might handle the greater system momentum results from a drop from a greater initial height. The
effect of this relative momentum scale is measured qualitatively in the mechanisms which provide the momentum
reduction during the final moments of downward COM velocity.

The next section lists the outcomes of each model design proposition, model sensitivity analysis, and finally virtual
biomechanics experimentation applications.

RESULTS

The multiphase drop landing model of the human body is beholden to phase specific measureables in each phase of
the drop landing task. In the flight phase, joint angles and end-effector vertical velocity are the important measures as
indicated by experimental studies. In the impact phase, the model aims to predict VGRF impulse, VGRF peak
magnitude, and body collapse. In the post-impact phase, the model strives for system stability, bringing the COM to
rest over the base of support, while mirroring the subject specific joint coordinations demonstrated in the experimental
chapter of this research. To provide weight to its outcomes, this model must also demonstrate that its predictions are
robust to modifications in those model parameters which are most uncertain. Namely, the dynamic outcomes should
be insensitive to variations in VGRF coefficients, impedance control coefficients, and joint constraint weights to
strengthen the weight of these predictions. Finally, the model is applied to several biomechanics investigations to offer
insights which might otherwise be inaccessible due to the inseparability of the human subject and those control
mechanisms which define them dynamically.

3.7.1 Open Loop Control

The NJMs derived from inverse dynamic analysis of each land-and-stop task provide sets of coordinated NJMs which
produce demonstrably successful landings. Thus, these NJM sets are applied to the simple rigid, multilink model to
determine if it is of sufficient complexity to produce similarly scaled VGRFs across different impact conditions. These
open loop control NJM-time curves serve as a proof of concept, showing that a simple multilink rigid body model is
capable of capturing task-objective based NJM technique modification effects on the VGRF. Using
MATLAB/Simulink the NJM-time curves are applied to each joint directly as they were calculated from inverse
dynamics, starting the simulation and the application of torque 0.05s prior to impact for consistency. The resulting
VGRFs for each set of NJMs are combined in Figure 91, for comparison.

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Figure 91. Simulated VGRF-time curves (blue-dash) for each set of NJMs corresponding to an experimental landing
condition: self-selected softer-than-normal, normal, and harder-than-normal show a scaling of peak GRF in the
same way as those observed experimentally for Subject 1. Using open-loop control, these NJMs would have to be
timed perfectly to impact, even if they are applied in an actual perfect human body model, in order to produce an
identical GRF-time curve. Instead, as expected, open-loop NJMs are insufficient for reproducing experimentally
measured VGRFs, exhibiting end-effector bouncing from improperly tuned NJM magnitudes / coordination.

3.7.2 Phase Metrics

The following subsections each address the phase specific measures by which model quality is validated.

3.7.2.1 Flight Phase Base Segment Angle, Joint Angles, and Ankle Pull-Up Velocity
Experimentally measured flight phase control manifests as adjustments of the absolute base segment angle, the relative
segment angle (joint angle) time histories, as well as ankle-COM relative vertical velocity changes. The proposed
model accurately captures subject specific flight phase landing technique modifications because the simulations match
the experiments for each of the flight phase metrics, for each subject, during each of the three landing conditions (soft,
normal, hard). The absolute angle of the simulated base segment (shank) matches that of the experiment within the
±10 ⁰ limit, Figure 92. Similarly, each set of simulated joint angles follow those observed experimentally within the
±10 ⁰ limit, Figure 93. Finally, the relative vertical velocity of the ankle with respect to the total body center of mass
in both the simulation and the experiment demonstrate pull-up landing preparation techniques which match each other
within 10% of the absolute COM vertical velocity, Figure 94.

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Figure 92. The absolute orientation of the simulated shank matches that of the experiment within 10 ⁰, a measure
born of the approximate accuracy of the human eye in identifying segment angle, which occurs via manual
digitization. This accuracy holds for both subjects, across all three landing conditions with the worst case matching
occurring in the soft landing of subject 1, at 5.34 ⁰.

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Figure 93. Simulated joint angle time histories of the simple rigid link, hinged model match those measured in
human subjects experimentally within the 10 ⁰ requirement by employing joint following control. Using the
experimental joint angle time histories as the reference trajectories, each joint uses PD control to scale the NJM
such that these joint angles are tracked within 1 degree of accuracy. The only outliers occur at the tail end of the
soft and hard landing conditions for subject 1 (2.37 ⁰ and 1.02 ⁰, respectively) which suggest slightly premature
ground contact in simulation.

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Figure 94. The relative vertical velocity of ankle with respect to the center of mass shows the capacity of the model
to mirror the pull-up techniques of the human subject. Notice, all the simulations match the relative pull-up velocity
of the subject’s ankle within 10% of the absolute downward velocity of the COM at contact (~0.25m/s).

In summary, this simple, rigid body model driven by closed-loop feedback control is sufficient to capture
experimentally measured dynamics outcomes of flight phase human body landing technique control.

3.7.2.2 Impact Phase VGRF Impulse / VGRF Peak / Body Collapse
If the proposed simple model is successful in capturing human body landing dynamics and controls, the simulation
results predict the total impulse during the impact phase for each landing objective, the peak VGRFs measured in each
landing objective, and the experimental COM vertical collapse kinematics within the field standard ±10%.

The impulse, arguably the most significant indicator of model impact accuracy, describes the degree to which the total
system vertical momentum is reduced during the initial spike in VGRF. Because the model to matches this measure,
it accurately captures the primary mechanical objective during the impact phase, Figure 95. This figure also shows the
peak ground reaction forces of all 3 conditions (softer-than-normal, normal, and harder-than-normal) for both subjects
are accurately predicted within the field standard, ±10%. A final feature of the waveform is the presence of the
prototypical dual-peak, toe-heel impact pattern, with a small first peak followed by a more prominent second peak.
The model capacity to match impact phase kinematics is similarly verified with the change in vertical COM position
across each trial matching the change in experimental collapse within 6cm, Figure 96.

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Time w.r.t. Impact [s] Time w.r.t. Impact [s] Time w.r.t. Impact [s]
Figure 95. Predicted VGRFs match those measured experimentally within 10% of both the impulse and peak VGRF
for both subjects (1 = top , 2 = bottom) under all three landing conditions: softer than normal, normal, and hard
(left to right). Notice, each of these waveforms manifest the fundamental dual-peak toe-heel pattern, with a small
first peak followed by a larger second peak.

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Figure 96. The degree of vertical collapse of each subject’s body during the impact phase of the drop landing
experiments, as measured by the change in the COM position, is overestimated in the corresponding simulation by
less than 4cm in all trials for all subjects. The exception to this rule is subject 2’s normal landing which matches
within 6cm. Notice, experimentally speaking, this is also the only case which deviates from the trend of decreasing
collapse for increasing landing intensity.

In summary, this simple, rigid body model driven by closed-loop feedback control is sufficient to capture
experimentally measured dynamics outcomes of impact phase human body landing technique control. The VGRF
kinetics match within the 10% field standard. The VGRF waveform exhibits the dual peak pattern common to toe-
heel foot strikes. Unfortunately, the body compression kinematics do not match as well (<6cm), but they do mirror
experimental trends which correlate reducing compression for increasing landing intensity.

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3.7.2.3 Post-Impact Phase Joint Coordination and System Stability
During the post impact phase, the main objective is to bring the body COM to rest over the base of support. As shown
in Ch. 2: Drop Landing Experiments, each subject accomplishes this task whilst gravitating to a unique ratio of hip
and knee coordination. Simulations of the simple rigid body, hinged model during the post-impact phase demonstrate
the model’s ability to successfully compensate for the residual momentum from the impact phase to bring the body to
rest over the base of support, Figure 97, whilst ensuring joint coordination matched to the subject specific coordination
ratio, Figure 98.

Figure 97. The COM of the simulated human body, stabilizes over the base of support as indicated by its resting
position at a percentage along the length of the foot, measured with respect to the heel. Notice, each simulation
executes until the running average of the total body kinetic energy is reduced below 1J, so they extends longer than
the experimental trials and do not have common time-scales. Most importantly, the simulated system stabilizes for
each unique kinetic energy initial condition presented from the previous phase demonstrating that this model, like
the human, is capable of bringing the system to rest over the base of support for both subjects in all three landing
conditions.

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Figure 98. The unique joint coordination exhibited by human subjects are mirrored by the simple hinged joint
model, as illustrated by the alignment of simulated joint coordination evolution with the subject specific joint
coordination ratio (dashed lines). The end of the simulation is denoted by a black dot, as time is difficult to perceive
in the state-space plane. Note, these coordination trends are easiest to see in the soft and normal landings which
undergo a greater range of motion during the simulation.

In summary, this simple, rigid body model driven by closed-loop feedback control is sufficient to capture
experimentally measured dynamics outcomes of post-impact phase human body landing technique control. The
modeled system stabilizes over the base of support for both subjects in all three landing conditions. Simultaneously,
the simulated joint coordination evolution matches the coordination ratio of the subject, with the state trajectories
following the dashed-coordination lines.

3.7.3 Sensitivity Analysis

Before this model may be utilized, its sensitivity must be analyzed in the context of the most uncertain parameters. In
essence, a model’s predicted measures should not be sensitive to those parameters which are not well known, or else
the outcomes are not well founded.

3.7.3.1 Impact Phase VGRF Sensitivity
The impact phase mechanical objective is the correct application of support force to resist ground reaction forces,
reducing the system momentum to a desired extent. The simulated should not VGRF predictions should not be
sensitive to modifications of the coefficients on either side of the ground-human interaction. The following figures
illustrate that normal landing simulation is insensitive to the ground coefficients, with the stiffening spring power
coefficient exercising the greatest influence over the resulting VGRFs, Figure 99. Similarly, 10% changes in most of
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the impedance coefficients result in small changes to the impact phase VGRF impulse and peak. However, some
coefficients, such as the initial impedance length of both LOI1 (ankle-COM) and LOI2 (heel-thigh) carry significant
influence over the dynamic outcomes, Figure 100.

% Stiffness (𝐶 𝑦 ) [N/m] Stiffening (𝐶
) Damping (𝐷 𝑦 ) [Ns/m]
75

90

110

125

Time w.r.t. Impact [s]

Time w.r.t. Impact [s]

Time w.r.t. Impact [s]
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Figure 99. The VGRF waveform is insensitive to 25% stiffness and damping modifications in the VGRF
coefficients. However, the ground spring stiffening power modifies the VGRF peak from the optimized peak error
of 6% to -47% and 12% error. Similarly, the impulse error modulates most with the stiffening spring power, from
the optimized error of 0.7%, to 3% and 0.9% error. In all of these cases, the impact phase VGRF impulse
requirements are still met, with all errors landing well below the 10% threshold. Interestingly, the stiffening term
has the largest effect of the quality of the waveform as well, showing signs of bouncing or excessive compression
in the 75% and 125% plots, respectively.

% 90 110
LOI1 K p

LOI1 K v

151

LOI1 t

LOI1 x 0

LOI1 x 1

152

LOI1 x 2

LOI2 K p

LOI2 K v

153

LOI2 t

LOI2 x 0

Simulation Failure
LOI2 x 1

154

LOI2 x 2

Figure 100. The simulated VGRFs for Subject 1’s normal landing condition show considerable robustness to
variations of 10% in most variables, with more variation occurring in the peak VGRF than the impulse, in general.
Large changes in the VGRF occur from differences in optimized subject ankle to COM stiffness and damping
(LOI1: K p, K v), the initial impedance length (LOI1: x 0), and initial impedance slope (LOI1: x 1). The second line of
impedance appears to be robust to all measures except the initial impedance length (LOI2: x 0) and the secondary
impedance slope (LOI2: x 2). In both lines of impedance, the initial impedance length is a highly sensitive quantity.
The sensitivity to these quantities may be problematic, as they are the variables which one would use to model the
human technique control modifications. There are many different coefficient sets which result in effective
impedance control solutions, so there may be a more robust method for finding the global optimum set.

In summary, this simple, rigid body model driven by closed-loop feedback control predicts experimentally measured
kinetics of the impact phase human body landing as influenced by technique control, while being robust to changes in
the ground reaction force floor model. In general, the modeled ankle-COM impedance of the system has a larger
influence over the predicted VGRFs than the heel-knee impedance. While the predictions are robust to most variables,
the model is sensitive to the ankle-COM stiffness, damping, initial impedance length and slope. In the second line of
impedance (heel-knee) the system is extremely sensitive to the initial impedance length, with a much smaller, though
significant sensitivity to the second impedance slope. These results suggest that the model is, as one would hope,
sensitive to the impedance coefficients, exactly as the experiment is sensitive to changes in the human control
technique.

3.7.3.2 Post-impact Phase Constraint Weight Sensitivity
Post-impact simulation stability outcomes are robust to 10% changes in constraint weighting. Both the joint
coordination (K coord) and joint limit (K lim) weights show negligible influence over the stability of the model, as
illustrated by the COM horizontal settling position along the length of the foot. Instead, one shows that the stability
of the system is completely insensitive to the joint limit constraint, and only sensitive to the coordination constraint in
a binary sense, Figure 101.

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% 0 50
K lim

K coord

Figure 101. COM position along the foot (base of support) as measured from the heel illustrates how the simulated
system settles with respect to how the experimental system settles. When the joint limit constraint is removed, the
system still remains stable, relying on the coordination constraint to keep the joints away from their limits.
However, when the coordination constraint is removed, the system becomes unstable and bounces off of its joint
limits before finally tipping over all together. Thus, the system is sensitive to the activation of the joint coordination
constraint for stability

Examining these outcomes through the lens of energy shaping, one evaluates of the angle of the COM with respect to
the base of support plotted against the angular velocity of this point over top of the base. These terms reflect the kinetic
energy of the system (angular velocity) and potential energy (angle from equilibrium), such that a stable system will
spiral to this plot’s center. As the angular velocity of the system reduces, the center of mass is simultaneously pulled
to a point over the base of support, resulting in the centered spiral patterns below, Figure 102. As observed in the
previous set of plots, the angular velocity of the system does not decrease sufficiently when the coordination constraint
is removed entirely, resulting in a run-away system which fails to stabilize (bottom left).

% 0 50
K lim

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K coord

Figure 102. Plotting the COM angular velocity over the base of support against the COM angle with respect to the
center of the base of support illustrates how the simulated system settles with respect to how the experimental
system settles. When the joint limit constraint is removed, the system still remains stable, relying on the
coordination constraint to keep the joints away from their limits. However, when the coordination constraint is
removed, the system becomes unstable and bounces off of its joint limits before finally tipping over all together.
Thus, the system is sensitive to the activation of the joint coordination constraint for stability.

In summary, the model is not sensitive to either of the joint constraint weights. It appears that there is a large amount
of room for variation of both weights, without having a noticeable influence over the ability of the model to stabilize.
Finally, completely disregarding any joint coordination resulted in an unstable system, demonstrating the importance
of this subject specific joint coupling.

3.7.4 Model Application Results

Having verified the insensitivity of the model to most of the system parameters, one may now confidently apply the
model to biomechanical studies to gain insight into the effects of previously inaccessible technique modifications.

3.7.4.1 Miscued Landing Stiffness Preparation
A scenario that humans encounter on a fairly regular basis is the technique shift which occurs to compensate for
interaction with different contact surfaces[13]. In this study, each ground reaction force coefficient is modulated by
25% increments to show how the NJM control torques would change as a function of changes in the stiffness and
damping of the floor if the impedance control was not properly re-optimized. Using the optimized normal landing
condition as the baseline, deviations in the NJM time-history indicate how the subject’s miscued impedance trajectory
drives the observable NJMs to non-optimal variations, Figure 103.

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Δ Stiffness (𝐶 𝑦 ) [N/m]

Δ Stiffening (𝐶
)

Δ Damping (𝐷 𝑦 ) [Ns/m]

Figure 103. These plots illustrate how changes in the VGRF coefficients affect the NJMs required by the impedance
control. The impedance control trajectory and stiffness were held constant, while the three ground reaction force
coefficients (Cy, Dy, and Cexp which describe floor stiffness, damping, stiffening, respectively) are altered. What
is plotted is the difference between the correctly optimized NJM set and that of the modified floor condition set.
By this measure, both the stiffness and damping cause NJM changes on the order of 20Nm at most. The stiffening
power term, however, causes changes to the baseline simulated NJM on the order of 100s Nm. This suggests that
the largest disparity between intended NJMs and actualized NJMs would occur if the ground stiffens at a different
rate than expected.

In summary, the impedance control is robust to significant (50%) changes in the stiffness or damping of the contact
surface. The largest control compensation occurs when the floor stiffening rate is faster or slower than expected, as
illustrated by the NJM deltas being an order of 10 higher in the right most column (C exp) than that of the left and
middle column (C y and D y, respectively).

3.7.4.2 Modified Impact Phase Joint Effort Distribution
During a landing involving impact, joint recruitment patterns may vary, as evidenced by the differences observed
between the joint pull-up techniques shown in the Observed Flight Phase Control: Kinematics Section. Therefore, an
important investigation into technique modification examines how the NJM patterns vary as a function of joint
weighting. These joint specific weights serve to distribute the control effort across the joints according to some internal
cost function built into their natural control scheme. Using the optimized normal landing condition as the baseline,
deviations in the NJM time-history indicate how the changes in subject joint specific weights (increments of 25%)
drive the observable NJMs to non-optimal variations, Figure 104.

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Δ Hip Weight

Δ Knee Weight

Δ Ankle Weight

Figure 104. These NJM comparisons show that there is a variety of techniques to achieve an overall momentum
reduction during the impact phase. Some of these techniques seem to require less effort than others. Interestingly,
the hip joint weighting NJM outcomes seem to be the exact inverse of the ankle joint weighting NJM outcomes.
Knee weighting seems to reduce NJMs in the hip and ankle simultaneously.

For reference, the resulting VGRF from these modified joint weights illustrate strong agreement with the original,
optimized NJM set, Figure 105. Importantly, none of the corresponding simulated VGRF-time curves violate the 10%
experimental impulse accuracy requirement described in the model validation section. While there are noticeable
differences in the NJM sets, their overall support effects serve to provide the same level of momentum reduction as
with the original weighting set. Some local differences between the VGRFs waveforms manifest as instantaneous
VGRF errors, which likely occur due to the control structure design with two lines of impedance. While modification
of joint weights for a single line of impedance change the null-space of the system, negligibly modifying the effective
impedance. The impedances defined in this research span different sets of joints. Thus, modification of a joint weight
may affect the internal-interaction of the impedances.

Δ Hip Weight

Δ Knee Weight

Δ Ankle Weight

Figure 105. These VGRF comparisons show that modifications in joint effort distribution, while internally affecting
the NJM at each joint, do not have linear influence over the resulting ground reaction forces. This makes sense, as
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the weighted Jacobian should not significantly scale the support forces, merely manipulate the null space of the
system to redistribute the internals. Of course, there are some instantaneous differences due to the definition of two
separate lines of impedance, whose relative trajectory timings also serve to govern instantaneous effort distribution.

In summary, there are a variety of ways to achieve vertical momentum reduction. Depending on the muscle distribution
of a subject, these NJMs may occur in any one of the NJM comparisons demonstrated by the weight distribution study.
Interestingly, increased hip weight seems to correlate with a reduced ankle weight, and vice versa. This model can be
used to predict how NJM patterns might shift with muscle capacity shifts, by approximating the relative muscle growth
as increases in the associated joint weights.

3.7.4.3 Joint Range Post-impact Flexibility Effects
The post-impact phase constraint weights for each joint are built on the associated joint range available to the subject.
In order to investigate how a subject’s flexibility affects the controlled NJM techniques, each joint range is expanded
and contracted by 10 ⁰ (5 ⁰ on each end), Figure 106. This study is useful for predicting how a solution space might
change given a slight increase or decrease in flexibility.

Δ Hip Range Δ Knee Range Δ Ankle Range

Figure 106. These simulations appear to be unaffected by the 20⁰ (±10⁰) expansion and contraction of each joint
range. This suggests that the simulated subject is not approaching the joint limits during this task (as simulated).
When the artificially diminished ankle joint limit is reached, the NJM sets show significant deviations in magnitude.

In summary, the joint flexibility range does not affect the resulting NJM sets until the range becomes narrow enough
to infringe on the range of motion during the normal landing condition. This suggests that both the hip and knee do
not get anywhere near their joint limits with the current definition of the energy-shaping control. Similarly, the ankle
range of motion is at least 5 ⁰ inside the normal ankle joint range, only showing NJM compensations once the range is
compressed by 10 ⁰ at either end.

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3.7.4.4 Characteristic Stabilization Force Stiffness Study
Different modes of stabilization occur for different degrees of initial kinetic energy. To understand how increased
initial kinetic energy might affect the stabilization process, the energy-shaping control gains are decreased. In effect,
this modifying the ratio of momentum-reduction-rate to initial-momentum conditions, offering insight into how the
body might handle greater system momentum. These insights are relevant to cases in which the body drops from a
greater height, or is for some reason unable to reduce the momentum sufficiently during the impact phase. Thus
changes to the stabilization process may offer insight into how humans adjust their stabilization technique to
compensate. Reducing the energy shaping control gains results in exaggerated body collapse during the post impact
phase before recovery, Figure 107. Interestingly, the time to system stability varies nonlinearly as a function of gain
scaling.

Figure 107. Reducing the energy shaping control gains from their original values results in a fundamental shift in
the constraint mechanism responsible for initial kinetic energy reduction. Unmodified energy shaping control relies
entirely on the joint coordination constraint to define the control effort distribution. As these gains are reduced, the
joint limit constraints play a larger role in redistributing these control efforts. At one fourth the original control
gains, the stabilization technique involves spending the entire range of joint motion, using the joint limits as a hard
stop to reduce the residual downward momentum. Show constraint specific activation patterns in each landing
condition.

In summary, the joint coordination mode dominates the control process when the energy-shaping stiffness is
sufficiently high to handle the momentum conditions. Halving the control gains results in a more stable model that
takes advantage of the increased horizontal stability achieved by lowering the center of gravity. At this point, the most
influential constraint remains the joint coordination term. However, further halving the control gains results in a
system which uses the full range of body collapse, riding on the joint limits to increase the rate of kinetic energy
reduction. This demonstrates a bifurcation in the constraint mechanism used to achieve overall system stability, relying
on joint limit support rather than simply the joint coordination manifold.

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DISCUSSION

A simple, rigid segment, closed-loop feedback control based dynamics model is built to capture the subject’s capacity
to modulate VGRFs via technique modification during this drop-landing task through net joint moment control.

Due to the simplifications and floor definition, simply applying the empirical NJMs to the model are insufficient to
recreate the dynamics observed experimentally. Thus, each phase-specific control loop feedback is designed to control
the model according to the mechanical objectives during each phase of a landing with impact. The flight phase uses a
following control to successfully match experimentally measured subject kinematics during impact preparation within
10 ⁰. The impact phase uses impedance control during impact to successfully capture how subject’s effective stiffness
determines the resulting VGRF impulse and peak within 10%. Finally, the post-impact phase uses passivity based
control to successfully capture the kinetic energy-dissipation and system stabilization process while incorporating
subject-specific joint coordination strategies.

During the impact phase, the model proves to be insensitive to the ground stiffness and damping coefficients, but is
highly sensitive to the floor stiffening power. Similarly, the post-impact phase simulations prove to be insensitive to
each of the constraint weights, only showing stabilization failure when the presence of the joint coordination constraint
is negligible.

Applying the normal landing condition model to under various ground definitions demonstrates how the model can
predict the NJM-level consequences of miscued impedance control expectations. Similarly, adjusting the relative
impedance control joint weights illustrates how specific joint capacities may influence the kinetic outcomes.
Modifying the joint limit ranges may offer insights into the role that joint limit avoidance plays in determining the
kinetic outcomes during stabilization. Reducing the post-impact energy-shaping control gains demonstrates how the
ratio of initial vertical momentum to control force gain shifts the constraint mechanisms which ultimately terminate
vertical motion.

3.8.1 Model Limitations

By definition, a model is a simplification of a much more complex dynamic system to extract the most essential
elements to better understand the fundamental phenomena which explain empirical measures. Before addressing the
model results, sensitivities, and application outcomes, one must first acknowledge the potential shortcomings of the
model.

A most obvious shortcoming of this model derives from the need to describe the interaction between the environment
and the subject in terms of kinematics. This model must capture the degree of contact force in terms of kinematics
because the experimental kinematics of the foot may remain in place while the kinetic measures vary significantly,
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leaving no physical indicator of the localized support force applied to each point along the foot. This simple definition
leads to the very non-physical penetration of the floor by the model’s foot segment (toe and heel). Due to this
penetration, the model will never perfectly match experimental kinematics, as the subject never breaches the contact
surface.

The drop landing model has also simplified the system into 4-rigid segments. A shortcoming of this definition derives
from the assumption of a rigid foot, as this segment very clearly flexes during the empirical landings. This inaccuracy
manifests in a few ways. First, the model tends to express a greater degree of pull-up during the flight phase to reduce
the relative toe velocity at contact, as the rate at which a rigid foot loading occurs must be temporally equivalent to
the rates observed in a compressible, flexible foot. Second, the ankle angle must be allowed to collapse much further
than the subject is physically capable of as a means for capturing how one might roll up onto the ball of the foot during
dorsiflexion saturation in the impact phase. Third, the floor model must absorb the lost flexibility of the foot in order
to captures the rate of force production generated by a flexible contact segment. A flexible foot segment generates
support force at a slower rate than a rigid segment, much like the crumple-zone of a car during impact. The modeling
result is an overly soft landing surface which has unintended side-effects as the model continues to be integrated in
time. First, the foot penetrates the floor significantly during impact, more so at the toe, but also measurably at heel
contact. Next, the as the model stabilizes, it may have a tipped foot segment as a function of the effective center of
pressure described in terms of two soft contact points, leading to an inaccurate ankle angle. Finally, this tipping results
in a model that takes longer to stabilize, as it is attempting to balance on a rocking flat foot. This rocking motion is a
problem because the control algorithms have been defined internal to the human body system (as it is not fixed to any
other surface). Thus, foot tipping rotates the reference frame of the entire system and the model attempts to bring the
body COM to the equilibrium point of the tipped reference frame. This tipped equilibrium chasing may further tip the
system leading to instability. A current work around is the addition of externally defined equilibrium point which
approximates to that of the human body system when the foot is flat. Thus, the model is also controlled to a pseudo-
inertially fixed (and overall more stable) equilibrium point.

A shortcoming of the impact phase is the way in which impedance control coefficients are derived. Because these
elements are largely a theoretical construct, the starting point for the optimization of these values is someone abstract.
Allowing the optimization process to begin from different seeding points results in a variety of impedance coefficients
which accurately capture the desired system kinetics. As a result, identification of trends within these coefficients
remains difficult.

A shortcoming of the post-impact phase control is a force definition which is satisfied in the least-norm sense. This
means that the desired control forces may not be generated perfectly if they conflict with system constraints. This
shortcoming is observable in the small offset error between the commanded equilibrium point and the one at which
the model settles.

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Additional assumptions are perfectly planar motion, with ideal hinge joints. These assumptions do not leave room for
the twisting of joints which may occur at the hip or ankle as the system flexes. Additionally, the NJM reported at each
joint describe the minimum necessary joint torque to achieve the observable motions. In reality, the subject may be
co-contracting several muscles resulting in joint torques which may internally cancel out.

3.8.2 Research Questions

This section of the dissertation addresses the results of the drop landing model, as well as each result’s implications
in the context of biomechanics.

3.8.2.1 Open Loop Control
The application of open loop NJM control to the simple rigid body model was, as expected, insufficient for
reproducing the successful human landing mechanics observed during the drop-landing experiment, despite using the
empirical NJM time-histories as inputs. Fundamentally, this outcome speaks to the idea that systems completing
complex tasks rely on feedback to correct for deviations from an intended outcome. The human body, for example,
uses feedback from the eyes, ears, skin, and other proprioceptors to understand the state of the body in space. Proof
of this reliance on feedback can be found by simply closing one’s eyes while balancing on one foot. This task becomes
measurably more difficult when feedback from the sense of sight is eliminated.

In the context of modeling, the approximations of mass distribution are enough render the system physically
inadequate to blindly follow control inputs to a successful end. In fact, the very assumptions which allow one to model
the system are fundamental discrepancies with the real system. Thus, driving the model by empirical inputs must, by
definition, result in some inequivalent behavior. This holds particularly true when the modelled system is in contact
with the modelled environment. Because the simulated environment allows the foot to breach the surface, the
simulated kinematics can never match those found empirically.

3.8.2.2 Model Sufficiency
The primary working hypothesis of this work is that a simple 2D model of the human body composed of rigid segments
connected by ideal hinge-joints is sufficiently complex to capture the empirical dynamics of the body across multiple
phases of landing, when driven by nonlinear feedback control. As a result of this research project, one can conclude
that this model is sufficiently complex to capture the mechanical objective based technique modifications observed in
the human body during drop-landings under various impact intensity conditions.

3.8.2.2.1 Floor Model
The first subject’s normal landing experimental data is used to train the model environment coefficients and trial
specific impedance coefficients simultaneously. The rest of the first subject’s trials (along with the second subject’s
trials) are used to validate the model’s ability to capture condition-specific technique modification using the
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aforementioned experiment trained environment across all subsequent trials. Because this one ground reaction force
model, when used in conjunction a human body model driven by different control coefficients, accurately reproduces
the VGRF waveforms for the 5 additional trials for which the floor was not trained, it is considered a sufficiently
accurate and robust definition.

3.8.2.2.2 Flight Phase Model
The accuracy of this human component of the model is determined by comparing phase specific empirical indicators
with those of the corresponding simulations. Studies suggest that kinematics prior to touch-down drive the kinetics
measured during impact phase[39], so the adherence to experimental kinematics is used to measure model quality
during the flight phase. The simulated joint-angle time-histories accurately follow the experimentally measured joint
angles throughout the flight phase, with a maximum following error much less than 10 ⁰. This outcome implies that a
human being attempting to follow a set of intended joint trajectories (sometimes referred to as “efferent copy”) could
sufficiently modify their flight phase techniques to achieve a variety of conditions at contact. This variation of contact
states allows the model to achieve the empirical variety of observable VGRF-waveforms during the impact phase.

3.8.2.2.3 Impact Phase Model
A primary mechanical objective of the impact phase is to reduce the total vertical momentum of the body to a
manageable degree for the post-impact phase. Consequently, the results from the impact phase of the model
demonstrate that the simple closed-loop controlled model accurately (within 10%) predicts both the vertical impulse
(aka: net vertical momentum reduction) and VGRF peak of 6 different drop landing experimental VGRF waveforms
modifying only the control coefficients. Similarly, the degree to which the modelled human body COM crumples
during the impact phase is accurate to its empirical counterpart within 6cm for all 6 experimental landings. The
agreement of the model with these two features implies that the human modification of NJM techniques during impact
phase of the drop-landing task are sufficiently modelled by the impedance control algorithms described in this work.
Specifically, the modulation of body stiffness and intended body trajectory both contribute to achieving the kinetic
outcome of scaled momentum reduction and scaled peak VGRF.

3.8.2.2.4 Post-impact Phase Model
Finally, the sufficiency of the post-impact phase model is qualified by how well the joint coordination patterns match
those of the subject, while momentum is reduced and the center of mass is constrained to reside over the base of
support. The results from the post-impact phase simulation demonstrate that a simple rigid-segment model is
sufficiently complex to capture the stabilization and kinetic energy reduction of human subjects under a variety of
landing conditions, when it is driven by closed-loop energy-shaping control. The most immediate evidence of this
sufficiency is that the model stabilizes the system COM over the base of support, despite being initialized from the 6
unique kinetic energy conditions which are products of the impact phase simulations.

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Stabilization encompasses two key elements: kinetic energy reduction of the system, and effective equilibrium
definition. Kinetic energy reduction, the more straightforward of the two, simply implies that the system can be
controlled such that it eventually stops moving. This objective is not difficult on its own, as it could be achieved
without any control input. However, when it is coupled with the equilibrium point shaping control, the kinetic energy
reduction timing must coincide with times when the COM is over the base of support. Second, the energy-shaping
control successfully draws the COM to a point over the base of support. Interestingly, this control objective may be
achieved by directly defining the forces which would place the COM at a location, then allowing the forward kinematic
transformation to convert these task space forces into joint space NJMs. This modelling success implies that the
human control system may act in a similar way, defining a balance point in space, and allowing lower order control
transformations to determine the specifics of achieving that objective.

These control objectives alone are insufficient to reproduce the subject specific kinematics, suggesting that there may
be additional drivers contributing to the empirical outcomes. Within this vein, the addition of joint limit constraints
and joint coupling constraints serve to provide the necessary solution manifold definition to recreate the joint
coordination patterns observed in experiment. Specifically, the success of weighting each joint torque by the inverted
Unified Weighted Least Norm method suggests that each subject may have a preferred solution space. While the
reason for these subject-specific solution spaces is unclear, their success demonstrates that not only must a plant be
properly sized a subject, but also their phase-specific control objectives must be shaped to each subject-specific
coordination technique preferences.

3.8.2.3 Sensitivity Analysis
As noted in previous sections, a model’s results are not trustworthy unless they are insensitive to those parameter
inputs which are most uncertain.

The model of the floor is a key element of this drop-landing model, with both kinetic and kinematic requirements. The
VGRF waveform predicted by the model shows considerable insensitivity to the stiffness and damping coefficients,
with coefficient modifications of ±25% resulting in the impulse and peak reaction force metrics ranging from 0.61-
0.88% error and 4.31-7.68% error, respectively. On the other hand, these VGRF metrics are much more sensitive to
the floor stiffening power coefficient, showing qualitative differences in the waveform (i.e. dual-peak to mono-peak
and multi-peak) as well as peak VGRF errors on the order of 7% to changes of only ±10%. The peak VGRF for a
±25% change in the stiffening power varies from 13-47%, respectively. Incredibly, even under these circumstances,
the VGRF impulse is well below the 10% accuracy threshold, maxing out in the -25% stiffening case around 3% error.
In summary, the momentum reduction objective (VGRF impulse) is very robust to changes in the VGRF definition
coefficients, while the peak VGRF metric is much more susceptible to inaccuracies in the floor stiffening parameter.

Another point of uncertainty within the model is the relative weight of the constraint gains during the post-impact
phase. Constraint weights serve to describe the relative importance of each constraint with respect to the overall control
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objective. As a constraint weight increases, the system becomes less likely to violate that constraint. The Unified
Weighted Least Norm method, applied in this research, automatically increases a weight as the system nears the
associated constraint boundary. However, the rate at which these constraint weights increase is defined by the
associated user-defined constraint gain. Varying these gains by 50% did not have a noticeable effect on the system’s
performance, so the range of variation was increased until the constraints were on or off. Regardless of the state of the
joint range limit constraint, when the joint coordination constraint is active, the joint range limits are not violated at
all in the normal landing condition. Thus, even when the joint limit constraint is eliminated, the system still achieves
the mechanical objectives of the phase. Alternatively, eliminating the joint coordination constraint leads to
stabilization failure even when the joint limit constraints are active. This failure is likely because the rate of kinetic
energy dissipation is insufficient to prevent the horizontal COM oscillations from extending beyond the base of
support. These sensitivities imply that subjects learn and utilize joint coordination patterns which inadvertently prevent
the violation of joint angles limits. It is also possible that the joint coordination constraint is only dominant due to the
steepness (stiffness) of the energy shaping control manifold, a trait which would prevent exploration of the full range
of motion.

Beyond these measures, outcome variations are a product of the modelled control definitions which intentionally
manipulate system behavior to mirror the technique modifications observed experimentally. Thus, these associated
outcome variations are not only expected, but they are the research objective of the project. They demonstrate how
variations in control strategy might influence the dynamic behaviors of the human body across the phases of a landing
task.

3.8.2.4 Application Analysis
This section addresses implications of the model as it is applied to the various research questions.

3.8.2.4.1 Miscued Landing Stiffness Preparation
Expectations may have a large effect of the dynamic behavior of the human body. In the event of mismatching between
expected environmental interactions and those which are actualized, humans may be put at risk of injury[40]. While
investigations into this biomechanical dynamic cannot be conducted in situ without risk of subject injury, the model
provides a means to test this disconnect. By modifying the VGRF model and maintaining the effective joint control
impedance schedule from the optimized normal landing condition, one is able to observe the NJM effects of an
improperly tuned control law.

In order to highlight the effects of the mismatch, the optimized NJM time-histories are used as a baseline and the
differences between these curves and those realized under different ground conditions are reported. These differences
represent the additional effort that each joint would need to carry as a result of the miscued interaction. Because the
differences from changes in stiffness and damping generally fall below 10Nm, these ground modifications may not
be likely to cause injury. However, the ground stiffening term causes differences in the ankle load on the order of
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100Nm’s, an error magnitude on par with the original NJM time-history itself. Thus, differences in ground stiffening
may double the load applied at the joints, a scenario which could conceivably result in injury. These results suggest
that the floor stiffening may be a more significant indicator of human injury during an impact, than other mechanical
measures.

3.8.2.4.2 Modified Joint Effort Distribution
The relative joint effort weighting serves to skew the distribution of task-space defined control forces across the joints
in the form of NJM efforts which rely more heavily on joint effort contributions from joints with higher relative effort
weights. The model is originally optimized with all three joint weights assigned unitary values. However, there is
likely a relative joint capacity scheme which should be taken into account for each subject. In order to understand how
this scheme might affect the affect the NJM distribution, each joint weight was individually increased and decreased
from 0.5 to 1.5 in increments of 0.25 while holding the other joint weights constant. The result demonstrates how the
entire body must compensate a changes to a single joint weight. Interestingly, the changes in hip and ankle NJMs due
to increases in the hip joint weight appear to be inverse of those which derive from an increases in the ankle joint
weight, as shown in the first and third columns of Figure 104. The knee NJM shows the largest swings in compensation
during variation of its joint effort weight as compared to that of the other two joints. This outcome suggests that
weakening or strengthening of the knee joint may have the largest effect on the overall power generation required to
accomplish the same effective stiffness during the drop landing task.

3.8.2.4.3 Joint Range Flexibility Effects
Contrary to expectations, modification of the joint flexibility ranges does not influence the NJM control sets until the
ankle range is reduced enough (-20 ⁰) to infringe on the free motion of the ankle during the drop-landing motion in
question. The Unified Weighted Least Norm method selectively activates the joint weight increases when the joint
state nears one of the associated joint constraints, so the joint limit constraint is not expected to modify the technique
until this is the case. Recall, the constraint margin definition is represented as 5% of the entire joint range on either
end of the range. Prior to simulation one expected the simulated range of joint motion to be such that the joint limit
margins are reached during a normal drop landing, as evidenced by the subject’s heel coming off the ground at
maximum ankle dorsiflexion during the experiment. The lack of joint limit influence may be a product of the stiffness
of the energy-shaping control as it is currently applied. If the shaping control is too stiff (large potential energy gain),
the body will not be allowed to move through the experimental range of motion, leading to a system which is solely
reliant on the joint coordination constraint to provide plausible, human-landing state-space solution manifolds. This
question is addressed in the following section

3.8.2.4.4 Characteristic Stabilization Force Stiffness Study
Reducing the implemented energy-shaping control gains (K p,K v) to varying degrees still results in self-stabilizing
system behavior, though it relies on different control mechanisms to do so. The implication here is that the post-impact
phase control law should also account for the vigor with which the subject attempts to bring the system to equilibrium.
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This rigidity is reflected in both the resistance to continued crumpling of the body and the rate of relative kinetic
energy dissipation during this final phase of landing.

Given the original control law used in this study, the system crumples less than 10cm. This reduced range of motion
means that it must handle the residual horizontal momentum at the top of its stance, resulting in the system rocking
back-and-forth due to the large lever arm of horizontal contact forces. It quickly reaches the vertical equilibrium point,
but because the foot has only two points of contact and stands on a soft surface, the body continues to oscillate until
the foot comes to rest. Despite reducing the relative momentum of the body with respect to the foot, this motion means
that the system kinetic energy is not reduced for upwards of three seconds.

Dividing the control outputs by factors of 2 (i.e. ½ and ¼) results in similarly stable system behaviors. However, the
model achieves this stability by utilizing an increasing range of collapse. Throughout this collapse the kinetic energy
is being reduced with the end-effector (COM) at a much lower vertical position. This results in a smaller effect on the
horizontal velocity of the system from the initial rocking of the foot, due to the much smaller lever arm for the
horizontal contact forces. Notice, this is similar to how humans are much more stable in a crouched position due to
their lower center of gravity. In addition to greater horizontal stability, this use of the full range of motion causes the
joint limits to come into play. In the ¼ scale case, the model uses the collision of segments (captured by lower bounds
on the joint limits) to enhance the momentum reduction of the system. This outcome implies that if a subject simply
reduces their post-impact phase energy-shaping control gains, they can utilize characteristically different landing
stabilization modes.

More importantly, this control gain reduction demonstrates how the system might behave given greater initial
downward momentum, which occurs during drop-landings from a greater height. In the case of a softer control system
(reduced control gains) the ratio of support force to initial body momentum is much lower. This lower ratio is also
achieved by full capacity control forces and greater initial momentum. This study highlights the bifurcation in
stabilization behavior which occurs when the control efforts alone are insufficient to terminate the downward
momentum, and joint limits must be used as hard stops for joint motion. Because scaling of control gains demonstrates
a tradeoff between rate of vertical kinetic energy reduction and horizontal stabilization, one hypothesizes that further
studies may illustrate a momentum specific optimal degree of collapse which both reduces the kinetic energy of the
system as fast as possible while taking advantage of the stability gained through a lower crouch point. Evidence of
this pattern is observable in the reduced settling time of the 50% control gain scale simulation.

CHAPTER SUMMARY

Techniques used to develop and validate a 2D multi-segment, rigid-link model of the human body driven by nonlinear
controls were described. Three phase-specific feedback control algorithms, following control, impedance control, and
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energy-shaping control were designed for use in flight phase, impact phase, and post-impact phase, respectively.
Model validation studies demonstrate that the simple 2D rigid-segment, connected by hinge joints, is sufficient for
capturing the empirical changes in human body dynamics across mechanically unique phases of a landing task
involving impact, when driven by nonlinear closed-loop feedback control. Control based model application studies
show the various ways control efforts might change as a function of tuned contact surface expectations, joint-specific
capacity modifications, joint flexibility, and scale of initial momentum. The next chapter summarizes the findings of
this research and addresses potential directions this work could be extended in the future.
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CHAPTER 4

CONCLUSIONS

The research presented in this work investigates the sufficiency of a simple experimentally validated 2D forward
dynamic model of the human body in predicting system dynamics as a function of landing task objectives using only
rigid-body segments, idealized joints, and feedback-based, coordinated net joint moment control to highlight the
fundamental control-based mechanisms which define the behavior of the human body during the three phases of
landings: flight-phase, impact-phase, and post-impact phase, as well as its ability to adapt those mechanisms according
to the task objective.

DROP LANDING EXPERIMENT

The drop-landing experiments covered in Ch. 2: Drop Landing Experiments, describe the collection and analysis of
kinetic and kinematic data from three drop landing experiments each, for two subjects. The tools and techniques used
to analyze this data were presented, including a novel method for smoothing kinematic data which enables multiphase
analysis across kinetically distinct task conditions. While this task is largely 2D, tasks involving higher-dimension
rotational motion may be analyzed using a novel functional joint axis parsing method which demonstrates how one
might understand a complex, multiplanar motion in the context of its underlying kinetic drivers. These techniques are
used in conjunction with inverse dynamics to illustrate the importance of human body control. The empirical body
control is characterized in terms of phase-specific mechanical objectives throughout the drop-landing task, for the
purposes of developing a sufficiently complex model of the human body to replicate these phenomena. The flight
phase is described through end-effector positioning and relative velocity control, the impact phase is defined in terms
of effective body stiffness and compression, and the post-impact phase shows momentum termination with center of
mass stabilization over the base of support.

DROP LANDING SIMULATION

The modeling world of biomechanics walks a precarious line between overly complex models which boast broad
applicability but offer limited mechanical insights, and overly simplistic models with offer more fundamental insights,
but offer limited model applicability. The work presented in this dissertation combines the best of the both of these
approaches, keeping the plant (body) extremely simplistic, but maintaining broad applicability through complex
control law application. The techniques used to develop and validate a 2D multi-segment, rigid-link model of the
human body driven by nonlinear controls have been described in Ch. 3: Controls Based Modeling and Simulation.

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Model validation studies demonstrate that the simple 2D rigid-segment, connected by hinge joints, is sufficient for
capturing the empirical changes in human body dynamics across mechanically unique phases of a landing task
involving impact, when driven by nonlinear closed-loop feedback control. Three phase-specific feedback control
algorithms, following control, impedance control, and energy-shaping control were designed for use in flight phase,
impact phase, and post-impact phase, respectively. Finally, model validation and control based model application
results were presented, illustrating the broad applicability of this model, as it is not limited to a particular phase or
task.

Humans demonstrate an impressive capacity for adaptation to environmental conditions during tasks with a clear
mechanical objective. The model presented in this work highlights the importance of this model flexibility, driving
the system with a variety of phase-specific control laws to achieve an overall task objective. Because this model uses
closed-loop feedback control to enhance the system dynamics, it is applicable to any lower-extremity, multiphase task
with well-defined mechanical objectives. The ability to predict biomechanical dynamics beyond a single task phase
opens the door to understanding how phase variant mechanical objectives influence each other, as well as how a
particular phase specific shortcoming may influence the overall objective outcome. With a better understanding of the
substantial, mutable role which closed-loop feedback control plays in human dynamics, biomechanicians may begin
to utilize the subject specific control manifolds to design performance improvement curricula to fit with an individual’s
control strategies. This paradigm shift would serve to compliment the already well-documented training regimen
which focuses on shaping the plant (human body) to the task objective through strength and agility training.

Figure 108. Describing the dynamics of the human body requires more than a complex plant which captures all of
the physical aspects of the system. Instead, the field of biomechanics has begun to focus on the control laws driving
this system, as they are as much a part of the dynamic outcomes of the system as the plant itself.

FUTURE WORK

The kinematic analysis presented in the beginning of this work relies on the simplistic phase adaptive smoothing
method, PWCS. These techniques leave room for further improvement, exploring such changes as gradual changes
between phase weights instead of the sharp step functions employed in this work. Recall, the maximum instantaneous
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filtered trajectory errors occurred immediately outside of the impact phase, suggesting that the abrupt weight change
causes initial deviations.

A simple modification of the model would be the introduction of a flexible foot with three points of contact. With this
small addition, ankle joint angles may more faithfully adhere to those measured experimentally. Recall this model
used an extended ankle joint angle range to account for the subject’s ability to balance on the balls of their feet during
the post-impact phase.

As mentioned previously, built into the premise of this control-based model is an inherent flexibility, which extends
the model applicability to any lower-extremity task involving a bifurcation in dynamic system behavior. With the
control laws being the defining feature of each phase, this research could very well be extended to land-and-go tasks,
long-jump, or triple-jump tasks. Similarly, by applying the 4-element angular velocity functional axis descriptions,
the control laws can be extended to 3D stiffness manifolds, describing motions in terms of active motion and
stabilization planes.

Defining the body both in terms of physical components and functional feedback control opens the door to all manner
of control complexities. Current research suggests machine learning may one day be able to provide insight into new
sets of control laws which operate under a different constraint manifold, better describing the continuous multifaceted
feedback used by the human body. The research presented in this work has operated under several assumptions of
perfect actuators, perfectly 2D motion, and perfect hinge joints. Each joint currently has no limit on torque capacity,
though this was minimized through the post-impact phase weighting scheme. In the future, each of these assumptions
may be further investigated to show how they contribute to the empirical drop-landing outcomes.

From the biomechanics perspective, it may one day be possible to characterize the driving factors behind a subject’s
intended joint trajectories. With better knowledge of a subject intentions, one might be less reliant on empirical joint
angle time-histories for input to the system during flight phase. Shedding this data source allows the investigator to
broaden the range of model applicability to tasks which have not yet been conducted experimentally.

During the impact phase, there is a complex interaction between the impedance trajectories and their impedance gains,
which result in the overall stiffness of the system mirroring that of the experiment. Individually however, each of these
subcomponents may relate more directly to a particular aspect of the intended interaction. If the optimization process
were more rigorous such that one could guarantee global coefficient set convergence, trends in trajectory or control
gains might offer new insights into mechanical objective performance. For example, if the second impedance line
shows consistent softening of the impedance control gains to capture empirically softer VGRF interactions, then one
could point to muscle tension as the main predictor of the impending impact intensity. However, further investigation
into this concept is required.

173

Further investigation into the sensitivity of the impedance control to the reference frame from which it is defined may
offer better stability outcomes. Recall, this model assumes a flat foot, an assumption which is hampered by the
flexibility of the contact surface. One hypothesizes that defining control with respect to the head/torso segment may
provide more biologically realistic results in that it is the most massive segment, making it the dynamic system center
upon which all extremity motion is based, as well as holding the control center of the human body (the head).

Finally, developing a better understanding of the contribution of pain to the human body control system may allow
investigators to predict how injury compensations may influence dynamic control outcomes. As an example, if
subjects experience pain when the VGRF load exceeds a threshold, then they may be more likely to soften their
impedance control gains to avoid this outcome. This constraint could then be built into the control law definition in a
similar way as the joint coordination patterns of the post-impact phase.
174

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177

APPENDIX A FORWARD KINEMATIC END-EFFECTOR DESCRIPTIONS
Full description of Jacobian calculations utilized in impedance and energy shaping controls:
Variable Description
g : Pointing in the negative y direction i.e. gravity = [0 -9.80665 0 0 0 0]
T
[m/s
2
]
m1 : Mass shank (constant) [kg]
m2 : Mass thigh (constant) [kg]
m3 : Mass torso (constant) [kg]
m4 : Mass foot (constant) [kg]
I1z : Moment of inertia about z-axis of the shank (constant) [kg*m
2
]
I2z : Moment of inertia about z-axis of the thigh (constant) [kg*m
2
]
I3z : Moment of inertia about z-axis of the torso (constant) [kg*m
2
]
I4z : Moment of inertia about z-axis of the foot (constant) [kg*m
2
]
θ1 : Ankle angle [rad/s]
θ2 : Knee angle [rad/s]
θ3 : Hip angle [rad/s]
θ4 : Foot angle [rad/s]
θ5 : Angle from foot toe to foot COM (constant) [rad]
θ6 : Angle from foot toe to ankle (constant) [rad]
q : Joint angles: state of the system [theta1 theta2 theta3]
T
[rad]
L1 : Longitudinal length of shank (constant) [m]
L2 : Longitudinal length of thigh (constant) [m]
L3 : Longitudinal length of torso (constant) [m]
L4 : Longitudinal length of foot from heel to toe (constant) [m]
L5 : Thickness of compressible shoe sole (constant) [m]
L6 : Vertical distance bottom of heel to ankle (constant) [m]
L7 : Horizontal distance from heel to ankle (constant) [m]
Lcom1 : Longitudinal position of Shank COM w.r.t. Knee (constant) [m]
Lcom2 : Longitudinal position of Thigh COM w.r.t. Hip (constant) [m]
Lcom3 : Longitudinal position of Torso COM w.r.t. Head (constant) [m]
Lcom4 : Longitudinal position of Foot COM w.r.t. Heel (constant) [m]
Lcom5 : Vertical position of Foot COM w.r.t. Heel (constant) [m]

178

𝑒 𝑒 2
= (
−𝐿 1
sin( 𝑞 1
)−
𝐿 2
sin( 𝑞 1
+ 𝑞 2
)
20
−
𝐿 2
sin( 𝑞 1
+𝑞 2
)
20
0
𝐿 2
cos( 𝑞 1
+𝑞 2
)
20
+𝐿 1
cos( 𝑞 1
)
𝐿 2
cos( 𝑞 1
+𝑞 2
)
20
0
)

𝑒 𝑒 1
=
(

−
𝑚 2
( 𝐿 1
sin( 𝑞 1
)+ sin( 𝑞 1
+ 𝑞 2
)( 𝐿 2
− Lcom
2
) )+ 𝑚 3
( 𝐿 1
sin( 𝑞 1
)+sin( 𝑞 1
+𝑞 2
+𝑞 3
)( 𝐿 3
−Lcom
3
)+ 𝐿 2
sin( 𝑞 1
+𝑞 2
) )+ 𝑚 1
sin( 𝑞 1
)( 𝐿 1
− Lcom
1
)
𝑚 1
+𝑚 2
+ 𝑚 3
+ 𝑚 4
−
𝑚 3
( sin( 𝑞 1
+ 𝑞 2
+ 𝑞 3
)( 𝐿 3
− Lcom
3
)+ 𝐿 2
sin( 𝑞 1
+𝑞 2
) )+𝑚 2
sin( 𝑞 1
+𝑞 2
)( 𝐿 2
− Lcom
2
)
𝑚 1
+ 𝑚 2
+ 𝑚 3
+𝑚 4
−
𝑚 3
sin( 𝑞 1
+𝑞 2
+𝑞 3
)( 𝐿 3
−Lcom
3
)
𝑚 1
+𝑚 2
+ 𝑚 3
+ 𝑚 4
𝑚 2
( cos( 𝑞 1
+𝑞 2
)( 𝐿 2
−Lcom
2
)+𝐿 1
cos( 𝑞 1
) )+𝑚 3
( cos( 𝑞 1
+𝑞 2
+ 𝑞 3
)( 𝐿 3
−Lcom
3
)+𝐿 2
cos( 𝑞 1
+𝑞 2
)+ 𝐿 1
cos( 𝑞 1
) )+ 𝑚 1
cos( 𝑞 1
)( 𝐿 1
−Lcom
1
)
𝑚 1
+𝑚 2
+ 𝑚 3
+ 𝑚 4
𝑚 3
( cos( 𝑞 1
+ 𝑞 2
+ 𝑞 3
)( 𝐿 3
− Lcom
3
)+ 𝐿 2
cos( 𝑞 1
+𝑞 2
) )+ 𝑚 2
cos( 𝑞 1
+ 𝑞 2
)( 𝐿 2
−Lcom
2
)
𝑚 1
+ 𝑚 2
+ 𝑚 3
+𝑚 4
𝑚 3
cos( 𝑞 1
+𝑞 2
+ 𝑞 3
)( 𝐿 3
−Lcom
3
)
𝑚 1
+𝑚 2
+𝑚 3
+ 𝑚 4

)

End Effector Jacobians:
𝐿 𝑒 𝑒 2
= (
𝐿 4
+
𝐿 2
cos( 𝑞 1
+𝑞 2
)
20
+𝐿 1
cos( 𝑞 1
)
𝐿 1
sin( 𝑞 1
)+
𝐿 2
sin( 𝑞 1
+𝑞 2
)
20
)
𝐿 𝑒 𝑒 1
=
(

L
com4
𝑚 4
+ 𝑚 1
( 𝐿 4
+ cos( 𝑞 1
)( 𝐿 1
−L
com1
) )+𝑚 2
( 𝐿 4
+cos( 𝑞 1
+𝑞 2
)( 𝐿 2
− L
com2
)+ 𝐿 1
cos( 𝑞 1
) )+𝑚 3
( 𝐿 4
+cos( 𝑞 1
+𝑞 2
+ 𝑞 3
)( 𝐿 3
−L
com3
)+𝐿 2
cos( 𝑞 1
+ 𝑞 2
)+ 𝐿 1
cos( 𝑞 1
) )
𝑚 1
+ 𝑚 2
+𝑚 3
+ 𝑚 4
− 𝐿 4
L
com5
𝑚 4
+𝑚 2
( 𝐿 1
sin( 𝑞 1
)+sin( 𝑞 1
+𝑞 2
)( 𝐿 2
− L
com2
) )+ 𝑚 3
( 𝐿 1
sin( 𝑞 1
)+ sin( 𝑞 1
+ 𝑞 2
+ 𝑞 3
)( 𝐿 3
−L
com3
)+𝐿 2
sin( 𝑞 1
+ 𝑞 2
) )+ 𝑚 1
sin( 𝑞 1
)( 𝐿 1
−L
com1
)
𝑚 1
+ 𝑚 2
+𝑚 3
+ 𝑚 4 )

End Effector Forward Kinematics:
179

%% SYMBOLIC REFORMULATION OF IMPEDANCE MODEL FOR CONTROL AND ANALYSIS: 3x3
% Coded by Edward Wagner
% Modified: 1/27/17
close all
clear all
clc

%% Model definition:
% This is a 2D model defined in the XY-plane. The model is composed of 4
% links connected at the segment endpoints by ideal hinge joints.

%% SUMMARY:
% The purpose of this code is to automate the calculation of the impedance
% equations in terms of the joint angles. For completeness, we will leave
% the 4th angle (total body orientation) in the equations, as it can simply
% be removed for the simple case where the foot is assumed in flat contact
% with the ground.

%% Functions required:
% pdiff - calculates the symbolic partial derivative

%% Subject measurements For reference:
% mass = 66.7; %[kg]
% tot_height = 1574.8; %[mm]

%% Variable Definitions:

% Gravity:
% g -- Pointing in the negative y direction i.e. gravity = [0 -9.80665 0 0 0 0].'
variable_names = {'g'};
variable_descr = {'Pointing in the negative y direction i.e. gravity = [0 -9.80665
0 0 0 0]^T [m/s^2]'};
% Masses:
% m1 = mass shank (constant)')
variable_names = {variable_names{:},'m1'};
variable_descr = {variable_descr{:},'Mass shank (constant) [kg]'};
% m2 = mass thigh (constant)
variable_names = {variable_names{:},'m2'};
variable_descr = {variable_descr{:},'Mass thigh (constant) [kg]'};
% m3 = mass torso (constant)
variable_names = {variable_names{:},'m3'};
variable_descr = {variable_descr{:},'Mass torso (constant) [kg]'};
% m4 = mass foot (constant)
variable_names = {variable_names{:},'m4'};
variable_descr = {variable_descr{:},'Mass foot (constant) [kg]'};
% Moment of inertia:
% I1z = moment of inertia about the z axis of the shank
variable_names = {variable_names{:},'I1z'};
variable_descr = {variable_descr{:},'Moment of inertia about z-axis of the
shank (constant) [kg*m^2]'};
% I2z = moment of inertia about the z axis of the thigh
variable_names = {variable_names{:},'I2z'};
variable_descr = {variable_descr{:},'Moment of inertia about z-axis of the
thigh (constant) [kg*m^2]'};
% I3z = moment of inertia about the z axis of the torso
variable_names = {variable_names{:},'I3z'};
variable_descr = {variable_descr{:},'Moment of inertia about z-axis of the
torso (constant) [kg*m^2]'};
% I4z = moment of inertia about the z axis of the foot
variable_names = {variable_names{:},'I4z'};
variable_descr = {variable_descr{:},'Moment of inertia about z-axis of the
foot (constant) [kg*m^2]'};
syms g m1 I1z m2 I2z m3 I3z m4 I4z real
% Joint space:
180

% theta1 = ankle angle
variable_names = {variable_names{:},'theta1'};
variable_descr = {variable_descr{:},'Ankle angle [rad/s]'};
% theta2 = knee angle
variable_names = {variable_names{:},'theta2'};
variable_descr = {variable_descr{:},'Knee angle [rad/s]'};
% theta3 = hip angle
variable_names = {variable_names{:},'theta3'};
variable_descr = {variable_descr{:},'Hip angle [rad/s]'};
% theta4 = foot angle
variable_names = {variable_names{:},'theta4'};
variable_descr = {variable_descr{:},'Foot angle [rad/s]'};
% theta5 = angle from foot toe to foot COM (constant)
variable_names = {variable_names{:},'theta5'};
variable_descr = {variable_descr{:},'Angle from foot toe to foot COM
(constant) [rad]'};
% theta6 = angle from foot toe to ankle (constant)
variable_names = {variable_names{:},'theta6'};
variable_descr = {variable_descr{:},'Angle from foot toe to ankle (constant)
[rad]'};
syms theta1(t) theta2(t) theta3(t) theta4(t) theta5 theta6 ang real
% Joint angle states of the system:
q = [theta1 theta2 theta3].'; % theta4].';
variable_names = {variable_names{:},'q'};
variable_descr = {variable_descr{:},'Joint angles: state of the system [theta1
theta2 theta3]^T [rad]'};
% theta1_dt(t) theta2_dt(t) theta3_dt(t) ...
% Segment lengths:
% L1 = shank_l (constant)
variable_names = {variable_names{:},'L1'};
variable_descr = {variable_descr{:},'Longitudinal length of shank (constant)
[m]'};
% L2 = thigh_l (constant)
variable_names = {variable_names{:},'L2'};
variable_descr = {variable_descr{:},'Longitudinal length of thigh (constant)
[m]'};
% L3 = torso_l (constant)
variable_names = {variable_names{:},'L3'};
variable_descr = {variable_descr{:},'Longitudinal length of torso (constant)
[m]'};
% L4 = foot_l (constant)
variable_names = {variable_names{:},'L4'};
variable_descr = {variable_descr{:},'Longitudinal length of foot from heel to
toe (constant) [m]'};
% L5 = sole (thickness of the bottom of the foot) (constant)
variable_names = {variable_names{:},'L5'};
variable_descr = {variable_descr{:},'Thickness of compressible shoe sole
(constant) [m]'};
% L6 = foot_height (constant)
variable_names = {variable_names{:},'L6'};
variable_descr = {variable_descr{:},'Vertical distance bottom of heel to ankle
(constant) [m]'};
% L7 = horizontal distance from heel to ankle
variable_names = {variable_names{:},'L7'};
variable_descr = {variable_descr{:},'Horizontal distance from heel to ankle
(constant) [m]'};
syms L1 L2 L3 L4 L5 L6 L7 real
% Longitudinal position of Segment COM (relative to proximal end)
% Lcom1 = shank COM w.r.t. knee (constant)
variable_names = {variable_names{:},'Lcom1'};
variable_descr = {variable_descr{:},'Longitudinal position of Shank COM w.r.t.
Knee (constant) [m]'};
% Lcom2 = thigh COM w.r.t. hip (constant)
181

variable_names = {variable_names{:},'Lcom2'};
variable_descr = {variable_descr{:},'Longitudinal position of Thigh COM w.r.t.
Hip (constant) [m]'};
% Lcom3 = torso COM w.r.t. head (constant)
variable_names = {variable_names{:},'Lcom3'};
variable_descr = {variable_descr{:},'Longitudinal position of Torso COM w.r.t.
Head (constant) [m]'};
% Lcom4 = foot COM w.r.t. heel (constant)
variable_names = {variable_names{:},'Lcom4'};
variable_descr = {variable_descr{:},'Longitudinal position of Foot COM w.r.t.
Heel (constant) [m]'};
% Lcom5 = foot COM w.r.t. heel vertically (sole) (constant)
variable_names = {variable_names{:},'Lcom5'};
variable_descr = {variable_descr{:},'Vertical position of Foot COM w.r.t. Heel
(constant) [m]'};
syms Lcom1 Lcom2 Lcom3 Lcom4 Lcom5 real

% Display table of definitions:
clc
disp('Notation:')
defn_vars = table(variable_names.',variable_descr.');
defn_vars.Properties.VariableNames = {'Variable' 'Description'};
disp(defn_vars)
writetable(defn_vars,'VariableDefns.txt')
% Simple function for definition of rotation about the z axis:
rotz = @(ang) [cos(ang) -sin(ang); sin(ang) cos(ang)];

%% Define forward transformations:
% Foot:
R4 = rotz(0); % Rotation of foot reference frame w.r.t. inertial (heel)
% R4 = rotz(theta4); %  Could add a line for foot not oriented norizontally
d4 = [0 0].'; % Displacement of reference frame relative to inertial
% d4 = [x_toe y_toe].'; %  Unnecessary for system defined in relative terms
T4 = [R4 d4; 0 0 1]; % Transformation from inertial to toe (on foot) reference frame
% Position:
p_heel_4 = [0 0 1].'; % Position of heel in foot reference frame
p_ank_4 = [L4 0 1].'; % Position of ankle in foot reference frame
p_footCOM_4 = [Lcom4 Lcom5 1].'; % Position of footCOM in foot reference frame
p_toe_4 = [Lcom4 Lcom5 1].'; % Position of toe in foot reference frame
p_heel_0 = T4*p_heel_4; % Position of heel in inertial reference frame
p_ank_0 = T4*p_ank_4; % Position of ankle in inertial reference frame
p_footCOM_0 = T4*p_footCOM_4; % Position of footCOM in inertial ref. frame
p_toe_0 = T4*p_toe_4; % Position of toe in inertial reference frame
% Orientation:
theta_foot_0 = 0; % Orientation of foot reference frame

% Shank:
R1 = rotz(theta1-pi); % Rotation of shank reference frame w.r.t. the foot
d1 = p_ank_4(1:2); % Displacement of ref. frame (ankle) w.r.t. toe in foot ref. frame
T1 = [R1 d1; 0 0 1]; % Transformation from ankle (shank) to toe (foot) ref. frame
% Position:
p_knee_1 = [-L1 0 1].'; % knee in shank reference frame
p_shankCOM_1 = [-(L1-Lcom1) 0 1].'; % shankCOM in shank ref. frame
p_knee_0 = T4*T1*p_knee_1; % knee w.r.t. inertial reference frame
p_shankCOM_0 = T4*T1*p_shankCOM_1; % shankCOM w.r.t. inertial ref. frame
% Orientation:
theta_shank_0 = [theta1-pi].'; % Orientation of shank reference frame

% Thigh:
R2 = rotz(theta2); % Rotation of thigh reference frame w.r.t. shank
d2 = p_knee_1(1:2); % Displacement knee ref. frame w.r.t. ankle in shank ref. frame
T2 = [R2 d2; 0 0 1]; % Transformation from knee (on thigh) to ankle (on shank)
reference frame
182

% Position:
p_hip_2 = [-L2 0 1].'; % hip in thigh reference frame
p_thighCOM_2 = [-(L2-Lcom2) 0 1].'; % thighCOM in thigh reference frame
p_hip_0 = T4*T1*T2*p_hip_2; % hip in inertial reference frame
p_thighCOM_0 = T4*T1*T2*p_thighCOM_2; % thighCOM in inertial reference frame
% Orientation:
theta_thigh_0 = [(theta1-pi)+theta2].'; % Orientation of thigh reference frame

% Torso:
R3 = rotz(theta3); % Rotation of torso reference frame w.r.t. thigh
d3 = p_hip_2(1:2); % Displacement hip ref. frame w.r.t. knee in thigh ref. frame
T3 = [R3 d3; 0 0 1]; % Transformation from hip (torso) to knee (on thigh) ref. frame
% Position:
p_head_3 = [-L3 0 1].'; % Position of head in torso reference frame
p_torsoCOM_3 = [-(L3-Lcom3) 0 1].'; % Position of torsoCOM in torso ref. frame
p_head_0 = T4*T1*T2*T3*p_head_3; % Position of head in inertial ref. frame
p_torsoCOM_0 = T4*T1*T2*T3*p_torsoCOM_3; % torsoCOM in inertial ref. frame
% Orientation:
theta_torso_0 = [(theta1-pi)+theta2+theta3].'; % Orientation of torso ref. frame

%% L(t): Collecting terms into forward kinematic transformation  vector of formulas
% p_heel_temp = formula(p_heel_0); <-- already a double
p_toe_temp = formula(p_toe_0);
p_ank_temp = formula(p_ank_0);
p_knee_temp = formula(p_knee_0);
p_hip_temp = formula(p_hip_0);
p_head_temp = formula(p_head_0);
p_footCOM_temp = formula(p_footCOM_0);
p_shankCOM_temp = formula(p_shankCOM_0);
p_thighCOM_temp = formula(p_thighCOM_0);
p_torsoCOM_temp = formula(p_torsoCOM_0);

% converts formulas 1:3 into symbolic functions with independent variable 't'
% p_heel_0 = simplify(symfun(p_heel_temp(1:2),t));
p_toe_0 = simplify(symfun(p_toe_temp(1:2),t));
p_ank_0 = simplify(symfun(p_ank_temp(1:2),t));
p_knee_0 = simplify(symfun(p_knee_temp(1:2),t));
p_hip_0 = simplify(symfun(p_hip_temp(1:2),t));
p_head_0 = simplify(symfun(p_head_temp(1:2),t));
p_footCOM_0 = simplify(symfun(p_footCOM_temp(1:2),t));
p_shankCOM_0 = simplify(symfun(p_shankCOM_temp(1:2),t));
p_thighCOM_0 = simplify(symfun(p_thighCOM_temp(1:2),t));
p_torsoCOM_0 = simplify(symfun(p_torsoCOM_temp(1:2),t));
p_COM_0 = (m1*p_shankCOM_0 + m2*p_thighCOM_0 + m3*p_torsoCOM_0)/(m1+m2+m3);

clearvars p_heel_temp p_toe_temp p_ank_temp p_knee_temp p_hip_temp p_head_temp ...
p_footCOM_temp p_shankCOM_temp p_thighCOM_temp p_torsoCOM_temp

% L(t) = [p_footCOM_0; theta_foot_0;
% p_shankCOM_0; theta_shank_0;
% p_thighCOM_0; theta_thigh_0;
% p_torsoCOM_0; theta_torso_0];

% Center of mass inverse kinematics:
Lee1(t) = p_COM_0 - p_ank_0; % Takes in angKnee, angHip and outputs XY-COM w.r.t.
Ankle
Lee2(t) = (p_hip_0 + 19/20*(p_knee_0 - p_hip_0)) - p_heel_0(1:2,:); % Takes in
angAnkle, angKnee, angHip and outputs XY-COM
Lee1_latex = latex(Lee1);
Lee2_latex = latex(Lee2);
% $$Lee1_latex

% Each segment's center of mass:
L1(t) = [p_shankCOM_0];
183

L2(t) = [p_thighCOM_0];
L3(t) = [p_torsoCOM_0];

%% J(t): Determine the Jacobian of the forward transformation matrix

% End effector Jacboian:
Jee1(t) = symfun(simplify(pdiff(Lee1(t),q(t))),t)
Jee2(t) = symfun(simplify(pdiff(Lee2(t),q(t))),t)
% Segment specific Jacobians:
J1(t) = symfun(simplify(pdiff(L1(t),q(t))),t);
J2(t) = symfun(simplify(pdiff(L2(t),q(t))),t);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Partial Derivative
% Coded by Edward Wagner
% This function takes the partial derivative of a symfun with respect to
% another symfun (a function not currently available in MATLAB)
%
% INPUT FORMAT:
% df/dx(t) = pdiff(f, x)
% where f is the function of which the derivative is taken
% x is the variable with respect to which the derivative is taken
%
% NOTE:
% 1. if you desire an output which itself is a function of your main
% independent variable (in my example it will be t), enter f simply as "f":
% i.e. dfdx(t) = pdiff(f,x) <-- output is a symfun
% 2. if you desire an output which is handled as a sym
%
% REF: http://stackoverflow.com/questions/27085362/how-to-implement-a-derivative-of-a-
symbolic-function-by-a-symfun-in-matlab

function df = pdiff(f,x,num)
narginchk(2,3)
if length(x) > 1
switch nargin
case 3
for ii = 1:length(f)
for jj = 1:length(x)
xx = sym('abcd');
df(ii,jj) = subs(diff(subs(f(ii),x(jj),xx),xx,num),xx,x(jj));
end
end
case 2
for ii = 1:length(f)
for jj = 1:length(x)
xx = sym('abcd');
df(ii,jj) = subs(diff(subs(f(ii),x(jj),xx),xx),xx,x(jj));
end
end
otherwise
end
% Do nothing
else
switch nargin
case 3
for ii = 1:length(f)
for jj = 1:length(x)
xx = sym('abcd');
df(ii,jj) = subs(diff(subs(f(ii),x,xx),xx,num),xx,x);
end
184

end
case 2
for ii = 1:length(f)
for jj = 1:length(x)
xx = sym('abcd');
df(ii,jj) = subs(diff(subs(f(ii),x,xx),xx),xx,x);
end
end
otherwise
end
end

185

APPENDIX B FORCE PLATE RESOLUTION ANALYSIS
KISTLER plates generate electrical signals using piezoelectrics, whose charge is measured in pico-
Coulombs (pC). The force gain is selected by adjusting the amplifier gain, the number of pC across which
the +/-10V signal is distributed. For this work, the gain is set to 1000pC/V as it involves impacts which
could saturate higher gains. Because these piezoelectrics are analog, their resolution is limited only by the
digital to analog converter of the DAQ Card (NI-6071E). The card has 12 bit resolution, meaning that the
+/-10V signal is spread across 2
12
bits.
1 𝑏𝑖𝑡 ∙
20𝑉 2
12
𝑏𝑖𝑡 = 0.005𝑉
The output from the DAQ is converted to Newtons by the amplifier setting conversion factors. In this
case, with the op-amp set to 1000pC/V, the conversion gains are given in Table 8.
Table 8. Conversion factors from amplifier output (V) to force measurements (N)
for KISTLER 9281B plates in B10, when using Gain Option 3 (1000pC/V).
Plate X [N/V] Y [N/V] Z [N/V]
1 128.2051 128.7001 261.0966
2 127.2265 128.0410 260.4167

Applying the DAQ conversion limitation to these amplifier gains, the resolution of the force plates is, at
best, 1.3N in the vertical (Z) direction.
0.005𝑉 ∙ 260.4167
𝑁 𝑉 = 1.3𝑁

186

APPENDIX C EVIDENCE OF PIXEL SPATIAL RESOLUTION DISCRETIZATION

When tracking ultra-highspeed marker trajectories with the digitization program DLTdv5/6 MATLAB from the
Hedrick lab[42], the pixelated content of the videos becomes evident in discrete jumps in spatial position. The
histogram shown below illustrates the most common spatial jumps between tracking frames. Since there are some
jumps which occur more often, one can infer there are discrete steps between pixels which will cause noise in higher
order derivatives of the data. This chattering illustrates the need for kinematic smoothing to produce realistic
calculations of velocity and acceleration.

Figure 109. Spatial resolution of the kinematic data is shown by the common discrete jumps in centroid location.
Digitization software (DLTdv5, MATLAB) determines which pixel contains the centroid of a tracking marker,
leading to discrete steps in space. By showing the most common discrete steps, the metric size of a pixel is
determined.

187

APPENDIX D FILMSTRIP KINEMATICS FOR ALL TRIAL CONDITIONS

The beginning of the impact and post-impact impact phases are denoted by a circle above the head.
Notice, because the figure includes a downsampled subset of kinematic frames, the impact and post
impact frame may not be shown explicitly.

Figure 110. This kinematic filmstrip of the the segments of both subjects during all three landing conditions overlaid for comparison,
shows how the initial conditions during flight phase are nearly identical until shortly before touchdown when the effects of subject joint
control prepares the body for impact. The solid circles indicate the beginning of the impact phase (black) and trial specific post-impact
phase (red=softer than normal, blue=normal, green=harder than normal)
Position Y [m] Position Y [m]

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Asset Metadata

Creator Wagner, Edward Vaughn (author)

Core Title Using nonlinear feedback control to model human landing mechanics

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Mechanical Engineering

Publication Date 02/08/2018

Defense Date 01/11/2018

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag 2D,biomechanics,Control,energy shaping,filtering,forward dynamics,impact,impedance,kinematics,Landing,lower-extremity,multiphase,net joint moment,nonlinear,OAI-PMH Harvest,passivity-based,reaction force,simulation

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Flashner, Henryk (committee chair), McNitt-Gray, Jill (committee member), Shiflett, Geoffrey (committee member)

Creator Email ed.wagner88@gmail.com,evwagner@alumni.usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c40-472327

Unique identifier UC11268131

Identifier etd-WagnerEdwa-6018.pdf (filename),usctheses-c40-472327 (legacy record id)

Legacy Identifier etd-WagnerEdwa-6018.pdf

Dmrecord 472327

Document Type Dissertation

Rights Wagner, Edward Vaughn

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA