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Simultaneous center of mass estimation and foot placement selection in complex planar terrains for legged architectures

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Simultaneous center of mass estimation and foot placement selection in complex planar terrains for legged architectures

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Content SIMULTANEOUS CENTER OF MASS ESTIMATION AND FOOT
PLACEMENT SELECTION IN COMPLEX PLANAR TERRAINS FOR
LEGGED ARCHITECTURES
by
Luenin Barrios
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulllment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(COMPUTER SCIENCE)
December 2017
Copyright 2017 Luenin Barrios
Epilogue
\So little do we see before us in the world, and so much reason have we
to depend cheerfully upon the great Maker of the world, that He does
not leave His creatures so absolutely destitute, but that in the worst cir-
cumstances they have always something to be thankful for, and some-
times are nearer deliverance than they imagine; nay, are even brought
to their deliverance by the means by which they seem to be brought to
their destruction."
- Daniel Defoe, Robinson Crusoe
i
Acknowledgements
So many people are to be thanked for contributing to this journey, that I lack
the words to express my deep appreciation and gratitude to each and everyone of
them. My absolute and sincere thanks to my advisor, mentor and friend, Prof.
Wei-Min Shen. This would not have been possible without your prudent insight,
your constant guidance and your unwavering support. You gave me the freedom
to choose my topic of interest and the encouragement to explore and pursue it to
its greatest depths. It has been a pleasure and great of accomplishment mine to
have worked beside you, and it is something I will cherish forever.
I would also like to thank my labmates at the Polymorphic Robotics Lab who
were equally instrumental in helping me complete this odyssey. Joseph Chen, for
your warmth, kindness and sense of humor which served as a great motivation
to keep working and always stay positive. T.J. Collins, for your genuineness and
gentle spirit and for our many philosophical discussions which reminded me to stay
humble and seek truth. And to Nadeesha Ranasinghe, a jack of all trades whose
patience, wit, and knowledge were invaluable to me during the program.
I am also indebted to the members of my committee: Prof. Stephan Haas,
Prof. Aiichiro Nakano, Prof. Aristides Requicha and Greg Ver Steeg for their
friendly advice and their valuable insight and feedback. It has been a privilege to
have received the counsel and instruction of such great and wonderful minds.
ii
And to my anc e and best friend, Alice. Thank you for patiently suering
with me through those dark days when the end seemed so far away. Your all
encompassing love was a source of motivation and comfort through every peak
and valley. You encouraged me and made me feel special and rich. We didn't have
much but we had each other. That's all we needed. That's all we'll ever need.
Lastly, I would like to thank my parents and family. Throughout my life you
have been a pillar of love and support. From my childhood rst steps to my early
adulthood, you have been there with me every step of the way. This work is as
much yours as it is mine. I have only tried to make you proud and to give back
through my eorts what you have given me in overwhelming aection and support.
With all my heart and soul, thank you.
iii
Contents
Epilogue i
Acknowledgements ii
List of Tables vi
List of Figures vii
Abstract xi
1 Introduction 1
1.1 Exordium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3.1 Sensor Restrictions . . . . . . . . . . . . . . . . . . . . . . . 3
1.3.2 Subject to Kinematic Constraints . . . . . . . . . . . . . . . 4
1.3.3 Complex Terrains . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.4 Non-linear Dynamics . . . . . . . . . . . . . . . . . . . . . . 5
2 Overview of Framework 6
2.1 Outline of Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . 8
3 CoM Construction and Optimization of Foot Placements 10
3.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Outline of Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.3 Optimized Geometric Hermite Curve . . . . . . . . . . . . . . . . . 14
3.4 Single Contact Point Model . . . . . . . . . . . . . . . . . . . . . . 16
3.4.1 State Space Behavior . . . . . . . . . . . . . . . . . . . . . . 18
3.5 Path Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.6 Contact Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.7 Foot Placement Optimization . . . . . . . . . . . . . . . . . . . . . 23
3.7.1 Experimental Results . . . . . . . . . . . . . . . . . . . . . . 24
iv
4 CoM Estimation: A Geometric Approach 29
4.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.2 Outline of Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2.1 Virtual Steps . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2.2 Forward Progress Angles . . . . . . . . . . . . . . . . . . . . 35
4.2.3 Construction of CoM Estimate . . . . . . . . . . . . . . . . 37
4.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.3.1 Experimental Setup and Data Collected . . . . . . . . . . . 40
4.3.2 Cross-validation . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.3.3 CoM State-Space Behavior . . . . . . . . . . . . . . . . . . . 44
5 CoM Estimation: Load Carriage Extension 48
5.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.2 Load Carriage Eect . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.2.1 Load Virtual Step Model . . . . . . . . . . . . . . . . . . . . 53
5.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.3.1 Experimental Setup and Data Collected . . . . . . . . . . . 56
5.3.2 Cross-validation . . . . . . . . . . . . . . . . . . . . . . . . . 58
6 Foot Placement Selection: A Phase Space Planning Approach 61
6.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.1.1 Objective of Study . . . . . . . . . . . . . . . . . . . . . . . 65
6.1.2 Foot Placement Analysis: Overview of Approach . . . . . . . 66
6.2 Terrain Traversal Case Studies . . . . . . . . . . . . . . . . . . . . . 66
6.2.1 Experimental Setup and Data Collected . . . . . . . . . . . 66
6.2.2 General Protocol . . . . . . . . . . . . . . . . . . . . . . . . 67
6.2.3 Data Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
7 Conclusion 92
7.1 Summary and Contributions . . . . . . . . . . . . . . . . . . . . . . 92
7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Reference List 95
v
List of Tables
4.1 Subject Attributes Data. . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2 RMSE Est. vs Measured CoM Vertical Position . . . . . . . . . . . 42
5.1 Load Carriage Subject Characteristics Data. . . . . . . . . . . . . . 57
5.2 Load Carriage RMSE Est. vs Measured CoM Vertical Position . . . 58
vi
List of Figures
3.1 OGH curves with dierent vector tangent angles. The center and
right plots show COH curves constructed using piecewise OGH seg-
ments. The angles and are both measured counterclockwise with
respect to the vectorP
0
P
1
, with corresponding to the pointP
0
and
to P
1
. Note the diversity of complex yet smooth paths possible. . 15
3.2 Walking prole schematic showing the CoM position and accelera-
tion and feet centers of pressure (CoP) and reaction forces. . . . . . 16
3.3 CoM paths generated using COH curves for dierent CoM range
constraints over various terrains. The left and middle plot demon-
strate how the environment and the selection of tangent angles
aects the CoM paths. The right plots displays a complete CoM
path consisting of piecewise OGH/COH curves with minimum cur-
vature and length to traverse the terrain. . . . . . . . . . . . . . . . 20
3.4 OGH curve with = 10 and = 340. The phase diagrams of sagittal
and vertical CoM velocities are shown in the middle and right plots
respectively. The CoP is located at (0; 0). Various boundary apex
CoM velocities are shown. . . . . . . . . . . . . . . . . . . . . . . . 22
3.5 CoM saggital and vertical phase diagrams for two contact feet at
(0:2; 0) and (0:4; 0:25) with the step transition point shown. The
apex velocity of the CoM over each contact point is 1m=s. . . . . . 22
3.6 OGH CoM paths with constant apex velocity v = 1m=s and dis-
tinct tangent and position endpoint conditions traversing dierent
terrains are shown in (a) and (d). In (b) and (e) several second
foot contact locations and the corresponding phase space behavior
of the CoM are shown demonstrating the aects of foot locations.
The strain energy cost landscape for each terrain is shown in (c)
and (f) respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.7 Enery cost landscape for CoM paths and terrains of Fig 3.6 at step
1 and step 2 shown in (a-b) and (d-e) respectively. The cost is the
strain energy of the vertical CoM phase between adjacent potential
foot locations. The maximum foot placement for each robot is 0:35m
and 0:4m. (c) and (f) are the full step sequences. . . . . . . . . . . 27
vii
3.8 Strain energy cost landscape for Fig 3.6 (a). CoM constant apex
velocities of v = 0:75m=s and v = 1:5m=s. . . . . . . . . . . . . . . 28
3.9 Strain energy cost landscape for Fig 3.6 (d). CoM constant apex
velocities of v = 0:6m=s and v = 1:2m=s. . . . . . . . . . . . . . . . 28
4.1 Sagittal plane view showing step geometric model. . . . . . . . . . . 33
4.2 Four step walking sequence through terrain. The virtual steps and
forward angles (vectors) are also shown. . . . . . . . . . . . . . . . 36
4.3 Example OGH curves (red) for rst three steps and forward angles
of Fig 4.2. At each step, the curve with minimum SE is selected as
the CoM path (black). . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.4 Sagittal motion capture of human traversal over terrain. . . . . . . 41
4.5 (Left) a six step walking sequence over terrain 1. (Right) a ve step
walking sequence over terrain 2. The estimated versus measured
CoM vertical position rmse is 0.0244m and 0.0405m respectively. . . 43
4.6 (Left) a seven step walking sequence over terrain 1. (Right) an eight
step walking sequence over terrain 2. The estimated versus mea-
sured CoM vertical position rmse is 0.0307m and 0.0252m respectively. 43
4.7 Subject 4 sagittal and vertical phase curves using CoM estimate for
terrain 1 from Fig 4.5 (left). . . . . . . . . . . . . . . . . . . . . . . 45
4.8 Subject 4 sagittal and vertical phase curves using CoM estimate for
terrain 2 from Fig 4.5 (right). . . . . . . . . . . . . . . . . . . . . . 45
4.9 Subject 6 sagittal and vertical phase curves using CoM estimate for
terrain 1 from Fig 4.6 (left). . . . . . . . . . . . . . . . . . . . . . . 46
4.10 Subject 6 sagittal and vertical phase curves using CoM estimate for
terrain 2 from Fig 4.6 (right). . . . . . . . . . . . . . . . . . . . . . 46
5.1 The vertical CoM displacement during a loaded step (middle) and
its equivalent geometric representation with longer stride (right). . . 52
5.2 The CAP Barbell vest with 1.815kg weight pods. . . . . . . . . . . 56
5.3 Sagittal snapshots of human traversal through terrain. . . . . . . . 57
5.4 Subject 1 performing 6 step loaded walking sequences through ter-
rain 1 while carrying 7.26kg (left )and 14.5kg (right). The rmse of
estimated versus measured CoM for each is 0.0380m and 0.0419m
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.5 Subject 1 performing 7 step loaded walking sequences through ter-
rain 2 while carrying 7.26kg (left )and 14.5kg (right). The rmse of
estimated versus measured CoM for each is 0.0274m and 0.0272m
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
viii
5.6 Subject 3 performing 5 step loaded walking sequences through ter-
rain 1 while carrying 7.26kg (left )and 14.5kg (right). The rmse of
estimated versus measured CoM for each is 0.0362m and 0.0425m
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.7 Subject 3 performing 5 step loaded walking sequences through ter-
rain 2 while carrying 7.26kg (left )and 14.5kg (right). The rmse of
estimated versus measured CoM for each is 0.0415m and 0.0354m
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6.1 The measured CoM path versus the estimated CoM path for three
steps (two known and a candidate CoP location) is shown in (a).
The sagittal phase portrait with the inclusion of the multi-contact
phase is shown in (b) for the same candidate CoP location using the
estimated CoM path from (a). The vertical phase portrait (multi-
contact phase included) and vertical acceleration are depicted in (c)
and (d) respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6.2 Subject 4, terrain 1: Candidate CoP locations (yellow) and their
geometric CoM paths (black). The measured CoM traversal is
shown in (red). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.3 CoM vertical phase portraits for candidate CoP locations of Fig6.2(a)-
(j). In (g), the least deviation from the natural walking vertical
phase portrait is reached. Note the correlation of estimated vs.
measured CoM paths in Fig6.2(g). . . . . . . . . . . . . . . . . . . . 71
6.4 Energy cost landscape of step transitions of candidate CoP locations. 72
6.5 Energy cost landscape of step transitions of next CoP candidates. . 73
6.6 Infeasible vertical CoM acceleration prole of a candidate CoP. The
area under the dashed line represents nonviable acceleration values. 74
6.7 CoM vertical phase portraits for next set of candidate CoP locations
using CoM paths from Fig 6.8 (a)-(j). In (g)-(i), the least deviation
from the natural walking vertical phase portrait is reached. . . . . . 75
6.8 Subject 4, terrain 1. Next set of candidate CoP locations (yellow)
and their geometric CoM paths (black). The measured CoM traver-
sal is shown in (red). . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.9 Left: Energy cost landscape of 5th step candidate CoPs. Middle:
CoM vertical phase portrait of selected CoP. Right: CoM path esti-
mate of selected CoP. . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.10 Left: Energy cost landscape of 6th step candidate CoPs. Middle:
CoM vertical phase portrait of selected CoP. Right: CoM path esti-
mate of selected CoP. . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.11 CoM vertical phase portrait for subject 4 through terrain 1 as mea-
sured using motion capture video data. . . . . . . . . . . . . . . . . 78
ix
6.12 The measured (red) and geometric (black) CoM paths. Note the
unreachable location of the 6th CoP candidate solution. . . . . . . . 79
6.13 Two step candidate CoP (yellow) lookahead and its geometric CoM
path (blue). Note the increased clearance (C) resulting from antic-
ipating the obstacle. . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.14 Simulated subject 4, terrain 1 foot paths and trajectories. . . . . 81
6.15 Top: snapshots of simulated traversal of subject 4 through terrain
1. Bottom: sagittal view of the traversal run is shown with inclusion
of the left and right foot trajectories. . . . . . . . . . . . . . . . . . 82
6.16 Top: snapshots of simulated traversal of subject 4 through terrain
2. Bottom: Sagittal view of foot paths and feet trajectories. . . . . 83
6.17 Left: Motion capture CoM vertical phase. Middle: computed CoM
vertical phase portrait. Right: CoM path estimate (black) using
selected CoPs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.18 Top: snapshots of simulated traversal of subject 6 through terrain
1. Bottom: Sagittal view of foot paths and feet trajectories. . . . . 85
6.19 Left: Motion capture CoM vertical phase. Middle: computed CoM
vertical phase portrait. Right: CoM path estimate (black) using
selected CoPs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.20 Top: snapshots of simulated traversal of subject 6 through terrain
2. Bottom: Sagittal view of foot paths and feet trajectories. . . . . 86
6.21 Left: Motion capture CoM vertical phase. Middle: computed CoM
vertical phase portrait. Right: CoM path estimate (black) using
selected CoPs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.22 Sagittal and vertical CoM phase portraits are shown in (a) and (b)
respectively. The CoM estimate and selected foot CoP locations are
given in (c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.23 Top: snapshots of simulated traversal through terrain 2 for female
measuring 1m in height. Bottom: Sagittal view of foot paths and
feet trajectories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.24 Top: Snapshots of simulated traversal for a 2m tall female subject.
Bottom: Sagittal and vertical CoM phase portraits are shown in
(a) and (b) respectively. The CoM estimate and selected foot CoP
locations are given in (c). . . . . . . . . . . . . . . . . . . . . . . . . 90
x
Abstract
Center of Mass (CoM) path planning and foot placement selection in complex
and rough terrains remains an important goal in the development of motion plans
for legged robots. Precise CoM measurements and percipient foot placements are
essential in understanding the behavior of a system, for example in gait selection
or in extreme locomotion maneuvers. However, operating and maneuvering in
dicult terrains has remained a challenging problem due to the diversity of envi-
ronments and the complex interplay of foot placements and CoM motions. These
locomotion maneuvers involve complex forces and movements that make analysis
of CoM behavior a challenging task. Nevertheless, understanding CoM dynamics
remains pivotal in locomotion planning for both humans and robots. Indeed, the
critical element in robot and human motion planning revolves around the abil-
ity to accurately measure and describe the CoM. But given the cyclopean space
of natural terrains available and the large number of kinematic shapes and sizes
possible, the question arises: Is it conceivable to create a generalized framework
for CoM construction and estimation with optimal foot placement selection that
incorporates the large variety of kinematic architectures and terrains? The work
described in this research addresses this issue by presenting a generalized geometric
framework from which accurate CoM estimates are produced for the case of bipedal
locomotion in complex planar terrains. This framework allows for the simultaneous
xi
treatment of CoM estimation and foot placement selection in legged architectures
in an ecient and straightforward manner. This is a marked change from current
methods for CoM position estimation that rely heavily on expensive and ungainly
tools, for example force plates and motion capture video. These render CoM anal-
ysis impractical and time consuming and serve as an impediment to understanding
locomotion maneuvers in uneven terrains. To tackle these challenges, this work
proposes a reliable geometric approach for CoM estimation that delivers accu-
rate CoM behavior in complex planar terrains. The geometric approach depends
only on terrain geometry information and essential kinematic data of the moving
body. Using this key information in conjunction with an Optimized Geometric
Hermite (OGH) curve, a model is developed that produces accurate CoM position
and phase space behavior. This phase space behavior is simultaneously optimized
during CoM estimation to nd candidate foot locations that produce an overall
plan with minimum energy. This provides a way to synthesize complex maneu-
vers in rough terrains and to develop accurate CoM estimates and foot placement
plans. Various human case studies were analyzed to validate the eectiveness of
the approach. The results show that for natural walking over complex planar ter-
rains, the geometric approach generates accurate CoM path approximations and
state space trajectories and is a powerful tool for understanding CoM behavior and
foot placements in irregular planar terrains.
xii
Chapter 1
Introduction
1.1 Exordium
The planning of a body's CoM and the sequencing of foot placements in rough and
complicated terrains are key elements to creating agile and versatile motion plans
for legged robots. In animals and robotic systems control of the CoM plays a vital
role in the fundamental behavior of the body, for example in maintaining static
balance or in the performance of extreme maneuvers [56]. However, considering
the large magnitude of natural terrains and the overwhelming number of available
motion plans that exist therein, it is evident that creating locomotion plans that
balance successful CoM control, terrain navigation and foot placement selection, is
a very dicult task. In nature, solutions to the problem have evolved to produce
distinct taxonomical characteristics and diversity in patterns of movement among
animals [2]. Indeed, an animal's performance in locomotion is directly tied to its
survivability, and thus developing an ecient motion plan takes on the utmost
importance. For example, some animals prefer speed or acceleration and have
evolved long and nimble limbs for increased
exibility in feet and body placement.
Others favor stability and security of foot contact, and have evolved sturdy and
stumpy limbs [2]. Even within a single species, the range of physical sizes and
shapes is so varied that creating models that unify their physical representations
is a daunting task. Indeed, the enormous scale of size of physical shapes and
locomotor mechanisms coupled with the breath of environments possible conspire
1
to make the problem highly intractable [9]. But it is precisely these criteria that
must be considered when planning across complex terrains.
Robots, like their biological brethren, also suer from the same enormity of
scale. Like animals and humans, they also come in myriad shapes and congura-
tions. Bipedal robots, for instance, have been designed to function like humans and
also bear the same magnitude of disparity in shapes as their human counterparts.
Furthermore, to function like humans, these robots must achieve the same level
of maneuverability and economy of motion while executing movements at slow or
high speeds. Humans alone are capable of maneuvering across extreme terrains,
performing a wide range of motions while varying speed, contact locations, and
direction and location of their CoM. These complex maneuvers are governed by
nonlinear dynamics with no closed form solution that signicantly impede CoM
behavior analysis. Specically, determining accurate CoM position paths, contact
transitions and foot placements across varying terrains while traveling at dier-
ent speeds has proved moiling for motion planners to incorporate. If appropriate
models and strategies can be developed, then their application in motion planning
and control could open up new avenues in the development of extreme locomotion
maneuvers for robotic systems.
1.2 Motivation
Inspired by the importance and need for a unifying general methodology that
incorporates the anatomical dierences in animals and robots as well as the char-
acteristics of the terrain, the research presented in this dissertation explores the
creation of a general geometric framework for CoM construction and estimation
that encapsulates these kinematic variations while simultaneously nding optimal
2
foot locations in the environment. Generalized motion planners that account for
these kinematic and environmental variations have so far been dicult to generate,
however developing models and control policies of the dynamic behavior of motion
is extremely consequential in planning through complex terrains. Specically, this
work uses essential body kinematic constraint information in tandem with knowl-
edge of the terrain to develop eective abstractions that allow for accurate bipedal
CoM position construction and estimation and foot placement selection. Develop-
ment of a general framework is important because it allows for fast and reliable
production of CoM behavior for whole classes of kinematic architectures with-
out the need for detailed subject specic data or calibration. Furthermore, the
approach is generalized to diverse types of terrains, namely, complex planar ter-
rains, and therefore provides greater scope for CoM behavior analysis. Thus the
general geometric approach allows for production of locomotion plans that incor-
porate the bodys kinematic constraints and the terrain traversed in an accurate
and eective manner.
1.3 Challenges
Developing a general geometric framework for bipedal CoM position construction
and estimation that also permits calculation of optimal foot placements includes
the following challenges:
1.3.1 Sensor Restrictions
The generalized approach undertaken in this research has no external or unnatu-
ral sensors. Non-reliance on sensor information means that real-time feedback is
unavailable and that the CoM must be calculated using only subject and terrain
3
information. Previous work has depended heavily on the use of continuous data
from sensors, e.g. force and video data, Wii balance board and Kinect. These
sensors are placed in the terrain or attached directly to the subject and allow for
reconstruction of the subjects's CoM. Independence from external sensors renders
the problem more dicult since CoM behavior must be retrieved directly from the
body's kinematic structure and the terrain.
1.3.2 Subject to Kinematic Constraints
A geometric framework for accurate CoM estimation that accounts for the large
variations of possible body kinematic architectures has so far not been treated.
Techniques for subject-specic CoM estimation have been developed, but these
require external sensors and lengthy calibration periods that adjust for each sub-
ject. Independence from sensors means that accurate abstractions must be devel-
oped that incorporate and describe the diverse space of body kinematic architec-
tures. Development of such abstractions is a non-trivial task.
1.3.3 Complex Terrains
The number of possible terrains is immense and must be accounted for when mod-
eling CoM behavior. Current methods lack tolerance to complex and rough ter-
rains and attempt to constrain the CoM to xed trajectories. However, accurate
modeling must consider both irregular CoM paths and varied rough terrains. Con-
sideration of these factors greatly hinders CoM estimation and phase behavior and
should be addressed by the generalized model.
4
1.3.4 Non-linear Dynamics
There is an intricate interdependency between the CoM and foot placement loca-
tions that makes determining contact transitions and foot placements across vary-
ing terrains while traveling at dierent speeds dicult for motion planners to
incorporate. This is because locomotion maneuvers are governed by nonlinear
dynamics that lack a closed form solution and make extraction of CoM behavior
very dicult. However, these critical factors must be addressed to determine how
they interrelate to produce suitable and optimal foot locations.
5
Chapter 2
Overview of Framework
The geometric framework presented in this work allows for 1) CoM path construc-
tion, 2) accurate CoM estimation for bipedal locomotion, 3) accurate CoM esti-
mation for ambulatory load carriage and 4) a method for optimal foot placement
selection in the terrain. Each aspect of the framework relies solely on geometry
information of the environment and essential knowledge of the kinematic body.
The work is built incrementally, starting from a model for path planning based on
algebraic geometry to which mathematical extensions are derived that provide for
additional features and generalizations.
In the rst stage of the framework, the geometric approach is used to plan the
motion of the CoM through complex planar terrains. The CoM plans produced
meet both the kinematic constraints of the body as well as the tangential require-
ments specied by the user. The freedom to select the endpoint CoM positions
and tangent angles endows the planner with the
exibility and power to plan com-
plex yet smooth CoM paths over any terrain while respecting the conditions of
the environment and staying within the limits of the robot's architecture. Next,
construction of the CoM path allows for the determination of potentially viable
foot placements in the terrain. To determine these potential locations, minimum
strain energy cost landscapes of CoM vertical phase behavior for step transitions
are created for the environment. This is accomplished using single contact model
dynamics for state-space approximations of center of mass behavior. The cost
landscapes provide information regarding foot locations that minimize the cost
6
between step transitions. Thus, this initial stage allows for generation of smooth
and optimal paths for CoM motion and provides a mathematical method based on
energy cost minimization to identify potentially viable foot placements.
In the second stage, the geometric approach is reworked to produce accurate
CoM estimates of natural walking human locomotion over irregular planar ter-
rains. This is accomplished using body anthropomorphic abstractions of human
subjects to determine the kinematic constraints and through dynamic calculation
of the endpoint tangent vectors. In contrast with the rst stage where the tangen-
tial requirements were specied by the user, in this stage the tangent vectors are
dynamically calculated. Using these dynamic tangent vectors, information of the
terrain, the contact foot locations as well as the subject's kinematic constraints,
the model is transformed to produce accurate CoM position estimates for human
natural walking locomotion over complex planar terrains.
Thirdly, after having presented the geometric CoM estimation technique, the
approach is extended to accommodate load carriage behavior. This is achieved
through a revamp of the mathematical formulations rst laid out for CoM esti-
mation. The resulting model expands the scope of the work to include sagittal
mass osets. Inclusion of sagittal mass osets further generalizes the framework
to describe behaviors such as back carrying of babies, backpacks, and other loads.
Finally, leveraging the ability to perform accurate CoM estimation and iden-
tify potentially viable foot placements, the simultaneous study of CoM behavior
and foot placement selection is investigated. A benecial consequence of the CoM
estimatation method is its dependence on foot contact point location. This depen-
dence, in tandem with analysis of CoM phase portraits, is exploited to develop
strategies that pinpoint suitable and optimal foot locations in the terrain. In this
7
manner, simultaneously and reliably performing CoM estimation and foot place-
ment selection is achieved. Thus, the geometric approach can be used to accurately
study CoM paths while contemporaneously examining foot locations for a wide
variety of bipedal legged architectures through complex planar terrains.
2.1 Outline of Dissertation
This dissertation is structured as follows: Chapter 3 presents CoM construction
and foot placement optimization through a terrain for a given kinematic architec-
ture. Specically, it presents the fundamental mathematics used to develop and
understand CoM dynamics. These include descriptions of cubic polynomial OGH
curves, the single contact point model and prismatic inverted pendulum dynam-
ics. The chapter concludes with an energy minimization method to locate suitable
contact foot placements.
Chapter 4 explains how the method for CoM construction can be modied to
develop a geometry based approach for CoM estimation. To accomplish this, the
concepts of virtual steps and forward progress angles are introduced and dened.
These describe the CoM double support phase and angular behavior respectively,
and combined with subject kinematic constraints and terrain geometry, allow for
accurate description of CoM behavior.
Chapter 5 describes an extension to the geometric CoM estimation technique
that incorporates load carriage behavior. Specically, sagittal mass osets and
load carrying are discussed, as well as the modied load virtual step model. The
chapter details the revised triangular representation for loaded walking and lastly,
validates the approach through comparisons with human case studies.
8
Chapter 6 presents an approach for foot placement selection in complex planar
terrains. The foot contact transition energy minimization and center of mass
estimation techniques from chapters 3 and 4 are combined into a unied framework
that allows for analysis of candidate foot locations. From this, selection strategies
for foot locations are developed that produce natural foot plans. The chapter
includes analysis of human traversal case studies and concludes with simulation
results for verication.
Chapter 7 summarizes the contributions of this thesis. This includes an eec-
tive alternative for CoM estimation based on a geometric model and a foot place-
ment selection approach rooted in CoM state space behavior. The chapter ends
with a discussion of possible extensions to the current work and future areas for
research in locomotion planning through complex terrains.
9
Chapter 3
CoM Construction and
Optimization of Foot Placements
The objective of this section is to develop CoM motion and foot placement plans
that take into account the kinematic architecture and the characteristics of the
terrain. The hallmarks of the approach outlined here are the ability to generate
CoM paths directly. This is done by creating abstractions of the robot's architec-
ture. The non-reliance on given CoM paths provides greater
exibility in planning
across robots with dierent kinematic constraints. Additionally, characteristics of
the terrain are used to optimize the locations of foot contacts to produce minimum
energy foot plans. This provides a mechanism to identify prospective foot contact
locations. The simultaneous study of CoM planning and optimal foot placement
selection has so far not been treated, especially not with respect to rough pla-
nar environments. Previous work has been limited in its scope and lacked the
malleability to deal with these terrains.
3.1 Related Work
Limit cycle based techniques were introduced in [44] where control of foot place-
ment is implicit and the result of passive dynamic walking. Because the gait is
powered naturally by gravity, passive limit cycle based robots are stable to small
perturbations and require less energy. Although these techniques provide stable
10
walking, their passive nature disallows CoM planning and restrict selection of foot
placements. Moreover, these techniques have shown poor performance on rough
terrains. Such terrains are replete with varying heights which require the ability to
plan CoM paths and contact foot placements for successful transit. To compensate,
further research has immixed biological elements with limit cycle techniques. In
[65] and [35] articial muscles and foot model analysis was investigated to enlarge
the stable walking range of biped robots. However these methods still lack toler-
ance to rough terrains and oer crude CoM and foot placement planning.
Another popular approach is Zero Moment Point based control. In ZMP the
robot's conguration is adapted to satisfy predened foot placement locations [71].
The drawback of such methods lies in their restriction of the robot to limited
motions that meet the foot placement conditions. Consequently, robots imple-
menting ZMP based control lack robustness against perturbations, demonstrate
poor performance on uneven terrains, and oer reduced energy eciency. Thus,
ZMP research has focused on methods that improve adaptivity such as auxiliary
ZMP based control [37] and adaptation of reference trajectories to large distur-
bances [45] [24] [48]. In [46] and [31], the authors simultaneously planned CoM
and ZMP trajectories using parametrized polynomials and handled small modi-
cations of foot placement. However, the CoM path was calculated only indirectly
as a result of the ZMP path and the foot contact locations were also preplanned in
advance. Such methods limit the choice of CoM path and perform no discrimina-
tory paring of contact foot locations resulting in reduced
exibility over terrains.
Lastly, Capturability based approaches use gait models to approximate capture
regions that allow planning of foot contacts [21] [53] [26]. Although eective, the
simplied gait models developed have only been demonstrated to work on level
terrains. Furthermore, the simplications presented discard valuable information
11
such as height variations of the CoM during locomotion. In contrast, the method
described here handles uneven terrains and accounts for CoM height changes, thus
generalizing to larger classes of environments and robots.
More recently, the authors in [62, 57, 60, 78] have made signicant contributions
in understanding the multi-contact dynamics of extreme locomotion maneuvers
in rough terrains. This work utilizes and extends the models they developed to
include planning of CoM paths that are contingent upon the robot's kinematics and
the terrain being traversed. This is consequently used to nd optimized contact
foot transitions and foot placements. Unlike all the previously mentioned work
and in particular that given in [46], [53] and [34], this framework allows for greater
freedom in the selection of both the CoM path and the foot contact locations. CoM
paths that respect the unique architecture of the robot can be specied directly.
These in turn are used as a guideline to nd optimal foot locations and contact
transitions in the environment. This endows the robot with greater versatility in
navigating environments. With this framework in hand, locomotion plans across
variable terrains can be eectuated for any robot.
3.2 Outline of Method
CoM path and optimal contact foot placement planning is achieved as follows: (1)
an optimized geometric Hermite curve with minimum curvature and length is used
to plan a path for the robot's CoM. The curve satises conditions dependent on
the robot's architecture and encapsulates the idea of size and functional variations
among dierent robots. (2) Prismatic inverted pendulum dynamics and pertur-
bation theory are used to obtain phase curves of CoM behavior. The behavior of
the CoM is critically dependent on the foot contact locations. The foot contact
12
locations themselves are conditional on the environment being traversed. (3) This
dependency is exploited to generate strain energy cost landscapes of vertical CoM
phase behavior for contact foot locations. The landscape is used to identify poten-
tially optimal foot center of pressure (CoP) locations. (4) Through iteration of
the ensuing foot across the terrain, the optimal foot location is chosen from the
cost landscape. Selection is made to meet the kinematic constraints of the robot
while selecting a location that maximizes progress and minimizes cost. (5) The
process is repeated starting from the newly selected foot CoP location to generate
successive foot contact locations. In this manner, optimal foot placement and CoM
path plans are accomplished. The overall method is shown in Algorithm 1.
Algorithm 1 Optimal CoM and Foot Placement Planner
1. Given a robot architecture k and a terrain q, determine the range of viable
CoM vertical positions r, for k in q.
2. Initialize rst foot placementF
0
and CoM positionP
0
and tangent vectorV
0
.
loop
3. Using Hermite endpoints P
i
, V
i
i = 1:::n, P
i
2r representing CoM
position and tangent vector, generate optimal geometric Hermite curves
with minimum curvature g
i
(x), i = 1:::n.
4. Select the curve g
i
(x) with minimum length.
5. Let CoM path be g
i
(x).
6. Select candidate secondary foot placement F
i
.
for each pair of F
0
and potential F
i
do
a. Use g
i
(x) to nd combined CoM phase curve and contact transition
point for F
0
and F
i
.
b. Find strain energy cost, C
i
of resulting phase curve.
c. Choose next foot placement candidate F
i
by iterating along the
terrain q.
d. Select F
i
yielding the lowest cost C
i
.
end for
7. F
0
=F
i
, P
0
=P
i
, V
0
=V
i
end loop
13
3.3 Optimized Geometric Hermite Curve
The pivotal element in robot locomotion essentializes to the ability to maintain
control over the position and velocity of the robot's CoM. The breadth of viable
CoM paths is vast but depends critically on the terrain the robot is traversing
and the foot placements therein. CoM dynamics can be analyzed by studying the
interaction and interplay between these components. In this work, an optimized
geometric Hermite (OGH) curve is used to plan CoM paths through the terrain.
OGH curves lend themselves well to CoM path planning because they are both
mathematically and geometrically smooth. In addition, the endpoint tangent vec-
tors can be specied allowing for both directional and velocity control of CoM
paths.
OGH curves have been optimized using various criteria including: minimum
length, minimum strain energy and minimum curvature [15] [75] [14]. This work
uses OGH curves with minimum curvature rst dened in [14] since they render
smooth and
uid curves that are often the required basis for planning over variable
terrain.
Denition 1. Given two endpoints P
0
and P
1
, and two endpoint tangent
vectors V
0
and V
1
, a cubic polynomial curve P (t), t2 [t
0
;t
1
], is called an opti-
mized geometric Hermite (OGH) curve with respect to the endpoint conditions
P
0
;P
1
;V
0
;V
1
if it has the smallest curvature variation among all cubic Hermite
curves P (t), t2 [t
0
;t
1
] satisfying the following condition:
P (t
0
) =P
0
;P (t
1
) =P
1
;P
0
(t
0
) =
0
V
0
;P
0
(t
1
) =
1
V
1
(3.1)
14
Figure 3.1: OGH curves with dierent vector tangent angles. The center and right
plots show COH curves constructed using piecewise OGH segments. The angles
and are both measured counterclockwise with respect to the vector P
0
P
1
, with
corresponding to the point P
0
and to P
1
. Note the diversity of complex yet
smooth paths possible.
where
0
and
1
are arbitrary real numbers, P
0
(t) is the rst derivative of P (t),
and the cubic Hermite curveP (t);t2 [t
0
;t
1
] satisfying the constraints in equation
2.1 can be expressed as
P (t) = (2s + 1)(s 1)
2
P
0
+ (2s + 3)s
2
P
1
+
(1s)
2
s(t
1
t
0
)
0
V
0
+ (s 1)s
2
(t
1
t
0
)
1
V
1
(3.2)
where s = (tt
0
)=(t
1
t
0
). The objective function to optimize is dened as
E =
Z
t
1
t
0
[P
000
(t)]
2
dt (3.3)
and is the approximate curvature variation of the curve P (t).
The tangent angle requirements are specied with respect to and where
is the counterclockwise angle from the vectorP
0
P
1
toV
0
, is the counterclockwise
angle from the vector P
0
P
1
to V
1
, and and are called the tangent angles. To
guarantee both mathematical and geometric smoothness, the tangent angles must
meet angle constraints. For angles outside these regions, a composite optimized
15
geometric Hermite (COH) curve is constructed. A COH is a piecewise cubic poly-
nomial composed of 2 or 3 segment pieces wherein each piece is an OGH curve. For
complete details regarding angle constraints and construction of OGH and COH
curves, see [14]. Fig 3.1 depicts several example OGH and COH curves.
3.4 Single Contact Point Model
Figure 3.2: Walking prole schematic showing the CoM position and acceleration
and feet centers of pressure (CoP) and reaction forces.
The OGH curve dened above lays the groundwork for planning smooth CoM
paths over volatile and inconstant terrains. However, the behavior (state space
trajectory) of the CoM is fundamentally reliant on both the CoM path and the
locations of the foot contacts. As the robot maneuvers and negotiates its envi-
ronment it makes contact with the terrain via its feet which determine the overall
dynamics of the system (see Fig 3.2). This study treats the single contact point
support paradigm explicated in [62, 57, 60] to gain insight into the dynamic behav-
ior of the CoM under these conditions. The principle of dynamic equilibrium states
16
that the sum of all moments acting on a moving system is equal to the net inertial
moment. For the single contact paradigm, the balance of moments is given by:
p
cop
k
f
r
k
=p
com
(f
com
+Mg) +m
com
(3.4)
wherek is the contact limb,p
cop
k
is the limb's center of pressure (CoP) or contact
point,M is the total mass of the system,g is the gravitational constant,p
com
is the
location of the CoM with respect to the coordinate origin, andf
r
k
,f
com
, andm
com
are the 3-dimensional vectors of reaction forces, center of mass inertial forces and
moments, respectively. The dynamic equilibrium of forces under the single contact
model can be expressed as f
r
k
= f
com
+Mg which substituted into equation 2.4
leads to:
(p
com
p
cop
k
)f
r
k
=m
com
(3.5)
Equation 2.5 can be expressed in vectorial form as:
0
B
B
B
@
0 f
r
[kz]
f
r
[ky]
f
r
[kz]
0 f
r
[kx]
f
r
[ky]
f
r
[kx]
0
1
C
C
C
A
0
B
B
B
@
p
com[x]
p
cop
k
[x]
p
com[y]
p
cop
k
[y]
p
com[z]
p
cop
k
[z]
1
C
C
C
A
= 0 (3.6)
in which the inverted pendulum model is assumed to be a single point mass such
that inertial moments about the center of mass can be ignored, i.e., m
com
= 0
[37] [36]. If a planar model is considered consisting of only the vertical and frontal
directions [57] [60] and if Newton's second law is used to express force as a function
of acceleration and mass, namely, f
r
[kx]
= Ma
com[x]
, f
r
[ky]
= Ma
com[y]
, and f
r
[kz]
=
M(a
com[z]
+g), equation 2.6 can be rewritten as:
a
com[x]
=
(p
com[x]
p
cop
k
[x]
)(a
com[z]
+g)
p
com[z]
p
cop
k
[z]
(3.7)
17
for the dynamic behavior of the CoM in the sagittal direction. Similar expressions
can be obtained for the lateral and vertical directions.
3.4.1 State Space Behavior
Equation 2.7 gives the acceleration prole of the CoM as a function of the position
of the CoM and the position of the foot contacts. As discussed in [62] [57], the
trick is to nd a dependency by seeding a preselected geometric CoM path such
that a
comx
can be represented as a function of only p
comx
, p
copx
, and p
copz
. Under
such circumstances, the general form of equation 2.7 reduces to:
a
comx
= (p
comx
p
cop
kx
) (p
comx
;v
comx
;p
cop
kx
;p
cop
kz
) (3.8)
where (;;;) is a non-linear function and has no closed form solution. The
work presented in this dissertation uses an OGH curve with minimum curvature
constrained by the robot's architecture to determine the preselected geometric CoM
path. Now using perturbation theory and supposing x, p
com
[x]
and x, a
comx
,
equation 2.8 can be represented as x = f(x; _ x). Over small iterations of x, the
velocity and the acceleration of the CoM ( _ x and x respectively) remain constant
and can be used to approximate the behavior at neighboring points, i.e., CoM
velocity versus its position (state-space trajectory). Thus, given initial conditions
for (x
k
; _ x
k
), the change in CoM position over a time-step (the perturbation), can
be approximated as:
_ x
k+1
_ x
k
+ x
k
(3.9)
x
k+1
x
k
+ _ x
k
+ 0:5 x
k

2
(3.10)
18
Equation 2.9 can be used to nd an expression for the perturbation in terms of
velocity and acceleration to give ( _ x
k+1
_ x
k
)=( x
k
) which results in:
x
k+1

( _ x
2
k+1
_ x
2
k
)
2f(x
k
; _ x
k
)
+x
k
(3.11)
which is the state-space approximate solution that is sought. For a thorough
description outlining the process for nding state-space trajectories for the CoM,
see [62, 57, 60].
3.5 Path Planning
In the previous sections anent preselection of CoM paths, creation of the initial
CoM path was discussed as being dependent on the robot's architecture as well as
the environment being traversed. Clearly, the robot's kinematic constraints place
restrictions on viable CoM positions, as does the environment which channels the
robot along a conned space of paths. To characterize the kinematic variation
among robots, an abstraction is developed to encapsulate the robot's ability to
alter its CoM position. This is dened by an interval rangef(z
min
;z
max
)jz
min

p
comz
z
max
g, which represents the space of realizable CoM positions for a specic
body. Forsooth, this range will vary from body to body such that the paths
generated will be unique, e.g. a small robot kinematically constrained to travel
closer to the ground will have a range of viable CoM paths propinquant with the
terrain, while a larger robot will have them further removed. This interval range
is taken directly from the body's kinematic constraints, i.e. leg lengths, joint
limits etc, and in essence describes the body's ability to shift its CoM position
vertically. Additionally, in this study only planar environments are treated and
thus (z
min
;z
max
) will depend on the terrain such that8p
terz
2R :jz
min
p
terz
j =c,
19
Figure 3.3: CoM paths generated using COH curves for dierent CoM range con-
straints over various terrains. The left and middle plot demonstrate how the envi-
ronment and the selection of tangent angles aects the CoM paths. The right
plots displays a complete CoM path consisting of piecewise OGH/COH curves
with minimum curvature and length to traverse the terrain.
wherep
terz
is the vertical position of the terrain andc is a constant. In other words,
for each point in the terrain, there is a minimum threshold determined by the
body's kinematic constraints from which the CoM position is disallowed to enter.
Using this range, a family of OGH curves can be created to plan CoM paths through
the terrain. The process for performing this is as follows: (1) the interval range for
the robot is determined, (2) the endpoint tangent vectors are specied. These are
and explained previously and specify the direction the CoM should be traveling
in at particular locations over the terrain. (3) For these locations, OGH/COH
curves are traced out for eachp
comz
in (z
min
;z
max
). If at any point in its length the
OGH/COH curve exceeds the permissible CoM limits, it is rejected. This ensures
that the CoM path respects the conditions of the environment and stays within
the limits of the robot's architecture. (4) Lastly, the curve with minimum length
is selected from the family of curves generated. This yields a path with minimum
curvature and length. Several examples of OGH/COH CoM paths are shown in
Fig 3.3 illustrating the versatility of CoM path construction. The left and middle
plot show example curves generated for separate tangent angle, robot CoM interval
range, and terrain conditions. The tangent vectors at several points are specied
20
and allow for directional control of the CoM through the terrain. The plot on
the right shows curves generated at various terrain points. The nal CoM path is
constructed by connecting the minimum length OGH/COH curve at each interval
range to form the complete path. Note the variety of complex and smooth CoM
paths produced through selection of dierent endpoint CoM positions and tangent
angles and how the paths remain within the limits of the kinematic body. For a
detailed overview of the process, refer to Algorithm 1.
3.6 Contact Transition
The complete CoM path constructed above is chosen as the preselected geomet-
ric path and allows for the determination of foot placements in the terrain. To
determine optimal foot placement locations, strain energy cost landscapes of CoM
vertical phase behavior are created for the environment. This is accomplished
using the prismatic inverted pendulum model to describe single contact behaviors.
By using the CoM path derived above as the seeded equation into the system, the
model can be transformed into an inverted pendulum of variable height that tracks
the desired surface (OGH CoM path). The resulting vertical dynamics yields a sec-
ond order non-linear ODE which is solved via the numerical integration techniques
of section 3.4.1 to derive the state-space behavior of the CoM for given foot con-
tact locations and CoM velocities. Fig 3.4 depicts phase diagrams of CoM behavior
for an example OGH curve CoM path and various boundary conditions, namely,
the CoM velocities over the contact foot CoP location. Note that the state-space
behavior of the CoM depends on the sagittal and vertical placement of the foot
contact location, as well as the apex velocity of the CoM as it passes over the
foot CoP location. As will be discussed shortly, this important consequence can
21
Figure 3.4: OGH curve with = 10 and = 340. The phase diagrams of sagittal
and vertical CoM velocities are shown in the middle and right plots respectively.
The CoP is located at (0; 0). Various boundary apex CoM velocities are shown.
Figure 3.5: CoM saggital and vertical phase diagrams for two contact feet at
(0:2; 0) and (0:4; 0:25) with the step transition point shown. The apex velocity
of the CoM over each contact point is 1m=s.
be exploited to optimize the location of foot placements within a terrain. For the
nonce, it is observed that the predicted phase curves of CoM sagittal behavior can
be used to nd the step transitions between foot placements. This is done by nd-
ing the intersections of adjacent sagittal phase curves [57]. This ensures continuity
of position and velocity between adjacent CoM behaviors. To determine the exact
step transition point, a 6th degree polynomial is t to the sagittal CoM phases for
adjacent foot placements. The two polynomials are subtracted and the roots are
22
found, expunging any imaginary roots. The point of intersection is then obtained
by selecting the root that lies within the CoM range. Fig 3.5 demonstrates the
contact transition model for the OGH curve of Fig 3.4 with inclusion of a secondary
foot location and terrain. For each contact foot location, the CoM sagittal phase
space behavior is shown directly above it in green. The contact transition is then
the point of intersection of adjacent sagittal phase curves. Naturally, shifting the
foot contact location alters the resultant CoM state-space behavior and transition
model. Perforce, prudent selection of foot contact placements is required as it will
have considerable impact on the overall performance of the system. Lastly, it is
consequential to note that the switching time in the contact transition between the
x phase and the z phase as shown in Fig 3.5 is guaranteed to be synchronized. This
is because the preselected CoM path (OGH curve) creates a dependency between
the vertical CoM position and sagittal CoM position. The vertical CoM position
is a function of sagittal CoM position thus ensuring that the contact transition for
both occurs simultaneously.
3.7 Foot Placement Optimization
Previously, OGH curves were used to plan and ensure the smoothness of the CoM
path over terrain. However, this alone is insucient as the velocities of the CoM
are still susceptible to extreme changes. This is especially evident when traversing
over rugged and uneven terrain that cause jerkiness in the velocity proles of the
CoM, especially the CoM vertical phase. Indeed, an appealing quality of animal
and human locomotion is the ability to plan and perform complex actions and
maneuvers in smooth and steady manners, oftentimes in severely rugged environ-
ments. In this case, the most sensitive factor prone to jerkiness is the CoM vertical
23
phase behavior as it is more adversely aected by the terrain. Thus, attention is
now turned to diminishing the negative impact of the up and down motions of the
terrain on the CoM vertical phase behavior.
3.7.1 Experimental Results
Matlab was used to generate experimental results to study vertical phase behav-
ior and methods to temper vertical jerkiness. Earlier, the critical eects of foot
placements on CoM phase behavior were noted. Moving a contact foot location
up, down, left, or right causes changes not only in CoM sagittal and vertical phase
behavior, but also in the step transition point of the foot with its nearest neigh-
bors. Fig 3.6 depicts two example terrains and CoM paths and the eects on
CoM phase behavior resulting from changing the secondary foot contact location.
Additionally, the CoM apex velocity over the contact foot location also serves to
aect the phase space behavior of the CoM. A natural choice for traveling through
environments is to maintain a constant CoM apex velocity from contact foot to
contact foot, however, this framework allows for variable speeds in CoM travel as
desired by the planner. From Fig 3.5 it is observed that the complete vertical
phase of the CoM for adjacent foot contacts is composed of the forward vertical
phase segment for the back foot starting at its contact location, p
cop
x1
, up to and
including the step transition point x
tp
, and the backward vertical phase segment
for the front foot starting at its contact location, p
cop
x2
backwards to x
tp
. To gain
insight into the vertical phase behavior of the CoM, a 5th degree polynomial is t
to the aforementioned curve and dened in the range (p
cop
x1
;p
cop
x2
). Let this curve
beh(x). Thus, foot placements within the environment are sought such that h(x)
is optimized to reduce its volatility. To do this, the approximate strain energy
24
Figure 3.6: OGH CoM paths with constant apex velocity v = 1m=s and distinct
tangent and position endpoint conditions traversing dierent terrains are shown
in (a) and (d). In (b) and (e) several second foot contact locations and the cor-
responding phase space behavior of the CoM are shown demonstrating the aects
of foot locations. The strain energy cost landscape for each terrain is shown in (c)
and (f) respectively.
for the curve (also called the linearized bending energy) is used to measure its
mutability. The strain energy is dened as:
S =
Z
x
1
x
0
[h
00
(x)]
2
dx (3.12)
which returns a scalar that quanties the degree of bending in the curve. It is
precisely the bending and undulating nature of the CoM vertical phase that must
be diminished. Therefore, foot contact placements are desired that minimize this
quantity. To obtain the strain energy cost of all CoM vertical phase curves, h(x)
is integrated from x
0
= p
cop
x1
for each x
1
= p
cop
x2
where p
cop
x2
2 p
terx
. This is
25
equivalent to iterating through the terrain by placing the secondary foot contact
location at each point in the environment and calculating its strain energy. This
produces an energy cost landscape from which the minimum strain energy contact
foot location can be extrapolated based on the kinematic dynamics of the robot.
Because each robot architecture has dierent kinematic constraints, selection of
foot placements will vary from robot to robot. This process can subsequently be
used to plan foot locations and step transitions through the environment. From
Fig 3.6 it is observed that the terrain enforces restrictions on the space of available
foot contact locations as can be seen in (b) and (e). Furthermore, if the body's
kinematic constraints are included, then the space of potential foot placements is
further reduced to fall within a range dened by that particular body. Ergo, the
strain energy cost landscape can be used in conjunction with the robot's archi-
tecture to plan optimal foot placement strategies. For example, if the terrain and
CoM path from Fig 3.6 (a) were used with a robot with initial foot placement
at (0:5; 0) and kinematic limits for maximum foot placement of 0:35m, then an
ecient choice for optimizing progress and minimizing vertical phase velocity dis-
turbance would be to place the ensuing foot at (0:19;0:5). Similarly, starting
from that foot, the process can be repeated to generate the new energy cost land-
scape and plan yet another foot location. Naturally, situations may arise where
some of the optimal foot placements will be dicult to access and may be unreach-
able for the robot. However, this can be rectied by introducing additional robot
kinematics constraints into the framework. Example full step sequences for the
terrains and paths of Fig 3.6 are shown in Fig 3.7. In each, the strain energy
cost landscape is derived starting from the preceding foot to next potential foot
locations. This produces a guide for selection of subsequent optimal foot contact
locations.
26
Figure 3.7: Enery cost landscape for CoM paths and terrains of Fig 3.6 at step 1
and step 2 shown in (a-b) and (d-e) respectively. The cost is the strain energy of
the vertical CoM phase between adjacent potential foot locations. The maximum
foot placement for each robot is 0:35m and 0:4m. (c) and (f) are the full step
sequences.
Needless to say, the performance of the system is also ineluctably tied to the
choice of CoM apex velocity, and careful attention will have to be taken in its
selection. Nevertheless, it is the power and versatility of this framework that
despite the choice of velocity, foot placement locations and step transitions can
still reliably be obtained such that the robot stays within its kinematic constraints.
Returning to Fig 3.6 (a) and (d), Fig 3.8 and Fig 3.9 respectively show the energy
cost landscapes that would arise for each terrain and CoM path using dierent
CoM apex velocities. Note that increased velocities tend to reduce the cost of
foot placements in the terrain, whereas decreased velocities magnify them. This
agrees with intuition since slower velocities tend to exacerbate the conditions in
the environment, whereas faster velocities tend to diminish the eects. However,
the important consequence is that despite the velocity requirements for the robot,
27
Figure 3.8: Strain energy cost landscape for Fig 3.6 (a). CoM constant apex
velocities of v = 0:75m=s and v = 1:5m=s.
Figure 3.9: Strain energy cost landscape for Fig 3.6 (d). CoM constant apex
velocities of v = 0:6m=s and v = 1:2m=s.
foot placement and transition plans are derivable that ensure the robot traverses
the terrain in a manner that minimizes the eects on the CoM phase behavior
while traveling along a smooth CoM path. Thus, in conjunction with the CoM
paths, the energy cost landscapes can always be used to plan optimal foot contact
locations through the environment.
28
Chapter 4
CoM Estimation: A Geometric
Approach
The CoM construction process described above delivers smooth and optimal CoM
and foot plans that can be provided as input into a robot motion planner. Gen-
erally however, the behavior observed in natural systems may dier since animals
and humans operate while simultaneously maximizing several criteria, e.g. time of
travel, energy, step count etc. Therefore, accurate prediction and modeling of CoM
behavior of natural systems is important since it allows for development of motion
planners that are faithful to the CoM dynamics observed in the real world. This
necessity is paramount to robot motion planners aiming for performance equiva-
lence with human locomotion. Human locomotion is extremely varied and replete
with complex force interactions that makes CoM study very dicult. The problem
is exacerbated when planar and non-planar terrains are considered, each of which
aect CoM behavior in uniquely dierent ways and introduce additional param-
eters that must be addressed by the motion planner. Despite these challenges,
CoM estimation remains central to creating agile and versatile locomotion plan-
ners. To this end, models and tools have been designed to predict CoM behavior
in robots and humans, however current techniques for CoM estimation are encum-
bered by lengthy calibration periods requiring the use of specialized tools (force
plates, motion capture, etc). The work presented in this section oers an unob-
trusive and attractive alternative for CoM estimation that is devoid of force or
29
video sensors. A novel and straightforward geometric method for CoM estimation
is described that relies solely on geometry information of the environment and
essential knowledge of the kinematic body and delivers accurate approximations
of the CoM path for natural walking over complex planar terrains.
4.1 Related Work
Various approaches have been used to estimate the CoM in robots. Previous stud-
ies [64, 20, 3] employed a forward kinematics or a ZMP [47] based computation
approach for CoM state estimation. Such estimators proved accurate for modeling
CoM position, but required continuous body link attitude and force sensor infor-
mation. To further improve the accuracy of the forward kinematics paradigm, the
authors in [41] presented a novel Kalman lter for CoM estimation based on a
three scheme hybrid approach, and in [12] [13] the CoM position was estimated
using spectral analysis to lter multi-sensor data. However, the drawback to these
methods remained their continued reliance on expensive inertial and force sensors
for CoM estimation.
Similar issues have plagued CoM estimation in humans. The body segmental
method based on data from anthropomorphic tables [74, 76, 22] is a standard tech-
nique. Nevertheless, it requires constant body marker tracking and video data to
perform CoM position approximation. Newtonian techniques that use ground reac-
tion forces have also been used. The authors in [61] [38] attached force platforms
to the terrain and in [55] they were attached directly to footwear. But such data
acquisition methods are clumsy and impractical outside of laboratory settings.
In recent years the statically equivalent serial chain (SESC) method rst devel-
oped in [27] has been further rened in [17] [18] to limit the usage of external
30
sensors for human CoM estimation. The method represents a polyarticulated sys-
tem as a branched multi-link chain with the CoM position of the body specied
by the end-eector of the chain. In order to obtain subject specic data and
obviate dependence on force recordings, subjects undergo an extended calibration
phase during which stable postures and sizable force readings and motion capture
data are combined to determine the SESC for the subject. Next, sensor devices
are employed to nd the joint measurements from which the instantaneous CoM
position can be calculated. Naturally, real-time joint measurement is a non-trivial
task. To facilitate this operation the authors of [30] [29] switched to a Wii balance
board and Kinect for force and video data. Needless to say, serious questions anent
device cost and manageability remain, and in general, dependence on expensive
and intrusive equipment continues to impede CoM estimation in all but controlled
and non-realistic laboratory settings.
The sections that follow describe a geometric approach for performing approx-
imate CoM position estimation that is bereft of dependence on video or force data
and that is a natural extension of the geometric CoM construction method pre-
sented chapter 3. The inspiration behind the geometric approach is the concept
that the environment and the architecture of the kinematic body work in tan-
dem to impose limitations on the possible CoM positions in the terrain [5]. Thus,
knowledge of the terrain, body kinematic constraints, and foot contact locations,
provides insightful information that allows for accurate CoM position estimation
and phase space behavior during natural walking locomotion.
31
4.2 Outline of Method
The method for producing CoM estimates for natural walking over rough planar
terrains is as follows: (1) knowledge of the terrain, desired contact foot locations
and step durations are used to create virtual steps. Virtual steps are nonphysical
steps located in the terrain which function only as reference points to describe CoM
behavior during the double support phase-i.e., when two steps are simultaneously
in contact with the terrain. (2) The virtual steps, desired contact foot locations,
and the geometry of the terrain are then used to dene forward progress angles.
Forward progress angles are dynamically calculated endpoint tangent vectors and
are simply the direction of motion of the CoM at the apex of each step (virtual
and real). (3) An OGH curve with minimum curvature (MC) and strain energy
(SE) is used in conjunction with basic body kinematic parameters to construct the
CoM path. The path is constructed incrementally using the step (virtual and real)
locations and their respective forward progress angles. The nal CoM position
path is then formed by joining all the piecewise OGH segments.
4.2.1 Virtual Steps
Virtual steps dene nonphysical reference points in the environment that cap-
ture CoM behavior between successive foot locations. The concept is drawn from
observing that performance of successive steps by a subject results in CoM posi-
tion displacement in the negative vertical direction relative to the standing upright
position. This dip or vertical displacement
comz
allows for accurate description
of CoM behavior between consecutive foot centers of pressure (CoP). Hence, vir-
tual steps are dened as follows: a virtual step is a point in space (x
vs
;z
vs
) whose
32
Figure 4.1: Sagittal plane view showing step geometric model.
location lies between successive foot contact locations and whose value can be com-
puted directly using adjoining real CoP locations. This work adopts a geometric
approach to virtual step calculation and regards the center of the hip joint and
sequential CoP locations as forming three points of a changing and moving isosce-
les triangle. Under such a model, the displacement
comz
can be found directly
by considering the height h of the changing triangle. A geometric representation
of this model is given in Fig 4.1 with b dened as the length of the leg, d as the
distance taken from the subject's standing upright CoM to the hip joint center
as provided in [22][74], and a as the sagittal distance between adjacent CoPs.
Thus, two adjacent CoPs (x
cop
i
;z
cop
i
) and (x
cop
j
;z
cop
j
) determine the vertical CoM
displacement
ij
comz
as follows:

ij
comz
= (b +d) (h
ij
+d) (4.1)
33
and after expansion of terms

ij
comz
=b
s
b
2

(x
cop
j
x
cop
i
)
2
4
(4.2)
Lastly, the vertical component z
i
vs
of the ith virtual step is chosen as
z
i
vs
=min(z
cop
i
;z
cop
j
)
ij
comz
(4.3)
Note that
ij
comz
is always specied with respect to sagittal CoP locations (equation
3.2). By treating the leg conguration as constant during double support phases
(Fig 4.1), this obviates the need for real-time joint kinematic tracking and simplies
the model considerably.
The horizontal component value x
i
vs
is computed by considering adjacent CoP
height dierences as well as desired step durations. These factors were observed
to exercise the greatest in
uence on the sagittal location of the dip x
i
vs
. For two
consecutive CoP locations (x
cop
i
;z
cop
i
) and (x
cop
j
;z
cop
j
) with desired step durations
t
i
and t
j
, the height and time proportions of x
i
vs
are dened as
T =
max(t
i
; t
j
)
t
i
+ t
j
(4.4)
H =
v
u
u
t

xcop
j
xcop
i
2

2
+ (z
cop
j
z
cop
i
)
2
(x
cop
j
x
cop
i
)
2
+ (z
cop
j
z
cop
i
)
2
(4.5)
where T is a step duration bias and H is the geometric hypotenuse ratio of step
CoPs that characterizes height in
uence. Next, the weighted sum of the height
and time proportions is used to determine a horizontal oset
ij
x

ij
x
=

x
cop
i
x
cop
j

T +H
2

(4.6)
34
which is the sagittal displacement value. Finally, x
i
vs
is found by assigning
ij
x
to
the CoP of greater step duration.
x
i
vs
=
8
>
>
<
>
>
:
x
cop
i
+
ij
x
if t
i
t
j
x
cop
j

ij
x
if t
i
< t
j
(4.7)
Hence, locations of virtual steps always lie between sequential CoPs while possess-
ing vertical values z
i
vs
that lie below min(z
cop
i
;z
cop
j
).
4.2.2 Forward Progress Angles
As previously mentioned, forward progress angles are dynamically calculated and
specify the direction of motion of the CoM at the apex of each step, real and virtual.
The angles are found through the observation that CoM behavior at the current
step is closely related to the location and distance of the current step to both
the preceding and ensuing foot steps. Because the steps alternate between real
and virtual, knowing desired CoP and virtual step locations provides an angular
description of CoM behavior through the terrain. Furthermore, step CoP locations
depend on the terrain and thus knowledge of the environment and the location of
foot placements in the terrain provides key information that can be used to dynam-
ically determine the angular and tangential behavior of the CoM from step-to-step
locations. This is accomplished through an arctan angle calculation between steps.
The complete algorithm is presented in Algorithm 2 while an example terrain with
virtual steps, foot contact CoPs and forward angles is demonstrated in Fig 4.2.
Note that in general virtual step angles are dened with respect to the previous
real step CoP, while real step angles are dened based on the proximity and height
35
Algorithm 2 Forward Progress Angles
Let S=f1. . . ng be the steps with S odd being real steps and S even the
virtual steps.

ij
atan2d angle from step (x
i
;z
i
) to (x
j
;z
j
).

i
forward angle of each step.
Initial angle
1
for S=f1g is dened as

1
=
(
max(j
12
j;
13
) if z
3
z
1
min(j
12
j;j
13
j) if z
3
< z
1
for i = 2 to n do
if S=fig is a virtual step then

i
=
i1;i
else[real step]

i
=
(
max(
i1;i
;
i;i+2
) if z
i+2
z
i

i1;i
if z
i+2
< z
i
end if
end for
Figure 4.2: Four step walking sequence through terrain. The virtual steps and
forward angles (vectors) are also shown.
36
location of the ensuing real CoP. For example, from Fig 4.2 it can be seen that the
virtual step forward progress angles are set to the arctan angle from the previous
real step CoP location to the virtual step location. The calculation of forward
progress angles for real steps compares two arctan angles: the angle to the next
real step and the angle from the preceding virtual step. The forward progress angle
is ultimately chosen as the maximum value of the two.
Once determined, the forward progress angles can be used to build OGH with
MC curves that model CoM behavior. Whereas for CoM construction in section 3.3
the tangent angles were user dened, the inputs to the tangent angles for CoM esti-
mation are taken as the forward progress angles determined from Algorithm 2. This
is an important distinction to emphasize. Previously, the user specied the desired
tangents and it was these that were used to construct the CoM path. Henceforth,
for CoM estimation, the paths produced will be dependent on the step locations
undertaken in the terrain. Clearly, there is a duality of relationship between the
step locations performed and the resulting CoM position in the terrain. Therefore,
the usage of forward progress angles generates OGH curves with MC that capture
the CoM direction of motion reliably by developing the CoM path contemporane-
ously with the choice of step CoP location. Thus, curves are produced that ensure
delity to the step locations of the kinematic body as well as the step-to-step CoM
behavior across the terrain.
4.2.3 Construction of CoM Estimate
Broadly speaking, the following procedure is used to produce CoM estimates: 1)
two forward progress angles and step locations produce a family of OGH curves
with MC. The curves must meet the subject's kinematic constraints. 2) The curve
with minimum (SE) among all curves generated is chosen as the path and forms the
37
start location for the following step. 3) The next forward angle and step location
is processed. The above procedure continues until the last step is reached.
Similar to section 3.5, the subject's kinematic constraints are abstracted as a
vertical displacement rangef(z
min
+k;z
max
+k)j(z
min
+k)p
comz
(z
max
+k)g,
which species the range of viable CoM positions for the subject at terrain height
k. Note that the range is discretized into 0:01m intervals for eciency and that k
is simply the vertical component, z
i
, of step placement (x
i
;z
i
). This abstraction
allows for representation of human/robot kinematic variation and generates paths
that account for the unique limitations of a body. Furthermore, because only
complex planar terrains are treated, once again (z
min
+k;z
max
+k) are terrain
dependent such that8p
terz
2 R :j(z
min
+k)p
terz
j = c, with p
terz
being the
terrain's vertical position and where c is a constant. This places a lower bound
on possible CoM positions determined by the body's kinematic constraints. Thus,
CoM paths are created that are faithful to the body's motion over the terrain.
For human subjects,z
min
andz
max
are approximated directly from body anthro-
pomorphic data. Using the subject's height,z
max
is estimated as the upright stand-
ing posture CoM location plus the foot heel to ball distance, while z
min
is set to
the vertical CoM position during performance of a lunge. Note that only height
information and mean segment lengths [16][74] are required greatly facilitating the
computation of subject specic kinematic constraints. Furthermore, because the
CoM paths are generated step-to-step, the vertical location of each step (x
i
;z
i
)
identies the value for k. Perforce this allows for production of CoM paths that
follow the locomotion of a subject through the terrain. The algorithm for gener-
ating CoM paths is given in Algorithm 3 with an example presented in Fig 4.3.
38
Algorithm 3 CoM Path Construction
Input:
Steps S=f(x
1
;z
1
) . . . (x
n
;z
n
)g
Forward progress anglesf
1
. . .
n
g
Constraints z
min
and z
max
Initial CoM position (x
1
;p
com
z init
)
for i = 2 to n do
for p
comz
= (z
min
+z
i
) to (z
max
+z
i
) do
1. Generate OGH curve with minimum curvature using
(x
1
;p
com
z init
),
1
to (x
i
;p
comz
),
i
2. If OGH curve violates CoM lower bound threshold, reject curve
3. p
comz
= p
comz
+ 0:01m
end for
a. Select endpoint (x
i
;p
comz
) corresponding to curveG
`
with minimum (SE)
among all curves generated.
b. Set CoM path to G
`
c. x
1
=x
i
, p
com
z init
=p
comz
,
1
=
i
end for
return Complete CoM path, the piecewise composition of each curve G
`
.
Figure 4.3: Example OGH curves (red) for rst three steps and forward angles of
Fig 4.2. At each step, the curve with minimum SE is selected as the CoM path
(black).
39
4.3 Experimental Results
4.3.1 Experimental Setup and Data Collected
Two terrains composed of obstacles of variable heights were each navigated by
6 dierent subjects. Motion capture video data was gathered for each traversal
and analyzed using Matlab. To minimize lens distortion and obtain unadulterated
planar motion, the camera was placed at a suciently far distance from the human
traversal motion. The raw video was shot at 30 frames per second but only every
3
rd
frame was used to decrease data processing. A vectorial weighted sum using
14 body segment CoM locations and relative masses [74] was used to calculate
the CoM position for each trial run. Body segment markers were placed: one
for the head (head and neck), one for the torso (chest, abdomen and pelvis), and
one on each foot, calf, thigh, hand, forearm and upper arm. The resulting data
was interpreted using Matlab and a curve t produced using the cftool. Each
step's contact support point p
cop
k
was acquired from the video using Matlab's
ginput function and estimated initially as the surface midpoint of the contact foot
location. The duration of each step was acquired directly from the raw video while
terrain geometry information was provided as input to the system. The human
test subjects consisted of both male and female within healthy weight ranges. To
test across diverse architectures, the human samples varied in height from 1:55m to
1:78m and spanned a range of ages (24 to 35 yrs of age). This allowed for validation
of the geometric approach under dierent kinematic constraints (architectures)
and investigation of the eects of the terrain on the resulting CoM motion. The
complete subject attributes are provided in Table 4.1 while an example traversal
is shown in Fig 4.4.
40
Table 4.1:
Subject Attributes Data.
Subject Gender Height Weight
1 M 1.75m 68kg
2 M 1.75m 84kg
3 M 1.78m 82kg
4 M 1.72m 75kg
5 F 1.73m 56kg
6 F 1.55m 41kg
Figure 4.4: Sagittal motion capture of human traversal over terrain.
4.3.2 Cross-validation
The geometric approach was used to generate CoM position estimates for all sub-
jects on both terrains and then compared with CoM data obtained using motion
41
capture video. The automatically generated curves included an x-direction adjust-
ment of5cm to p
cop
k
. This attunement resulted in more accurate correlation
with the observed data and provides a reasonable adjustment bearing in mind the
physical limitations of the foot and ankle. Foot geometry and ankle use prevent
true human point contact behavior and perpetually shift the CoP. This disallows
measurement of p
cop
k
using only the video technique and justies small adjust-
ments to the initial estimate. The root mean squared error (rmse) between the
vertical CoM position (measured versus estimated) for all twelve runs is reported in
Table 4.2. Additionally, the CoM paths using motion capture (measured) and geo-
metric approach (estimated) for 2 subjects traversing terrains 1 and 2 are shown
in Fig 4.5 and Fig 4.6. The subjects averaged a rmse of 0.0333m on terrain 1,
Table 4.2:
RMSE Est. vs Measured CoM Vertical Position
Subject Terrain 1 Terrain 2
1 0.0229m 0.0401m
2 0.0273m 0.0306m
3 0.0579m 0.0444m
4 0.0244m 0.0405m
5 0.0364m 0.0273m
6 0.0307m 0.0252m
avg rmse 0.0333m 0.0347m
while for terrain 2, the average rmse was 0.0347m. The results indicate an average
performance across both terrains and for all subjects that is approximately similar
in value. This result is both remarkable and encouraging given the large disparity
of subject heights, weights, and step sequences (e.g., the 6 step walking sequence
shown in Fig 4.5 (left) and the 7 step walking sequence shown in Fig 4.6 (left)).
Moreover, the generated CoM estimates always displayed the same CoM pattern
42
Figure 4.5: (Left) a six step walking sequence over terrain 1. (Right) a ve step
walking sequence over terrain 2. The estimated versus measured CoM vertical
position rmse is 0.0244m and 0.0405m respectively.
Figure 4.6: (Left) a seven step walking sequence over terrain 1. (Right) an eight
step walking sequence over terrain 2. The estimated versus measured CoM vertical
position rmse is 0.0307m and 0.0252m respectively.
of behavior observed in the real world including in cases having a relatively poor
rmse, for example 0.0405m as shown in Fig 4.5 (right). In general, the results
exhibit signicant correlation between the measured and estimated CoM positions
and demonstrate the method's ability to encapsulate vital kinematic and terrain
information to produce accurate CoM estimates.
43
4.3.3 CoM State-Space Behavior
For further analysis of the correspondence between the measured and estimated
CoM paths, state-space trajectory comparisons were performed. The human traver-
sal sagittal and vertical phase trajectories were extracted directly from the human
traversal videos. State-space trajectories for the geometric approach were produced
using the numerical integration technique outlined in [58, 59, 60]. The numerical
integration technique uses perturbation theory to predict CoM phase curves in the
neighborhood of step contact locations using a desired kinematic CoM path. The
kinematic CoM path used here is the automatically generated path produced by
the geometric approach. This allows for conrmation that the CoM estimate gen-
erated by the geometric approach produces approximately similar CoM state-space
behavior as that obtained using video motion capture, and for validation that the
automatic step CoPs, p
cop
k
, match the real step locations of the human subject.
Sagittal and vertical phase curves using the measured and estimated CoM paths
from Fig 4.5 and Fig 4.6 are shown in Fig 4.7 through Fig 4.10. Note that for
clarity of presentation, the step CoPs were included along the x-axis. As can be
seen, the results display signicant correlation of phase curves between the human
and articial CoM phase portraits, supporting the CoM path estimates produced
by the geometric approach. Interestingly,p
cop
k
also matches the real step locations
as observed in Fig 4.7 wherep
cop
k
coincides with the heel contact point. The mea-
sured phase curve (red) then decelerates, creating a valley as the subject's weight
shifts towards the front of the foot. The same pattern occurs again when the next
step is taken. This behavior is shown for two steps in Fig 4.7 with blue arrows.
Similar behavior is observed in the sagittal phase curves for all subjects.
Lastly, it is important to highlight the alignment between the real and geometric
step CoP locations used. Specically, the geometric estimates that are produced
44
Figure 4.7: Subject 4 sagittal and vertical phase curves using CoM estimate for
terrain 1 from Fig 4.5 (left).
Figure 4.8: Subject 4 sagittal and vertical phase curves using CoM estimate for
terrain 2 from Fig 4.5 (right).
approximately match the observed real world results and it is not the case that
CoM estimates and phase curves were generated using vastly dierent values for
p
cop
k
. Rather, the geometric CoM estimates produced are derived using approx-
imate real world step CoP locations. Ergo, the geometric approach can be used
as a predictive model for CoM behavior estimation using terrain geometry, sub-
45
Figure 4.9: Subject 6 sagittal and vertical phase curves using CoM estimate for
terrain 1 from Fig 4.6 (left).
Figure 4.10: Subject 6 sagittal and vertical phase curves using CoM estimate for
terrain 2 from Fig 4.6 (right).
ject height information, and desired step CoP locations. This provides a reliable
and less cumbersome approach that greatly facilitates the study CoM dynamics
in complex planar terrains. These factors, along with the independence of force
or video sensors, contribute to make the geometric approach an innovative and
ecient alternative for estimation of CoM position in complex planar terrains. To
46
summarize: the geometric approach provides several important advantages includ-
ing: 1) independence from expensive and time prohibitive sensor equipment 2)
accurate CoM estimation and state-space trajectories that accounts for variations
in subject kinematic constraints and 3) the ability to handle complex planar envi-
ronments. These benets render the approach versatile and dependable for use
in non-laboratory settings and provide an eective tool to study natural walking
locomotion and CoM behavior in complex planar terrains.
47
Chapter 5
CoM Estimation: Load Carriage
Extension
The CoM estimation technique presented in chapter 4 provided a method for accu-
rate description of CoM behavior using a geometric model of subject kinematic
architecture and contact foot locations. By capitalizing on the interdependency
of step CoP locations and CoM vertical displacement, a geometric representation
of subject motion was developed that delineated CoM dynamics in the terrain.
The approach was then compared with data of male and female subjects within
healthy weight ranges, with the results demonstrating accurate estimation of CoM
behavior. A natural follow up to ask is if the approach can be extended to accom-
modate subjects outside these normal weight limits. This would include subjects
possessing weight distributions tilted towards specic extremes, for example due to
heavy load hauling. Indeed, the prevalence of everyday tasks requiring load bear-
ing or transfer makes CoM behavior analysis in such cases equally important for
locomotion planning. The work presented in this section explains how through a
revision of the geometric model for CoM estimation, load carriage behavior can be
incorporated into the framework. These sagittal mass osets place an additional
constraint on CoM displacement, however, the mathematical formulations can be
adapted by developing a relationship between the added weight and step CoPs.
Thusly, the geometric CoM estimation approach can be extended to model load
carriage behavior.
48
5.1 Related Work
Manual load carriage has been employed since the earliest of times to construct
dwellings, fortications or as in the ancient Maya, ceremonial pyramid struc-
tures [1]. Recently, interest in load carriage has grown due to its importance in
operating in complex environments. These inevitably involve performing actions
that alter the physical nature of the medium, interactions which usually take the
form of object manipulation and transfer. Frequently the objects must be carried
on the back and involve heavy loads that place tremendous strain on the body
and modify its behavior. Such behavior alterations are observed in military train-
ing where soldiers must carry heavy equipment loads leading to spatialtemporal
changes in posture and gait [4] [10]. These physiological eects, e.g., trunk incli-
nation and increased stride frequency [32], provide insight that can be used to
study the biomechanical mechanisms in
uenced during load carriage. Of these,
the CoM constitues the chief element since it describes the overall behavior of
the system with regards to motion performance and stability. Thus, methods have
been developed to understand load carriage in
uence on the CoM so as to mitigate
its eects.
The most common approach uses force sensors to measure CoM response during
load carriage. Devices such as those in [19, 66, 49] form an insole force sensory
system built into footwear to measure ground reaction forces and variations in CoP
under loads. These then form part of an exoskeleton-footwear framework able to
detect weight variations and trigger assistive forces to help during load handling,
such as the force and IMU based exoshoe [42] and the rehabilitation exoskeleton
Ekso [25]. Naturally, such systems are expensive to build and operate and are
encumbered by the large number of sensors needed as well as the lengthy subject
specic customization required.
49
To alleviate force sensor dependency, additional approaches have been devel-
oped. In [54], single external force plates were used to investigate CoP response
under loads, while CoM behavior for soldier combat loads was measured in [33].
However, the use of external force plates limits employment, causing neglection of
ambulatory CoM motion and conning the studies to standing postures.
Additional hybrid approaches have attempted to fuse various sensors to reduce
cost and extend portability. The authors of [7] combined insole force sensors,
accelerometer readings and video motion analysis to estimate CoM trajectories,
while mathematical models were developed in [8] to estimate CoM behavior dur-
ing walking based on accelerometer data. However, the increased portability of
the approaches was oset by the sheer number of sensors used, e.g., the 128 insole
force sensors deployed in [7]. Recently, body segmental type approaches [76, 22, 74]
using novel visual systems have returned to the forefront. A Wii Kinect and bal-
ance board was used in [30][29] for CoM estimation, while digitization of the human
body using a 3D scanner was performed in [11]. Clearly, to generate ambulatory
CoM estimates, these methods must perform continuous data gathering, a conse-
quence which restricts these techniques to controlled environments. Additionally,
although the number of sensors used has decreased, the sensors have remained
expensive and non-readily available. Indeed, the general reliance on cost pro-
hibitive and cumbersome sensors connes the methods discussed above to mostly
laboratory settings and limits their usage and versatility.
The research presented in the next sections continues and extends the geomet-
ric approach for CoM estimation described in [6] to include load carriage. Research
of load carriage eect on CoM behavior is vital since human and robot locomo-
tion is frequently performed under conditions requiring load bearing or transfer.
50
More recently, this has been accentuated in the design of exoskeletons and mil-
itary backpack systems aimed at assisting with weighty load handling. Despite
this importance, complicated body dynamics has stymied the study of CoM and
locomotion behaviors that arise under heavy load carrying. Current approaches
have focused mostly on external sensor use constraining the methods to expensive
and clumsy sensors. The extension provided here is a direct and accessible geo-
metric approach that can be used to produce CoM estimates of natural walking
locomotion with carried loads over rough planar terrains. The method shirks the
use of video and force sensors and instead builds on previous work that used an
Optimized Geometric Hermite (OGH) curve while relying only on essential body
kinematic knowledge, the terrain geometry, and load weight information. To val-
idate the accuracy of the approach, comparisons using motion capture video of
human subjects were performed. The results demonstrate an accurate estimate
of the CoM position path and behavior during loaded natural walking over rough
planar terrains.
5.2 Load Carriage Eect
To encompass load carriage eect on CoM behavior, extensions to the geometric
approach had to be performed that accurately described CoM behavior during
loaded natural walking but that maintained the method's independence on force
or video sensors. Thus, additional geometric relationships were needed between
the added weight, the terrain, and the subject's kinematic constraints that char-
acterized the resultant behavior.
Recent work in [69] has provided further insight into the mechanics and energet-
ics of loaded walking. The experimental data presented showed that 1) mechanical
51
Figure 5.1: The vertical CoM displacement during a loaded step (middle) and its
equivalent geometric representation with longer stride (right).
work performed on the body's CoM and 2) the metabolic energy expended, both
increased approximately linearly with added load mass. In light of these results,
the work presented in this paper also hypothesized a linear relationship between the
added load and the triangular formulation of the geometric approach. Specically,
the observation drawn here is that the execution of steps during load carriage aects
the intermediate dip between steps that is discerned during locomotion. This dip
or vertical displacement
comz
in the CoM was previously described using virtual
steps and was a function of subject leg length and the distancea between step CoP
locations. Therefore, it is possible to model the CoM vertical displacement during
load carriage with an equivalent CoM displacement produced using an updated
value for a. This model is shown graphically in Fig 5.1 for a loaded upward step
with its equivalent representation under the geometric approach using an increased
value for a, the distance between sequential step CoPs. Note that treatment of
load carriage eect on CoM displacement in this manner maintains the fundamen-
tal character of the geometric approach and avoids the introduction of force or
video sensors.
52
5.2.1 Load Virtual Step Model
As previously discussed in 4.2.1, virtual steps are immaterial points in the terrain
describing CoM behavior during double support phases and are composed of both
sagittal and vertical components (x
vs
;z
vs
). The vertical valuez
vs
is derived directly
from the CoM vertical displacement and depends on subject kinematics and the
distance between step CoP locationsa. However, to incorporate load carriage eect
into this model, it is necessary to modify z
vs
. Because z
vs
depends on the CoM
vertical displacement
comz
, this is accomplished by updating the value of a such
that the resultant dip in vertical CoM location during load carriage is accurately
modeled as was shown in Fig 5.1.
Returning to Fig 4.1, in [6] an isosceles geometric representation was used to
nd the vertical component z
i
vs
of the ith virtual step between adjacent CoPs.
This was done by calculating the vertical CoM displacement
ij
comz
which is the
distance between the location of the CoM during the upright standing posture and
its location during the performance of a step. The CoM height value h
ij
during
step locomotion is derived from the triangular formulation and is a function of a,
the distance between sequential step CoP locations (x
cop
i
;z
cop
i
) and (x
cop
j
;z
cop
j
)
and the subject's leg length b. Thus, the vertical displacement is calculated as:

ij
comz
= (b +d) (h
ij
+d) (5.1)
Using the Pythagorean theorem and expanding terms this reduces to

ij
comz
=b
r
b
2

a
2
4
(5.2)
53
The nal value for z
i
vs
of the ith virtual step is chosen as
z
i
vs
=min(z
cop
i
;z
cop
j
)
ij
comz
(5.3)
Note that a = (x
cop
j
x
cop
i
), the distance between sagittal CoP locations and
that
ij
comz
depends ona, equation 5.2. Perforce, by appropriately choosing a, the
equivalent load carriage eect on CoM behavior (Fig 5.1) can be produced. This
is achieved by developing a relationship between the added weight and the new
value for a.
Treatment of load carriage eect on locomotion behavior was performed by
considering three walking patterns: 1) planar 2) downward and 3) upward. Upward
walking was observed to be most adversely aected by load carriage due greater
energy expenditure caused by lifting weights onto higher surfaces. For planar
walking, the procedure outlined in [6] was used to nd the vertical component z
i
vs
of virtual steps while for downward walking, the observed eects depended mostly
on height dierences in terrain during descent. Therefore, for downward loaded
walking, equation 5.2 was used but weighted by a factor describing the eect of
height descent on the kinematic body, namely:

ij
comz
=
ij
comz

1 +
z
b

z 0:7b (5.4)
where z =j(z
cop
j
z
cop
i
)j is the height dierence between step CoPs and b is
the subject leg length. After calculating the new vertical displacement, the virtual
step z
i
vs
value is found as normal using equation 5.3.
For upward steps,
ij
comz
was found by considering the added weight, the sub-
ject's kinematic properties and the height dierence between adjacent step CoPs.
These factors were observed to have considerable aect on selection of a
0
, the
54
equivalent step CoP distance to model load carriage behavior. Thus, given a load
massm, a kinematic body with massM, subject leg lengthb and step CoP height
dierence z, the equivalent stride distance a
0
is dened as:
a
0
=a +

1 +
z
b

m
M

z 0:7b; m 0:3M (5.5)
Consequently, when a loaded upward step is performed, it is treated using its
analogue geometric representation with longer stridea
0
, and it is this value that is
used in place ofa in equation 5.2 to nd the vertical displacement. The nalz
i
vs
is
calculated as always using equation 5.3. In this manner, z
i
vs
is determined for all
virtual steps ensuring an accurate model of load carriage eect on natural walking
CoM behavior.
As a nal note, it is important to observe that the process for nding the
sagittal value x
i
vs
of virtual steps and the step (real and virtual) forward progress
angles remains unchanged from that depicted in [6] and described in section 4.2.2.
Additionally, the method for constructing the nal CoM estimate is identical and
involves applying Algorithm 3 from section 4.2.3. Only the CoM double support
phase behavior has been altered via the new virtual step calculation. However,
the change in virtual step vertical displacement has a ripple eect on the forward
progress angles calculated for real steps and the modied loaded virtual steps. This
thereupon aects OGH curve generation but has the important consequence of
establishing delity of the CoM estimate produced by Algorithm 3 to the observed
loaded walking behavior of the kinematic body through the terrain.
55
5.3 Experimental Results
5.3.1 Experimental Setup and Data Collected
Four subjects were asked to traverse two dierent terrains consisting of planar
obstacles of varying heights while supporting loads of 7:26kg and 14:5kg on their
back. The loads were carried by an adjustable weighted vest (CAP Barbell) with
standard shoulder straps and hip belt and secured tightly to minimize movement.
The vest (Fig 5.2) allows weights to be carried rmly on the back in increments of
1:815kg through the addition of weight pods. Kinematic data on subject traversals
Figure 5.2: The CAP Barbell vest with 1.815kg weight pods.
was obtained using Matlab and video motion analysis with the camera positioned
suitably far to ensure pure planar recordings with minimal lens distortion. Video
data was recorded at 30 frames per seconds but to facilitate data processing, only
every 3
rd
frame was used. The complete load carriage CoM for subject traversals
was calculated using a vectorial weighted sum of the CoM locations and relative
masses of 14 body segments [22][74] and 1 vest component. The 14 body segment
markers used were: 1 for the head (head and neck), 1 for the torso (chest, abdomen,
and pelvis), and 1 on each hand, forearm, upper arm, foot, shank, and thigh. The
vest marker was placed at the approximate CoM location of the vest with the
56
Figure 5.3: Sagittal snapshots of human traversal through terrain.
added weight. The resultant data was processed with Matlab and the cftool used
to compute an accurate t to the data. Lastly, each step's supporting contact point
p
cop
k
was procured with Matlab using ginput and originally taken as the midpoint
of the foot's contact surface. An example human traversal is shown in Fig 5.3 with
subject attributes recorded in Table 5.1. Note that females and males were tested
and that subject weights and heights spanned a range of values thereby providing
a diverse sample size.
Table 5.1: Load Carriage
Subject Characteristics Data.
Subject Gender Height Weight
1 F 1.68m 59kg
2 M 1.72m 65kg
3 M 1.75m 80kg
4 M 1.88m 93kg
57
5.3.2 Cross-validation
CoM estimates were produced using the extended geometric approach for all sub-
jects, terrains and load quantities for a total of 16 trials. The generated estimates
included a5cm sagittal adjustment to p
cop
k
. This adjustment was observed to
yield stronger correlations with the measured results and is a valid rectication
considering
exible foot tissues and joints that perpetually shift the foot CoP
and prevent direct treatment of p
cop
k
using the video motion capture approach.
Geometric estimates were generated for all trials and compared with their corre-
sponding video motion capture data (measured). Table 5.2 provides the root mean
squared error (rmse) of vertical CoM position between the estimated and measured
paths for all subjects, load quantities and terrain traversals while Fig 5.4 through
Fig 5.5 show the CoM position results (measured and estimated) for subject 1
trials and Fig 5.6 and Fig 5.7 show the results for subject 3 trials. The results
demonstrate accurate modeling of load carriage eect on natural walking CoM
behavior for all subjects over both terrains, with very similar average rsme results.
Indeed, the overall rmse for terrains 1 and 2 was 0.0409m and 0.0404m respectively,
showing comparable behavior in the generated CoM path esitmates. The results
are notable given the variations in carried loads, subject heights and weights, and
Table 5.2: Load Carriage
RMSE Est. vs Measured CoM Vertical Position
Subject Terrain 1 Terrain 2
7.26kg 14.5kg 7.26kg 14.5kg
1 0.0380m 0.0419m 0.0274m 0.0272m
2 0.0542m 0.0415m 0.0463m 0.0383m
3 0.0362m 0.0425m 0.0415m 0.0354m
4 0.0402m 0.0330m 0.0447m 0.0626m
avg rmse 0.0421m 0.0397m 0.0400m 0.0409m
58
Figure 5.4: Subject 1 performing 6 step loaded walking sequences through terrain
1 while carrying 7.26kg (left )and 14.5kg (right). The rmse of estimated versus
measured CoM for each is 0.0380m and 0.0419m respectively.
Figure 5.5: Subject 1 performing 7 step loaded walking sequences through terrain
2 while carrying 7.26kg (left )and 14.5kg (right). The rmse of estimated versus
measured CoM for each is 0.0274m and 0.0272m respectively.
step sequences performed (e.g. the 6 step walking sequence with 7:26kg carried
load in Fig 5.4 and the 7 step walking sequence with 14:5kg carried load in Fig 5.5).
Despite these variations, the extended geometric approach properly modeled load
carriage, terrain geometry and kinematic information to produce accurate natural
walking load carriage CoM estimates. The results conrm the
exibility of the
geometric model and how revisions of the triangular geometric representation can
59
Figure 5.6: Subject 3 performing 5 step loaded walking sequences through terrain
1 while carrying 7.26kg (left )and 14.5kg (right). The rmse of estimated versus
measured CoM for each is 0.0362m and 0.0425m respectively.
Figure 5.7: Subject 3 performing 5 step loaded walking sequences through terrain
2 while carrying 7.26kg (left )and 14.5kg (right). The rmse of estimated versus
measured CoM for each is 0.0415m and 0.0354m respectively.
be used to include additional CoM estimate and behavioral features. Because the
kinematic locomotion is charted in a step-by-step manner, sagittal mass osets
and potentially other types of behavior may be added to the framework, leaving
open multiple areas for exploration. These will ultimately increase the power and
scope of the geometric approach and streamline CoM estimation without relying
on burdensome or singular sensors.
60
Chapter 6
Foot Placement Selection: A
Phase Space Planning Approach
After having developed a method for accurate CoM estimation, the purpose of
this study was to investigate the factors involved in foot placement selection dur-
ing locomotion over complex planar terrains. This was accomplished by leveraging
the previous results into a unied framework that allows for analysis of contact
foot locations. This is consequently used to develop strategies that can gener-
ate natural foot placement plans in the environment. Building o the ability to
produce accurate CoM estimates as described in chapter 4, the geometric CoM
estimation approach is used as a predictive tool to fashion potential CoM paths
based on associated candidate foot locations. This sweep of the environment and
the generation of potential CoM paths provides a way to examine the eects of
the terrain and the choice of foot placements on the motion of the CoM. Addi-
tionally, by using the prismatic inverted pendulum model to derive sagittal and
vertical CoM phase portraits, a velocity assessment of the CoM motion can also be
performed to explore its role in foot placement selection. The combined analysis
reveals that foot location is selected in a manner that minimizes the deviations and
irregularities from the natural walking vertical phase portrait. Natural walking is
characterized by smooth and regular phase portrait cycles and foot selection in
complex terrains attempts to restore this modality. To evaluate the accuracy of
the framework, analysis of human subjects and simulations were performed. The
61
results demonstrate that during traversal of complex planar terrains, foot location
is chosen to minimize the divergence from the preferred smooth walking phase por-
trait. This framework provides a way to categorize the dierent strategies involved
in foot selection in the presence of obstacles and to synthesize human locomotion
in complex and rough planar terrains.
6.1 Related Work
The emergence of bipedality and its ability to locomote over complex and rough ter-
rains has been paramount in the development of biological systems. This physical
and behavioral adaptation has endowed select terrestrial organisms with the abil-
ity to simultaneously perform both locomotion and manipulation, a key advantage
for survivability under environmental changes and pressures. Among all mam-
mals, the biomechanics of human bipedalism are unrivaled for their dynamism and
behavioral characteristics [70]. These stem largely from the habitual orthograde
nature of human posture [67], a quality which has resulted in benecial biome-
chanical properties. These include, for example, the economic energy consumption
of human walking that permits greater distance coverage [28] and the energy e-
ciencies of the musculoskeletal and nervous system that limit muscle and neuron
commitment during locomotion [68]. The combined mechanical and neural ener-
getic proles are a direct consequence of the motor control strategies used during
leg motion. The leg operates as an anatomical inverted pendulum [70][39][63] lead-
ing to harmonic cycles of human locomotion which among other things, controls
the interchange of kinetic and potential energy in the mechanical structure [43].
62
Taken as a whole, the bipedal adaptation has proved vital in adjusting to vari-
ous temporal environments while providing greater dexterity of manipulation and
latitude from the restrictions of quadrupedal motion.
A clear description of the bipedal anatomical structure and its adaptive bene-
ts, however, is dierent from understanding the resultant motion patterns observed
in complex and irregular terrains. These quite often involve sloped and friction
varying surfaces, (e.g. a mountain), or obstacle lled and cluttered environments,
(e.g. a busy pedestrian sidewalk). Under these conditions, selection of appropriate
and safe footholds is primal to maintaining balance, avoiding obstacle collisions and
ensuring ecient travel through the terrain. Thus, the choice of foot contact loca-
tion can be understood in the context of these external environmental constraints,
factors which limit the space available foot locations. Naturally, to successfully
navigate the terrain, sensory input of the environment is required to ascertain its
status and character. Foremost among sensors is the visual system which provides
valuable terrain information of both immediate and approaching topographic con-
ditions. This information can be used for obstacle avoidance and clearance, to plan
anticipatory motion maneuvers, and to determine leg and foot motions leading to
secure footholds. Because of this importance, substantial research into the visual
system has been performed to determine its role in adaptive human locomotion.
Various human studies in the presence of obstacles have investigated the optical
in
uence on human movement planning. Initial investigations occluded to varying
degrees the visual input to understand the resulting behavioral characteristics. In
[23] the authors demonstrated that successful circumvention of obstacles was possi-
ble if the interval between visual impairment and the target was limited to 8s, and
in [51] moderate toe clearance over obstacles was observed through manipulation
of exproprioceptive input. These studies revealed that continuous visual guidance
63
was not necessary for moderately successful navigation and that only critical tem-
poral and spatial information was needed to perform complex interactions with
the environment.
With regards to the visual control of stride length and foot placements, sev-
eral landmark studies have been conducted. Long jumping was analyzed in [40]
revealing a two phase approach to jumping. This consists of an initial stereotyped
stride pattern followed by a nal approach sequence that corrects the cumulative
errors accrued in the rst stage. This allows long jumpers to run at consistent high
speeds before visual regulation of stride kicks in to ensure they hit the board with
minimal error. Additional investigations in [73] proposed a time-based model for
step length adjustment in the presence of targets, and in [50] strategies were devel-
oped for selection of alternate foot placements based on minimization of ongoing
locomotor muscle activity and changes to current movement.
Recent research has further rened the eects of visual information on walk-
ing performance. The importance of gaze xations and the need for online visual
regulation of foot placement was examined in [52] with the authors of [77] recon-
rming the coupling of current visual information to human movement control and
locomotion, e.g. steering in the presence of moving and stationary obstacles [72].
Lastly, visibility windows during natural walking were explored in [43] establishing
a visual information requirement of two ensuing step lengths. This allows for suc-
cessful foot placement guidance over complex terrains, maximizes energy eciency
and avoids terrain impediments.
The work presented in the following sections aims to advance the study of
foot placement selection in the context of complex and rough planar terrains, an
area heretofore little investigated. Through an analysis of the coupling of CoM
behavior and foot contact point location, a framework is developed that provides a
64
quantitative measure of suitable foot placement regions in the environment. This
is achieved through a multivariate approach that considers CoM dynamics, contact
foot location, and step terrain lookahead. In this manner, strategies for optimal
selection of subsequent steps are developed that provide guidance for percipient
foot placement and locomotion over irregular planar terrains.
6.1.1 Objective of Study
The majority of locomotion research in the area of foot placement planning has
so far centered on gait control in
at and planar terrains. These studies usually
ask participants to walk across a eld comprised of virtual obstacles, commonly
projected lighted areas [43]. The primary focus of this work has been in under-
standing the quantity of temporal and spatial visual information necessary to plan
anticipatory movements to circumvent and avoid obstacles. The study presented
here investigates gait control in more complex environments consisting of irregu-
lar planar surfaces. Research of step selection in these types of environments has
been little explored, particularly as it relates to foot center of pressure (CoP) and
the biomechanical factors involved in its appointment. The visual information is
here only indirectly treated. Instead, a comprehensive analysis of the exact foot
CoP chosen is conducted from a CoM path and phase space perspective as well as
through consideration of potential CoM two step obstacle clearance. This leads to
a general framework by which the cost of potential future steps can be quantied
and thereby minimized. Thus a planning methodology is formed for suitable foot
placement selection through the terrain.
65
6.1.2 Foot Placement Analysis: Overview of Approach
Determination of suitable foot placement locations in the terrain is accomplished by
analyzing potential candidate foot CoP locations. These candidate CoP positions
are examined as follows: 1) for each CoP, its corresponding CoM path estimate
is formed. To accomplish this, the geometric CoM estimation technique described
in section 4 is used to construct the CoM path. 2) The sagittal and vertical CoM
phase portraits are generated using the desired sagittal walking speed. The phase
portraits are transformed to include the mutli-contact (double support) phase. 3)
For each vertical phase portrait, its transition strain energy cost is calculated. This
produces a cost landscape that includes all candidate step CoPs and their corre-
sponding cost values. 4) The local minima of the cost landscape curve are found.
For each minima, the associated step CoP vertical phase portrait is compared with
the preferred smooth vertical phase portrait. The foot CoP resulting in the least
deviation from the smooth walking gait portrait is identied as the CoP solution.
5) Lastly, in the presence of obstacles, the CoM path using two step lookahead
(the foot CoP solution and subsequent CoP candiates) is explored. This require-
ment is necessary since optimal foot locations might be unreachable due to terrain
impediments. In these cases, the step CoP providing maximum obstacle clearance
is selected as the foot location.
6.2 Terrain Traversal Case Studies
6.2.1 Experimental Setup and Data Collected
The motion capture video data collected in section 4.3.1 (see Table 4.1) was used
for analysis. The data consists of terrain traversal runs for 6 subjects of dierent
66
heights and weights. All participants were asked to traverse two separate complex
terrains consisting of irregularly spaced and sized planar surface obstacles. Sub-
jects were asked to perform natural walking locomotion and were not given any
rules or policies regarding foot placement locations. This was purposefully done to
avoid introducing bias in behavior. It was important that the movement patterns
and foot locations selected be as natural as possible and free of contrived or arti-
cial constraints. Using the motion capture and body segmental approach, the CoM
paths for all subject runs was calculated. The human sagittal and vertical phase
portraits were acquired similarly from the video data. Additionally, the foot CoP
was once again estimated as the midpoint of the foot contact surface and obtained
directly from the video. The measured data thus consisted of subject CoM paths,
sagittal and vertical phase portraits, and foot CoP locations in the terrain.
6.2.2 General Protocol
The measured CoM paths and phase portraits were analyzed incrementally in a
step by step manner. For each step, subsequent CoP candidates were generated
in 0:05m intervals starting from half the stride length to the full stride distance.
These values were approximated using only subject heights and anthropomorphic
descriptions of step length versus height ratio [74]. For each CoP location, CoM
path estimates were produced using the geometric approach. Furthermore, the
sagittal apex velocity (velocity of the CoM at the apex of the foot) obtained from
the video was used to reconstruct CoM dynamics. This was accomplished using
the numerical integration technique described in section 3.4.1. This provides a
mechanism to derive phase space approximations of the CoM to compare with the
measured data. Fig 1 shows subject 4's known third step in the terrain, however,
to analyze step placement, the CoM path estimate and state space behavior for
67
Figure 6.1: The measured CoM path versus the estimated CoM path for three
steps (two known and a candidate CoP location) is shown in (a). The sagittal
phase portrait with the inclusion of the multi-contact phase is shown in (b) for
the same candidate CoP location using the estimated CoM path from (a). The
vertical phase portrait (multi-contact phase included) and vertical acceleration are
depicted in (c) and (d) respectively.
the rst two known steps and a third candidate CoP location are developed. The
behavior of the CoM during the planar two step natural walking stage serves as
the baseline for comparison of subsequent CoP locations.
The state space behavior approximations of the CoM give rise to discontinuities
in the sagittal and vertical phases that must be resolved to produce more natural
behavior. This is evident in (Fig 6.1 (b)) where the intersection of sagittal phase
curves for adjoining CoP locations produces sharp peaks in CoM behavior. Such
68
characteristics are undesirable since they denote an instantaneous velocity transi-
tion, something impossible to achieve in the natural world, and because they lead
to feet transitions at higher velocities which are likewise impractical. To moderate
these eects and achieve more natural behavior, a multi-contact phase was added
for step transitions. A process similar to [78] was used wherein a segment of the
phase curves was removed and replaced with a 5th degree polynomial dened by:
x(t) =
5
X
i=0
a
i
t
i
(6.1)
where the boundary and timing conditions for initial position, velocity and acceler-
ation are used to compute the polynomial coecients. In this study a multi-contact
phase that occurs during 30% of the time of a given step was used. This was found
to produce more natural and smooth CoM phase portraits and allows for more
accurate comparisons with the observed CoM behavior. Fig 6.1 (b) displays the
single contact dynamics with the addition of the multi-contact phase for the three
step walk shown in Fig 6.1 (a). To summarize: the measured data consists of the
motion capture video CoM path and sagittal and vertical phase portraits while
the computed data consists of the estimated CoM path (geometric approach) and
the calculated CoM state space behavior (single contact point dynamics including
multi-contact phase). In this manner, foot placement and CoP locations can be
studied with accuracy to determine their eects on CoM behavior and the mech-
anisms involved in their selection in the terrain.
6.2.3 Data Analyses
The complete snapshot series for the rst set of candidate CoP locations and their
corresponding geometric CoM paths for subject 4 traversing terrain 1 is shown in
69
Figure 6.2: Subject 4, terrain 1: Candidate CoP locations (yellow) and their
geometric CoM paths (black). The measured CoM traversal is shown in (red).
70
Figure 6.3: CoM vertical phase portraits for candidate CoP locations of Fig6.2(a)-
(j). In (g), the least deviation from the natural walking vertical phase portrait is
reached. Note the correlation of estimated vs. measured CoM paths in Fig6.2(g).
71
Fig 6.2 (a)-(j). The computed geometric CoM path is depicted in black, while the
measured CoM path is displayed in red. The last step (yellow) is at the edge of the
frontier and represents the candidate CoP location. As mentioned previously these
are explored in 0:05m intervals starting from half the normal subject stride distance
to the full stride length. Note also that the measured CoM path remains constant
but that for better comparison with the estimated CoM path, it is only shown up
to the candidate CoP location. Lastly, the vertical phase portraits corresponding
to each candidate CoP location are shown in Fig 6.3 (a)-(j) respectively.
Applying the technique for optimization of foot placements discussed in section
3.7, the complete strain energy cost landscape of CoM vertical phase behavior for
all CoP candidates is presented in Fig 6.4. The landscape suggests two areas for
investigation, one centered around x 2:2m and another at x 1:95m. Upon
Figure 6.4: Energy cost landscape of step transitions of candidate CoP locations.
examination of the vertical phase portraits for all candidate CoP locations, it is
observed that candidate CoP vertical phase valleys (marked by the red arrow in
Fig 6.3 (a)) begin a gradual descent until there is an approximate equivalence
72
reached when 2:05mx2:1m (see Fig 6.3 (g)). At that point, the CoM vertical
phase behavior reaches a similar value at both vales, during the initial walking cycle
and for the candidate step CoP on the terrain. If indeed this region for step CoP
location was selected by the human subject, there should also be a correspondence
of estimated and measured CoM paths. An inspection of this range of step CoP
locations and their generated CoM paths from Fig 6.2 (g) and (h) conrms the
correlation of CoM paths and corroborates this foot placement CoP area as the
selected location.
The same process can be applied to the next foot to analyze its behavior.
Continuing from the previously selected foot CoP location, the ensuing set of can-
didate CoP locations is generated. The new strain energy cost landscape for these
candidates is given in Fig 6.5. Note the red colored cost values depicted in the
cost graph. An additional benet of using a phase space planning approach to
study foot placement selection is the ability to produce vertical direction accelera-
tion proles. This provides a way to examine the feasibility of step CoP locations
Figure 6.5: Energy cost landscape of step transitions of next CoP candidates.
73
Figure 6.6: Infeasible vertical CoM acceleration prole of a candidate CoP. The
area under the dashed line represents nonviable acceleration values.
given the apex velocity and CoM path boundary conditions. Since the subject's
foot is never fastened or axed to the ground, it is impossible to ever achieve
vertical acceleration values for the CoM that are more negative than gravity. An
example of this for a candidate CoP is shown Fig 6.6 where the area under the
dashed line represents infeasible CoM z-acceleration values. Candidate CoPs with
CoM acceleration proles that fall below this line can be ruled out since they are
impossible to reach. Thus, greater renement of candidate CoP locations can be
performed to further narrow the space of CoP candidates to those leading only
to viable force proles. With this in mind and considering only feasible foot CoP
candidates, the cost landscape of Fig 6.5 points to a single suitable area centered
around x 2:75m. Turning once more to the vertical phase portraits for all can-
didate CoP locations (Fig 6.7), this time there is a gradual ascent in the transition
vertical phase valleys of the CoP candidates (red arrow in Fig 6.7 (a)), until a
74
Figure 6.7: CoM vertical phase portraits for next set of candidate CoP locations
using CoM paths from Fig 6.8 (a)-(j). In (g)-(i), the least deviation from the
natural walking vertical phase portrait is reached.
75
Figure 6.8: Subject 4, terrain 1. Next set of candidate CoP locations (yellow)
and their geometric CoM paths (black). The measured CoM traversal is shown in
(red).
76
plateau is reached in (g)-(i). If this region is the suitable foot CoP location, there
should again be a correspondence in the estimated and measured CoM paths. An
observation of Fig 6.8 (g)-(i) reveals the agreement of CoM paths and validates
this region as the selected foot placement location.
A similar procedure can used to determine the remaining suitable foot CoP
locations. The nal two selected step CoPs, their CoM vertical phase portraits
and generated CoM paths, as well as the complete strain energy cost landscape,
are shown in Fig 6.9 and Fig 6.10. As can be seen from Fig 6.10 (right) the nal
computed terrain traversal using the above strategy and analysis results in very
similar CoM path behavior as the observed human run. Furthermore, the loca-
tion of the step CoPs as computed using this framework and those observed in the
Figure 6.9: Left: Energy cost landscape of 5th step candidate CoPs. Middle: CoM
vertical phase portrait of selected CoP. Right: CoM path estimate of selected CoP.
Figure 6.10: Left: Energy cost landscape of 6th step candidate CoPs. Middle: CoM
vertical phase portrait of selected CoP. Right: CoM path estimate of selected CoP.
77
measured data are also approximately equal. This step CoP analysis was discussed
in section 4.3.3 and was accomplished by studying the CoM state-space behavior
of the estimated CoM path and the CoM measured through motion capture. The
resulting curves were found to have approximately coincident CoP locations, vali-
dating the CoM estimation approach and the choice of step CoP. For comparison,
the human traversal vertical phase portrait is provided in Fig 6.11. Note the strong
correlation with the nal CoM vertical phase portrait (Fig 6.10 (middle)) using the
framework presented here for a phase space planning approach to foot placement
selection.
Figure 6.11: CoM vertical phase portrait for subject 4 through terrain 1 as mea-
sured using motion capture video data.
The preceding analysis suggests that for a specic desired sagittal apex velocity
(sagittal velocity of the CoM at the apex of the CoP) for natural walking loco-
motion through the terrain, there is a preferred region for placement of the foot
CoP that is based on minimization of the dierences in CoM vertical phase behav-
ior from the natural smooth walking cycle. It is important to note however, that
sometimes these regions may be unattainable due to terrain obstacles or imped-
iments. In these cases, an alternate location for step CoP must be selected. An
78
example occurrence of this scenario is presented in Fig 6.12 for subject 6 on terrain
2. After performing the above analysis on the previous steps, the suitable location
for the 6th CoP in the terrain is found to be x 2:99m. However, a glance of the
resulting CoP reveals that it would be impractical and kinematically impossible to
reach that location. Under these conditions, the suitable location of the foot CoP
Figure 6.12: The measured (red) and geometric (black) CoM paths. Note the
unreachable location of the 6th CoP candidate solution.
Figure 6.13: Two step candidate CoP (yellow) lookahead and its geometric CoM
path (blue). Note the increased clearance (C) resulting from anticipating the
obstacle.
79
was found by retreating from the optimal location in 0:05m intervals until a cor-
responding CoM path was found that oered the greatest clearance of the terrain
obstacle. This was accomplished by performing two step CoP lookahead at each
iteration and generating the resultant CoM estimate. Two examples of this retreat
are shown in Fig 6.13 where the greater the anticipatory step CoP location prior to
the terrain obstacle, the greater the maximum clearance of the CoM from terrain.
The results indicate that in the presence of extreme and unavoidable constraints
exerted by the terrain, that compromises are made in the suitable region of step
CoPs in order to overcome the terrain obstacles. This analysis agrees with pre-
vious results, for example [43] where two step visual information was needed to
successfully navigate the terrain. As a nal note, it is worth mentioning again that
the analysis presented here is only possible due to the ability to generate accurate
CoM estimates for given step CoPs using the geometric approach. This allows for
straightforward generation of CoM paths and provides a method to simultaneously
study step CoP locations and CoM dynamics in complex planar terrains.
6.3 Simulation Results
Simulations of male and female subjects (Table 4.1) were performed in Matlab and
compared with the human traversal data for validation. Step CoP locations and
CoM position estimates were found using the aforementioned process and analyzed
on the same terrains used during testing of human subjects. The measured data
was used to obtain the desired sagittal apex velocity of each step CoP. These apex
velocities were used for creation of candidate CoP phase space portraits. Note that
this human data is not required by the planner, however they were used here for
80
direct comparison of the foot CoP plans produced through this framework and the
actual foot placement plans performed in the real world.
Feet trajectories were calculated using the forward progress angles dened in
section 4.2.2. An additional benet of the angular description of CoM motion
described in this work is its basis on foot placements in the terrain. Because the
foot CoP locations determine the tangential components that direct CoM behav-
ior, these same tangents can be used in a step-to-step manner to construct foot
trajectories. This was accomplished by separating the step CoP locations found
using the phase space planning approach into left and right leg partitions. Next,
OGH curves with minimum SE were used to produce the foot paths. This involved
using the forward progress angles of two steps: one step CoP and one virtual step
(the posterior real step CoP and the preceding virtual step of the anterior real
step CoP). Note that this was done for each leg partition separately and that the
curves were generated ensuring a clearance of at least 0:1m with terrain obstacles.
Additionally, the desired step durations used to generate the CoM path estimates
were matched with each step CoP and interpolated using cubic splines to obtain
smooth leg motions for each foot's trajectory. Foot trajectories for the step CoP
locations shown in Fig 6.10 (right) are provided in Fig 6.14. Note that to develop
Figure 6.14: Simulated subject 4, terrain 1 foot paths and trajectories.
81
smooth feet and leg motions entering and departing the terrain, additional feet
trajectories were extrapolated using the normal walking stride length. An exam-
ple is shown in Fig 6.14 (left) where the right foot path (blue) was extrapolated
one more step in order to dismount the last terrain obstacle.
To visualize the foot CoP and CoM traversal plans, animations were used to
display the 2D dynamic walking results. The CoM position estimate was rst
converted into a time trajectory using the sagittal phase portrait. Next, for each
subject, thigh and shank leg lengths were approximated using body anthropomor-
phic data. Lastly, an inverse kinematics process was used to obtain joint angles
that t the legged architecture to the CoM trajectory and satised feet trajectories
Figure 6.15: Top: snapshots of simulated traversal of subject 4 through terrain
1. Bottom: sagittal view of the traversal run is shown with inclusion of the left
and right foot trajectories.
82
Figure 6.16: Top: snapshots of simulated traversal of subject 4 through terrain
2. Bottom: Sagittal view of foot paths and feet trajectories.
Figure 6.17: Left: Motion capture CoM vertical phase. Middle: computed CoM
vertical phase portrait. Right: CoM path estimate (black) using selected CoPs.
83
and contact conditions. The simulation traversal run for subject 4 on terrain
1 using the geometric CoM estimate and the selected foot CoP locations from
Fig 6.10 (right) and the foot trajectories shown in Fig 6.14 is displayed in Fig 6.15.
The simulation results for subject 4 locomoting through terrain 2 are shown in
Fig 6.16. The CoM position estimate and the nal CoM vertical phase portrait
for this run are also given in Fig 6.17. A comparison of the results demonstrates
the strong correlation between the step CoPs and CoM paths undertaken in the
real world and those computed using the geometric CoM estimation technique and
the foot selection approach. For example, comparing Fig 6.17 (left) and (middle)
it is observed that the measured and estimated CoM vertical phase portraits are
approximately equivalent, validating the locations of step CoPs. This is also the
case for the CoM path through the terrain in Fig 6.17 (right) where the CoM
estimate (black) and the measured CoM path (red) display the same pattern of
behavior. Together with the calculated foot trajectories, the combined terrain
traversals achieve extremely natural locomotion very similar to their human coun-
terparts. The kinematic motion of the feet and legs, as well the CoM motion
through the terrain, coordinate in a
uid and prevailing manner. This allows
for successful navigation of the terrain that is realistic and matches the observed
human traversal runs and is not encumbered by awkward motions.
An additional auspicious element of the framework presented here is the latitude
to examine dierent architectures. The simulated subject analyzed above was a
male measuring 1:72m. Due to the taller stature, the body had greater facility in
overcoming terrain obstacles. To further validate the approach, derived results for
a simulated female were compared with their real world counterparts. In Fig 6.18
and Fig 6.19 simulated results for a female of height 1:55m (comparable to subject
84
Figure 6.18: Top: snapshots of simulated traversal of subject 6 through terrain
1. Bottom: Sagittal view of foot paths and feet trajectories.
Figure 6.19: Left: Motion capture CoM vertical phase. Middle: computed CoM
vertical phase portrait. Right: CoM path estimate (black) using selected CoPs.
85
Figure 6.20: Top: snapshots of simulated traversal of subject 6 through terrain
2. Bottom: Sagittal view of foot paths and feet trajectories.
Figure 6.21: Left: Motion capture CoM vertical phase. Middle: computed CoM
vertical phase portrait. Right: CoM path estimate (black) using selected CoPs.
86
6) are shown across terrain 1 while in Fig 6.20 and Fig 6.21 the same simulated
subject is compared across terrain 2. Note once again the similarity of results in the
measured vs. estimated CoM vertical phase portraits, e.g., Fig 6.19 and Fig 6.21
(middle) and (left), and the natural feet and CoM trajectories and motions through
the terrain. Unlike subject 4, the simulated subject 6 required greater eort to
overcome certain terrain impediments. This was due to having shorter legs which
reduced the walking stride length and oered far limited potential placements for
ensuing steps. Ergo a greater number of steps were undertaken in the terrain.
However, because these constraints (stride length based on subject height) are
considered by this framework, generation and selection of foot CoPs is adapted
to each legged architecture to develop suitable and natural foot placement plans.
A prime example of this is displayed in Fig 6.20 where the step CoP locations
(see Fig 6.21 (right)) indicated a foot placement in the gap between obstacles at
x 2:5m. These step CoP locations were found using the phase space planning
approach and reconciled the simulated subject constraints. This led to the simu-
lated subject performing a step between obstacles while maintaining near natural
walking motion that allowed it to make progress through the terrain. A compar-
ison of the CoM phase portraits and position paths in Fig 6.21 reveals that this
behavior matched the observed motion capture human traversal.
Lastly, to investigate the locomotion behavior produced by the framework
under extreme conditions, simulations of idealized subject traversals were per-
formed. As mentioned previously, the sagittal CoM velocity at the apex of each
foot was taken from the motion capture video for more accurate comparisons with
the measured data. However, these apex velocities are an input to the model and
can be set as desired to study the foot CoP plans produced under various travel
velocities. Likewise, the legged architecture's height can be varied to determine
87
how the body responds and performs in the terrain in accordance with the above
apex velocities. Because of the interplay between subject height and stride length,
this will aect potential foot CoP locations in the terrain and consequently have
a direct impact on CoM behavior. To study these eects, simulations were per-
formed for hypothetical subjects measuring 1m and 2m in height. The subjects
were arbitrarily chosen be female, and thus female anthropomorphic data were used
to determine the values of thigh and shank length. Furthermore, the sagittal apex
velocities at all foot CoPs were set to a constant value of 0:7m/s. This corresponds
to approximately the normal walking speed and was a reasonable value to examine
natural walking locomotion. The sagittal and vertical CoM phase portraits for the
1m tall subject are shown in Fig 6.22 (a) and (b) respectively while the selected
CoPs and complete CoM path are given in (c). As expected the quantity of steps
performed (10 steps) in the terrain is large. This agrees with intuition since a 1m
Figure 6.22: Sagittal and vertical CoM phase portraits are shown in (a) and (b)
respectively. The CoM estimate and selected foot CoP locations are given in (c).
sized subject would possess very short limbs and would require more steps to tra-
verse the terrain. Moreover, the CoM estimate displays more extreme behavior, for
example atx 3m, where the short subject must overcome the tall planar obstacle.
Because the CoM is closer to the ground, surmounting the impediment warrants a
sharper ascent in the CoM path. Similarly, once the obstacle has been summitted,
descending onto the ground requires a sharp downward trajectory in the CoM. This
88
Figure 6.23: Top: snapshots of simulated traversal through terrain 2 for female
measuring 1m in height. Bottom: Sagittal view of foot paths and feet trajectories.
behavior is also illustrated in Fig 6.23 (top) where the simulated subject displays
the acute leg joint angles needed to overcome the terrain obstacle at x 3m. The
contrast in performance and behavior is most clearly visible when compared with
the taller 2m simulated subject in Fig 6.24 who navigates the same terrain with
little diculty. In fact, the 2m sized simulated subject completely bypasses the two
smaller terrain obstacles and only makes contact with the tallest terrain impedi-
ment as shown in the CoM path and step CoP locations in Fig 6.24 (c). Because of
this the overall CoM behavior is only minimally aected by the terrain, for exam-
ple, both the sagittal and vertical phase portraits maintain greater smoothness and
89
Figure 6.24: Top: Snapshots of simulated traversal for a 2m tall female subject.
Bottom: Sagittal and vertical CoM phase portraits are shown in (a) and (b) respec-
tively. The CoM estimate and selected foot CoP locations are given in (c).
consistency characteristic of the natural walking phase portrait. Finally, note also
the anticipatory foot CoP performed x 2:75m at the edge of the small planar
surface during the terrain traversal of the smaller subject, see Fig 6.23 (top). As
was described earlier, such steps are performed when the optimal CoP location is
impacted by terrain constraints. In these cases, alternate CoP locations must be
selected that allows the subject to continue with terrain traversal. The advantage
of the framework described here is that this behavior is incorporated by leveraging
the CoM estimation technique to perform two step lookahead. This provides a
90
method to identify anticipatory step locations that maximize obstacle clearance
and overcome terrain impediments. Overall, these last results re-substantiate the
fecund quality of the geometric CoM estimation technique. The method can be
used to produce natural walking CoM estimates, and equally important, it can also
be used as a predictive tool to investigate CoM behavior based on potential contact
foot locations. These foot placements, qualied based on step transition energy
cost minimization and also obstacle clearance, can be assessed for feasibility to
select the suitable contact foot locations. In short, this results in a comprehensive
framework that enables development of natural locomotion CoM and foot plans in
complex planar terrains for a range of bipedal legged architectures.
91
Chapter 7
Conclusion
7.1 Summary and Contributions
Locomotion planning over complex terrains is dicult due to the coarseness of
environments and the lack of representational models that incorporate the vari-
ance of shapes and sizes in kinematic architectures. The large variety of terrains
and body kinematic constraints combine to make CoM and foot placement analysis
highly intractable. However, in order to derive suitable motions plans, the simulta-
neous treatment of CoM dynamics and foot placement plans must be investigated
to understand how the CoM behaves with regards to the choice of contact foot
locations. The work presented in this dissertation describes a geometric frame-
work for CoM and foot placement motion planning that treats these issues for
bipedal kinematic architectures traveling through complex planar terrains. The
framework includes several novel contributions to the body of locomotion research.
These include the following: 1) The ability to plan smooth and optimal minimum
curvature and length CoM paths that meet both the kinematic constraints of the
architecture and the conditions specied by the user. 2) CoM paths that can be
generated for both planar and irregular planar terrains. 3) A general geomet-
ric approach for natural walking locomotion that accurately estimates the CoM
position through the terrain. The method is sensor restricted and possesses min-
imal sensor requirements. Furthermore, it uses body kinematic abstractions to
describe and model dierent kinematic architectures. 4) A geometric extension
92
to estimate CoM position through the terrain during ambulatory load carriage.
Specially, CoM behavior for sagittal mass osets placed on the subject's back is
accurately described. 5) The ability to nd suitable foot placement locations using
variable speeds in CoM behavior while traveling in complex planar terrains. This is
achieved by performing in parallel CoM position estimation and phase space anal-
ysis of multi-contact dynamics to determine suitable foot placement locations. 6)
A method for generating foot trajectories of terrain traversals based on an angular
description of CoM forward progress angles and desired step durations. 7) Lastly,
the geometric method operates in real-time. This delivers fast and reliable CoM
and foot plans that can be used to study locomotion maneuvers in complex planar
terrains.
The novel approach described here oers several advantages. Firstly, the non-
reliance on motion capture or force measurement hardware renders the method
immensely practical and portable. Such an advantage obviates the need for costly
and cumbersome devices requiring time consuming and uncouth calibration peri-
ods. This greatly facilitates the estimation of body specic CoM paths. Secondly,
the successful application of this method to complex planar terrains makes the geo-
metric approach suitable for a wide range of test environments, increasing its power
and versatility. Lastly, the method requires minimal input information regarding
the kinematic architecture and the terrain, greatly reducing the complexity of its
usage. Overall, this provides an important tool that can be used to explore the
eects on CoM behavior of various foot CoP locations and determine suitable plans
for terrain locomotion.
93
7.2 Future Work
The contributions made in this work have oered important resources to study
complex locomotion maneuvers, nevertheless, further areas remain open for inves-
tigation. Specically, the study of more disparate kinematic architectures could be
explored to determine how they aect virtual step locations and forward progress
angle calculations. This could lead to ner optimizations that account for the het-
erogeneity of large populations. Expansion to more complex terrains could also
be examined, including extensions to sloped terrains, 3D environments and CoM
behavior estimation in the lateral direction. These would require estimation of 3D
CoM path surfaces, an interesting endeavor and one which would further generalize
the approach. Lastly, load carriage eects at dierent locations, e.g. frontal load
transport, might be considered. Incorporation of these elements could increase the

exibility and expand the application of the method. These extensions, along with
the auspicious results presented in this dissertation, would increase the functional
scope of the method and build upon the work undertaken thus far.
94
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Creator Barrios, Luenin (author)

Core Title Simultaneous center of mass estimation and foot placement selection in complex planar terrains for legged architectures

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Computer Science

Publication Date 09/22/2017

Defense Date 05/25/2017

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

Tag center of mass,foot placement selection,locomotion planning,OAI-PMH Harvest,phase space behavior,rough terrains

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Shen, Wei-Min (committee chair), Haas, Stephan (committee member), Nakano, Aiichiro (committee member)

Creator Email luenin@gmail.com,lueninba@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c40-431613

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