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Understanding reactive balance control strategies in non-disabled and post-stroke gait

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Understanding reactive balance control strategies in non-disabled and post-stroke gait

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UNDERSTANDING REACTIVE BALANCE CONTROL STRATEGIES
IN NON-DISABLED AND POST-STROKE GAIT

by

Chang Liu

A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(BIOMEDICAL ENGINEERING)

August 2021

Copyright 2021 Chang Liu
ii
Dedication

I dedicate this work to my parents and grandparents
for their unconditional love and support.

iii

Acknowledgments
I would first thank my advisor, Dr. James Finley, for his patience, encouragement,
support, and guidance throughout my five years of Ph.D. I have learnt so many things from him.
Not only did he train me how to do research and how to raise scientific questions, he also taught
me about scientific writing, presentations, and enjoying time during conferences. I would say Dr.
Finley is the best advisor one can ask for, and I have learnt so much about how to be a good
mentor and a good teacher from him.
I would also like to thank my committee members: Dr. Kristan Leech, who gave me very
detailed comments and valuable advice related with clinical research and encourage me to think
about the clinical implications of my study; Dr. Jill McNitt-Gray, who always guided me to think
step by step about dynamics and helped me formulate the measures that I used for part of my
thesis; and Dr. Francisco Valero-Cuevas who introduced me to the area of neuromechanics when
I attended his class during my senior year of college and provided useful suggestions to help
improve my research.
I would also like to thank Dr. Nicolas Schweighofer. I have known him since the start of
my Ph.D program as I did my rotation in his lab. Dr. Schweighofer has taught me a lot about
motor learning, computation modeling, and neurorehabilitation in the upper extremities.
Throughout the years that I interacted with him, he always challenged me to learn new skills and
never give up pursuing a research question.
I also want to express my gratitude to Dr. Natalia Sanchez. I met Dr. Sanchez when I
visited the lab five years ago. After I joined the lab, she taught me everything about running and
designing experiments, and we had a great time sharing the office and decorating the walls. She
also gave me so much emotional support outside the lab.
iv
I would like to thank my present and former lab members: Sungwoo, Aram, Tom (the
honorary lab member), Cathy, Russell, Pouria, Shreya, Ryan, Sarah, Catherine, and Isaiah, who
all provide me with valuable feedbacks during presentations, manuscript writing, discussion
about scientific ideas, and have fun outside lab (such as video games). Aram and I always went
on morning coffee runs and talked about projects, life, and science. These walks and talks have
become an essential part of my PhD life. Aram always helped me with statistics, peer-reviewed
my writing and even provided perspectives of clinical research. Sungwoo and I did the stroke
project together. Whenever we ran into trouble during the experiments, Sung was always super
calm to debug the equipment/program. Since he is one of the founding members of the lab, I
have benefited from the Visual3D model he built and his template of Matlab codes. Tom, it was
a great time during his visit in LA and doing research with him, and I hope we will meet again in
future in-person conferences. Cathy and I performed experiments when the school restarted
human-related research during COVID. It was a very challenging time full of uncertainties and
thanks to her that I could collect preliminary data for my dissertation. Pouria, he taught me about
the simulations and we were the BME buddy in the Health Science Campus. Russell and I shared
the office for more than a year. He always gave me very detailed edits and suggestions for my
writing. I also appreciate the help from all other lab members and everyone who are/were in the
journal club: Julia, Lily, Octavio, Dr. Chris Laine, Nadir, Victor, Yannick, Chunji, Wayne
(Wenyang), and everyone who have helped me during my PhD.
I also like to thank my dear friends: Hua Weiwei who cheered me up with numerous late
night boba drinks, Li Chen, Dong Sijia, and Dr. Shu who made Pasadena my second home in LA
and went through the challenging time of COVID together, my friends since high school Lin
Yuan and Hu Lanqing, Jade He whose welcomed me in Chicago along with her cute cats, Liu Jia
v
who gave me a lot of support throughout my Ph.D, Peiyu and Qiong who took me to a lot of food
adventures and made delicious dumplings during Chinese New Years, and Esolene who always
listened to my concerns and gave me ‘life’ suggestions.
I want to acknowledge funding organizations that helped fund this research, including
National Institute of Health and Clinical and Translational Science Institute. I would also thank
my participants without whom this dissertation will not be possible.
My utmost gratitude is for my parents Tian Hui and Liu Chenghang, my grandparents,
my cousins Wei Sijia and Tian Shiyi, and my uncles and aunts. I could not complete this journal
without the tremendous support from all my family members.

vi
T able of Contents
Dedication ....................................................................................................................................... ii
Acknowledgments.......................................................................................................................... iii
List of Figures ................................................................................................................................. x
List of Tables ............................................................................................................................. xxiii
Abstract ...................................................................................................................................... xxiv
Chapter 1. Literature Review .......................................................................................................... 1
1 Balance control during walking .......................................................................................... 1
1.1 Feedforward - proactive control of balance ............................................................ 1
1.2 Feedback - reactive control of balance ................................................................... 2
2 Metrics to quantify dynamic balance .................................................................................. 3
2.1 Whole-body angular momentum ............................................................................ 3
2.2 Maximum Floquet multiplier .................................................................................. 7
2.3 Margin of stability................................................................................................... 7
3 Mechanisms to stabilize gait following external disturbances ........................................... 9
3.1 Foot placement mapping derived from walking ................................................... 11
4 Post-stroke change in the lower extremities ..................................................................... 12
4.1 Gait asymmetries in post-stroke gait ..................................................................... 13
4.2 Impaired reactive balance in people post-stroke ................................................... 15
4.3 Measures of dynamic balance during post-stroke gait .......................................... 17
5 Significance and potential clinical implications ............................................................... 18
Chapter 2. Choice of Reference Axis for Computing Angular Momentum Affects Inferences
about Dynamic Balance ................................................................................................................ 20
Abstract .................................................................................................................................. 20
1 Introduction ....................................................................................................................... 21
2 Methods............................................................................................................................. 24
2.1 Participant characteristics ..................................................................................... 24
2.2 Experimental protocol ........................................................................................... 24
2.3 Data Acquisition ................................................................................................... 26
2.4 Data processing ..................................................................................................... 26
2.5 Segmental Angular Momentum and Whole-body Angular Momentum .............. 27
2.6 Sparse Principal component analysis (sPCA) ....................................................... 28
2.7 Comparison of intersegmental coordination patterns ........................................... 29
2.8 Statistical Analysis ................................................................................................ 30
3 Results ............................................................................................................................... 31
3.1 Changes of whole-body angular momentum in response to treadmill
perturbations ......................................................................................................... 31
3.2 Effects of perturbation amplitude on WBAM ...................................................... 32
3.3 Variance Accounted for (VAF) by PCs ................................................................ 34
vii
3.4 Segmental coordination patterns differed based on the reference axis chosen
for WBAM ............................................................................................................ 35
4 Discussion ......................................................................................................................... 39
4.1 Limitations ............................................................................................................ 42
5 Conclusion ........................................................................................................................ 42
6 Acknowledgements ........................................................................................................... 43
Chapter 3. Conservation of Reactive Stabilization Strategies in the Presence of Step Length
Asymmetries during Walking ....................................................................................................... 44
Abstract .................................................................................................................................. 44
1 Introduction ....................................................................................................................... 46
2 Methods............................................................................................................................. 49
2.1 Participant characteristics ..................................................................................... 49
2.2 Experiment protocol .............................................................................................. 50
2.3 Data acquisition .................................................................................................... 52
2.4 Data processing ..................................................................................................... 53
2.5 Statistical Analysis ................................................................................................ 56
3 Results ............................................................................................................................... 57
3.1 Modulation of Whole-body Angular Momentum in Response to Treadmill
Perturbations ......................................................................................................... 59
3.2 Effects of Asymmetry on the Reactive Control of Whole-body Angular
Momentum ............................................................................................................ 61
3.3 Orbital Stability ..................................................................................................... 63
4 Discussion ......................................................................................................................... 65
4.1 Conservation of reactive response ........................................................................ 65
4.2 Effects of Limb-dominance on Reactive Control of Balance ............................... 66
4.3 Orbital Stability during Asymmetric Walking ...................................................... 67
4.4 Limitations ............................................................................................................ 69
5 Acknowledgements ........................................................................................................... 70
6 Author Contributions ........................................................................................................ 70
Chapter 4. Asymmetric Gait Patterns Alter the Reactive Control of Intersegmental
Coordination Patterns in the Sagittal Plane during Walking ........................................................ 71
Abstract .................................................................................................................................. 71
1 Introduction ....................................................................................................................... 73
2 Methods............................................................................................................................. 76
2.1 Participant characteristics ..................................................................................... 76
2.2 Experimental protocol ........................................................................................... 76
2.3 Data Acquisition ................................................................................................... 77
2.4 Data processing ..................................................................................................... 78
2.5 Segmental Angular Momentum and Whole-body Angular Momentum .............. 78
2.6 Principal component analysis (PCA) .................................................................... 80
2.7 Comparison of intersegmental coordination patterns ........................................... 82
2.8 Statistical analysis ................................................................................................. 83
3 Results ............................................................................................................................... 84
3.1 Patterns of intersegmental coordination when walking with equal step lengths .. 85
3.2 Effects of perturbations on patterns of intersegmental coordination .................... 86
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3.3 Effects of step length asymmetry on patterns of intersegmental coordination ..... 88
4 Discussion ......................................................................................................................... 93
5 Acknowledgments............................................................................................................. 97
6 Author Contributions ........................................................................................................ 97
Chapter 5. Paretic deficits prevent the execution of reactive control strategies during walking .. 98
Abstract .................................................................................................................................. 98
1 Introduction ....................................................................................................................... 98
2 Methods........................................................................................................................... 103
2.1 Participants .......................................................................................................... 103
2.2 Experimental protocol ......................................................................................... 105
2.3 Data Acquisition ................................................................................................. 106
2.4 Data Processing ................................................................................................... 107
2.5 Whole-body angular momentum ........................................................................ 107
2.6 Measures of reactive control strategies ............................................................... 108
2.7 Statistical Analysis .............................................................................................. 111
3 Results ............................................................................................................................. 113
3.1 Whole-body angular momentum ........................................................................ 114
3.2 Changes in the stance-phase forward pitch impulse during the perturbation
(∆LStance) .............................................................................................................. 118
3.3 Changes in the collision angular impulse during the recovery step (∆Lcol) ........ 121
3.4 Changes in the push-off angular impulse during the recovery step (∆LPush ) ..... 122
3.5 Changes in the net angular impulse during the recovery step (∆LNet) ................ 123
3.6 Association between reactive stabilization strategies and clinical measures ...... 124
4 Discussion ....................................................................................................................... 125
4.1 Limitations .......................................................................................................... 128
5 Acknowledgements ......................................................................................................... 129
Chapter 6. Reducing step length asymmetry does not improve reactive control of balance
during walking for people post-stroke ........................................................................................ 130
Abstract ................................................................................................................................ 130
1 Introduction ..................................................................................................................... 131
2 Methods........................................................................................................................... 134
2.1 Experiment Protocol ........................................................................................... 135
2.2 Data Acquisition ................................................................................................. 138
2.3 Data Processing ................................................................................................... 139
2.4 Measures of Dynamic Stability ........................................................................... 140
2.5 Statistical Analysis .............................................................................................. 141
3 Results ............................................................................................................................. 142
3.1 Whole-Body angular momentum changes in response to treadmill
perturbations ....................................................................................................... 142
3.2 Associations between angular momentum and clinical assessments of balance 143
3.3 Participants modified step length asymmetry using visual feedback ................. 144
3.4 Changes in SLA magnitude do not affect angular momentum ........................... 145
4 Discussion ....................................................................................................................... 147
4.1 Association between clinical balance assessments and angular momentum ...... 147
ix
4.2 Reduction in step length asymmetry does not improve reactive balance ........... 148
4.3 Limitations .......................................................................................................... 149
5 Acknowledgments........................................................................................................... 150
Chapter 7. Evidence of a nonlinear mapping between body state and foot placement during
perturbed walking ....................................................................................................................... 151
Abstract ................................................................................................................................ 151
1 Introduction ..................................................................................................................... 152
2 Method ............................................................................................................................ 154
2.1 Participant characteristics ................................................................................... 154
2.2 Experimental protocol ......................................................................................... 154
2.3 Data Acquisition ................................................................................................. 154
2.4 Data Processing ................................................................................................... 154
2.5 Statistical Analysis .............................................................................................. 157
3 Results ............................................................................................................................. 158
3.1 Foot placement mapping during unperturbed walking ....................................... 158
3.2 Foot placement mapping during perturbed walking ........................................... 159
4 Discussion ....................................................................................................................... 162
4.1 Limitations .......................................................................................................... 165
Chapter 8. Discussion and future work ....................................................................................... 167
1 Findings from our studies ............................................................................................... 168
1.1 Referencing whole-body angular momentum to different axes .......................... 168
1.2 Differed reactive control strategies in people post-stroke .................................. 169
1.3 Association between step length asymmetry and reactive balance .................... 169
1.4 Mapping between body’s state and foot placement ............................................ 170
2 Future work ..................................................................................................................... 171
References ................................................................................................................................... 175
Appendices .................................................................................................................................. 201
Appendix A: Adaptation of proactive adjustments during repeated, treadmill-induced
perturbations of walking ...................................................................................................... 201
Appendix B: Reduced sensitivity of foot placement control to center-of-mass velocity
during steady-state post-stroke walking .............................................................................. 225

x
List of Figures
Figure 1: Normalized whole-body angular momentum (L) during gait cycle starting with the
right foot strike for one representative non-disabled young participant for 5 min of normal
walking. Blue: Mean whole-body angular momentum, Gray shaded area: standard deviation of
the L. 0% and 100% gait cycle correspond to foot strike of the same foot. Vertical line: the
contralateral foot strike. .................................................................................................................. 5
Figure 2: Diagram showing that people post-stroke can have step length asymmetries in two
directions, either taking shorter paretic or shorter non-paretic steps. The top row (A) shows that
people post-stroke take longer paretic steps than non-paretic steps due to the deficits at the
paretic propulsion to advance the trunk forward at non-paretic step. The bottom row (B) shows
that people post-stroke take shorter paretic steps than non-paretic steps by placing the paretic leg
closer to the body CoM at the paretic foot strike. ......................................................................... 14
Figure 3: People use multiple reactive control strategies to reduce the effect of forward fall.
Green arrows here indicate the forward angular momentum induced by the perturbation. GRF:
ground reaction force. ................................................................................................................... 16
Figure 4: Whole-body angular momentum with respect to the mediolateral axis projecting
through (A) the BoS and (B) CoM during pre-perturbation steps, perturbation steps, and the
following recovery steps for one representative participant. Dashed lines: angular momentum
referenced to BoS and CoM during pre-perturbation walking. Colored lines: angular momentum
during perturbation steps and the first recovery steps. Arrows indicated the approximate mid-
swing phase for each step. ............................................................................................................ 32
Figure 5: Association between perturbation amplitude and mean peak whole-body angular
momentum referenced to CoM axis and BoS axis across participants (N = 11). (A-B) Maximum
xi
LCoM and LBoS were negatively correlated with perturbation amplitude. (C) Minimum LCoM
was not correlated with perturbation amplitude. (D) Minimum LBoS were negatively correlated
with perturbation amplitude. Each dot represents one participant. ............................................... 33
Figure 6: VAF for the first three PCs across all levels of perturbations and pre-perturbation steps.
Dark shaded: referenced to the axis through CoM; lightly shaded: referenced to the axis through
BoS. (*p<0.05). ............................................................................................................................. 35
Figure 7: Principal components (PC) extracted from segmental angular momentum referenced to
the axis through CoM (left panel A, C) during perturbation steps. Principal components (PC)
extracted from segmental angular momentum referenced to the axis through BoS (right panel B,
D) during perturbation steps. (N=11). The segments include: RTH (right thigh), RSH (right
shank), RFT (right foot), LTH (left thigh), LSH (left shank), LFT (left foot), H (head), PEL
(pelvis), THX (thorax). All PC weights for arm segments (forearms and upper arms) are close to
zero, so they are not included in this figure to improve clarity. Error bars represented standard
deviation. ....................................................................................................................................... 37
Figure 8: (A) Boxplot shows the scalar product between the PC1CoM and the most similar
PCBoS extracted from segmental angular momentum during pre-perturbation steps and
perturbation steps at different levels of speed change (*p<0.05). (B) Boxplot shows the scalar
product between the PC1CoM and the most similar PCBoS extracted from segmental angular
momentum during pre-perturbation steps and perturbation steps at different levels of speed
change. Horizontal dashed line indicates the critical r value = 0.684 for the two PCs to be similar
in the 13-dimensional space. (C) PC1CoM and its matched PCBoS (D) during perturbation steps
with -0.3m/s speed change. Error bars represented standard deviation. ....................................... 38
xii
Figure 9: (A) Experiment protocol. Participants completed a total of six trials. Participant’s
baseline step length asymmetry was collected during the first three-minute baseline trial without
visual feedback. Then, they were instructed to complete a randomized sequence of five six-
minute trials with target step length asymmetries of 0%, ±10%, and ±15%. During each visual
feedback trial, the participant first practiced with feedback for one minute, then 10 perturbations
were randomly applied at foot strike on each side. (B) Visual feedback for three of the five trials
of step length asymmetry are shown. (C) Experimental setup. Participant were instructed to walk
on the split-belt treadmill. A “success” message would appear on the screen when step length
was within the three standard deviations of the desired target. .................................................... 50
Figure 10: Example of time series data from an unperturbed and perturbed step. (A) Treadmill
belt velocity, (B) vertical ground reaction force, and (C-E) whole-body angular momentum for a
representative perturbation step and recovery stride. The gray traces indicate the time series data
for an unperturbed stride while the black traces indicate a perturbation stride. Each stride begins
at heel strike. Black vertical lines correspond to the time of foot strike and gray vertical lines
correspond to time of toe-off. Solid lines and dashed lines represent contralateral legs. ............. 52
Figure 11: (A) Example of a 3D projection of the angular momentum trajectory recorded during
baseline walking for one representative participant. (B) Illustration of a hypothetical
perpendicular slice of the angular momentum trajectory as a Poincare section. S* represents the
fixed point which is the average of pre-perturbation strides. ........................................................ 56
Figure 12: (A) Raw step length asymmetry data for one representative participant. Each data
point represents the step length asymmetry. The target asymmetries for this example followed
the order of 10%, -10%, 0, 15%, -15%. Each target asymmetry is represented by a different
color. BSL: baseline step; PTB: perturbation step; REC: recovery step. (B) Achieved step length
xiii
asymmetry versus target step length asymmetry for all participants (N=19). Achieved step length
asymmetry is calculated as the average of all pre-perturbation strides and tends to undershoot the
target at 15% and -15%. The green dots represent individual data. Horizontal bars indicate the
median across all participants. ...................................................................................................... 58
Figure 13: Averaged integrated angular momentum over the step cycle for all participants about
the (A) pitch, (B) roll, and (C) yaw axes for perturbations that occurred on the non-dominant (left
column) and dominant side (right column). These results represent the 0% asymmetry condition
(N=19). The first step (B1) corresponds to the non-dominant limb for the left column and the
dominant limb for the right column. Subsequent steps alternate between non-dominant and
dominant. B: Baseline; PTB: Perturbation; R: Recovery. The horizontal bars and corresponding
stars indicated whether the difference in integrated angular momentum between two steps was
significant (**p<0.001, * p<0.05). The data are represented as boxplots such that the lower and
upper edges of the box indicate the 25th and 75th percentile of the data, respectively. The
horizontal line within each box indicates the median. The whiskers extend to the furthest data
point beyond the lower or upper edges of the box that is within a distance of 1.5 times the middle
50th percentile of the data. Points that lie beyond the whiskers denote outliers. ......................... 60
Figure 14: Box plot of integrated angular momentum about the (A) pitch, (B) roll, and (C) yaw
axes at baseline step (B2), perturbation step (PTB), and recovery steps (R1 and R2) across each
level of achieved asymmetry (N=19) for perturbations on the non-dominant (left column) and
dominant (right column) sides. ..................................................................................................... 63
Figure 15: (A) Variation in the magnitude of the maximum Floquet multiplier (FM) across the
gait cycle for five levels of target asymmetry (N=17). The shaded area indicates the 95%
confidence interval. (B) FMMax across all levels of asymmetry for (N=17) participants. .......... 64
xiv
Figure 16: (A) Sagittal plane angular momentum (Lx) for 13 segments during one representative
baseline stride (black) and one perturbation stride (grey). The segments included the thigh,
shank, foot, forearm, and upper arm, bilaterally as well as the head, pelvis, and thorax. The
duration of each trace is one full stride from 0 to 100% of the stride cycle. (B) Schematic of
principal component analysis (PCA) of segmental angular momentum. The organization of the
data used as input to the PCA is illustrated to the left. PCA extracts weighting coefficient as
intersegmental coordination patterns or principal components (PC1 and PC2) and time series
scores of each PC (Filled bar plots: PC1; Open bar plots: PC2). ................................................. 81
Figure 17: Principal components (PC) extracted from segmental angular momentum during (A)
baseline right steps, (B) baseline left steps, (C) perturbation steps, (D) recovery left steps, and
(E) recovery right steps when walking symmetrically (N=19). Blue: Right step; Pink: Left step;
Filled bars: PC1; Unfilled bars: PC2. The 13 segments include: RTH (right thigh), RSH (right
shank), RFT (right foot), LTH (left thigh), LSH (left shank), LFT (left foot), LFA (left forearm),
RFA (right forearm), LUA (left upper arm), RUA (right upper arm), H (head), PEL (pelvis),
THX (thorax). ............................................................................................................................... 86
Figure 18:Included angle between PCs extracted during each step relative to baseline steps
during symmetric walking (** p<0.001,* p<0.05). The horizontal bars and corresponding stars
indicate significant differences in the included angle. The data are represented as boxplots such
that the lower and upper edges of the box indicate the 25th and 75th percentile of the data,
respectively. The horizontal line in each box indicates the median. The whiskers extend to the
furthest data point beyond the lower or upper edges of the box that is within a distance of 1.5
times the middle 50th percentile of the data. Dots that lie beyond the whiskers indicate outliers.
Blue: Right step; Pink: Left step; Filled box plots: PC1; Non-filled box plots: PC2. The black
xv
line indicates the mean of the permutated angle distribution of baseline steps and the shading
indicates the standard deviation. ................................................................................................... 88
Figure 19: The first intersegmental coordination pattern (PC1) and the second coordination
pattern (PC2) during (A) baseline right step, (B) perturbation step, and (C) the second recovery
step with -15%, 0% and 15% step length asymmetry. The colored bars indicate the mean value
across all participants (N=19), and the black lines indicate the standard deviation. .................... 89
Figure 20:Included angle between PCs extracted during asymmetrical walking (5%, 10%, and
15%) and symmetrical walking for each step (*** p<0.001, ** p<0.01, * p<0.05) for (A) PC1
and (B) PC2. (C) Integrated whole-body angular momentum during asymmetrical walking
relative to symmetrical walking for each step. Blue: Right step; Pink: Left step; Filled box plots:
PC1; Non-filled box plots: PC2. The shaded gray area indicated the standard deviation of
permutated included angle for each step, and the black line indicated the mean of the
distribution. ................................................................................................................................... 92
Figure 21: Graphical illustration of hypothesis for whole-body angular momentum, angular
impulse during perturbation steps, and the leading limb angular impulse and the trialing limb
angular impulse during the first recovery steps. ......................................................................... 103
Figure 22: Diagram of computed angular impulse about the body CoM by the leading and
trailing leg during the perturbation (PTB) step and the first recovery step (R1). The PTB ∆LStance
is computed during the phase from midstance of the PTB step until the foot strike of the R1 step.
R1 ∆Lcol and R1 ∆Lpush are computed during the collision phase of the R1 steps. R1 ∆LNet is
computed as the sum of R1 ∆Lpush and R1 ∆Lcol .FS: Foot strike; FO: Foot-off. The arrows (+/-)
indicate the backward and forward moments by the GRF about CoM, respectively. ................ 110
xvi
Figure 23: Whole-body angular momentum in the sagittal plane and ground reaction force for
one representative control participant (A) and a stroke participant during a paretic perturbation
(B) and non-paretic perturbation (C) for both a pre-perturbation stride and a perturbation stride.
Each stride began at foot strike. The gray traces indicate the time series data for a pre-
perturbation stride while the black or colored traces indicate a perturbation stride. Negative
values of angular momentum represent forward rotation while positive values represent
backward rotation. Ground reaction force (% body weight) in the vertical and anterior-posterior
directions for the perturbed and the contralateral limb when perturbations occurred on the
dominant side for the control participant (A), or on the paretic (B) or non-paretic sides (C) for the
stroke participant. For the control participant, black lines indicated the perturbed side and the
dashed lines indicated the contralateral side. For the stroke participant, pink and blue lines
represent the paretic leg and non-paretic leg during the perturbation stride, respectively. Gray
dashed lines represent non-paretic leg and paretic leg during the pre-perturbation stride,
respectively. Black dashed vertical lines correspond to the time of foot strike. Gray shaded
vertical box corresponds to the double support phase from the time of foot strike to the
contralateral foot-off. Pre-PTB: pre-perturbation stride; PTB: perturbation stride. ................... 113
Figure 24: Median integrated angular momentum in the sagittal plane over the step cycle relative
to the corresponding pre-perturbation steps (ΔLint) for all participants. Steps alternated between
paretic and non-paretic for stroke participants. PTB: Perturbation; R: Recovery. The asterisks on
top of the boxplots indicate whether the difference in Lint from pre-perturbation steps was
significantly different from zero (*p < 0.05) and the # indicated that the ΔLint was different
between groups. Note that for people post-stroke, if the non-paretic leg was perturbed, the R1
xvii
steps were paretic steps, and Pre-PTB step and PTB steps were non-paretic steps and vice versa
for the paretic perturbations. ....................................................................................................... 116
Figure 25: Time series trajectories of the PTB ∆LStance, R1∆Lcol, R1∆LPush and the corresponding
trajectories during the pre-perturbation steps for the representative control participant, one
paretic perturbation, and one non-paretic perturbation for one representative stroke participant.
The left column shows the PTB ∆LStance (A), R1 ∆Lcol (D), R1 ∆LPush (G) for the representative
control participant (black) and the corresponding pre-perturbation trajectories (gray). The middle
column shows the PTB ∆LStance (B), R1∆Lcol (E), R1 ∆LPush (H) for the representative stroke
participant during the paretic perturbation (darker red) and the corresponding pre-perturbation
trajectories for comparison (lighter red). The right column shows the PTB ∆LStance (C), R1∆Lcol
(F), R1 ∆LPush (I) for the representative stroke participant during the non-paretic perturbation
(darker blue) and the corresponding pre-perturbation trajectories for comparison (lighter blue).
For the middle and right column, the solid line indicates the non-paretic leg and the dashed line
indicates the paretic leg. FS: Foot strike, FO: Foot-off, MST: Midstance. Vertical lines indicated
gait events and the solid vertical line indicated the ipsilateral limb while the dashed vertical
indicated the contralateral limb. .................................................................................................. 117
Figure 26: Median forward pitch impulse ∆Lstance during pre-perturbations and changes in ∆Lstance
following perturbations across control and stroke participants. Median ∆Lstance (A) during pre-
perturbation steps across control participants (N = 13) and stroke participants (N=38) during
paretic and non-paretic steps. Gray: Control, Pink: Paretic step, Blue: Non-paretic step. (B)
Median changes in ∆Lstance during the PTB steps compared to measured during pre-perturbation.
The asterisks (*) on top of the boxplots indicated whether the difference in these variables during
each step from pre-perturbation steps was significantly different from zero (**p<0.001). Gray:
xviii
Control, Pink: Paretic perturbation, Blue: Non-paretic perturbation. Solid box: Non-paretic leg,
Dash box: Paretic leg. ................................................................................................................. 119
Figure 27: Median ∆Lcol, ∆Lpush, and ∆LNet about CoM during pre-perturbations and changes of
those variables following perturbations across participants. The left column shows median ∆Lcol
(A), ∆Lpush (C), ∆LNet (D) during pre-perturbation steps across control participants (N = 13) and
stroke participants (N=38) during paretic and non-paretic step. Gray: Control, Pink: Paretic step,
Blue: Non-paretic step. The right column panel shows median changes in ∆Lcol (B), ∆Lpush (D),
∆LNet (E) during the PTB steps and R1 steps compared to those measured during pre-
perturbation. The asterisks (*) on top of the boxplots indicated whether the difference in these
variables during each step from pre-perturbation steps was significant (*p<0.05, **p<0.001,
***p<0.0001). The hashes (#) and the black lines indicated whether the comparison between two
groups were significantly different (#p<0.05, ##p<0.001). Gray: Control, Pink: Paretic
perturbation, Blue: Non-paretic perturbation. Solid box: Non-paretic leg, Dash box: Paretic leg.
Note that for people post-stroke, if the non-paretic leg was perturbed, the R1 steps were paretic
steps, and Pre-PTB step and PTB steps were non-paretic steps and vice versa for the paretic
perturbations. .............................................................................................................................. 120
Figure 28 Associations between deviation in R1∆LPush in the sagittal plane and clinical
assessments. Deviation of R1 ∆LPush from pre-perturbation steps was positively associated with
(A) Fugl-Meyer score and (B) the Functional Gait Assessment only following perturbations
occurred at the paretic limb. FM: Fugl-Meyer; FGA: Functional Gait Assessment. ................. 124
Figure 29: A) Experimental protocol. Participants first completed a series of clinical assessments
and familiarization with the treadmill. Then, participants completed trials with perturbations
under three conditions: without visual feedback (BASE), with baseline asymmetry feedback
xix
(BASE+FBK), and symmetry feedback (SYM+FBK). These trials were randomized. For the
visual feedback trials, we provided online visualization of the ankle markers during the swing
phase as a black point. The red lines represented the target range of the step lengths which was
equal to three standard deviations of their baseline step length. We also provided scores on the
top left or right corner of the display to encourage the participants to achieve the target step
lengths. ........................................................................................................................................ 138
Figure 30 An example of time series whole-body angular momentum in the sagittal plane for a
pre-perturbation and perturbation stride when perturbations occurred on the non-paretic (A) or
paretic side (B). The gray traces indicated the time series data for a pre-perturbation stride while
the black traces indicate a perturbation stride. Each stride began at foot strike. Negative values
of angular momentum represent forward rotation while positive values represent backward
rotation. Black dashed vertical lines correspond to the time of foot strike for the first recovery
step. Pre-PTB: pre-perturbation stride; PTB: perturbation stride. .............................................. 143
Figure 31: Associations between deviation in integrated whole-body angular momentum during
the perturbation step in the sagittal plane (ΔLint) and clinical balance assessments. ΔLint was
positively associated with (A) Berg Balance Scale, and (B) the Functional Gait Assessment, (C)
the Activity-based Confidence Scale, and (D) Fugl-Meyer score only when perturbations
occurred at the paretic limb. BBS: Berg Balance Scale, FGA: Functional Gait Assessment, ABC:
Activity-based Confidence Scale, FM: Fugl-Meyer. .................................................................. 144
Figure 32: A) Raw step length asymmetry data for one representative participant who took
longer paretic steps. Each data point represents the step length asymmetry for a stride. B)
Magnitude of step length asymmetry for all participants with baseline |SLA| > 0.05 during
BASE, BASE+FBK, and SYM+FBK. *p<0.05. ........................................................................ 145
xx
Figure 33: (A) Changes in the 𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 in the sagittal plane with changes in SLA magnitude relative
to BASE+FBK during perturbation step when perturbations occurred at the paretic side during
the Pre-PTB steps (A), PTB steps (B), and R1 steps (C). Pink: Paretic perturbations, blue: non-
paretic perturbations, filled: participants who took longer non-paretic steps, unfilled: participants
who took longer paretic steps. .................................................................................................... 146
Figure 34: Diagram of the simplified model describing the CoM motion (S) and foot positions.
CoM state included the CoM position and velocity in the fore-aft and mediolateral direction.
Blue: swing leg, Red: stance leg). CoM position and the position of the swing foot were
referenced to the stance foot. The gray dashed trajectory represents the nominal (average) CoM
trajectory. The gray solid trajectory represents one measured trajectory. 𝛥𝛥𝛥𝛥 and 𝛥𝛥𝛥𝛥 described the
step-to-step fluctuation of the CoM state and foot placement. ................................................... 155
Figure 35: Scatter plots showing the foot placement during unperturbed walking and following
perturbation responses for a representative participant and goodness of fit for each model. (A)
Left foot placement relative to the right perturbed stance foot during unperturbed steps and
perturbed steps; (B) Actual foot placement v. fitted foot placement in the anteroposterior
direction during perturbed walking at midstance using the mapping derived during unperturbed
walking. Colored dots indicate foot placement following perturbations. Gray dots represent foot
placement during unperturbed walking. (C & D) AIC of deviations in foot placement after
applying the mapping derived at midstance from unperturbed walking to perturbed steps
(UNPTB), applying the linear mapping derived from perturbed walking to perturbed steps (LR),
and applying the piecewise linear regression model to perturbed steps (PLR). (C) Mediolateral
direction. (D). Anteroposterior direction. (*p<0.05, **p<0.001,***p<0.0001). Dots represent
each participant. .......................................................................................................................... 161
xxi
Figure 36: The estimated coefficients of the partial derivative of the foot position with respect to
CoM state at midstance in the Jacobian matrix. Boxplot shows coefficient estimates across
participants. Gray: estimates during unperturbed walking, Green: estimates from piecewise linear
regression for forward perturbations, Blue: estimates from piecewise linear regression for
backward perturbations. Dots represented individual estimates of coefficients by summing the
random effects and the fixed effects from the mixed effect models (*p<0.05,
**p<0.001,***p<0.0001). ........................................................................................................... 162
Figure 37: Diagram of MoS in the fore-aft direction for the forward and backward MoS. We
calculated the forward MoS and the backward MoS using the toe and heel markers, respectively
at foot strike. ............................................................................................................................... 207
Figure 38 Spatiotemporal patterns throughout unperturbed trial and perturbation trials for one
representative participant. (A) Step length, (B) Step width, and (C) double support phase. Blue
dots: unperturbed baseline trial, gray dots: perturbed trials, red cross: the first perturbation.
Vertical lines indicate No PTB, INIT, FINAL intervals and each intervals included 100 steps. 209
Figure 39 Average (A) step length, (B) step width, and (C) double support phase for all
participants (N = 11) during unperturbed baseline walking (No PTB), initial phase, and final
phase of perturbed walking. ........................................................................................................ 210
Figure 40: Example gait cycle from one participant showing XCoM position, edge of base of
support (BoS) position, and MoS in AP (A) and ML (B) direction during unperturbed baseline
walking. Shaded area: double support phase from initial contact to the contralateral limb’s toe
off. Solid line: right step. Dashed line: left step. Note that we calculate FWD MoS and BWD
MoS at each foot strike by using the MoSToe of the leading limb and MoSHeel of the trailing limb,
respectively. ................................................................................................................................ 211
xxii
Figure 41: Average (A) forward MoS, (B) backward MoS, and (C) ML MoS for all participants
(N = 11) during unperturbed baseline walking (No PTB), initial phase, and final phase of
perturbed walking. ...................................................................................................................... 212
Figure 42: The average angular momentum (L) across the gait cycle for one participant during
unperturbed walking (No PTB), initial phase of perturbed walking, and final phase. (A) Angular
momentum in the sagittal plane. (B) Angular momentum in the frontal plane. Average peak-to-
peak range of whole-body angular momentum in the sagittal plane (C) and frontal plane (D) for
all participants (N = 11) during No PTB, initial phase, and final phase of perturbed walking. . 214
Figure 43: The coefficient of determination and coefficients from the foot placement mapping
for control participants and stroke participants during both paretic and non-paretic steps. (A)
Fraction of foot placement variance (R
2
) explained by the CoM state throughout the step cycle
during unperturbed walking for control participants (Gray), post-stroke participants during non-
paretic steps (Red) and paretic steps (Blue) in mediolateral and anteroposterior direction. Dashed
vertical line: Midstance of the gait cycle. Darker lines: mean value across participants. Shaded
area: standard deviation across participants in each group; (B) Fraction of foot placement
variance explained by CoM state at midstance. (C) Coefficients of the foot placement mapping in
the mediolateral direction and anteroposterior direction. Bar height: mean values across
participants. Error bars: standard deviation (*p<0.05). .............................................................. 232

xxiii
List of Tables
Table 1 Statistical results from the ANOVA examining the effects of asymmetry and
perturbation side on integrated whole-body angular momentum for each step type. B2: Baseline
step, PTB: Perturbation Step, R1: First Recovery Step, R2: Second Recovery Step. .................. 62
Table 2 Variance accounted for (VAF) for PC1, PC2, and PC3 during baseline steps,
perturbation steps, and recovery steps. ......................................................................................... 85
Table 3 Statistical results from the ANOVA examining the effects of asymmetry and direction on
the included angle for each step type. ........................................................................................... 91
Table 4 Participant demographics for both control and stroke participants. .............................. 104
Table 5 – Participant demographics and clinical assessments (N=38). ...................................... 135
Table 6: Parameter Values for Model Relating ΔLint during B1, PTB, and R1 steps to
Spatiotemporal Variables. ........................................................................................................... 146
Table 7 Statistical results of the association between coefficient of determination

of the foot
placement mapping and the clinical assessment scores for people post-stroke .......................... 233

xxiv

Abstract
One of the primary challenges for human locomotion is to maintain balance when faced
with internally generated or externally imposed perturbations. The ability to maintain balance in
response to such unexpected perturbations during walking is largely mediated by reactive control
strategies, which involve the use of feedback about the body’s state to generate balance
correcting responses. Overall, we are interested in study the reactive balance control strategies in
response to treadmill-induced perturbations during walking in both healthy populations and
people post-stroke. We also assessed whether modifications to people’s gait pattern such as
reducing the spatiotemporal gait asymmetries for people post-stroke would affect the reactive
control of balance. One common way to characterize the resulting perturbation recovery
strategies is the measure of whole-body angular momentum, which reflects the contribution of all
body segments to the body’s rotation about a given axis. In this dissertation, we present six
chapters that aim to understand the reactive control of balance during walking for both healthy
individuals and people post-stroke.
Chapter 1 provided an overview of the literature on the balance control strategies during
walking and metrics commonly used to characterize dynamic balance during walking. In
addition, we reviewed impairments in gait and balance post-stroke and provided rationales to
justify the importance of our research interest in the relationship between spatiotemporal gait
asymmetries and reactive control of balance.
Chapter 2 investigated whether referencing whole-body angular momentum to different
axes would affect our interpretation of reactive control of balance during walking. Two reference
axes were used for computing whole-body angular momentum in the sagittal plane: a
mediolateral axis projecting through 1) the body center of mass or 2) the leading edge of the base
xxv
of support. Non-disabled control participants walked on a dual-belt treadmill at their self-
selected speed. Sudden treadmill accelerations of different magnitude and direction were
remotely triggered at foot strike, and belt speed returned to the self-selected speed during swing.
Peak backward angular momentum during the perturbed steps referenced to both axes was
positively associated with perturbation speed change. In addition, the low-dimensional
intersegmental coordination patterns extracted using the principal component analysis referenced
to the two axes had one similar component and one dissimilar principal component during
perturbation steps. Computing whole-body angular momentum using a reference axis that
projects through the center of mass helped to identify the degree of segmental angular
momentum cancellation during perturbation responses as perfect limb cancellation would result
in zero angular momentum. Whereas, analyzing coordination pattern referenced to axis through
edge of support may provide more insights about how the upper body responded to sudden loss
of balance.
Chapter 3 and 4 addressed an important question for clinical researchers: whether there is
a causal relationship between changes in gait asymmetry and the ability to maintain dynamic
balance during walking. Our objective was to investigate if modulating people’s natural
spatiotemporal gait asymmetry would impair people’s reactive control of balance during walking
for healthy young participants. Our results suggested that although walking with asymmetrical
gait patterns changed the limb coordination during perturbation responses, there was no
association between step length asymmetry and impaired reactive control of balance in the
absence of neuromotor impairments. These results also indicated that our nervous system may
allow for variability in limb coordination to regulate a higher-order performance variable such as
the whole-body angular momentum.
xxvi
Chapter 5 characterized differences in reactive control strategies for people post-stroke
following the forward-falling, slip-like perturbations at paretic and non-paretic sides and
differences between people post-stroke and control participants. We used whole-body angular
momentum to assess the reactive control of balance in response to perturbations at either side
and for both populations. We also characterized the effect of ground reaction forces on the body
dynamics using angular impulse. Following paretic perturbations, people post-stroke tended to
fall forward more as demonstrated by a higher increase in the forward integrated whole-body
angular momentum compared to that following non-paretic perturbations and compared to
control participants. In addition, unlike control participants, people post-stroke did not reduce the
forward angular impulse during the second half of the stance phase using their paretic limb. They
also did not use their paretic limb to increase collision angular impulse or decrease the forward
push-off angular impulse during the collision phase at the first recovery steps. The reduction in
push-off angular impulse decreased the forward whole-body angular momentum, which could be
important to help restore balance during the slip-like forward falls. These results could provide a
better understanding of the impaired reaction stabilization strategies for people post-stroke that
might contribute to the high risk of falls.
Chapter 6 assessed whether modifying step length asymmetry influenced whole-body
angular momentum for people post-stroke. Recent work suggested that people post-stroke still
retained the capacity to reduce their step length asymmetry voluntarily. We used a biofeedback
paradigm that allowed participants to reduce their natural step length asymmetries while slip-like
perturbations randomly occurred at foot-strike. We found that reducing asymmetry did not affect
whole-body angular momentum in response to perturbations and thus did not immediately
xxvii
improve their ability to recover from imposed perturbations for people post-stroke during
walking.
Chapter 7 determined the mapping between the body’s state and foot placement in
healthy adults. We first derived the foot placement strategies during both unperturbed and
perturbed walking. The foot placement mapping derived from unperturbed walking could not
capture the balance corrective response when external perturbations occurred. A nonlinear
piecewise linear regression was needed to account for the directional response in foot placement
following slip-like and trip-like perturbations during walking.
Chapter 8 provided an overall discussion of all chapters, limitations of our work, and
future directions. Lastly, Appendix I provides a short analysis of proactive strategies that people
adopted during long repeated exposure to perturbations. We assessed the changes in
spatiotemporal gait patterns prior to the perturbations over the time course of repeated exposure
treadmill-induced perturbations. Appendix II determined and compared the mapping between the
body’s state and foot placement during unperturbed walking for people post-stroke and healthy
age-matched controls. We found that the foot placement mapping and the coefficient of
determination of the mapping for people post-stroke during unperturbed walking was different
from that derived from age-matched controls.

1
Chapter 1. Literature Review

Bipedal locomotion is inherently unstable due to the small base of support (BoS), long
single-limb support times, and sensorimotor transmission delays (Winter, 2009). For example,
when walking in the anteroposterior direction, the body’s center of mass (CoM) constantly falls
outside the BoS during the single support phase, which poses a challenge to maintain dynamic
balance. As a result, we must frequently generate corrective responses to maintain balance in
response to both internal and external perturbations. Two balance control strategies are generally
used during locomotion: proactive and reactive control of balance (Patla, 1993). While proactive
or feedforward control involves the use of predictions of impending perturbations to avoid
falling, reactive control of balance involves the use of feedback about the body’s state to
generate balance correcting responses (Tang et al., 1998).

People tend to adopt proactive strategies to reduce the risk of falls when they expect to
encounter slips or trips during walking. For example, people shorten their step lengths in
anticipation of a backward loss of balance either when they were informed of a slippery surface
ahead (Cham and Redfern, 2002) or a slip that would be induced by a forward translation of the
movable platform during walking (Cham and Redfern, 2002; Espy et al., 2010). People could
shift CoM more anteriorly at foot strike and increase the distance between the CoM state (the
combination of position and velocity) and the posterior BoS pre-perturbation when anticipating
backward loss of balance induced by the forward translation of platform during walking (Bhatt et
al., 2006). Similarly, during repeated obstacle-tripping perturbations causing forward rotation of
the body, people increased the margin between the anterior edge of the BoS and CoM state when
2
anticipating the obstacle that would obstruct the swing leg after the initial exposure to the
obstacle tripping (Wang et al., 2012). Other parameters, such as increasing the step width, may
also be beneficial to balance recovery (Tang et al., 1998). When people are informed about an
upcoming slippery surface before they step, they tend to step on the surface with an enlarged
mediolateral distance between CoM and foot placement (Marigold et al., 2003). Contributions to
adopting proactive control strategies to maintain balance could depend on prior experiences
(Heiden et al., 2006), anticipation of perturbations (Pater et al., 2015), or metabolic/energetic
cost related with the proactive strategies (Donelan et al., 2004).

One of the primary ways to study reactive control of balance is by applying perturbations
during walking and characterizing the resulting recovery strategies. To date, various
experimental methods such as moveable floor platforms (Wang et al., 2012), changes in ground
surface compliance (Bierbaum et al., 2010), and treadmill accelerations/decelerations (Liu et al.,
2018; Martelli et al., 2013; Sloot et al., 2015) have been used to perturb people’s normal gait and
trigger reactive responses. In response to these sudden disturbances to the regular walking
pattern, people adopt reactive control strategies using feedback information about the body’s
state to generate balance stabilization strategies. The feedback sensory information could be an
integration of vision, vestibular information, or proprioception. The perturbation-related afferent
information would trigger corrective responses which involve muscle reflexes to help maintain
balance. Reflex muscle response to stumbling perturbation during human walking have been
widely studied (Berger et al., 1984; Dietz et al., 1986; Sloot et al., 2015). When sudden
acceleration or deceleration occurs during treadmill walking, only medium or long-latency reflex
were observed. These medium or long-latency reflexes, reported to have a latency time of ~60-
3
100ms, are believed to be more flexible than the monosynaptic responses (latency ~20-
40ms)(Dietz et al., 1986). The long-latency reflexes involve some supraspinal pathway, mediated
by the brainstem or transcortical pathways, and also account for the biomechanical properties of
the limb rather than just responding to local muscle stretch reflex (Kurtzer, 2015). Overall,
people rely on the reactive control of balance to produce coordinated intersegmental responses to
help maintain balance when perturbations occur.

Several metrics have been used to quantify balance during locomotion including
measures of variability (Stergiou and Decker, 2011), measures derived from nonlinear dynamics
such as the maximum Lyapunov exponent (Dingwell et al., 2001; Dingwell and Cusumano,
2000) and long-range correlations (Hausdorff et al., 1996), and biomechanical measures such as
dynamic margins of stability (Hof, 2008; Hof et al., 2005). Bruijn et al. (Bruijn et al., 2013) has
provided a detailed review of metrics used to assess dynamic stability during gait. In the
following sub-sections, we describe the various measures of dynamic stability used in the study
of human locomotion.

While each of these methods is useful for characterizing features of control in the
presence of instability, we are particularly interested in measures that directly capture whole-
body dynamics. One such measure is whole-body angular momentum (WBAM) (Eqn. 1)
computed as the sum of all segmental angular momenta which were composed of segmental
rotation about the body’s CoM and rotation of each segment about its own CoM (Silverman and
Neptune, 2011)(Figure 1). It is used to capture the body’s response to perturbations(Liu et al.,
2018; Martelli et al., 2013), and it reflects the net contribution of all body segments to the body’s
4
rotation about a given axis. Biomechanists typically choose to calculate angular momentum with
respect to axes that project through the CoM. WBAM about CoM is highly regulated during
normal human locomotion as the peak-to-peak range of WBAM is much smaller than the angular
momentum of single segments due to momentum cancellation between the limbs (Herr and
Popovic, 2008; Popovic et al., 2004). In the sagittal plane, cancellation of segmental angular
momentum is achieved between sides by the anti-phase movement of the contralateral limbs. In
this case, the time point when the peak flexion for the segments on one side coincides with the
time point when peak extension occurs for the contralateral side (Park et al., 2021). In the frontal
plane, the cancellation of segmental angular momentum is achieved by the anti-phase movement
of the trunk segment and legs during the double support phase of the gait cycle (Herr and
Popovic, 2008). Such regulation of whole-body angular momentum indicates that segment
rotations about the CoM are coordinated in a way that whole-body angular momentum remains
small.
𝐿𝐿 � ⃗
= �[ 𝑚𝑚 𝑖𝑖 � 𝑟𝑟
� � ⃗
𝐶𝐶𝐶𝐶 − 𝑖𝑖 𝑖𝑖 × 𝑣𝑣 ⃗
𝐶𝐶𝐶𝐶 − 𝑖𝑖 𝑖𝑖 � + 𝐼𝐼 𝑖𝑖 𝜔𝜔 𝑖𝑖 ]
𝑖𝑖
(1)
Here, m is segmental mass, r is the distance from segment to the body COM, I is the segmental
moment of inertia, 𝜔𝜔 is segmental angular velocity, and the index i corresponds to individual
limb segments. In the sagittal plane, negative values of angular momentum represented forward
rotation, while positive values represented backward rotation.
5

Figure 1: Normalized whole-body angular momentum (L) during gait cycle starting with the
right foot strike for one representative non-disabled young participant for 5 min of normal
walking. Blue: Mean whole-body angular momentum, Gray shaded area: standard deviation of
the L. 0% and 100% gait cycle correspond to foot strike of the same foot. Vertical line: the
contralateral foot strike.

People can regulate WBAM about CoM under various conditions. WBAM is investigated
during walking at different speeds (Bennett et al., 2010), when walking with different step
lengths (Thielemans et al. 2014), walking with additional weights (Thielemans et al. 2014),
during incline/decline walking (Silverman et al. 2012), and during stair ascent/descent
(Silverman et al. 2014). Studies have found that WBAM can capture perturbation responses
which resulted in dramatically increased angular momentum from that measured during
unperturbed walking (Liu et al., 2018; Martelli et al., 2013; Pijnappels et al., 2004, 2005;
Potocanac et al., 2014). This deviation captures the features of body rotation that, if not arrested,
would lead to a fall. To restore balance and avoid falling, our central nervous system needs to
6
activate muscles to accelerate body segments and restore angular momentum across multiple
recovery steps (Liu et al., 2018; Martelli et al., 2013).
Previous investigators have used peak values of WBAM (Martelli et al., 2013; Park et al.,
2020), peak-to-peak range (Herr and Popovic, 2008) ,the rate of change (Nott et al., 2014), and
integrated angular momentum (Liu et al., 2018; Potocanac et al., 2014) to describe the whole-
body rotational behavior. The peak values of WBAM in either direction (positive/negative) can
be used as an indirect measure of the amount of effort needed to restore the body toward the
equilibrium position during falling. The peak-to-peak range of WBAM can be used to infer how
well WBAM is regulated during different tasks. The rate of change in angular momentum equals
to the sum of external moment about CoM. And lastly, the integrated WBAM during each step
indicates the amount of body rotation and how well the segmental angular momenta cancel
during each step.
While biomechanists mostly compute angular momentum about an axis projecting
through the CoM to quantify balance during walking (Herr and Popovic, 2008; Liu et al., 2018;
Martelli et al., 2013), computing angular momentum about the axis through the CoP or the edge
of the BoS may provide additional information about CoM dynamics. Referencing angular
momentum to the axis through the foot contact point with the ground can capture the forward
walking dynamics in the sagittal plane as an inverted pendulum model. In this case, the superior
segments with larger mass such as the trunk show larger magnitude of segment angular
momentum about an axis that projects through contact point between the foot and the ground
(Gaffney et al., 2017). Additionally, computing WBAM about the foot contact axis can be used
to predict the range of the next foot placement region that will result in a static standing or
continuous walking for a biped model (Millard et al., 2009; Wight et al., 2008). Thus, computing
7
WBAM about the CoM axis is not the only appropriate choice. It may be possible to gain further
insights about how humans control balance by computing the angular momentum with respect to
the axis that project through the edge of BoS.

Another metric to characterize dynamic stability that is derived from mechanics is the
maximum Floquet multiplier (FM). This metric is commonly used to assess the rate of
divergence/convergence from a fixed point or steady state in response to small perturbations
(Dingwell and Kang 2007; Hurmuzlu and Basdogan 1994; Kuo 1999). This measure is based on
the fact that human walking is strongly periodic and can be characterized as a limit cycle
attractor that human would return to steady-state of walking following small perturbations
(Dingwell and Kang, 2007). Previous studies have established that the maximum FM remains
below one during unperturbed walking (Dingwell and Kang 2007; Hurmuzlu and Basdogan
1994; Granata and Lockhart 2008; Bruijn, et al., 2009), which indicates that small perturbations
always converge toward a limit cycle. The maximum FM increases when walking in
destabilizing environments, but still remains below one as people are able to use proactive and
reactive control to maintain balance (McAndrew et al., 2012). WBAM and the maximum FM
capture different aspects of reactive control during walking, and together provide a detailed
description of the control of dynamic balance.

In addition to the measures that characterize whole-body dynamics, we assess the
relationship between the base of support and CoM dynamics using Margin of Stability (MoS).
The metric is based on the inverted pendulum model of human walking behavior and describes
the relationship between CoM dynamics and the contact point between the foot and ground. In
8
this model, we use the point mass of the pendulum to represent the whole-body CoM and
characterize the relationship between CoM and BoS. Traditionally, the relationship between
CoM and BoS assessed during standing shows that the projection of CoM on the ground has to
be within the BoS region for the body to remain balanced (Kuo, 1995; Patla et al., 1990).
However, this relationship does not take into account the CoM velocity which is particularly
important for walking because walking is not static, and the projection of the CoM is typically
outside the BoS during the single support phase in the sagittal plane. Thus, we have to take into
account both the position and velocity of CoM relative to BoS (Hof, 2008; Hof et al., 2005; Pai
and Patton, 1997).
The MoS is defined as the distance between the edge of BoS and the ‘extrapolated center
of mass (XCoM)’ in both anteroposterior direction and mediolateral direction (Hof et al., 2005) ,
and accounts for both CoM position and velocity derived from a linearized inverted pendulum
(Eqn. 2-4). The XCoM is derived from the zero-energy orbit and can be computed as the sum of
CoM position (x) and a distance term proportional to the horizontal CoM velocity (v).
Theoretically, the model will come to a stop in the upright position by placing the CoP exactly at
the XCoM (Derivation can be found in Hof 2008). Thus, when progressing forward, the CoP
should be posterior and lateral to XCoM to maintain lateral balance and keep walking forward.

𝑋𝑋𝑋𝑋𝑋𝑋 𝑋𝑋 � � � � � � � � � � � � ⃗
= 𝑥𝑥 ⃗ +
𝑣𝑣 ⃗
𝜔𝜔 0

(2)

𝜔𝜔 0
= �
𝑔𝑔 𝑙𝑙
(3)

𝑋𝑋𝑋𝑋 𝛥𝛥
� � � � � � � � � � ⃗
= 𝐵𝐵 𝑋𝑋 𝛥𝛥 � � � � � � � ⃗
− 𝑋𝑋𝑋𝑋𝑋𝑋 𝑋𝑋 � � � � � � � � � � � � ⃗

(4)
Here, l is leg length, g is the gravity constant equal to 9.81. BoS is commonly defined in
the mediolateral direction using the 5th metatarsal position to calculate mediolateral MoS, in the
9
anteroposterior direction using 1st distal phalanx of the leading limb to calculate the forward
MoS, and using the heel of the trailing limb to calculate the backward MoS (Buurke et al., 2020).
The MoS is defined to be proportional to the minimum impulse needed to unbalance
people so that XCoM will fall out of the BoS (Hof et al., 2005). The MoS thus implies that,
given the position and velocity of the CoM, how far the inverted pendulum would move beyond
the BoS leading to instability so that a stepping response is necessary to maintain balance (A. L.
Hof et al. 2005). MoS can be directly modulated by modifying the spatiotemporal gait
parameters (Buurke et al., 2019; Hak et al., 2012). Hak et al. found that increase cadence would
increase ML MoS and shortening step length would increase backward MoS (Hak et al., 2012).
Also, a decrease in single support time or an increase in double support time would decrease the
time for the CoM to fall before foot placement to reduce the sway of CoM. MoS in the
mediolateral direction would thus be increased by widening the steps and by increasing the
double support time (Buurke et al., 2019).
One potential goal of walking may be to maintain a desired MoS that balances trade-offs
be between energy cost and stability, even during perturbed walking. MoS in the mediolateral
direction is consistent when walking on smooth and rough surfaces (Curtze et al., 2011) although
people increased step width when negotiating with more uneven surfaces. In the contrary, during
destabilized conditions, subjects walked with a higher step width in the lateral directions but
smaller MoS in the AP direction (McAndrew Young et al., 2012).

Several reactive strategies are generally established to stabilize gait (van den Bogaart et
al., 2020). One of the strategies to control balance is referred to as the foot placement strategy,
and it involves adjusting the placement of the swing leg relative to the body. When perturbations
10
occur, an adequate adjustment in foot placement needs to be implemented so that the resulting
reaction forces can help restore balance. For example, when a person is falling toward the right,
the next foot placement also needs to be placed toward the right of its baseline position to
produce a compensating leftward force to redirect the CoM and avoid a fall. A second
mechanism is the ‘ankle strategy’, and this involves shifting the center of pressure of the ground
reaction force of the stance leg with respect to the projection of both the CoM’s position and
velocity. This strategy involves actively modulating ankle moments of the stance foot (Winter,
2009). Another potential mechanism to stabilize gait is by actively rotating the body segments to
change the angular momentum of segments about the axis projecting through the center of mass
(Herr and Popovic, 2008; Hof, 2007). In this case, the interlimb responses to perturbations can
restore stability by generating changes in angular momentum that counteract the body's rotation
toward the ground.
Human can flexibly coordinate multiple strategies to restore balance when encountering
perturbations. For example, when people trip over an obstacle during mid-swing, the swing leg
foot placement usually occurs much later than when the stance limb starts to reduce forward
momentum during the push-off phase (Pijnappels et al., 2004). The latency between the tripping
perturbation and EMG onset at the stance leg can be as quick as ~70ms in healthy young adults
so that the support limb can increase the stance time to help the swing leg to regulate foot
placement (Pijnappels et al., 2004, 2005). Moreover, the stance limb can generate backward
angular momentum about the body CoM during the push-off phase to help arrest the forward fall.
Additionally, people can develop compensatory strategies during walking when forced to change
walking dynamics (Vlutters et al., 2018). Vlutters et al. first showed that people changed their
foot placement and ankle moments during pelvis perturbations during normal walking. Then,
11
participants put on a rigid ankle orthosis that reduced the base of support by attaching a pin
under the orthosis and restricted ankle modulation during perturbation response. In this case,
people mostly relied on foot placement adjustment for recovery during pelvis perturbations when
the ability to modulate ankle moment modulation became ineffective. Overall, healthy
individuals can coordinate multiple reactive stabilization strategies to maintain balance.

The foot placement variability observed during walking may reflect how humans respond
to externally or internally generated perturbations as our central nervous system integrates
sensory information to make adjustments in motor commands. For example, to recover from a
perturbation in the anterior-posterior direction, one simple way to correct for this perturbation is
to place the foot away from the body CoM so that the leading leg force has a greater fore-aft
component to arrest excessive body rotation and restore balance. Thus, modulating foot
placement from step-to-step is an important strategy for humans to maintain balance.
Step-to-step balance corrective strategies can be derived by relating foot placement to the
body’s state using a data-driven approach. This has been done for both healthy young
participants (Joshi and Srinivasan, 2019; Rankin et al., 2014; Roden-Reynolds et al., 2015; Wang
and Srinivasan, 2014) and populations who are at risk for falls such as older adults and people
post-stroke (Arvin et al., 2018; Dean and Kautz, 2015). Given a nominal trajectory, the deviation
of CoM state from this trajectory explains the deviation of the next foot placement location
through a linear mapping for healthy people during steady walking on the treadmill (Wang and
Srinivasan, 2014). These experimentally derived foot placement mappings can explain ~80% of
the variance in foot placement in the mediolateral direction and ~30% of the variance in the
anteroposterior direction at the midstance (Wang and Srinivasan, 2014). The high predictive
12
power, especially in the mediolateral direction, may indicate that our central nervous system uses
information about the body’s state to actively control the next foot placement during unperturbed
walking. The foot placement mapping derived from older adults who are at risk of falls shows
less predictive power compared to that for young adults (Arvin et al., 2018). Although a few
studies have investigated foot placement control strategies during external perturbations in
healthy populations (Joshi and Srinivasan, 2019; Vlutters et al., 2016), it remains to be seen
whether foot placement mapping derived from normal walking can be generalized to foot
placement during perturbation responses.

A stroke occurs when a blood vessel is either blocked by a clot (ischemic stroke) or
ruptured (hemorrhagic stroke). Each year almost 800,000 individuals suffer a stroke in the
United States, 87% of which are ischemic and 10% are hemorrhagic stroke (Virani et al., 2020).
Interruption of blood flow leads to neuronal death and disrupts the descending motor pathways
that control body movement. Following damage to the central nervous system, a sequence of
motor recovery is typically observed (Brunnstrom, 1970). Immediately after stroke, individuals
will go through a flaccid period during which movement cannot be initiated. During the initial
recovery stage, patients typically lose independent joint control (Dewald et al., 1995).
Movements that normally involve a specific joint require the involvement of a group of joints
instead. Later, patients can regain more independent control of joints (Brunnstrom, 1970). These
deficits are most prominent on the paretic side which is contralateral to the brain lesion. Other
motor deficits include muscle weakness due to weaker descending commands, fewer functioning
motor units, changes in muscle fiber type, and decreased motor unit firing rate (Arene and
Hidler, 2009). Additionally, there may be persistent spasticity or hyperexcitability of the stretch
13
reflex on the paretic side after stroke (Hsu et al., 2003). These changes in the lower extremities
lead to functional deficits at the paretic side.

Sensorimotor impairments after stroke often lead to spatiotemporal asymmetries during
walking. These asymmetries may result from difficulty advancing the paretic leg or standing on
the paretic leg. For example, a reduction in ankle plantar-flexor moment in the paretic limb
might lead to a reduction in paretic propulsion (Allen et al., 2014; Lauzière et al., 2015). If
people are not able to generate enough propulsion at the paretic leg, they cannot progress the
trunk forward during the paretic stance phase, so the non-paretic step lengths become shorter
than the paretic step lengths (Roerdink and Beek, 2011)(Figure 2A). On the other hand, people
post-stroke may also have decreased hip flexor moments, and thus have difficulty swinging their
paretic limb forward (Balasubramanian et al., 2007) (Figure 2B). As a result, they may take
shorter paretic steps. Therefore, people post-stroke can have step length asymmetries in two
directions, either taking shorter paretic or shorter non-paretic steps, depending on their
impairments (Allen et al., 2011; Sánchez and Finley, 2018).
14

Figure 2: Diagram showing that people post-stroke can have step length asymmetries in two
directions, either taking shorter paretic or shorter non-paretic steps. The top row (A) shows that
people post-stroke take longer paretic steps than non-paretic steps due to the deficits at the
paretic propulsion to advance the trunk forward at non-paretic step. The bottom row (B) shows
that people post-stroke take shorter paretic steps than non-paretic steps by placing the paretic
leg closer to the body CoM at the paretic foot strike.

One common objective of rehabilitation post-stroke is to reduce gait asymmetries
(Patterson et al., 2015; Reisman et al., 2013). Due to sensorimotor deficits post-stroke such as
muscle weakness (Chen et al., 2005; Olney et al., 1996) and abnormal muscle coordination
patterns (Cruz and Dhaher, 2008; Sánchez et al.), up to half of the stroke survivors in their
chronic phase exhibited spatiotemporal asymmetries during walking (Patterson et al., 2008).
Lewek et al. have found that step length asymmetries were correlated with scores on the Berg
Balance Scale, suggesting that gait asymmetries are associated with high fall risk in these
individuals (Lewek et al., 2014). Although this study has demonstrated an association between
15
asymmetry and measures of balance control, it remains uncertain if spatiotemporal asymmetry
alone is causally associated with balance control.

Among people post-stroke, up to 70% of them fell at least once every year (Sackley et al.,
2008; Watanabe, 2005), and most of the falls are related to walking (Weerdesteyn et al., 2008).
The high fall risk reduces the ease and confidence for people post-stroke to ambulate safely in
everyday life, especially when walking in challenging conditions. Using reactive control
strategies to maintain balance is essential for people post-stroke as falls often occur when people
are unable to generate corrective response to recover from trips or slips. However, due to
functional deficits in the lower extremities, people post-stroke commonly demonstrate delayed
muscle activation to external perturbations (Kirker et al., 2000; Marigold et al., 2004), abnormal
muscle activation patterns (Higginson et al., 2006), and muscle weakness (Olney and Richards,
1996), especially at the paretic side. These sensorimotor deficits after brain injury may contribute
to an increased risk of falls. Although improving reactive balance control is a common objective
in post-stroke gait rehabilitation, we do not fully understand how post-stroke sensorimotor
deficits would influence the regulation of whole-body dynamics during reactive stepping
responses.
Non-disabled people use multiple strategies to reduce the effects of external perturbations
(Mathiyakom and McNitt-Gray, 2008). For example, when perturbations occur at the heel strike
as posteriorly-directed slips, these perturbations will induce forward rotation of body and
generate forward momentum about the center of mass (CoM) (Figure 3). One way to recover
from falling is to first increase the trailing limb ankle push-off to increase the swing speed of the
trailing limb (Maki and McIlroy, 2006; Zelik and Adamczyk, 2016). Swinging the leg forward
16
could generate backward momentum about the body CoM and cancel out the forward
momentum due to this perturbation. Another effective way to reduce the forward body rotation
induced by this perturbation is to place the swing foot more anteriorly relative to the body’s
center of mass. This foot placement and the corresponding ground reaction force would generate
backward moments at the CoM to help arrest the forward rotation of the body. Meanwhile,
people could modulate the trailing limb ground reaction force primarily in the anterior direction
during the push-off phase of the gait cycle to reduce forward angular momentum following the
forward falls (Pijnappels et al., 2005). These strategies coordinate together to counteract the
effect of external perturbations that induced forward falls.

Figure 3: People use multiple reactive control strategies to reduce the effect of forward fall.
Green arrows here indicate the forward angular momentum induced by the perturbation. GRF:
ground reaction force.

However, these reactive control strategies are impaired in people post-stroke. For
instance, if a slip-like perturbation occurs during the paretic stance and the paretic leg is too
weak to support the body, the non-paretic swing leg may not have sufficient time to swing
forward and arrest the body’s angular momentum. Conversely, when the perturbation occurs on
the non-paretic side, the paretic leg may have difficulty initiating a successful stepping response
17
to help restore the balance as the paretic leg demonstrates decreased propulsion (Allen et al.,
2014; Chen et al., 2005; Lauzière et al., 2015) and decreased hip flexor activity (Rybar et al.,
2014) compared with their non-paretic side. As a result, for example, people post-stroke
demonstrate shorter paretic compensatory step lengths than non-paretic step lengths following
the backward slips induced by a forward movable platform translation during walking near foot
strike (Kajrolkar and Bhatt, 2016). Thus, people post-stroke will likely show differences in
reactive stabilization strategies at their paretic and non-paretic sides to restore balance following
perturbations.

While non-disabled people are capable of regulating their WBAM within a small range,
people post-stroke who usually have gait asymmetries are shown to have an increased range of
WBAM in both the sagittal plane (Honda et al., 2019) and the frontal plane (Nott et al., 2014)
compared to age-matched controls. Increases in angular momentum may result from impaired
interlimb and intralimb coordination for people post-stroke. During normal walking for non-
disabled people, the swing leg and stance leg rotates in the opposite direction about the CoM so
that the angular momentum for the two legs are in anti-phase. With the presence of gait
asymmetries, this interlimb cancellation will be lessened, and thus the magnitude of WBAM
would increase. Additionally, smaller values of WBAM in the sagittal plane and frontal plane
during normal walking are associated with better balance assessment scores in people post-stroke
(Park et al. In review), suggesting that gait asymmetries in people post-stroke may contribute to
impaired balance control. Therefore, if we modify their walking patterns to reduce spatial
asymmetries, reduction in step length asymmetry may improve limb angular momentum
cancellation and thus improve balance control during perturbation responses.
18

The high fall risk in post-stroke population reduces the ease and confidence for people
post-stroke walk safely in everyday life, especially when walking in challenging conditions.
Non-disabled people can rapidly and adequately respond to perturbations by selecting the most
appropriate strategies to restore balance. However, brain injury after stroke disrupts the
descending motor pathways and leaves people post-stroke with limited motor control options to
restore balance during perturbation responses. Thus, we aim to identify the contributors to
impaired reactive balance control. One common assumption is that this increased fall risk may be
associated with deviations in some specific features from the normal gait, such as
spatiotemporally asymmetric gait patterns as opposed to almost symmetric ones observed for
healthy walking (Lewek et al., 2014). As a result, rehabilitation practices have focused on
reducing asymmetric walking. However, whether there is a causal relationship between gait
asymmetries and the ability to maintain balance in response to trips or slips during walking has
yet to be established.
Moreover, trainings that are effective to reduce fall risk for older adults, such as balance
and functional training, have not been successful in reducing rate of falls among people post-
stroke (Batchelor et al., 2012; Chang et al., 2004; Ng et al., 2019; Verheyden et al., 2013). The
lack of effectiveness calls for more task-specific perturbation training and mechanistic studies
about the specific contributors to impaired reactive control of balance among people post-stroke.
We also expect to identify individual differences in their reactive responses to perturbations due
to the heterogeneous functional impairments and the compensatory walking pattern developed
after stroke. We expect that the differences in reactive response will be associated with clinical
assessments of balance control and motor impairment levels. Knowledge of individual
19
differences could lead to the design of better personalized balance training to reduce fall risks.
Moreover, our findings will inform the design of rehabilitation programs that seek to improve
balance control and mobility for other patient populations with asymmetric walking patterns,
such as amputees and people with Parkinson disease who also tend to have impaired ability to
correct for perturbations due to sensorimotor deficits.

20
Chapter 2. Choice of Reference Axis for Computing Angular Momentum
Affects Inferences about Dynamic Balance

One common measure of dynamic balance in human walking is whole-body angular
momentum (WBAM). To compute WBAM, one must specify a reference axis about which
momentum is calculated. Momentum-based controllers for humanoid robots may use axes that
project through either the center of mass (CoM) or the center of pressure. However,
biomechanists primarily compute angular momentum about an axis projecting through the CoM
to quantify balance during walking. Here, we asked if the choice of the reference axis influences
interpretations of how dynamic balance is controlled during perturbed walking. Eleven healthy
young individuals walked on a dual-belt treadmill at their self-selected speed. Sudden treadmill
accelerations of different magnitude and direction were remotely triggered at foot strike, and belt
speed returned to the self-selected speed during swing. Two mediolateral reference axes were
used for computing WBAM in the sagittal plane: an axis projecting through 1) the CoM or 2) the
leading edge of the base of support (BoS) as estimated by a marker on the first phalanx. Both
axes were defined as positive to the person’s right. We also performed principal component
analysis to characterize the intersegmental coordination patterns during the imposed
perturbations and assessed whether the choice of reference axis influences our interpretations of
how intersegmental coordination patterns contribute to regulation of angular momentum during
perturbation recovery. We found that the peak WBAM scaled with perturbation size and the
slope of this relationship did not differ between reference axes. We also found that one
advantage of using a reference axis that projects through the CoM is that it allowed us to more
easily identify the degree of segmental angular momentum cancellation during perturbations.
21
Analysis of coordination patterns referenced to an axis projecting through the leading edge of
support may provide more insights about how the upper body responded to sudden loss of
balance.

Human locomotion is inherently unstable due to the small base of support, long single-
limb support times, and sensorimotor transmission delays (Winter, 1995; Woollacott and Tang,
1997). When walking in the anteroposterior direction, the body center of mass (CoM) constantly
moves out of the base of support (BoS) during the single support phase, which poses a challenge
for maintaining dynamic balance. One way to generate corrective responses to maintain balance
in response to both internal and external perturbations is through the reactive control of balance,
which involves the use of feedback about the body’s state to generate balance correcting
responses (Patla, 1993; Tang et al., 1998). For example, people can actively rotate both the upper
and lower limbs in a coordinated way to help restore stability by generating changes in angular
momentum to counteract the body’s rotation toward the ground.
One common measure to capture dynamic balance is whole-body angular momentum
(WBAM). This measure reflects the net contribution of all body segments to the body’s rotation
about a specified reference axis. To compute WBAM in the sagittal plane, biomechanists
typically choose to calculate angular momentum about the axis projecting through the CoM
(Bennett et al., 2010; Herr and Popovic, 2008; Thielemans et al., 2014). WBAM about CoM axis
is highly regulated during normal human locomotion as the peak-to-peak range of WBAM is
much smaller than the angular momentum of single segments due to momentum cancellation
between the limbs (Herr and Popovic, 2008; Popovic et al., 2004). This tight regulation indicates
that segment rotations about the CoM are coordinated in such a way that WBAM remains small
22
during walking. People can regulate WBAM about CoM well under various conditions such as
walking at different speeds (Bennett et al., 2010), walking with different step lengths
(Thielemans et al. 2014), or during stair ascent/descent (Silverman et al., 2014). WBAM about
an axis projecting through the CoM can also capture changes in body dynamics during
perturbation responses and subsequent compensatory reactions, which typically result in a sharp
increase in angular momentum from that measured during unperturbed walking (Liu et al., 2018;
Martelli et al., 2013; Pijnappels et al., 2004, 2005; Potocanac et al., 2014). These deviations in
WBAM capture the features of body rotation that could lead to fall during perturbations.
While biomechanists mostly compute angular momentum about an axis projecting
through the CoM to quantify balance during walking (Herr and Popovic, 2008; Liu et al., 2018;
Martelli et al., 2013), computing angular momentum about the axis through the edge of the BoS
may provide additional information about CoM dynamics. Referencing angular momentum to
the axis through the foot contact point with the ground can capture the forward walking
dynamics in the sagittal plane similar to the inverted pendulum model of walking (Cavagna et
al., 1977; Kuo et al., 2005; Lee and Farley, 1998). In this case, the superior segments with larger
mass, such as the trunk, show larger magnitudes of segmental angular momenta about an axis
that projects through the point where the foot contacts the ground during walking (Gaffney et al.,
2017). Additionally, computing WBAM about the foot contact axis and using the conservation of
angular momentum at the foot contact transition can predict the range of the next foot placement
region that will allow the biped model to continue steady-state walking (Millard et al., 2009;
Wight et al., 2008). Thus, there is no guarantee that computing WBAM about CoM is the only
appropriate choice to gain insights about how human control balance during walking.
23
Furthermore, we need to investigate how people coordinate their limb segments to
counterbalance the variations in the whole-body angular momentum.
Dimensionality reduction techniques, such as principal component analysis (PCA), are
commonly used to capture how the central nervous system coordinates multiple limb segments
during reactive control of balance (Aprigliano et al., 2017; Chiovetto et al., 2018; Liu and Finley,
2020; Martelli et al., 2013). PCA reduces high-dimensional data into a lower-dimensional set of
latent variables to capture the maximum variance(Zou et al., 2006). Studies used PCA of
segmental angular momenta to infer the segmental cancellation from the extracted coordination
patterns during walking. For example, as the right and left limbs rotated in anti-phase in the
sagittal plane about CoM when calculating WBAM referenced to CoM axis, PCA could capture
such momentum cancellation between the left and right limbsby showing the weighting
coefficients of the left and right limbs in the extracted coordination patterns in opposite signs
(Herr and Popovic, 2008; Liu and Finley, 2020). During the perturbation response, the degree of
segmental cancellation decreases so that WBAM increases sharply. However, it is unclear if
whether extracting segmental coordination patterns using measures of angular momentum
referenced to the edge of the BoS would provide similar interpretations about how humans
coordinate body segments as would occur if measures of momentum were referenced to the
CoM.
The objective of this study is to determine whether using different reference axes to
compute WBAM would influence our interpretations of dynamic balance control during
treadmill-elicited perturbations. We hypothesized that the peak WBAM referenced to either axis
would be positively associated with the change in speed during a perturbation as WBAM
captures how much people are perturbed. We tested this hypothesis by imposing both
24
posteriorly-directed and anteriorly-directed perturbations on a dual-belt treadmill. We also
hypothesized that intersegmental coordination patterns extracted using segmental angular
momenta referenced to CoM axes would provide us intuition about the segmental cancellation of
distal segments while referenced to BoS would provide insights about the upper-body segments’
dynamics following perturbations. Ultimately, our findings will extend our understanding of how
the healthy central nervous system coordinates intersegmental dynamics to maintain balance
during perturbation responses during walking.

A total of 11 healthy young individuals (5M, 27±3yrs old, 67.9±17.2kg, self-selected
walking speed = 1.0±0.1m/s) with no musculoskeletal or gait impairments participated in this
study. Participants self-reported the right side as their dominant limb when asked which leg they
would use to kick a ball. The study was approved by the Institutional Review Board at the
University of Southern California, and all participants provided informed consent before
participating. All aspects of the study conformed to the principles described in the Declaration of
Helsinki.

Participants walked on an instrumented, dual-belt treadmill with force plates underneath
(Bertec, USA) for six separate trials and reacted to accelerations of the treadmill belts throughout
the experiment. Their self-selected walking speed was first determined using a two-alternative
forced-choice staircase method (Dal et al., 2010; Farrens et al., 2020; García-Pérez, 1998). We
first determine the participants’ overground walking speed using the 10-meter walk
test(Bohannon, 1997). Then we used 50% of their overground walking speed as a starting point
25
to determine their self-selected walking speed on a treadmill. Treadmill speed was incrementally
increased and decreased by 0.02m/s, and the participant was asked if they were walking at their
self-selected speed after each change until their self-selected walking speed was found using a
custom Matlab program. For the first trial, participants walked on the treadmill for five minutes
with their self-selected walking speed. We informed the participants that no treadmill-induced
perturbations would occur during this trial. Then, for five subsequent trials (two participants
completed four perturbation trials), we informed the participants that treadmill perturbations of
different magnitude and direction would occur at random foot strikes. Each trial consisted a total
of 24 perturbations with 12 on each belt. The perturbations had magnitudes at ±0.3m/s, -0.4m/s,
±0.5m/s, 0.7m/s, where a positive values indicated an increase in speed relative to the
participant’s self-selected walking speed, and negative values corresponded to reductions in the
participant’s self-selected walking speed. The order of these perturbations was randomized. Foot
strike was computed as the point when vertical ground reaction forces reached 80 N. Each
perturbation was remotely triggered by customized Matlab code (Matlab 2020b, Mathworks,
USA). The perturbation was characterized by a trapezoidal speed profile in which the treadmill
accelerated at foot strike to the target belt speed at an acceleration of 3 m/s
2
(or -3 m/s
2
if the
target speed was less than their walking speed), held this speed for 0.7 s, and then returned back
to their self-selected walking speed at an acceleration of -3 m/s
2
(or 3 m/s
2
). The perturbations
were randomly triggered to occur within a range of 15 to 25 steps after the previous perturbation
to prevent participants from anticipating perturbation timing. We also selected this range of steps
to provide participants with sufficient time to reestablish their baseline walking pattern.
26

A ten-camera motion capture system (Qualisys AB, Gothenburg, Sweden) recorded 3D
marker kinematics at 100 Hz and ground reaction forces at 1000 Hz. We placed a set of 14 mm
spherical markers on anatomical landmarks to create a 13-segment, full-body model (Havens et
al., 2018; Song et al., 2012). We placed marker clusters on the upper arms, forearms, thighs,
shanks, and the back of heels. Marker positions were calibrated during a five-second standing
trial. We removed all joint markers after the calibration.

We post-processed the kinematic and kinetic data in Visual3D (C-Motion, Rockville,
MD, USA) and Matlab 2020b (Mathworks, USA) to compute variables of interest. Marker
positions and ground reaction forces were lowpass filtered by 4
th
order Butterworth filters with
cutoff frequencies of 6 Hz and 20 Hz, respectively (Kurz et al., 2012; Reisman et al., 2009;
Winter, 2009). We defined foot strike as the point when the vertical ground reaction force
reached 80N. We also examined the timing of perturbations relative to foot strike post-hoc to
remove the perturbations that occurred ~150ms after the foot-strike (Buurke et al., 2020). We
categorized pre-perturbation (Pre-PTB) steps as the two steps before the perturbation occurred
and perturbation (PTB) steps as the step during which the perturbation was applied. For treadmill
perturbations that elicited a backward loss of balance, some participants generated an aborted
step during the PTB steps (Bhatt et al., 2005). We defined an aborted step as one in which the
trailing leg unloaded and landed again after the perturbation was initiated without the
contralateral perturbed leg being lifted off of the ground (Bhatt et al., 2005). In this case, we
defined the perturbation steps to end with trailing foot landing on the treadmill. For each
participant, we included 9 ± 1 perturbations for each perturbation amplitude per side. We also
27
focused our analysis on angular momentum in the sagittal plane as this was the direction in
which the most prominent changes in WBAM were observed.

We created a 13-segment, whole-body model in Visual3D and calculated the angular
momentum of each segment about the body’s center of mass. Segmental angular momenta ( 𝐿𝐿 𝑠𝑠 i
)
captured how the rotational and translational behavior of each body segment changed in response
to the treadmill perturbations. The model included the following segments: head, thorax, pelvis,
upper arms, forearms, thighs, shanks, and feet. The limb segments’ mass was modeled based on
anthropometric tables (Dempster, 1955), and segment geometry was modeled based on the
description in Hanavan (1964). Segmental linear and angular velocity were computed using Eqn.
1 (Silverman and Neptune, 2011).

𝐿𝐿 𝑠𝑠 i
= 𝑚𝑚 𝑖𝑖 � 𝑟𝑟 𝑅𝑅𝑅𝑅𝑅𝑅 − 𝑖𝑖 𝑖𝑖 × 𝑣𝑣 𝑅𝑅𝑅𝑅𝑅𝑅 − 𝑖𝑖 𝑖𝑖 � + 𝐼𝐼 𝑖𝑖 𝜔𝜔 𝑖𝑖
(1)

Here, mi is segmental mass, rRef-i is the displacement from the segment's COM to the
reference axis, vRef-i is the velocity of each segment’s COM relative to the reference axis, I
i
is the
segmental moment of inertia, ω
𝑖𝑖 is segmental angular velocity, and the index i corresponds to
individual limb segments. Two reference axes were used for computing WBAM: a mediolateral
axis projecting through 1) the CoM (LCoM) or 2) the leading edge of the base of support (BoS) of
the stance limb as estimated by a marker on the first phalanx (LBoS). Both axes were defined as
positive to the person’s right. LBoS was calculated only during the single support phase for each
step since we were interested in the pendular mechanics captured by LBoS.
Whole-body angular momentum was calculated as the sum of all segmental angular
momenta using Eqn. 2.
28

𝐿𝐿 =
∑ 𝐿𝐿 𝑠𝑠 i
𝒊𝒊 𝑋𝑋 𝑀𝑀𝑀𝑀
(2)
We normalized momentum by the participant’s mass (M), self-selected treadmill velocity
(V), and the participant’s height (H) (Eq. 2) following previous literature (Herr and Popovic,
2008; Honda et al., 2019; Liu and Finley, 2020). The convention for measuring angular
momentum was defined such that positive values represented backward rotation. We used the
maximum and minimum value of LCOM and LBoS during the perturbation step to quantify the
effect of perturbations(Martelli et al., 2013). Thus, the maximum value of L indicated peak
backward rotation and the minimum value of L indicated peak forward rotation. Lastly, we could
use the parallel axis theorem (Eqn.3) to assess the relationship between LBoS and LCOM.
Here, M is the participant’s mass, 𝑟𝑟 ⃗
𝐵𝐵𝐵𝐵 𝐵𝐵 − 𝐶𝐶 𝐵𝐵 𝐶𝐶 represents the displacement from the leading edge of
BoS to the body CoM, and the 𝑣𝑣 ⃗
𝐵𝐵𝐵𝐵 𝐵𝐵 − 𝐶𝐶 𝐵𝐵 𝐶𝐶 represents the velocity of the leading edge of BoS
relative to the body CoM.

We used sparse principal component analysis (sPCA) to extract intersegmental
coordination patterns for each step cycle (Chiovetto et al., 2018). Sparse principal component
analysis (SPCA) is a variation of principal component analysis to allow for better interpretability
for the principal components (PCs) by setting the weighting coefficients of segmental angular
momenta with small variance to be zero(Zou et al., 2006). Such a parsimonious model is
obtained via a sparsity promoting regularizer using a combination of lasso (L1) and ridge (L2)
regression (Erichson et al., 2020). Before performing sPCA, we first time normalized the
segmental angular momenta to 100 points for each step cycle. Then, for each participant, we
generated a 𝐿𝐿 𝑠𝑠 matrix for each step type (Pre-PTB and PTB steps). We only included
L
B oS
= L
C oM
+ 𝑋𝑋 𝑟𝑟 ⃗
𝐵𝐵𝐵𝐵 𝐵𝐵 − 𝐶𝐶 𝐵𝐵 𝐶𝐶 × 𝑣𝑣 ⃗
𝐵𝐵𝐵𝐵 𝐵𝐵 − 𝐶𝐶 𝐵𝐵 𝐶𝐶 (3)
29
perturbations of the right (dominant) side for this analysis. The 𝐿𝐿 𝑠𝑠 matrix had dimensions of
n_steps*100 rows and 13 columns, one for each segment. We created 7 matrices per participant,
including Pre-PTB steps and perturbation steps for each perturbation amplitude (6 levels). We
then standardized each matrix to have zero mean and performed sPCA to extract subject-specific
coordination patterns using the spca function in R (version 3.6.1). Using sPCA, we decomposed
the segmental angular momenta data into 1) a weighting coefficient matrix consisting of PCs
ordered according to their variance accounted for (VAF) and 2) time series scores which
represented the activation of each PC throughout the step cycle. PCs were ranked by how much
variance in the data was explained by each of them. The sparsity controlling parameter was
selected based on the cumulative variance explained by the first three PCs. We retained the
number of PCs necessary to account for at least 90% (95% CI) of variance in 𝐿𝐿 𝑠𝑠 .

To assess the similarity between PCs across different participants, we first sorted the PCs
to match the similar PCs across participants (Chiovetto et al., 2018). Each participant’s PCs were
matched to that of one reference participant by calculating the scalar product (r) of the two PCs
as this is a common method to compare the similarity between vectors in a high-dimensional
space. The scalar product of the unit vectors was between 1 (parallel and identical) and 0
(orthogonal and most dissimilar). The PC that was the most similar to the first PC of the
reference participant (scalar product closer to 1) was assigned as the first PC for the participant.
The PCs of the first best-matching pair were then removed from the PCs and we repeated the
same procedure to match the next pair of PCs and so forth.
30
Then, we assessed the similarity between PCs referenced to two different axes by
computing the scalar product to match the most similar PCs extracted from the segment angular
momenta referenced to axes projecting through the CoM and BoS. This analysis allowed us to
examine whether the coordination patterns referenced to two axes were similar. We first
identified the most similar PCBoS to PC1CoM within subject for each perturbation level. Then we
repeated the same process for PC1BoS. A pair of PCs were considered “similar” if r > 0.684,
which corresponded to the critical value of r
2
for vectors in a 13-dimensional space (since we
have 13 segments) at p = 0.01 (Allen et al., 2017). Otherwise, a pair of PCs was considered
‘dissimilar’.

We first determined if the minimum and maximum WBAM referenced to the axis
through BoS and CoM during the perturbation steps was associated with the perturbation
amplitude (Amplitude) and side of the perturbation (Side). Linear regression analyses were
performed in Matlab 2020b. We performed linear mixed-effects regression analyses to examine
the relationship between independent variables Speed, Side, and interaction between Speed and
Side, and each dependent variable of peak angular momentum (Max LCoM, Max LCoM, Min LBoS,
Min LBoS). For each regression model, we determined if a model with random effects provided a
better fit than a model with only fixed effects by comparing both models using Akaike
Information Criterion (AIC)(Akaike, 1981). We selected the model with a lower AIC. Based on
this analysis, we included random effects in all models to account for the individual differences
between subjects.
We also examined whether differences in variance explained were present between the
PCs extracted from segmental angular momenta referenced to the different axes by performing a
31
two-sample Welch’s t-test. To examine the similarity between PCs extracted from segmental
angular momenta referenced to two axes, we tested if the scalar product between the most similar
pairs of PCBoS and the first PCCoM and pairs of PCCoM and the first PCBoS was higher than critical
r value using a one-tailed t-test. A scalar product that was higher than the critical value indicated
that the two PCs were statistically similar (Allen et al., 2017). In this case, rejecting the
hypotheses indicated that the PCs were not similar. These analyses were performed in R. We
checked normality using the Shapiro-Wilk normality test. If any of the data were not normally
distributed, we log-transformed the data to ensure that they were normally distributed before
performing the two-sampled Welch’s t-test. We reported normally distributed values as mean ±
standard deviation of the corresponding mean and non-normally distributed data as median with
interquartile range (IQR), [25% IQR, 75% IQR]. Significance was set at p<0.05 level.

During pre-perturbation walking, LCoM in the sagittal plane was most negative during the
transition from the swing phase to the stance when the peak forward momentum occurred
(Figure 4A). Then, LCoM increased to become positive until mid-swing before becoming negative
again during late stance. The peak backward momentum occurred during mid-swing of each
step, which corresponds to ~25% and ~75% of the gait cycle.
For LBoS, during pre-perturbation walking, LBoS was the most negative at the beginning of
single support phase or at the end of the single support phase (Figure 4B). Participants increased
LBoS to be more positive until the mid-swing phase and then becomes more negative during the
late stance. LBoS magnitude was higher than LCOM as a result of the larger distance between most
segments and the reference axis.
32
Measures of angular momentum varied in response to changes in perturbation magnitude.
During treadmill accelerations, angular momentum about both axes became more negative as the
body rotated forward. Participants then generated more positive angular momentum during the
recovery step and initiated backward rotation. During treadmill deceleration, angular momentum
initially became more positive as the body rotated backward. Angular momentum deviated more
from the unperturbed walking when the treadmill speed changes became larger. Similar to what
was observed during the pre-perturbation steps, the peak backward angular momentum still
occurred around mid-swing and the peak forward angular momentum occurred during the
transition from swing to stance phase during the treadmill perturbations.

Figure 4: Whole-body angular momentum with respect to the mediolateral axis projecting
through (A) the BoS and (B) CoM during pre-perturbation steps, perturbation steps, and the
following recovery steps for one representative participant. Dashed lines: angular momentum
referenced to BoS and CoM during pre-perturbation walking. Colored lines: angular momentum
during perturbation steps and the first recovery steps. Arrows indicated the approximate mid-
swing phase for each step.

Peak backward (i.e. maximum) angular momentum in the sagittal plane was negatively
associated with perturbation amplitude regardless of the reference axis. The maximum LCoM (F
(1,128) = 492, p <0.001) and LBoS (F (1, 128) = 1923, p <0.001) during the perturbation steps
33
were both negatively correlated with perturbation amplitude (Figure 5A & B). The peak forward
(i.e. minimum) angular momentum during the perturbation steps were only negatively correlated
with the perturbation amplitude for LBoS (F(1,128) = 1084, p <0.001) but not for LCoM (p = 0.99)
(Figure 5C & D). The reason that we did not observe an association between peak forward LCoM
and perturbation amplitude could be that the minimum values usually occurred at the foot strike
before the perturbations had any effect on body dynamics. We did not find any significant
effects of Side or interaction between Side and Speed on any of the dependent variables.

Figure 5: Association between perturbation amplitude and mean peak whole-body angular
momentum referenced to CoM axis and BoS axis across participants (N = 11). (A-B) Maximum
LCoM and LBoS were negatively correlated with perturbation amplitude. (C) Minimum LCoM was
34
not correlated with perturbation amplitude. (D) Minimum LBoS were negatively correlated with
perturbation amplitude. Each dot represents one participant.

On average across all perturbation levels, two principal components accounted for more
than 90% of the variance in segmental angular momentum (Figure 6). PC1 explained 74 ± 6% of
the variance, PC2 explained 22 ±5% of the variance, and PC3 accounted for 3 ±2% of the
variance for angular momentum referenced to the CoM. The first two PCs accounted for 96%
(95%CI: 91 - 93%) of the variance. For segmental angular momentum referenced to the BoS,
PC1 explained 74 ±14% of the variance, PC2 explained 18 ±11% of the variance, and PC3
accounted for less than 5±3% of the variance. First two PCs accounted for 92% (95% CI: 91%,
93%) of the variance on average. Therefore, we retained two PCs for segmental angular
momentum referenced for both axes. We then tested if there was a difference in the variance
explained in segmental angular momentum based on the reference axis. The variance accounted
in segmental angular momentum for PCCoM was no different than that explained for PCBoS for
PC1 (p = 0.97), but more variance was explained for PC2 (t(109) = 2.6, p = 0.009) and PC3
(t(121) = -5.2, p < 0.001).
35

Figure 6: VAF for the first three PCs across all levels of perturbations and pre-perturbation
steps. Dark shaded: referenced to the axis through CoM; lightly shaded: referenced to the axis
through BoS. (*p<0.05).

We only included perturbations that occurred at the right (dominant) leg for the principal
component analysis and thus, we refer to the right limb as the ipsilateral perturbed limb and the
left limb as the contralateral limb. For LCoM, contributions from the lower extremities were
typically dominant in the first PC, while contributions from the arms, pelvis, trunk, and head
were negligible (Figure 7A). During perturbed steps, the contralateral limb was in the swing
phase and generated more positive momentum about the body's COM while the ipsilateral
perturbed limb generated negative momentum. Thus, the signs of the weighting coefficients for
the contralateral leg segments (left thigh, shank, and foot) were opposite to the weighting
coefficients the ipsilateral leg segments. Differences in the composition of each PC between pre-
perturbation steps and perturbations increased with perturbation magnitudes. Overall, the first PC
36
captured the opposing momenta of the two legs resulting from differences in the direction of
rotation relative to the body's CoM. For PC2CoM, the weighting coefficients for distal lower
extremity segments were also larger than the weighting coefficients for proximal segments,
although the coefficient for the thorax and head increased compared to that in PC1CoM (Figure
7C).
For LBoS, contributions from the proximal segments such as pelvis, thorax, and head, were
typically dominant in the first PC, while contributions from the lower extremities were much
lower than what we observed in PC1CoM (Figure 7B). Since the angular momenta were
referenced to the perturbed leg’s leading edge of support such that the entire body rotated similar
as an inverted pendulum, angular momentum for the upper body segments were positively
correlated during both pre-perturbation and perturbation steps, indicated by the positive
weighting coefficients of all these segments. For PC2BoS, contributions from the contralateral leg
segments were dominant (Figure 7D).
37

Figure 7: Principal components (PC) extracted from segmental angular momentum referenced to
the axis through CoM (left panel A, C) during perturbation steps. Principal components (PC)
extracted from segmental angular momentum referenced to the axis through BoS (right panel B,
D) during perturbation steps. (N=11). The segments include: RTH (right thigh), RSH (right
shank), RFT (right foot), LTH (left thigh), LSH (left shank), LFT (left foot), H (head), PEL
(pelvis), THX (thorax). All PC weights for arm segments (forearms and upper arms) are close to
zero, so they are not included in this figure to improve clarity. Error bars represented standard
deviation.
38

Figure 8: (A) Boxplot shows the scalar product between the PC1CoM and the most similar PCBoS
extracted from segmental angular momentum during pre-perturbation steps and perturbation
steps at different levels of speed change (*p<0.05). (B) Boxplot shows the scalar product
between the PC1CoM and the most similar PCBoS extracted from segmental angular momentum
during pre-perturbation steps and perturbation steps at different levels of speed change.
Horizontal dashed line indicates the critical r value = 0.684 for the two PCs to be similar in the
13-dimensional space. (C) PC1CoM and its matched PCBoS (D) during perturbation steps with -
0.3m/s speed change. Error bars represented standard deviation.

Lastly, we compared the similarity between intersegmental coordination patterns (i.e.
PCs) extracted from angular momentum referenced to two different axes. The majority (87%) of
the PCBoS that were similar to the first PCCoM were the second PCBoS. The scalar products
between the first PCCoM and the most similar PCBoS extracted during pre-perturbation steps and
39
perturbations at different levels of speed change were not significantly lower than critical r value
at 0.684 except when the speed change was 0.7m/s (t(10) = -1.8, p = 0.05) (Figure 8A). On the
other hand, the scalar product between the first PCBoS and the most similar PCCoM extracted
during pre-perturbation steps and perturbations at different levels of speed change were all
significantly lower than r = 0.684 (p<0.001) except for pre-perturbation steps (Figure 8B). Thus,
in general, the first PC that captured variance in segmental angular momenta about the BoS was
not similar to either of the PCs extract when momentum was computed relative to the CoM. To
summarize, PCCoM and PCBoS had one similar PC and one dissimilar PC for all levels of
perturbation steps except when the speed change was 0.7m/s.

The objective of this study was to investigate whether referencing WBAM to different
axes would influence our interpretations of dynamic balance control during treadmill-induced
perturbations. Consistent with our hypothesis, peak backward angular momentum during the
perturbed steps was positively associated with perturbation amplitude regardless of the reference
axis used to define angular momentum. In addition, the low-dimensional intersegmental
coordination patterns extracted when referenced to axes projecting through the BoS and CoM
had one similar component and one dissimilar component during perturbation steps, indicating
that these methods provide complementary information about how healthy people coordinate
their segments to maintain WBAM during perturbation responses.
The observation that the directions of the association between perturbation amplitude and
the peak backward LCoM or LBoS were the same between methods reflects the parallel axis
theorem (Eqn. 3). We observed that participants’ peak backward angular momentum in the
sagittal plane was negatively associated with changes in perturbation speed suggesting that
40
participants fell more forward as the treadmill speed suddenly increased and participants fell
more backward as the treadmill speed suddenly decreased. The second term in the Eqn.3
( 𝑋𝑋 𝑟𝑟 ⃗
𝐵𝐵𝐵𝐵 𝐵𝐵 − 𝐶𝐶 𝐵𝐵 𝐶𝐶 × 𝑣𝑣 ⃗
𝐵𝐵𝐵𝐵 𝐵𝐵 − 𝐶𝐶 𝐵𝐵 𝐶𝐶 ) encapsulated the contribution of body CoM dynamics to whole-body
angular momentum referenced to the axis through the edge of support so that increases in the
velocity term would result in increases in LBoS. The intercept and slope magnitudes of the two
regression analyses relating peak backward angular momentum and perturbation amplitude were
different due to the large distance between CoM and the axis through BoS. The difference in the
slopes between the two regression analyses was approximately 1, which could be explained by
the normalized term of 𝑋𝑋 𝑟𝑟 ⃗
𝐵𝐵𝐵𝐵 𝐵𝐵 − 𝐶𝐶 𝐵𝐵 𝐶𝐶 × 𝑣𝑣 ⃗
𝐵𝐵𝐵𝐵 𝐵𝐵 − 𝐶𝐶 𝐵𝐵 𝐶𝐶 by participant’s weight, height, and walking
speed.
The whole-body response to perturbations during normal walking and perturbation steps
can be characterized by low-dimensional patterns that indicate the coordination between
segments. We identified how the body segments co-varied to control the angular momentum to
avoid falling toward the ground by extracting intersegmental coordination patterns from
segmental angular momentum. Although the first two PCCoM explained more variance than PCBoS
on average, the first two PCCoM and PCBoS both explained more than 90% of the variance of the
segmental angular momenta referenced to the axis through CoM and edge of support. Thus, our
results suggest that changing the reference axis of angular momentum calculation does not alter
the hypothesis that the central nervous system could coordinate the movement of body segments
using a combination of coordination patterns represented in a lower dimensionality (Chiovetto et
al., 2018; Martelli et al., 2013).
However, we have observed several differences between the coordination patterns
extracted referenced to two axes. First, the trunk, pelvis, and head contributions to whole-body
41
angular momentum were small when referenced to CoM as the distances between these segments
and the axis through CoM were minimal. In contrast, these segments that located far away from
the axis dominated the PC weights when referenced to BoS, which was in line with the previous
studies (Chiovetto et al., 2018; Gaffney et al., 2017). Controlling the rotation of upper-body
segments around the hip joints to maintain upright position in the sagittal plane is important as
the upper-body segments have high inertia, especially the trunk. Both abdominal and back
muscles needed to respond fast following trips to stiffen the trunk and thus reduce the excessive
forward rotation in the sagittal plane (Van Der Burg et al., 2005). Therefore, analyzing
coordination pattern referenced to an axis through edge of support may provide more insights
about how the upper body responds to losses of balance.
Secondly, we could infer how the contributions from the lower limbs counteracted the
whole-body angular momentum during unexpected perturbations only from the extracted
coordination pattern in PC1 CoM. PC1CoM indicated that the swing leg and perturbed leg generated
angular momentum in opposite directions about the CoM. Moving contralateral limb segments in
anti-phase helped to counteract the variation in whole-body angular momentum. These results
were consistent with previous literature demonstrating that the perturbed leg and the contralateral
swing leg were in anti-phase during walking and in response to perturbations (Herr and Popovic,
2008; Liu and Finley, 2020; Martelli et al., 2013). In contrast, the PCBoS paired with PC1CoM did
not provide the same information about leg segmental angular momentum cancellation even
though the PCBoS and PCCoM were considered to be similar. Instead, the paired PCBoS only
revealed how the segments within the swing leg covaried during the perturbation response. Thus,
one advantage of referencing to the CoM axis was that we could easily identify the degree of
segmental angular momentum cancellation during walking and perturbation responses. For
42
example, people with gait asymmetries such as people post-stroke and people with lower limb
amputation whose contralateral limb segments had difficulty moving in anti-phase usually
showed increased LCoM (Nott et al., 2014; Vistamehr et al., 2016). Therefore, we could infer the
degree of limb cancellation for people with gait asymmetries from LCoM. However, for LBoS, we
needed to compare the values with the angular momentum generated by the CoM about the
contact point with the ground using an inverted pendulum model to quantify the degree of limb
cancellation. Overall, coordination patterns extracted from angular momentum referenced to
different axes provide distinctive information yet complementary information about how people
coordinated their body segments in response to unexpected perturbations.

There might be learning effects during the course of perturbation trials that people
became better at balance control after repetitive practice. Although previous studies reported that
people adapted by reducing the CoM velocity during repetitive obstacle tripping during over
ground walking (Wang et al., 2012), we did not find any significant changes in CoM velocity at
foot landing of the contralateral swing foot or whole-body angular momentum during the
perturbation steps between the first perturbations and the last perturbations. One reason is that
our perturbations were randomized in timing and perturbation direction so that participants could
not make specific adjustments such as increasing the stability margin between their CoM and the
edge of support to prepare for the upcoming perturbations.

Our study showed that computing angular momentum referenced to different axes instead
of the commonly used axis through the CoM may provide additional and complementary insights
about how people reactively control balance during trip-like and slip-like treadmill induced
43
perturbations. For future analysis, we may extend this analysis to people with balance deficits,
such as aging population and people post-stroke. Several studies have used angular momentum
as a quantitative measure to assess dynamic balance for clinical populations with gait and
balance deficits (Nott et al., 2014; Vistamehr et al., 2016). Thus, we may be able to identify the
differences in coordination patterns following perturbations between healthy populations and
people with a higher fall risk, which may provide more insights into developing effective
rehabilitation protocols to improve balance control for clinical populations.

We would like to thank Cathy Broderick, Ryan Novotny, and Catherine Yunis for their
help during data collection. We also thank Aram Kim and Cathy Broderick for their help with
manuscript editing.

44
Chapter 3. Conservation of Reactive Stabilization Strategies in the Presence of
Step Length Asymmetries during Walking

Authors: Chang Liu, Lucas De Macedo, and James M. Finley (published in Frontiers Human
Neuroscience)
The ability to maintain dynamic balance in response to unexpected perturbations during
walking is largely mediated by reactive control strategies. Reactive control during perturbed
walking can be characterized by multiple metrics such as measures of whole-body angular
momentum, which capture the rotational dynamics of the body, and through Floquet analysis
which captures the orbital stability of a limit cycle attractor. Recent studies have demonstrated
that people with spatiotemporal asymmetries during gait have impaired control of whole-body
dynamics as evidenced by higher peak-to-peak ranges of angular momentum over the gait cycle.
While this may suggest that spatiotemporal asymmetries could impair stability, no studies have
quantified how direct modification of asymmetry influences reactive balance control. Here, we
used a biofeedback paradigm that allows participants to systematically adopt different levels of
step length asymmetry to test the hypothesis that walking asymmetrically impairs the reactive
control of balance. In addition, we tested the hypothesis that perturbations to the non-dominant
leg would cause less whole-body rotation due to its hypothesized role in weight support during
walking. We characterized reactive control strategies in two ways. We first computed integrated
angular momentum to characterize changes in whole-body configuration during multi-step
responses to perturbations. We also computed the maximum Floquet multipliers across the gait
cycle which represent the rate of convergence back to limit cycle behavior. Our results show that
integrated angular momentum during the perturbation step and subsequent recovery steps, as
45
well as the magnitude of maximum Floquet multipliers over the gait cycle, do not change across
levels of asymmetry. However, our results showed both limb-dependent and limb-independent
responses to unexpected perturbations. Overall, our findings suggest that there is no causal
relationship between step length asymmetry and impaired reactive control of balance in the
absence of neuromotor impairments. Our approach could be used in future studies to determine if
reducing asymmetries in populations with neuromotor impairments, such people post-stroke or
amputees improves dynamic stability.
46

One of the primary challenges for human locomotion is to maintain balance when faced
with internally generated or externally imposed perturbations. Two balance control strategies are
generally used during locomotion: proactive and reactive control of balance (Patla, 1993). While
proactive or feedforward control involves the use of predictions of impending perturbations to
avoid falling, reactive control of balance involves the use of feedback about the body’s state to
generate balance correcting responses (Patla, 1993; Tang et al., 1998). One of the primary ways
in which the reactive control of balance is studied is by applying perturbations during walking
and characterizing the resulting perturbation recovery strategies.
Several metrics have been used to quantify balance during locomotion including
measures of variability (Stergiou and Decker, 2011), measures derived from nonlinear dynamics
such as the maximum Lyapunov exponent (Dingwell et al., 2001; Dingwell and Cusumano,
2000) and long-range correlations (Hausdorff et al., 1996), and biomechanical measures such as
dynamic margins of stability (Hof, 2008; Hof et al., 2005). For a detailed review of metrics used
to assess dynamic stability during gait, see Bruijn et al. (Bruijn et al., 2013). While each of these
methods is useful for characterizing features of control in the presence of instability, we are
particularly interested in measures that directly capture whole-body dynamics. One such
measure, whole-body angular momentum (WBAM), can be used to capture the body’s response
to perturbations and reflects the net contribution of all body segments to the body’s rotation
about a given axis. WBAM is highly regulated during normal human locomotion (Herr and
Popovic, 2008; Popovic et al., 2004) as the peak-to-peak range of WBAM about the body’s
center of mass is much smaller than the angular momentum of single segments due to
momentum cancellation between the limbs (Herr and Popovic, 2008). In a recent study, Martelli
47
et al. used WBAM to characterize recovery strategies in response to multidirectional
perturbations during walking in healthy individuals (Martelli et al., 2013). They found that
perturbations resulted in increased angular momentum and subsequent compensatory reactions.
Another metric used to characterize dynamic stability is the maximum Floquet multiplier
(FM) which is commonly used to assess the rate of divergence/convergence from a fixed point,
characterized by a kinematic state vector, in response to small perturbations (Dingwell and Kang
2007; Hurmuzlu and Basdogan 1994; Kuo 1999). This measure is based on the fact that human
walking is strongly periodic and can be characterized as a limit cycle attractor. Previous studies
have established that the maximum FM remains below one during unperturbed walking
(Dingwell and Kang 2007; Hurmuzlu and Basdogan 1994; Granata and Lockhart 2008; Bruijn, et
al., 2009) which indicates that small perturbations always converge toward a limit cycle. The
maximum FM increases when walking in destabilizing environments, but still remains below one
as people are able to use proactive and reactive control to maintain balance (McAndrew et al.,
2012). Both WBAM and the maximum FM capture a different aspect of reactive control during
walking and together provide a detailed description of the control of dynamic balance.
The ability to successfully restore balance is vital for populations with gait asymmetries
such as people post-stroke (Allen et al., 2011; Balasubramanian et al., 2007; Chen et al., 2005),
unilateral amputees (Barth et al., 1992; Underwood et al., 2004; Zmitrewicz et al., 2006) and
patients with ACL reconstruction (Winiarski and Czamara, 2012). However, these populations
are known to have balance deficits during walking. For example, Lewek et al. examined the
relationship between spatiotemporal gait asymmetry and balance in people post-stroke and
showed that step length asymmetries were correlated with scores on the Berg Balance Scale,
suggesting that gait asymmetries are associated with fall risk in these individuals (Lewek et al.,
48
2014) . In addition, recent studies have demonstrated that people post-stroke have impaired
control of whole-body dynamics as captured by higher peak-to-peak ranges of whole body
angular momentum (WBAM) (Nott et al., 2014; Vistamehr et al., 2016) and reductions in local
and orbital stability (Kao et al., 2014). Likewise, unilateral amputees demonstrate a greater range
of angular momentum during the half of the gait cycle from foot contact of the residual limb to
contact of the intact limb and a smaller range of angular momentum during the second half of the
gait cycle due to reduced leg propulsion in the sagittal plane (Silverman and Neptune, 2011).
Although these studies have demonstrated an association between asymmetry and measures of
stability, it remains to be seen if spatiotemporal asymmetry alone is causally associated with
stability in the absence of neuromotor impairments.
In addition to an effect of asymmetry, the reactive control of stability may also be
impacted by limb dominance. There is evidence suggesting that the dominant leg generates more
propulsion during walking while the non-dominant leg preferentially provides support (Sadeghi
et al., 1997). Ounpuu and Winter (1989) found that the normalized EMG amplitude of most
plantar flexor muscles was greater in the dominant limb, which may reflect its preferential role in
propulsion generation. Also, Martelli et al. (Martelli et al., 2013) demonstrated that recovery of
WBAM about the roll-axis in response to a perturbation depends on the side of the perturbation.
Specifically, the reactive responses to perturbations on the non-dominant side, as captured by the
principal components of the segmental angular momenta, were more similar to pre-perturbation
behavior than responses following perturbations to the dominant side. Thus, the nondominant leg
may be better suited for maintaining stability in response to perturbations.
The objectives of this study are to quantify how direct modification of spatiotemporal
asymmetry influences the reactive control of balance during walking and to determine whether
49
the reactive control of balance is influenced by limb dominance. We hypothesized that 1)
modifications of step length asymmetry will impair the control of whole-body rotation and
increase the maximum FM during unexpected perturbations which, together, would indicate that
the reactive control of balance is compromised by asymmetry, and 2) that perturbations of the
non-dominant leg would produce less whole-body rotation due to this limb’s proposed role in
providing stability during locomotion (Sadeghi et al., 1997). Here, we chose to use the maximum
FM to quantify orbital stability because it allowed us to capture differences in stability
throughout the gait cycle. These hypotheses were tested by using visual feedback to induce
changes in step length asymmetry during walking and imposing slip-like perturbations on a dual-
belt treadmill. Our findings may inform our understanding of how interventions aimed at
improving symmetry in populations with neuromotor impairments may impact balance control.

A total of 19 healthy young individuals (10M, 24 ± 4 yrs old) with no musculoskeletal or
gait impairments participated in this study. Lower limb dominance was determined by asking
participants which leg they would use to kick a ball. The study was approved by Institutional
Review Board at the University of Southern California, and all participants provided informed
consent prior to participating. All aspects of the study conformed to the principles described in
the Declaration of Helsinki.
50

Figure 9: (A) Experiment protocol. Participants completed a total of six trials. Participant’s
baseline step length asymmetry was collected during the first three-minute baseline trial without
visual feedback. Then, they were instructed to complete a randomized sequence of five six-minute
trials with target step length asymmetries of 0%, ±10%, and ±15%. During each visual
feedback trial, the participant first practiced with feedback for one minute, then 10 perturbations
were randomly applied at foot strike on each side. (B) Visual feedback for three of the five trials
of step length asymmetry are shown. (C) Experimental setup. Participant were instructed to walk
on the split-belt treadmill. A “success” message would appear on the screen when step length
was within the three standard deviations of the desired target.

The purpose of this study was to assess whether changes in step length asymmetry affect
the reactive control of balance during walking. Participants completed six separate trials walking
on an instrumented, dual-belt treadmill at 1.0 m/s (Bertec, USA) and reacted to unexpected
accelerations of the treadmill belts throughout the experiment (Figure 9A). For the first trial,
participants walked on the treadmill for three minutes (Baseline) to obtain their natural level of
step length asymmetry. Then, for subsequent trials, visual feedback indicating the desired step
lengths was provided to assist participants in actively modifying their asymmetry relative to their
51
natural step length asymmetry. Participants completed a randomized sequence of five six-minute
trials with target step length asymmetries (SLA, Eq.1) of 0%, ±10%, and ±15% where 0%
represents each participant’s baseline SLA.
𝛥𝛥𝐿𝐿 𝑆𝑆 = 100 ∗
𝐵𝐵 𝐿𝐿 𝑙𝑙𝑙𝑙𝑙𝑙 𝑙𝑙 − 𝐵𝐵 𝐿𝐿 𝑟𝑟𝑟𝑟𝑟𝑟 ℎ 𝑙𝑙 𝐵𝐵 𝐿𝐿 𝑙𝑙𝑙𝑙𝑙𝑙 𝑙𝑙 + 𝐵𝐵 𝐿𝐿 𝑟𝑟𝑟𝑟𝑟𝑟 ℎ 𝑙𝑙 (1)
Participants viewed the step length targets on a computer monitor attached to the
treadmill post (Figure 9B). During each trial, participants first practiced walking at a given SLA
with visual feedback for one min before experiencing any perturbations (Figure 9C). A “success”
message would appear on the screen when the achieved step length was within the three standard
deviations of the desired target length. The standard deviation for each target was determined on
an individual basis from each participant’s baseline step length variability. Participants were
encouraged to maintain the desired SLA and get as many success messages as possible.
Step length was estimated during the experiment as the anterior/posterior distance
between the center of pressure on the left and right force plates at foot strike. Foot strike was
defined as the point when the vertical ground reaction force became greater than 150N. For the
trials with visual feedback, ten unexpected perturbations, where the treadmill accelerated to 1.5
m/s, were randomly applied to each side (right or left). Each perturbation was remotely triggered
by preprogrammed Python code such that the participants could not anticipate when the
perturbation would occur. During pilot testing, we found that there was approximately a 200ms
time delay between when the perturbation signal was sent to the treadmill and when the treadmill
began accelerating. Thus, during the first minute practice for each trial, we calculated each
individual’s average right and left step times. We then triggered the perturbations 200ms before
the predicted foot strike of the perturbed leg so that the acceleration of treadmill would coincide
with foot strike. All perturbations were characterized by a trapezoidal speed profile in which the
speed increased at foot strike to 1.5 m/s at an acceleration of 1.6 m/s
2
, was held for 0.3s, and then
52
decelerated back to 1.0 m/s at 1.6 m/s
2
during swing phase of the perturbed leg (Figure 10A).
These parameters were selected based on results from a series of pilot tests which demonstrated
that these perturbations were sufficient to elicit both changes in step length asymmetry and
changes in whole-body angular momentum. The belt speed was held for 0.3s to ensure that the
belt speed did not decelerate before the toe-off of the perturbed leg. The perturbations randomly
occurred within a range of 20 to 30 steps after the previous perturbation to provide participants
with enough time to reestablish their normal walking pattern.

Figure 10: Example of time series data from an unperturbed and perturbed step. (A) Treadmill
belt velocity, (B) vertical ground reaction force, and (C-E) whole-body angular momentum for a
representative perturbation step and recovery stride. The gray traces indicate the time series
data for an unperturbed stride while the black traces indicate a perturbation stride. Each stride
begins at heel strike. Black vertical lines correspond to the time of foot strike and gray vertical
lines correspond to time of toe-off. Solid lines and dashed lines represent contralateral legs.

A ten-camera Qualysis motion capture system (Qualysis AB, Gothenburg, Sweden)
recorded 3D marker kinematics at 100 Hz and ground reaction forces at 1000 Hz. A set of 19mm
spherical markers were placed on anatomical landmarks to create a 13-segment, full-body model
53
(Havens et al., 2018; Song et al., 2012). Marker clusters were placed on the upper arms,
forearms, thighs, shanks, and the back of heels. At the beginning of each trial, marker positions
were calibrated during a five-second standing trial. All joint markers were removed after the
calibration.

Kinematic and kinetic data were post-processed in Visual3D (C-Motion, Rockville, MD,
USA) and Matlab 2017a (Mathworks, USA) to compute variables of interest. Marker position
data and ground reaction forces were low-pass filtered by 4
th
order Butterworth filters with cutoff
frequencies of 6 Hz and 20 Hz respectively. The type of filter and cut-off frequency were
selected based on previous literatures (Kurz et al., 2012; Reisman et al., 2009; Winter, 2009).
The timing of each perturbation relative to foot strike was reexamined in Matlab. If the treadmill
belt did not accelerate during the 300ms window around foot strike (from 150ms before to
150ms after foot strike), the perturbation was excluded from analysis. On average, approximately
four of 20 perturbations were excluded for each trial.
In order to account for differences between the target and achieved SLA, we calculated
achieved SLA as follows: first, we calculated the mean SLA of four strides before each
perturbation and then distributed them into 5 equally spaced bins centered at -15%, -10%, 0,
10%, 15% with bin width equal to 5%. We used this re-categorized SLA as the independent
variable in our statistical analyses instead of target SLA.

Whole-body angular momentum was computed to determine how the rotational behavior
of the body changed in response to the treadmill perturbations. In order to calculate whole-body
angular momentum, a 13-segment full-body model was first created using Visual 3D (C-Motion,
54
Rockville, MD, USA). The segments of the model included the head, trunk, pelvis, upper arms,
forearms, thighs, shanks, and feet. Each limb segment’s mass was modeled based on
anthropometric tables (Dempster, 1955). Segment geometry was modeled based on the
description in Hanavan (1964). The trunk and pelvis were modeled as elliptical cylinders, the
head was modeled as an ellipsoid, and all the other segments were modelled as circular cones.
All segments had six degrees of freedom, and no connecting joints (i.e. constraints between
segments) were defined. Segmental linear and angular velocity were computed in Visual 3D
using the filtered marker position data. Whole-body angular momentum (L) was then computed
as the sum of all segmental angular momenta which were composed of segmental rotation about
the body’s center of mass and rotation of each segment about its own center of mass (Silverman
and Neptune, 2011). Then, L was normalized by the participant’s mass (M), treadmill velocity
(V), and the participant’s height (H) (Eq.2).
𝐿𝐿 � ⃗
=
∑ [ 𝑚𝑚 𝑟𝑟 � 𝑟𝑟
� � ⃗
𝐶𝐶𝐶𝐶 − 𝑟𝑟 𝑟𝑟 × 𝑣𝑣 � ⃗
𝐶𝐶𝐶𝐶 − 𝑟𝑟 𝑟𝑟 � + 𝐼𝐼 𝑟𝑟 𝜔𝜔 𝑟𝑟 ]
𝑟𝑟 𝐶𝐶𝑀𝑀 𝑀𝑀 (2)
Here, m is segmental mass, r is the distance from segment to the body COM, I is the
segmental moment of inertia, ω is segmental angular velocity, and the index i corresponds to
individual limb segments. The coordinate system for analyses of angular momentum was defined
as follows: the x-axis was the pitch axis and positive to the right, the y-axis was the roll axis and
positive in the anterior direction, and the z-axis was the yaw axis and positive in the vertical
direction. Whole-body angular momentum for each step cycle (from the foot strike on one side to
the subsequent foot strike on the same side) was normalized to 100 points. In addition, integrated
whole-body angular momentum ( 𝐿𝐿 𝑖𝑖 𝑖𝑖𝑖𝑖
) was computed as the area under the curve of the WBAM
trajectory for each step cycle to quantify the degree to which the body rotates about its center of
mass across a step cycle.
55

We used Floquet analysis (Bruijn et al., 2013; Dingwell and Kang, 2007; Hobbelen and
Wisse, 2007; Hurmuzlu et al., 1994, 1996; Kuo, 1999; Kurz et al., 2012) to determine how
orbital stability is affected by walking with different levels of SLA. Two participants were
excluded from this analysis as there was a break in one of their trials and therefore, we did not
have continuous data for the analysis. For this analysis, we used whole-body angular momentum
data as computed in section 2.4.1. First, state vectors (S) at each time point in the normalized
step cycle were constructed from the whole-body angular momentum signal and its first
derivative (Eq. 3). Then, Poincare maps (Eq. 4) were defined at each section of the gait cycle.
S = � 𝐿𝐿 𝑥𝑥 𝐿𝐿 𝑦𝑦 𝐿𝐿 𝑧𝑧 𝐿𝐿 𝑥𝑥 ̇ 𝐿𝐿 𝑦𝑦 ̇ 𝐿𝐿 𝑧𝑧 ̇ �
𝑇𝑇 (3)
S
k + 1
= 𝐹𝐹 (𝛥𝛥 𝑘𝑘 ) (4)
Here, k is the stride number and Sk are the state vectors of the system.
For each trial, we defined the fixed points (S*) (Eq. 5) at each Poincare section by
averaging all sets of angular momentum trajectory during the four strides before each
perturbation occurred (Figure 11). Stride-to-stride fluctuations about the fixed point allowed us
to examine the persistence of deviations from the mean trajectory.
𝛥𝛥 ∗
= 𝐹𝐹 (𝛥𝛥 ∗
) (5)
Orbital stability at each Poincare section was assessed by linearly approximating the
effects of perturbations that cause deviations from the fixed point (Eq. 6). For all trials, we used
250 strides to compute Floquet multipliers (FM) based on previous literature which established
that at least 150 strides were necessary to precisely measure FM (Bruijn et al., 2009).
[S
k + 1
− 𝛥𝛥 ∗
] ≈ 𝐽𝐽 (𝛥𝛥 ∗
)[𝛥𝛥 𝑘𝑘 − 𝛥𝛥 ∗
] (6)
56
Here, J is the Jacobian matrix estimated using the pseudo inverse at each Poincare section
(Kurz et al., 2012). [ 𝛥𝛥 𝑘𝑘 − 𝛥𝛥 ∗
] represents deviations from the fixed point. FM were calculated as
the eigenvalues of Jacobian matrix � 𝐽𝐽 ( 𝛥𝛥 ∗
) � , and we selected the maximum value of the FM
(FMmax) to assess orbital stability (Dingwell and Kang, 2007). If the magnitude of FMmax < 1, the
system is orbitally stable, otherwise, the system is unstable (Bruijn et al., 2013; Dingwell and
Kang, 2007; Hurmuzlu et al., 1994). We computed FMmax for each Poincare section (each % gait
cycle) to determine how stability changes over a stride cycle. We also determined FMmax across
the entire gait cycle for further statistical analysis as this value represents the most unstable point
during the gait cycle (Dingwell and Kang, 2007).

Figure 11: (A) Example of a 3D projection of the angular momentum trajectory recorded during
baseline walking for one representative participant. (B) Illustration of a hypothetical
perpendicular slice of the angular momentum trajectory as a Poincare section. S* represents the
fixed point which is the average of pre-perturbation strides.

All statistical analyses were performed in Matlab R2017a (Mathworks, Natick, MA,
USA). Repeated measures analysis of variance (RM-ANOVA) was used to determine if values
of integrated angular momentum about each axis for the steps after the perturbation differed from
57
values during baseline steps. Post-hoc comparisons used the Tukey-Kramer correction for
multiple comparisons.
Linear mixed-effect models were fit to examine the relationship between independent
variables achieved SLA (Asym) and side of perturbation (Side) and dependent variables L
in t

about each axis to determine how the effect of perturbations varied with asymmetry and between
limbs. This model included main effects for Asym and Side as well as an interaction between
Asym and Side to determine whether the effect of asymmetry depends on the side of the
perturbation. The linear mixed-effect models were fit for four consecutive steps (Baseline,
Perturbation, Recovery 1, and Recovery 2) for each axis. The integrated angular momentum for
the trial with an SLA of zero was selected as the reference level. We used a mixed effect model
instead of a RM-ANOVA for this analysis because the number of observations at each level of
achieved asymmetry was unequal.
Similarly, a linear mixed effect model was fit to represent the relationship between target
asymmetry (independent variable) and the FMmax (dependent variable) in order to see if the
orbital stability was associated with target asymmetry. For both sets of analyses, models
including random intercepts and/or slopes were compared against a model with only fixed effects
and the most parsimonious model was chosen based on the results of a likelihood ratio test.

Participants were able to update and maintain their SLA for the full duration of each trial
(~6 minutes) using visual feedback and recover from deviations in SLA resulting from the
perturbations (Figure 12A). If the perturbation occurred on the right leg, the left leg would step
further forward to recover from the perturbation and SLA would increase based on Equation 1.
On the other hand, if the perturbation was occurred on the left leg, SLA for the perturbation
58
stride became more negative. The achieved SLA (Figure 12B) was calculated relative to baseline
asymmetry of 1.7±3%. There was considerable variability in the achieved asymmetry across
participants, especially when the target SLA was large. The average residual between achieved
SLA and target SLA (|SLAtarget – SLAachieved|) across all participants was 4.7±2.9%, 2.9±1.7%,
2.4±1.6%, 4.3 ± 2.8%, 7.7±3.5% for -15%, -10%, 0, 10%, 15% target SLA respectively. Our
analysis used participants’ achieved asymmetry rather than target asymmetry to better reflect
their actual performance. The total number of perturbations in each step length asymmetry bin
were as follows: 110 perturbations for -15%, 314 perturbations for -10%, 265 perturbations for -
5%, 294 perturbations for 0%, 285 perturbations for 5%, 191 perturbations for 10%, and 69
perturbations for 15% SLA.

Figure 12: (A) Raw step length asymmetry data for one representative participant. Each data
point represents the step length asymmetry. The target asymmetries for this example followed the
order of 10%, -10%, 0, 15%, -15%. Each target asymmetry is represented by a different color.
BSL: baseline step; PTB: perturbation step; REC: recovery step. (B) Achieved step length
asymmetry versus target step length asymmetry for all participants (N=19). Achieved step length
asymmetry is calculated as the average of all pre-perturbation strides and tends to undershoot
the target at 15% and -15%. The green dots represent individual data. Horizontal bars indicate
the median across all participants.

59

Measures of WBAM varied systematically across trials. We measured the WBAM about
three axes to better understand how participants react to the perturbations. The rapid acceleration
of the belts caused consistent, immediate effects on WBAM and triggered multi-step balance
recovery responses. The immediate effect was most obvious along the direction of perturbation
(pitch axis, Figure 10C). During the perturbation step, angular momentum became more negative
as the body rotated forward (- pitch). In order to compensate for the perturbation, participants
increased the length of the subsequent step to generate positive angular momentum and initiate
backward rotation (+ pitch). Deviations in body rotation about the roll and yaw axes relative to
unperturbed walking were less prominent (Figure 10D & E).
To quantify the effects of the perturbations on whole-body configuration, we computed
the integrated angular momentum across the step cycle (Figure 13). When walking
symmetrically, the integrated angular momentum was relatively small during baseline walking
and showed little step-to-step variability about the pitch axis. For the roll axis, positive and
negative values of Lint correspond to transitions from the right to left leg and from the left to right
leg, respectively. For the yaw axis, positive and negative values of Lint correspond to transitions
from the left to the right leg and from the right to left leg, respectively. During the perturbation
step, there was a significant increase in integrated angular momentum about the pitch axis which
reflected the increase in the body’s forward rotation. A repeated measures ANOVA was used to
determine when the participants recovered from the perturbation when walking symmetrically.
We found a main effect of step number on integrated angular momentum for the pitch axis (RM-
ANOVA, F=185.5, p<0.001), and a significant interaction between step number and perturbation
60
side for roll axis (F=58.7, p<0.001), and yaw axis (F = 434.76, p<0.001). For the pitch axis, post
hoc analysis revealed that L
in t
differed from baseline during the perturbation (PTB) step
(p<0.001 both sides), recovery (R) steps R1 (p<0.001 both sides), R4 (p=0.006 Dominant side, p
= 0.002 Non-dominant side) and R6 (p = 0.013 Dominant side, p = 0.001 Non-dominant side).
About the roll axis, significant differences in L
in t
were found during the PTB step (p=0.001) and
R3 (p=0.001), but only on the dominant side. Lastly, about the yaw axis, a significant difference
was found at perturbation step (p<0.001) for both sides.

Figure 13: Averaged integrated angular momentum over the step cycle for all participants about
the (A) pitch, (B) roll, and (C) yaw axes for perturbations that occurred on the non-dominant
(left column) and dominant side (right column). These results represent the 0% asymmetry
61
condition (N=19). The first step (B1) corresponds to the non-dominant limb for the left column
and the dominant limb for the right column. Subsequent steps alternate between non-dominant
and dominant. B: Baseline; PTB: Perturbation; R: Recovery. The horizontal bars and
corresponding stars indicated whether the difference in integrated angular momentum between
two steps was significant (**p<0.001, * p<0.05). The data are represented as boxplots such that
the lower and upper edges of the box indicate the 25th and 75th percentile of the data,
respectively. The horizontal line within each box indicates the median. The whiskers extend to
the furthest data point beyond the lower or upper edges of the box that is within a distance of 1.5
times the middle 50th percentile of the data. Points that lie beyond the whiskers denote outliers.

Next, we asked whether walking with asymmetric step lengths would have negative
effects on participant’s reactive control of balance (Figure 14). We fit a linear mixed effect
model relating asymmetry to the integrated angular momentum at each step (Final Baseline Step
(B2), PTB, R1, and R2) and selected the simplest model based on a likelihood ratio test. An SLA
of zero was selected as the reference level. The model indicated that a random intercept was
necessary to account individual differences between participants (p<0.001). There were no
significant main effects found for Asymmetry, Side (pitch axis) or the interaction between
Asymmetry and Side during baseline (B2) steps (Table 1). Similarly, we examined whether
asymmetry influenced measures of integrated angular momentum at the PTB step and found no
significant main effects for Asymmetry, Side or the interaction between Asymmetry and Side
during any of these steps (Table 1). Lastly, we examined whether asymmetry influenced
integrated angular momentum during the first or second recovery steps (R1 and R2). There were
no significant main effects found for Asymmetry, Side or the interaction between Asymmetry
and Side during first or second recovery steps (Table 1). The significance found for Side in roll
and yaw axis was due to differences in the direction of body rotation at each step. Overall, these
results indicate that imposed asymmetry does not have a systematic effect on the reactive control
of balance as assessed by measures of WBAM.
62
Table 1 Statistical results from the ANOVA examining the effects of asymmetry and perturbation
side on integrated whole-body angular momentum for each step type. B2: Baseline step, PTB:
Perturbation Step, R1: First Recovery Step, R2: Second Recovery Step.
Step
Type
Axis Factor DF1 DF2 F Value p Value

B2

Pitch
Asymmetry 6 210 0.10 0.99
Side 1 210 2.84 0.09
Asymmetry:Side 6 210 0.18 0.98

Roll
Asymmetry 6 210 0.31 0.93
Side 1 210 76.1 <0.001
Asymmetry:Side 6 210 0.24 0.96

Yaw
Asymmetry 6 210 0.72 0.63
Side 1 210 397 <0.001
Asymmetry:Side 6 210 0.96 0.45

PTB

Pitch
Asymmetry 6 211 0.25 0.96
Side 1 211 0.30 0.59
Asymmetry:Side 6 211 0.59 0.77

Roll
Asymmetry 6 211 0.39 0.89
Side 1 211 85 <0.001
Asymmetry:Side 6 211 0.41 0.87

Yaw
Asymmetry 6 211 0.38 0.89
Side 1 211 577 <0.001
Asymmetry:Side 6 211 1.33 0.25

R1

Pitch
Asymmetry 6 211 0.51 0.80
Side 1 211 0.31 0.60
Asymmetry:Side 6 211 0.76 0.60

Roll
Asymmetry 6 211 0.36 0.90
Side 1 211 2.85 0.09
Asymmetry:Side 6 211 0.39 0.89

Yaw
Asymmetry 6 211 1.00 0.42
Side 1 211 491 <0.001
Asymmetry:Side 6 211 1.57 0.16

R2

Pitch
Asymmetry 6 211 0.90 0.5
Side 1 211 2.70 0.1
Asymmetry:Side 6 211 0.21 0.97

Roll
Asymmetry 6 211 0.65 0.69
Side 1 211 6.39 0.01
Asymmetry:Side 6 211 0.00 0.98

Yaw
Asymmetry 6 211 0.47 0.83
Side 1 211 386 <0.001
Asymmetry:Side 6 211 0.92 0.48

63

Figure 14: Box plot of integrated angular momentum about the (A) pitch, (B) roll, and (C) yaw
axes at baseline step (B2), perturbation step (PTB), and recovery steps (R1 and R2) across each
level of achieved asymmetry (N=19) for perturbations on the non-dominant (left column) and
dominant (right column) sides.

To further investigate how asymmetry impacts dynamic stability, we performed Floquet
analysis to determine if asymmetry influenced measures of orbital stability during perturbed
walking. Our results show that the FMMax from all five trials was less than one indicating that
participants remained orbitally stable in spite of the perturbations that occurred while walking
64
(Figure 15). The range of the FM computed across the stride cycle at 0% SLA was 0.41± 0.09
across all participants, which is similar, but slightly smaller than that reported by Dingwell et al.
(~0.5) when participants walked on the treadmill in the absence of perturbations (Dingwell and
Kang, 2007). Similar to our results for integrated angular momentum, there was no association
between the SLA and measures of orbital stability (F (4, 80) = 0.86, p = 0.5). The FMMax across
all asymmetries were 0.61 ± 0.09, 0.63 ± 0.13, 0.57 ± 0.11, 0.62 ± 0.13, and 0.59 ± 0.09 for
target asymmetries of -15%, -10%, 0, 10%, and 15%, respectively.

Figure 15: (A) Variation in the magnitude of the maximum Floquet multiplier (FM) across the
gait cycle for five levels of target asymmetry (N=17). The shaded area indicates the 95%
confidence interval. (B) FMMax across all levels of asymmetry for (N=17) participants.
65

This study asked the question of how step length asymmetry affects the reactive control
of balance during walking. Previous studies have demonstrated that people with gait
asymmetries have impairments in dynamic balance leading to the possibility that asymmetry
itself is sub-optimal for balance control. Here, we hypothesized that asymmetry would impair the
reactive control of balance, and we tested this hypothesis by imposing different levels of step
length asymmetry and characterizing participants’ response to perturbations. Although we
consistently elicited reactive responses to regain balance, we rejected our primary hypothesis that
asymmetry impairs the reactive control of balance as no significant difference in WBAM was
found across levels of asymmetry. In addition, Floquet analysis revealed that orbital stability was
well maintained and did not vary systematically with different levels of asymmetry. These results
indicate that reactive control of stability may be well controlled by healthy people even when
they change their preferred walking pattern to walk asymmetrically.
A potential explanation for the discrepancy between our hypothesis and the observed
results is that participants may have chosen a more conservative strategy due to the novel study
demands. In other words, participants may have taken more cautious, wider steps to increase
their base of support (Woollacott and Tang, 1997) when asked to walk asymmetrically and
subsequently improved the proactive control of stability during the task. However, we found no
difference in step width between levels of target asymmetry. Thus, there does not seem to be
strong evidence that participants’ recovery strategies were biased by use of a more conservative
pattern of movement.
66
Another possible reason why we did not observe an effect of asymmetry on the reactive
control of stability is that reactive responses to unexpected perturbations may be mediated by
neural pathways that generate stereotypical reactive responses that remain invariant across tasks.
Previous work by Aprigliano et al. used principal component analysis (PCA) to analyze
coordination between the shank, foot, and thigh in response to slip-like perturbations and found
that there was no difference in coordination between fall-prone elderly people and healthy young
adults (Aprigliano et al., 2017). Similarly, Martelli et al. used PCA of segmental angular
momentum to show that the intersegmental coordination patterns observed during compensatory
steps are highly correlated with the patterns observed during unperturbed walking. This
conservation of whole-body momentum across asymmetries suggests that reactive responses may
result from pre-programmed, stereotypical actions that are sufficient to restore the stability
(Martelli et al., 2013).

We also hypothesized that perturbations of the non-dominant leg would produce less
whole-body rotation due to the non-dominant limb’s role in maintaining balance. Previous work
has shown that the non-dominant leg may preferentially be used to support body weight while
the dominant leg may generate more propulsion (Sadeghi et al., 1997). Consistent with this idea,
whole-body angular momentum about the roll axis did not differ across strides for the non-
dominant leg while, in contrast, we observed significant differences in L
in t
about the roll axis
between unperturbed, perturbed, and recovery strides for the dominant leg. This suggests that the
non-dominant leg may be better at maintaining medial-lateral balance. However, we also found
that there was no significant effect of the side of the perturbation on our measures of integrated
angular momentum. This presence of both limb-specific and limb-independent recovery
67
responses requires further investigation to establish the effect of limb dominance on balance
recovery.

In general, populations with high incidence of falls are shown to have increased orbital
instability relative to unimpaired controls as characterized by a larger maximum Floquet
Multiplier (Granata and Lockhart, 2008; Hurmuzlu et al., 1996; Kurz et al., 2012). In our study,
we were interested in determining whether walking with spatiotemporal asymmetry would
modulate orbital stability. As the results demonstrated, all participants in our study had orbitally
stable walking patterns, regardless of the level of SLA. Also, no difference in FMMax was found
during asymmetrical walking compared with symmetric walking. This is in contrast to previous
work which shown that voluntarily changing step length reduces orbital stability of human
walking (McAndrew Young and Dingwell, 2012). One possible explanation for the discrepancy
between the current study and previous work is that our study provided visual feedback for
regulating SLA, which may decrease the step length variability compared to Young et al (2012).
In addition, we chose to maintain a consistent stride length while varying asymmetry whereas the
Young et al study (2012) involved significant increases in stride length. As a result, Floquet
analysis may be not sensitive enough to detect the destabilizing effects of SLA in the absence of
changes in stride length. Our findings also contrast previous studies which showed that walking
on a pseudo-randomly oscillating treadmill reduced orbital stability (Beurskens et al., 2014;
McAndrew et al., 2010). A potential reason for this difference is that our study perturbed the
participants with discrete mechanical perturbations at foot contact whereas previous work used
continuous perturbations throughout the gait cycle. It is possible that the effect of these discrete
68
perturbations dissipates quickly for healthy participants resulting in negligible changes in orbital
stability due to imposed SLA.
Although this study showed that the whole-body reactive response was not affected by
the presence of step length asymmetry in healthy participants, it is possible that this result was
influenced by the fact that participants were instructed to walk on the treadmill with a fixed
speed, which might alter the strategies participants use to generate the desired asymmetries.
While Nagano et al. showed that there were no differences in step lengths when young adults
walked on a treadmill or over ground at a given speed, temporal parameters such as double
stance time and swing time did differ (Nagano et al., 2011). In addition, people produce reduced
dorsiflexor moments, reduced knee extensor moments, and greater hip extensor moments in the
sagittal plane during treadmill walking (Lee and Hidler, 2008), which could also affect the
strategies used to modify symmetry on the treadmill. As a result, it would be interesting for
future studies to compare the effects of asymmetry on reactive control balance during over
ground versus treadmill walking.
Aside from reactive responses, proactive control also plays a part in maintaining balance
on the treadmill in the presence of slip perturbations. Although participants voluntarily changed
their SLA to match the visual feedback, modification of SLA did not impair their whole-body
balance during unperturbed steps (Figure 14). This might reflect the fact that healthy individuals
used novel proactive strategies to maintain a consistent range of WBAM in the presence of step
length asymmetry. Another possibility is that after the initial exposure to the treadmill
perturbations, participants may have adopted strategies to improve stability and to prepare
themselves for the perturbations. Previous work has shown that the central nervous system can
adjust its control strategy based on the prior experience to produce a more cautious gait and
69
reduce the risk of balance loss by altering muscle activation and the resulting interaction between
the foot and the surface (Heiden et al., 2006). In addition, previous studies have showed that
people were able to reduce backward balance loss with exposure to multiple slip perturbations
using proactive adjustments during sit-to-stance task and over ground walking (Bhatt et al.,
2006; Pai et al., 2003). Analysis of interlimb coordination during baseline trials could be useful
to reveal how participants were able to adopt an invariant control of WBAM despite the presence
of marked step length asymmetries.
Lastly, despite that we found that symmetric walking doesn’t necessarily bring benefits
for reactive control of balance for healthy subjects, this does not necessarily mean it holds true
for fall-prone populations with sensory or motor impairments. These impairments may
significantly affect one’s ability to recover from unexpected perturbation. Fall-prone populations
such as people post-stroke may suffer from sensorimotor deficits, which prevent them from
appropriately sensing perturbations and planning and executing effective responses to regain
balance. Although a number of recent studies have shown that it is possible to improve
spatiotemporal symmetry in people post-stroke (Awad et al., 2016; Reisman et al., 2009), it
remains to be seen if reductions in symmetry improve dynamic balance. The approaches used in
this study may help separate the effects of asymmetry on balance from the effects of neuromotor
deficits and lead to better informed locomotor training for people post-stroke.

This study had a few limitations. First, there was inconsistency in performance such that
the achieved step length tended to undershoot the target at the larger SLA. This is likely because
larger asymmetries are energetically costly and may also reflect biomechanical constraints that
prevent substantial extension of the hip beyond that observed during normal walking (Sánchez et
70
al., 2017). Another factor that could have affected our analysis of reactive control is that
participants were protected by a harness, which may have restricted forward trunk rotation.
However, since the harness was slack during the full experiment, we think this is unlikely.
Lastly, calculation of Floquet Multipliers requires a linear approximation to compute the effects
of perturbations from stride to stride, but the large perturbations in our study may introduce some
non-linearity. We think this effect is likely to be negligible as the Floquet multipliers we
computed were consistent with previous work that assesses stability during walking in
destabilizing environments (McAndrew et al., 2012).

We thank Natalia Sanchez, Ph.D. for her insights during the design of this experiment,
Aram Kim for her assistance with the statistical analysis, and Sungwoo Park for his help with
Matlab coding.

C.L designed the experiment, collected data, analyzed data, and wrote the manuscript.
L.dM designed the experiment, collected preliminary data, and helped develop the procedure for
data processing and analysis. J.M.F conceived of the experiment, advised in data analyses and
edited the manuscript.

71
Chapter 4. Asymmetric Gait Patterns Alter the Reactive Control of
Intersegmental Coordination Patterns in the Sagittal Plane during Walking

Authors: Chang Liu, and James M. Finley (published in PLOS ONE)
Recovery from perturbations during walking is primarily mediated by reactive control
strategies that coordinate multiple body segments to maintain balance. Balance control is often
impaired in clinical populations who walk with spatiotemporally asymmetric gait, and, as a
result, rehabilitation efforts often seek to reduce asymmetries in these populations. Previous
work has demonstrated that the presence of spatiotemporal asymmetries during walking does not
impair the control of whole-body dynamics during perturbation recovery. However, it remains to
be seen how the neuromotor system adjusts intersegmental coordination patterns to maintain
invariant whole-body dynamics. Here, we determined if the neuromotor system generates
stereotypical coordination patterns irrespective of the level of asymmetry or if the neuromotor
system allows for variance in intersegmental coordination patterns to stabilize whole-body
dynamics in the sagittal plane. Nineteen healthy participants walked on a dual-belt treadmill at a
range of step length asymmetries, and they responded to unpredictable, slip-like perturbations.
We used principal component analysis of segmental angular momenta to characterize
intersegmental coordination patterns before, during, and after imposed perturbations. We found
that two principal components were sufficient to explain ~ 95% of the variance in segmental
angular momentum during both steading walking and responses to perturbations. Our results also
revealed that walking with asymmetric step lengths led to changes in intersegmental coordination
patterns during the perturbation and during subsequent recovery steps without affecting whole-
body angular momentum. These results suggest that the nervous system allows for variance in
72
segment-level coordination patterns to maintain invariant control of whole-body angular
momentum during walking. Future studies exploring how these segmental coordination patterns
change in individuals with asymmetries that result from neuromotor impairments can provide
further insight into how the healthy and impaired nervous system regulates dynamic balance
during walking.
73

Bipedal locomotion is inherently unstable due to the small base of support, long single-
limb support times, and sensorimotor transmission delays (Winter, 1995). As a result, we must
frequently generate corrective responses to maintain balance in response to both internal and
external perturbations (Tang et al., 1998; Winter, 2009). For example, to recover from
unexpected perturbations such as slips or trips while walking, the nervous system generates
reactive control strategies involving simultaneous, coordinated responses of both the upper and
lower limbs (Marigold et al., 2003; Wang et al., 2012). These reactive, interlimb responses to
perturbations can restore stability by generating changes in angular momentum that counteract
the body's rotation toward the ground.
One conventional method to capture whole-body rotational dynamics during perturbation
responses is to compute whole-body angular momentum (WBAM). WBAM reflects the net
influence of all the body segments’ rotation and translation relative to a specified axis, which is
commonly taken to project through the body's center of mass (Liu et al., 2018; Martelli et al.,
2013; Pijnappels et al., 2005). WBAM is highly regulated as its value remains close to zero
during normal, unperturbed walking (Herr and Popovic, 2008; Popovic et al., 2004). During
perturbed walking, angular momentum dramatically deviates from that measured during
unperturbed walking (Liu et al., 2018; Martelli et al., 2013), and this deviation captures the
features of body rotation that, if not arrested, would lead to a fall. To regain balance when
encountering unexpected perturbations, the central nervous system activates muscles to
accelerate body segments and restore angular momentum across multiple recovery steps
(Pijnappels et al., 2004; Simoneau and Krebs, 2000).
74
Angular momentum can also capture balance impairments in populations with gait
asymmetries and sensorimotor deficits such as amputees and stroke survivors. These individuals
often have a higher peak-to-peak range of angular momentum than healthy controls (Honda et
al., 2019; Nott et al., 2014; Silverman and Neptune, 2011; Vistamehr et al., 2016), and the
presence of gait asymmetries may contribute to balance impairments in these populations. An
important question for clinical researchers is whether there is a causal relationship between gait
asymmetry and the ability to maintain balance in response to perturbations during walking.
Previous work demonstrated that whole-body dynamics, as measured by WBAM, do not change
in response to imposed gait asymmetries in healthy individuals (Liu et al., 2018). However, the
strategy that the central nervous system uses to stabilize whole-body dynamics remains to be
determined.
There are two distinct hypotheses capable of explaining the negligible influence of
asymmetry on whole-body angular momentum. First, the central nervous system may generate
stereotypical, invariant intersegmental coordination patterns in response to perturbations,
irrespective of the level of asymmetry. Alternatively, the nervous system could use reactive
control strategies that covary with asymmetry in a manner that would lead to invariant control of
whole-body momentum. This would be consistent with the uncontrolled manifold (UCM)
hypothesis, which predicts that the nervous system allows for variability in segmental angular
momenta to stabilize a higher-order performance variable such as whole-body angular
momentum (Latash et al., 2006).
Dimensionality reduction techniques, such as principal component analysis (PCA), are
commonly used to capture how the central nervous system coordinates multiple limb segments
(Aprigliano et al., 2017; Martelli et al., 2013). PCA reduces the high-dimensional, multi-
75
segmental time series data into a lower-dimensional set of latent variables capable of capturing
the variance in the overall behavior. Some studies used PCA of segmental angular momentum to
show that the intersegmental coordination patterns observed during recovery from slip-like
perturbations are highly correlated with the patterns observed during unperturbed walking
(Aprigliano et al., 2016; Martelli et al., 2016). Aprigliano et al. used PCA to show that there is no
difference in intersegmental coordination patterns between fall-prone older adults and healthy
young adults in response to slip-like perturbations (Aprigliano et al., 2017) despite that the older
adults commonly showed reduced ability to arrest forward body rotation during obstacle tripping
due to age-related changes in muscle strength and muscle activation onset time (Pijnappels et al.,
2005). In this case, the intersegmental coordination during perturbation response captured by
PCA is similar for both young and older adults while the time activation of coordination patterns
might differ across populations. Nevertheless, these studies suggest that the PCA analysis can
capture the preprogrammed and invariant response adopted by the central nervous system to
perturbation recovery across different tasks and populations.
Here, our objective was to determine how the presence of step length asymmetries
influences patterns of intersegmental coordination during slip-like perturbations. Since it has
previously been demonstrated that step length asymmetry does not influence the magnitude of
whole-body angular momentum, we aimed to determine if this was because the neuromotor
system generates stereotypical intersegmental coordination patterns across levels of asymmetry
or because the neuromotor system generates patterns of intersegmental coordination that covary
with spatiotemporal asymmetry. Ultimately, our findings extend our understanding of how the
healthy central nervous system coordinates intersegmental dynamics to maintain balance during
walking.
76

A total of 19 healthy young individuals (10M, 24 ± 4 yrs old) with no musculoskeletal or
gait impairments participated in this study. Lower limb dominance was determined by asking
participants which leg they would use to kick a ball. The study was approved by the Institutional
Review Board at the University of Southern California, and all participants provided informed
consent before participating. All aspects of the study conformed to the principles described in the
Declaration of Helsinki.

Data used here were collected as part of a previous study (Liu et al., 2018), and we
provide a summary of the procedures and setup below. Participants walked on an instrumented,
dual-belt treadmill with force plates underneath (Bertec, USA) at 1.0 m/s for six separate trials
and reacted to accelerations of the treadmill belts throughout the experiment. Although 1 m/s
was slower than the reported average self-selected speed during treadmill walking (Plotnik et al.,
2015), we chose this speed to be consistent with other investigations of the role of asymmetry
during healthy gait (Finley et al., 2013; Reisman et al., 2005; Sánchez et al., 2017). For the first
trial, participants walked on the treadmill for three minutes (Baseline) to obtain their natural
level of step length asymmetry. Then, for subsequent trials, participants were instructed to
modify their step lengths according to visual feedback provided via a display attached to the
treadmill, and we informed them that random slip-like perturbations would occur during these
trials. The visual feedback displayed the target step length for both right and left legs. A
“success” message would appear on the screen if the participants were able to step within three
standard deviations of the target step length. The standard deviation was determined on an
77
individual basis from each participant’s baseline step length variability. Participants completed a
randomized sequence of five, six-minute trials with target step length asymmetries (SLA, Eq.
1Error! Bookmark not defined.) of 0%, ±10%, and ±15% where 0% represents each
participant’s baseline SLA.
𝛥𝛥𝐿𝐿 𝑆𝑆 = 100 ∗
S L
l eft
− S L
r igh t
S L
l eft
+ S L
r igh t
(1)
SL
lef t
represents left step length and SL
rig ht
represents the right step length. Each trial
consisted of one-minute of practice walking without any perturbations, and then a total of 20
perturbations were applied (10 to each belt) during the remainder of the trial. Foot strike was
computed as the point when vertical ground reaction forces reached 150 N. Each perturbation
was remotely triggered by Python code and was characterized by a trapezoidal speed profile in
which the treadmill accelerated at foot strike to 1.5 m/s at an acceleration of 1.6m/s
2
, held this
speed for 0.3 s, and then decelerated back to 1.0 m/s at a deceleration of 1.6m/s
2
during the
swing phase of the perturbed leg. Participants were aware that they would experience
perturbations during the experiment, but the perturbations were randomly triggered to occur
within a range of 20 to 30 steps after the previous perturbation to prevent participants from
precisely anticipating perturbation timing. This range of steps was also selected to provide
participants with sufficient time to reestablish their walking pattern to match with the visual
feedback.

A ten-camera motion capture system (Qualisys AB, Gothenburg, Sweden) recorded 3D
marker kinematics at 100 Hz and ground reaction forces at 1000 Hz. We placed a set of 19 mm
spherical markers on anatomical landmarks to create a 13-segment, full-body model (Havens et
al., 2018; Song et al., 2012). We placed marker clusters on the upper arms, forearms, thighs,
78
shanks, and the back of heels. Marker positions were calibrated during a five-second standing
trial at the beginning of each trial. We removed all joint markers after the calibration.

We post-processed the kinematic and kinetic data in Visual3D (C-Motion, Rockville,
MD, USA) and Matlab 2017a (Mathworks, USA) to compute variables of interest. Marker
positions and ground reaction forces were low-pass filtered by 4
th
order Butterworth filters with
cutoff frequencies of 6 Hz and 20 Hz, respectively. We selected the type of filter and cut-off
frequency based on previous literature (Kurz et al., 2012; Reisman et al., 2009; Winter, 2009).
We calculated the achieved SLA as follows: first, we calculated the mean SLA of the four strides
before each perturbation and then distributed these mean values into five equally spaced bins
centered at -15%, -10%, 0, 10%, 15% with bin width equal to 5%. We used this achieved SLA
instead of target SLA as the independent variable in our statistical analyses. We categorized
Baseline (BSL) steps as the two steps before the perturbation occurred, perturbation (PTB) steps
as the step during which the perturbation was applied, and recovery (REC) steps as the four steps
that followed the perturbation. Since we did not find any differences between left and right
perturbations, our current analysis includes only perturbations of the right limb (Liu et al., 2018).
We also focused our analysis on angular momentum in the sagittal plane as this was the direction
in which the most prominent changes in WBAM were observed. Only minor deviations in
WBAM in the frontal and transverse plane occurred during the perturbation and recovery steps
(Liu et al., 2018).

We created a 13-segment, whole-body model in Visual3D and calculated the angular
momentum of each segment about the body’s center of mass. Segmental angular momenta ( 𝐿𝐿 𝑠𝑠 i
)
79
captured how the rotational and translational behavior of each body segment changed in response
to the treadmill perturbations. The model included the following segments: head, thorax, pelvis,
upper arms, forearms, thighs, shanks, and feet. The limb segments’ mass was modeled based on
anthropometric tables (Dempster, 1955), and segment geometry was modeled based on the
description in Hanavan (Hanavan, 1964). All segments were modeled with six degrees of
freedom, and we did not define any constraints between segments. Segmental linear and angular
velocity were computed using Eq. 2 (Silverman and Neptune, 2011).
𝐿𝐿 𝑠𝑠 i
=
𝑚𝑚 𝑟𝑟 � 𝑟𝑟 𝐶𝐶𝐶𝐶 − 𝑟𝑟 𝑟𝑟 × 𝑣𝑣 𝐶𝐶𝐶𝐶 − 𝑟𝑟 𝑟𝑟 � + 𝐼𝐼 𝑟𝑟 𝜔𝜔 𝑟𝑟
𝐶𝐶 𝑀𝑀𝑀𝑀
(2)
Here, mi is segmental mass, rCM-i is a vector from the segment's COM to the body's COM,
vCM-i is the velocity of each segment’s COM relative to the body’s COM, I
i
is the segmental
moment of inertia, ω
𝑖𝑖 is segmental angular velocity, and the index i corresponds to individual
limb segments.
Whole-body angular momentum was calculated as the sum of all segmental angular
momenta using Eq. 3. Integrated angular momentum In addition, integrated whole-body angular
momentum was computed as the area under the curve of the WBAM trajectory for each step
cycle to quantify the degree to which the body rotates about its center of mass across a step
cycle.
𝐿𝐿 � ⃗
= ∑ 𝐿𝐿 𝑠𝑠 i
𝒊𝒊 (3)
Lastly, we normalized momentum by the participant’s mass (M), baseline treadmill
velocity (V), and the participant’s height (H) (Eq. 2 and Eq. 3) following previous literature (Herr
and Popovic, 2008; Honda et al., 2019). Since our statistical analysis used a within-subject
design, the choice of variables used for normalization should not affect the statistical results. The
80
convention for measuring angular momentum was defined such that positive values represented
backward rotation.

We used principal component analysis (PCA) to extract intersegmental coordination
patterns for each step cycle. Before performing PCA, we first time normalized the time series of
segmental angular momenta to 100 points for each step cycle. Then, for each participant, we
generated an 𝐿𝐿 𝑠𝑠 matrix for each achieved SLA (±15%, ±10%, ±5%, %0) and step type (BSL1,
BSL2, PTB, REC1, REC2, REC3, REC4) with n_steps*100 rows and 13 columns. On average,
we created 6 (achieved SLA) by 7 (step types) matrices per participant as not all participants
achieved each desired level of asymmetry. We then standardized each matrix to have zero mean
and performed PCA to extract subject-specific coordination patterns using the pca function in
Matlab’s Statistical and Machine Learning Toolbox. Using PCA, we decomposed the segmental
angular momenta data into 1) a weighting coefficient matrix consisting of principal components
(PCs) ordered according to their variance accounted for (VAF) and 2) time series scores which
represented the activation of each PC throughout the step cycle (Figure 1). We retained the
number of PCs necessary to account for at least 90% of variance in 𝐿𝐿 𝑠𝑠 .
81

Figure 16: (A) Sagittal plane angular momentum (Lx) for 13 segments during one representative
baseline stride (black) and one perturbation stride (grey). The segments included the thigh,
shank, foot, forearm, and upper arm, bilaterally as well as the head, pelvis, and thorax. The
duration of each trace is one full stride from 0 to 100% of the stride cycle. (B) Schematic of
principal component analysis (PCA) of segmental angular momentum. The organization of the
data used as input to the PCA is illustrated to the left. PCA extracts weighting coefficient as
intersegmental coordination patterns or principal components (PC1 and PC2) and time series
scores of each PC (Filled bar plots: PC1; Open bar plots: PC2).

82

To investigate how intersegmental coordination patterns changed after each perturbation,
we compared the PCs extracted from the perturbation and recovery steps to the PCs extracted
from baseline steps. We computed the included angle ( θ
st ep
, Eq. 4) between each pair of PCs as
this is a common method to compare the similarity between vectors in a high-dimensional space.
The included angle of the unit vectors was between 0° (parallel and identical) and 90°
(orthogonal and most dissimilar) (Valero-Cuevas et al., 2016).
θ
st ep
= cos
− 1
(𝑃𝑃 𝑋𝑋 𝑏𝑏𝑏𝑏 𝑠𝑠 𝑅𝑅𝑏𝑏 𝑏𝑏 𝑖𝑖 𝑅𝑅 � � � � � � � � � � � � � � � � � � � � ⃗
∙ 𝑃𝑃 𝑋𝑋 𝑝𝑝 𝐵𝐵 𝑠𝑠 𝑖𝑖 � � � � � � � � � � � � � ⃗
) (4)
We then determined if the included angle between perturbation steps and baseline steps
was outside the distribution of included angles observed during unperturbed baseline walking.
To this end, we performed a permutation test that randomly and repeatedly selected two groups
of ten baseline steps for each participant. For each permutation, we first performed PCA for each
group of 10 steps and then calculated the included angle between the two PCs. We repeated this
shuffling process 10000 times for each participant. We used the median of this distribution as the
reference level in the statistical model to determine if the included angle for post-perturbation
values was greater than what would be expected from step-to-step variance.
Similarly, we computed the included angle between PCs extracted during walking at
different levels of asymmetry to those extracted from symmetrical walking to investigate how
asymmetry influenced intersegmental coordination patterns. (Eqn. 5).
θ
asy m
= cos
− 1
(𝑃𝑃 𝑋𝑋 𝑠𝑠 𝑦𝑦𝑚𝑚
� � � � � � � � � � � � ⃗
∙ 𝑃𝑃 𝑋𝑋 𝑏𝑏𝑠𝑠𝑦𝑦𝑚𝑚 � � � � � � � � � � � � � � � ⃗
) (5)
We also determined if the differences in coordination observed during walking with
different levels of asymmetry were above the level of variance observed during symmetrical
walking. As described above, we obtained a reference distribution of included angles from
83
symmetric walking to determine if the included angle for each level of asymmetry was greater
than would be expected from natural, step-to-step variance.

All statistical analyses were performed in R (3.4.3) using linear mixed-effects (LME)
models. We used the lme4 package to fit the model, the multcomp comparison for multiple
comparisons (Hothorn et al., 2008), and lmerTest package to calculate p-values (Kuznetsova et
al., 2017). Residual normality was confirmed using the Shapiro-Wilk test. When computing p-
values, we used the Satterthwaite approximation for the degrees of freedom based on differences
in variance between conditions. We used the Bonferroni correction for multiple comparisons for
all post-hoc analyses. For each model, we determined if a model with random effects provided a
better fit than a model with only fixed effects by comparing both models using a likelihood ratio
test and choosing the most parsimonious model. Based on this analysis, we ultimately included
random effects in all models to account for the individual differences between subjects.
Significance was set at p<0.05 level.
We first determined if the PCs extracted from the recovery steps differed from the PCs
extracted from the baseline steps during symmetrical walking. Here, the independent variable
was step type, and the dependent variable was θ
st ep
. The models were fit for both PC1 and PC2.
We performed a log transformation of the dependent variable ( θ
st ep
) to ensure that the residuals
were normally distributed. The full model structure was defined as follows: log( θ
st ep
) ~ 1 +
Step_Type + (1|Subject). To provide further insight into how the weighting coefficients for
individuals segments varied across steps, we used linear mixed effect model for each segment to
determine if the segment’s weighting coefficients (Wsegment) during perturbation and recovery
steps differed from the baseline steps. For this analysis, the independent variable was step type,
84
and the dependent variable was each weighting coefficient. The model was fit for each segment
and for both PC1 and PC2. The reference level was set to be right baseline step (BSL1) for all
right steps and left baseline steps (BSL2) for all left steps. The full model structure was defined
as follows: Wsegment ~ 1 + Step_Type + (1|Subject).
Then, we determined if intersegmental coordination patterns during asymmetrical
walking differed from those during symmetrical walking. For this analysis, we used Welch’s t-
test to evaluate if the included angle between the PCs extracted from the asymmetrical trials and
those extracted from symmetric walking were greater than what would be expected by chance.
We used Welch’s t-test because the included angle was not normally distributed.
Lastly, we determined if the included angle between each asymmetric trial and symmetric
walking varied with the magnitude or direction of asymmetry. For this analysis, the independent
variables were the magnitude of asymmetry (|Asym|), the direction of asymmetry (Direction),
and the interaction between asymmetry magnitude and direction, and the dependent variable was
θ
asym
. We performed a log transformation of the dependent variable ( θ
asym
) to ensure that the
residuals were normally distributed. The full model structure was defined as follows:
log( θ
asym
) ~ 1 + |Asym|*Direction + (1|Subject). We fit separate linear mixed-effect models for
each of five steps (Baseline1, Baseline2, Perturbation, Recovery 1 and Recovery 2) and each PC.

For all steps, two principal components accounted for ~95% of the variance in segmental
angular momentum (Table 2). On average, PC1 explained 74 ± 4% of the variance, and PC2
explained 22± 1% of the variance, while PC3 accounted for less than 3% of the variance. Thus,
the remaining analysis focuses on the first two PCs.

85
Table 2 Variance accounted for (VAF) for PC1, PC2, and PC3 during baseline steps,
perturbation steps, and recovery steps.
Step Type PC1 PC2 PC3 Sum
Baseline steps 74±4% 22±5% 2±1% 98±1%
Perturbation steps 75±5% 20±5% 3±1% 98±1%
Recovery steps 74±3% 21±4% 3±1% 98±1%
All steps 74±4% 22±4% 2±1% 98±1%

Contributions from the lower extremities were typically dominant in the first PC, while
contributions from the arms, pelvis, thorax, and head were less prominent (Figure 2). During
right steps, the left leg was in the swing phase and generated more positive momentum about the
body's COM, while the right leg generated negative momentum. Thus, the weighting coefficients
for the left leg segments (left thigh, shank, and foot) were positive while the coefficients for the
right leg segments were negative. Similarly, during a left step, the right leg was in the swing
phase and generated more positive momentum about COM, while the left leg generated negative
momentum. Thus, the weighting coefficients were positive while the coefficients for the left leg
segments were negative. Overall, the first PC captured the opposing momenta of the two legs
resulting from differences in the direction of rotation relative to the body's center of mass.
86

Figure 17: Principal components (PC) extracted from segmental angular momentum during (A)
baseline right steps, (B) baseline left steps, (C) perturbation steps, (D) recovery left steps, and
(E) recovery right steps when walking symmetrically (N=19). Blue: Right step; Pink: Left step;
Filled bars: PC1; Unfilled bars: PC2. The 13 segments include: RTH (right thigh), RSH (right
shank), RFT (right foot), LTH (left thigh), LSH (left shank), LFT (left foot), LFA (left forearm),
RFA (right forearm), LUA (left upper arm), RUA (right upper arm), H (head), PEL (pelvis), THX
(thorax).

For PC2, weighting coefficients for distal segments were also larger than the weighting
coefficients for proximal segments, although the coefficient for the thorax (THX) increased
compared to that in PC1. During the right step, the left thigh and left shank’s momenta opposed
the momentum of the left foot. Similarly, during the left step, the right thigh and shank momenta
opposed the right foot momentum. Thus, PC2 captured intralimb cancellation of segmental
momenta.

During the perturbation step, there was a significant increase in the included angle, which
indicated that the intersegmental coordination patterns during perturbation steps differed from
87
the coordination patterns during baseline steps (Figure 3). For this analysis, the results of the log-
likelihood ratio test revealed that random intercept were necessary for the regression model. For
this analysis, the results of the log-likelihood ratio test revealed that random intercept were
necessary for the regression model. For PC1, we found that the intersegmental coordination
patterns were significantly different from the patterns during baseline walking for the
perturbation steps (t(80)=18.2, p<2e-16), first recovery steps (t(80)=11.8, p<2e-16), and second
recovery steps (t(80)=8.4, p=2.2e-8). Similarly, for PC2, intersegmental coordination differed
during perturbation steps (t(80)=10.8, p<2.0e-16), first recovery steps (t(80)=12.5, p<2e-16),
second recovery steps (t(80)=4.3, p=3.81e-5), and fourth recovery step (t(80)=2.3, p=0.02).
There was no significant difference between intersegmental coordination patterns during the
third recovery steps for either PC1 (p = 0.58) or PC2 (p = 0.22). Thus, participants generally
were able to restore their coordination patterns to baseline by the third recovery step.
We also observed a significant main effect of step type on segment weighting coefficients
for PC1 for all segments (all p<0.005) except for pelvis (p=0.02). Similarly, for PC2, we found a
significant main effect of step type on segment weighting coefficients for all segments (all
p<0.005). Random intercepts were necessary for all the regression models for this analysis based
on the results of the log-likelihood ratio test.
88

Figure 18:Included angle between PCs extracted during each step relative to baseline steps
during symmetric walking (** p<0.001,* p<0.05). The horizontal bars and corresponding stars
indicate significant differences in the included angle. The data are represented as boxplots such
that the lower and upper edges of the box indicate the 25th and 75th percentile of the data,
respectively. The horizontal line in each box indicates the median. The whiskers extend to the
furthest data point beyond the lower or upper edges of the box that is within a distance of 1.5
times the middle 50th percentile of the data. Dots that lie beyond the whiskers indicate outliers.
Blue: Right step; Pink: Left step; Filled box plots: PC1; Non-filled box plots: PC2. The black
line indicates the mean of the permutated angle distribution of baseline steps and the shading
indicates the standard deviation.

Although the general patterns of intersegmental coordination were similar across levels
of asymmetry, asymmetric walking patterns led to measurable changes in the contributions of the
distal lower extremity segments (Figure 4).
89

Figure 19: The first intersegmental coordination pattern (PC1) and the second coordination
pattern (PC2) during (A) baseline right step, (B) perturbation step, and (C) the second recovery
step with -15%, 0% and 15% step length asymmetry. The colored bars indicate the mean value
across all participants (N=19), and the black lines indicate the standard deviation.

90
As the magnitude of achieved asymmetry increased, we observed an increase in the
deviation of intersegmental coordination patterns from symmetrical walking (Figure 5A-B) while
there was no change in integrated whole-body angular momentum compared to symmetrical
walking (Figure 5C). Results of log-likelihood ratio tests showed that random intercepts were
required in the regression models. One outlier was removed before fitting the linear mixed model
for the perturbation step for PC2 because it was more than three standard deviations higher than
the median of the included angles. Excluding the outlier did not change the statistical outcome.
All included angles differed from the permutated estimate of included angles (p<0.05), indicating
that intersegmental coordination at each level of asymmetry differed from the coordination
pattern during symmetrical walking. For all steps, we observed a significant main effect of
asymmetry on the included angle between the PCs from the asymmetric trials and the symmetric
trial (Table 2).
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Table 3 Statistical results from the ANOVA examining the effects of asymmetry and direction on
the included angle for each step type.
Step Type PC Factor numDF denDF F-value P value
Baseline1 PC1 Asym 2 73 9.7 <0.001
Direction 1 77 2.4 0.13
Asym:Direction 2 74 0.6 0.55
PC2 Asym 2 72 14.4 <0.001
Direction 1 78 4.0 0.049
Asym:Direction 2 74 0.08 0.92
Baseline2 PC1 Asym 2 72 5.7 0.005
Direction 1 75 1.3 0.26
Asym:Direction 2 73 0.1 0.88
PC2 Asym 2 71 11.0 <0.001
Direction 1 74 0.007 0.93
Asym:Direction 2 72 2.2 0.12
Perturbation PC1 Asym 2 73 19.0 <0.001
Direction 1 75 1.9 0.18
Asym:Direction 2 73 0.5 0.59
PC2 Asym 2 72 8.7 <0.001
Direction 1 73 1.3 0.25
Asym:Direction 2 72 0.68 0.51
Recovery1 PC1 Asym 2 74 11.2 <0.001
Direction 1 78 0.1 0.74
Asym:Direction 2 75 1.3 0.29
PC2 Asym 2 72 9.1 <0.001
Direction 1 75 0.1 0.72
Asym:Direction 2 73 0.8 0.45
Recovery2 PC1 Asym 2 72 8.7 <0.001
Direction 1 75 1.8 0.18
Asym:Direction 2 73 0.4 0.67
PC2 Asym 2 73 8.5 <0.001
Direction 1 77 1.9 0.18
Asym:Direction 2 74 0.2 0.84

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Figure 20:Included angle between PCs extracted during asymmetrical walking (5%, 10%, and
15%) and symmetrical walking for each step (*** p<0.001, ** p<0.01, * p<0.05) for (A) PC1
and (B) PC2. (C) Integrated whole-body angular momentum during asymmetrical walking
93
relative to symmetrical walking for each step. Blue: Right step; Pink: Left step; Filled box plots:
PC1; Non-filled box plots: PC2. The shaded gray area indicated the standard deviation of
permutated included angle for each step, and the black line indicated the mean of the
distribution.

The included angle between the PCs extracted during asymmetric walking and symmetric
walking increased with the magnitude of achieved asymmetry (Figure 5 A,B). Specifically, the
difference between intersegmental coordination patterns was greater when walking with 15%
asymmetry compared to 5% asymmetry during right baseline steps (Bonferroni corrected
p<0.001), perturbation steps (Bonferroni corrected p<0.001), first recovery steps (Bonferroni
corrected p=0.03) and second recovery steps (Bonferroni corrected p=0.002) for PC1. The
difference in included angles was also significantly different from 5% asymmetry for PC2 when
walking with 15% asymmetry during baseline right steps (Bonferroni corrected p=0.01) and
perturbation steps (Bonferroni corrected p = 0.003) and second recovery steps (Bonferroni
corrected p=0.04). Lastly, there was only an effect of the direction of asymmetry for PC2 (F(1,
79), p=0.049) during the baseline right step (Baseline 1).

We investigated how step length asymmetry affected intersegmental coordination
patterns during responses to treadmill-based slip perturbations during walking. Our central
finding was that intersegmental coordination patterns observed during asymmetrical walking
differed from symmetrical walking during both unperturbed walking and perturbation recovery.
When combined with previous observations that the reactive control of overall WBAM is not
influenced by asymmetry (Liu et al., 2018), these results indicate that healthy people use a
flexible combination of intersegmental coordination patterns rather than invariant reactions to
maintain WBAM during perturbation responses when walking with asymmetric gait patterns.
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Variations in coordination patterns during asymmetrical walking likely resulted from
changes in the momentum generated by the lower extremities to reach the target asymmetry.
Since the distal segments of the lower limbs are relatively far from the body’s center of mass and
have a high velocity, they make the largest contribution to changes in intersegmental
coordination patterns. For example, to achieve a positive asymmetry, participants placed their
left foot further in front of the center of mass and increased the extension of their right hip so that
the right foot was further behind their COM at heel strike. To achieve this objective, participants
had to increase swing velocity. This likely explains why we observed increased weights of the
left foot as SLA increased during right steps in the first principal component since positive step
length asymmetries required longer left steps and faster foot swing.
The observation that reactive control of WBAM is consistent across levels of asymmetry
(Liu et al., 2018) despite the variation in intersegmental coordination observed here may indicate
that WBAM is a task-level variable that is stabilized by the nervous system during perturbation
recovery. This is consistent with the framework proposed by the uncontrolled manifold
hypothesis (UCM), which argues that the central nervous system allows for variability over a
manifold of solutions that all successfully stabilize a higher-level performance variable (Domkin
et al., 2002). Here, WBAM would serve as a high-level performance variable that is stabilized
through covariation of elemental, segmental-level momenta. For example, Papi et al.
demonstrated a similar concept when they found no differences between people post-stroke and
healthy individuals in COM displacement during the stance phase of walking despite between-
group differences in lower extremity joint kinematics (Papi et al., 2015). Therefore, it is possible
that when dynamic stability is challenged during walking, the central nervous system carefully
regulates WBAM while allowing variance in lower-level, intersegmental coordination patterns.
95
In this study, we provided visual information about the desired and actual step lengths at
each foot-strike throughout all trials, including the perturbation and recovery steps. Participants
were encouraged to achieve the target step lengths for as many steps as possible, and therefore
participants may have relied on this feedback during perturbation recovery to return to their pre-
perturbation walking patterns faster than they otherwise would without visual feedback.
However, participants’ reactive response is unlikely to influence measures of momentum until
late into the first recovery step as the step length information was only shown after the foot-strike
of the first recovery step. It remains to be seen if patterns of interlimb coordination would differ
in the presence of asymmetries that are not guided by online visual feedback.
Although the reactive intersegmental coordination patterns were significantly different
from those observed during unperturbed locomotion, the overall patterns were qualitatively
similar across steps. Taken together, these results may reflect two keys aspects of coordination
during perturbed walking. First, the qualitative similarity between pre- and post-perturbation
patterns observed here and in previous work (Aprigliano et al., 2016) may reflect the dominant
coordination patterns that characterize both unperturbed and perturbed bipedal walking. In
contrast, the statistical differences between pre- and post-perturbation coordination patterns may
reflect the changes in coordination necessary to maintain balance in response to perturbations.
Patterns of intersegmental coordination observed during responses to external perturbations
during walking likely capture a combination of passive limb dynamics, stereotypical pattern
generation, and reactive balance control responses (Cappellini et al., 2006).
We observed that the upper limbs’ contribution to the control of angular momentum in
the sagittal plane was negligible compared with lower limb segments during perturbation
recovery. Since a stepping response is sufficient to restore balance from the treadmill
96
accelerations used in this study, increases in momentum from the lower extremities may have
been sufficient to restore sagittal plane WBAM. Consistent with our findings, Pijnappels et al.
also found that arm movements had a small effect on body rotation in the sagittal plane during
tripping over obstacles which elicits excessive forward rotation similar to the current study
(Pijnappels et al., 2010). However, during larger perturbations that trigger backward falls, the
arms elevate to shift the body’s center of mass back within the base of support (Marigold et al.,
2003). This difference in the role of the arms across studies of perturbation recovery may result
from the use of a larger velocity and displacement of the foot in the Marigold et al. (Marigold et
al., 2003) study. However, it remains to be seen how systematic variation of the magnitude and
direction of external perturbations influences the role of the upper extremities during balance
recovery.
Our results may also have implications for understanding the potential effects of
interventions designed to reduce gait asymmetries in people post-stroke, as this is a common
rehabilitation objective in this population (Patterson et al., 2015). The data from the current study
illustrate how the intact neuromotor system modulates coordination between the upper and lower
extremities in response to changes in asymmetry. Based on the current results, we would
hypothesize that reducing asymmetry in people post-stroke would also affect their reactive
control strategies and potentially alter their interlimb coordination during perturbation responses.
However, people post-stroke have impaired reactive control of balance due to transmission
delays (Sharafi et al., 2016), muscle weakness (Gray et al., 2012) and decreased propulsion in the
paretic leg (Chen et al., 2005). These deficits may prevent the paretic leg from initiating a
successful stepping response to prevent excessive body rotation due to external perturbations.
Together, further investigation is necessary to determine if reductions in asymmetry would
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change their interlimb coordination during reactions to perturbations for people post-stroke and
affect the ability to restore WBAM during perturbation recovery in people post-stroke.

We thank Natalia Sanchez, Ph.D., for her insights during the design of this experiment
and Aram Kim for her assistance with the statistical analysis.

C.L designed the experiment, collected data, analyzed data, and wrote the manuscript.
J.M.F conceived of the experiment, advised in data analyses, and edited the manuscript.

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Chapter 5. Paretic deficits prevent the execution of reactive control strategies
during walking
Maintaining balance in response to unexpected perturbations during walking is often
mediated by reactive stabilization strategies. The objective of this study was to characterize the
differences in the regulation of whole-body dynamics following paretic and non-paretic
perturbations due to post-stroke deficits. Thirty-eight post-stroke participants and thirteen age-
matched speed-matched non-disabled participants walked on the dual-belt treadmill while
receiving slip-like perturbations on either limb at foot strike. We assessed the whole-body
angular momentum and the effect of ground reaction forces of both leading and trailing limbs on
the body dynamics using angular impulse in response to perturbations. We found that people
post-stroke needed one more recovery step to restore balance compared with controls.
Perturbations to the paretic side caused more whole-body rotation during the perturbation steps
versus non-paretic perturbations and that of control participants. In addition, we found that
unlike control participants, people post-stroke did not reduce the forward angular impulse during
the second half of the stance phase at their paretic side. They also did not use their paretic limb to
increase collision angular impulse or decrease the forward push-off angular impulse during the
collision phase at the first recovery steps. These post-stroke impairments in reactive stabilization
strategies, especially at their paretic side, may contribute to the high fall risk for people post-
stroke.

People post-stroke have an increased risk of falls, which could lead to injuries and reduce
their mobility during daily activities (Weerdesteyn et al., 2008). Using reactive control strategies
99
to maintain balance is essential for people post-stroke, since falls often occur when people are
unable to generate an adequate corrective response to recover from slips or trips. However, due
to functional deficits in the lower extremities, people post-stroke commonly demonstrate delayed
muscle activation to external perturbations (Kirker et al., 2000; Marigold et al., 2004), abnormal
muscle activation patterns (Higginson et al., 2006), and muscle weakness (Olney and Richards,
1996). These sensorimotor deficits after stroke likely contribute to the greater risk of falls in this
population.
One established metric used to characterize dynamic balance during walking is whole-
body angular momentum (WBAM). This measure reflects the net contribution of all body
segments to the body’s rotation about a given axis, and this axis is commonly specified as
extending through the body’s center of mass (CoM) (Herr and Popovic, 2008). WBAM in the
sagittal plane is highly regulated within a small range for people without disabilities during
unperturbed walking. Such regulation is achieved by coordinating the moment generated by the
ground reaction force (GRF) about the body CoM across the gait cycle. Typically, the leading
limb’s GRF generates a backward moment about CoM from foot-strike to midstance and the
trailing limb’s GRF generates a forward moment from midstance to toe-off. WBAM increases
sharply during sudden perturbations and subsequent recovery steps compared with unperturbed
walking (Liu et al., 2018; Martelli et al., 2013), and this increase captures body rotations that
might lead to a fall if not arrested. People post-stroke have a larger peak-to-peak range of
WBAM during the non-paretic step of unperturbed walking compared to controls (Honda et al.,
2019), and greater values of WBAM during walking are associated with worse performances in
clinical assessments of balance (Nott et al., 2014; Park et al., 2021).
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People without disabilities use rapid stepping to reduce the effects of external
perturbations during walking (Debelle et al., 2020; Mathiyakom and McNitt-Gray, 2008). One
common paradigm to study reactive response during walking is by increasing the posterior
velocity of one belt of the split-belt treadmill (Debelle et al., 2020; Liu et al., 2018; Martelli et
al., 2013; Sloot et al., 2015). Such posteriorly-directed slip-like perturbations will induce forward
rotation of body and generate forward momentum about the CoM. People need to coordinate the
net moment about the CoM generated by the ground reaction force of the legs. They typically
step quickly forward and place the leading foot further forward relative to the body’s CoM
compared to that during pre-perturbation steps (Maki and McIlroy, 2006). The corresponding
increase in the posteriorly-directed component of the GRF would generate backward moment
about the CoM and help arrest the forward rotation of the body. In addition, people could
modulate trailing limb push-off force which is the anterior-directed component of the GRF to
reduce forward angular momentum following the forward falls (Pijnappels et al., 2005). Thus, a
combination of responses in both the leading and the trailing limbs could counteract the forward
whole-body angular momentum caused by balance perturbations during walking.
However, sensorimotor deficits in people post-stroke may prevent them from executing
successful reactions in response to the forward fall induced by the increase of belt speed. During
paretic stance, the paretic leg may be too weak to support the body and, therefore, the non-
paretic swing leg may not have sufficient time to modify its trajectory and step further forward
and arrest forward momentum. Conversely, if such slip occurs during non-paretic stance, the
paretic leg may have difficulty initiating a successful stepping response to help restore balance as
the paretic leg demonstrates decreased propulsion (Allen et al., 2014; Chen et al., 2005; Lauzière
et al., 2015) and decreased hip flexor activity (Rybar et al., 2014) to step at the proper location.
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Among few studies that assessed the differences in reactive responses between side of
perturbations during walking for people post-stroke, post-stroke participants demonstrated
shorter paretic compensatory step lengths than non-paretic step lengths (Kajrolkar and Bhatt,
2016) when backward loss of balance was induced by forward translation of a platform at foot
strike. Additionally, there is increased paretic braking force (Turns et al., 2007), reduced
propulsive force (Bowden et al., 2006), and reduced vertical ground reaction force during the
paretic step (Kim and Eng, 2003; Lee et al., 2020). These deficits in foot placement and the force
generation at the end-effector affect the magnitude and orientation of GRF vectors relative to
CoM and may lead to impairments in the ability to generate sufficient corrective angular
impulses about the CoM. Thus, people post-stroke will likely show differences in the
stabilization strategies they use following perturbations to their paretic versus non-paretic limbs.
Here, our objective was to determine how stroke influences the mechanical consequences
of reactive control strategies to control whole-body angular momentum following sudden
treadmill accelerations (Figure 21). (1) We hypothesized that treadmill accelerations would lead
to more forward WBAM during the perturbation step for people post-stroke compared to healthy
controls regardless of the side of the perturbation. We expected that perturbations of the paretic
leg would lead to greater forward angular momentum than perturbations of the non-paretic side
due to the paretic leg being unable to support body weight and maintain balance during the
perturbation step. (2) We hypothesized that control participants could increase the backward
angular impulse during the perturbation step more than people post-stroke. (3) We hypothesized
that control participants would increase the net angular impulse during the collision phase of the
first recovery step by increasing the angular impulse generated by the leading limb and
decreasing the angular impulse generated by the trailing limb. People post-stroke would show a
102
greater increase in the angular impulse during the first recovery step following both paretic and
non-paretic perturbations compared to controls to compensate for greater forward momentum.
During the collision phase of the first recovery steps following paretic perturbations, they would
also generate a greater increase of backward angular impulse using the leading limb but lesser
decrease of forward angular impulse using their trailing leg compared to those measured
following non-paretic perturbations.
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Figure 21: Graphical illustration of hypothesis for whole-body angular momentum, angular
impulse during perturbation steps, and the leading limb angular impulse and the trialing limb
angular impulse during the first recovery steps.

We recruited 38 people post-stroke (Table 4) from the IRB-approved, USC Registry for
Aging and Rehabilitation, the USC Physical Therapy Associates Clinic, and from Rancho Los
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Amigos National Rehabilitation Center. Inclusion criteria for the stroke survivors were the
following: 1) a unilateral brain lesion 2) paresis confined to one side, 3) ability to walk on the
treadmill for five minutes without holding on to any support. Use of ankle-foot orthoses were
permitted during the experiment. We also recruited 13 age-matched controls from the
community. Exclusion criteria for healthy older adults were neurological, cardiovascular,
orthopedic, and psychiatric diagnoses. Study procedures were approved by the Institutional
Review Board at the University of Southern California and all participants provided written,
informed consent before testing began. All aspects of the study conformed to the principles
described in the Declaration of Helsinki.
Table 4 Participant demographics for both control and stroke participants.
Control
(N = 13)
Stroke
(N= 38)
p value
Age (yrs) 58 (29) 60 (11) 0.76
Female/Male 6/7 14/24 /
Treadmill speed (m/s) Matched: 0.6 (0.2) 0.6 (0.2) 0.62
Scaling factor √ 𝐠𝐠𝐠𝐠
(m/s)
3.08 (0.082) 3.04 (0.095) 0.13
Self-selected
Overground speed
(m/s)
1.3 (0.2) 0.8 (0.3) <0.0001
Berg Balance Scale 55 (2) 51 (6) 0.017
Activity-specific
Balance Confidence
Scale
97 (3.5) 77 (13) <0.0001
Falls Efficacy Scale 18 (2) 29 (12) 0.0025
Fugl-Meyer lower
extremity
/ 26 (5) /
Left/right side
hemiparesis
/ 15/23 /
Months after stroke / 83 (55) /
Functional Gait
Assessment
/ 21 (6) /

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The experimental protocol for post-stroke participants was described previously (Buurke
et al. 2020), and we provide a summary of the procedures and setup below. The complete
protocol consisted of a set of clinical assessments and walking trials on the treadmill. Before the
walking trials, we evaluated motor impairment using the lower extremity portion of the Fugl-
Meyer Assessment (FM) (Fugl-Meyer et al. 1975), static balance using Berg Balance Scale
(BBS) (Berg et al. 1992), static and dynamic balance during locomotion tasks using the
Functional Gait Assessment (FGA)(Leddy et al. 2011), and over-ground walking speed using the
10-meter walking test. Participants also completed questionnaires about balance confidence
using the Activity-Specific Balance Confidence Scale (ABC) (Powell and Myers 1995). Higher
scores on all these assessments indicated better balance control or higher balance confidence.
Lastly, we completed a Fall History Questionnaire for participants who experienced at least one
fall within the past year. After clinical evaluations, we instructed stroke participants to walk on
the dual-belt treadmill (Bertec, Columbus, OH, USA). A harness was provided to prevent the
participants from falling but no body weight support was provided. First, the participants walked
on the treadmill to familiarize themselves with the experimental set-up. We used the following
method to identify participants’ preferred walking speed on a treadmill (Park et al. In Review).
We started from 70% of the speed obtained from a 10-meter walking test and adjusted their
walking speed by 0.05m/s increments or decrements until the participants verbally indicated that
they achieved their preferred walking speed. For the next trial, the participants walked for three
minutes at their self-selected speed. Then the participants completed a familiarization trial with
at least two perturbations on each side characterized as sudden treadmill accelerations which
106
were triggered at foot-strike based on the ground reaction forces recorded by the treadmill’s
force plates. During each trial, we applied five accelerations to the treadmill belts at each side.
Control participants also completed a set of clinical assessments including ABC, FES,
BBS, and over-ground 10-meter walking test. We instructed the participants to walk at matched
speeds with a stroke participant of similar age. Control participants completed two walking trials:
one unperturbed walking trial with matched walking speed and one perturbed trial with matched
walking speed. For the perturbed trial, 10 perturbations occurred at each side.
For both groups, treadmill accelerations were triggered at random intervals within 15 to
25 steps after the previous perturbation to allow participants to reestablish their walking patterns.
Each perturbation was characterized by a trapezoidal speed profile in which the speed increased
by 0.2 m/s at an acceleration of 3 m/s
2
, was held for 0.7 s and then decelerated back to the self-
selected speed during the swing phase of the perturbed leg. Between each trial, stroke
participants had breaks of at least three minutes to minimize fatigue while control participants
had breaks if needed. Participants did not hold on to handrails while walking on the treadmill.

A ten-camera motion capture system (Qualisys AB, Gothenburg, Sweden) recorded 3D
marker kinematics at 100 Hz and ground reaction forces at 1000 Hz. We placed a set of 14 mm
spherical markers on anatomical landmarks to create a 13-segment, full-body model (Havens et
al., 2018; Song et al., 2012). For stroke participants, the pelvis segment was modeled to be
rigidly connected to the trunk because they wore an extra harness that blocked the markers
necessary to track the pelvis accurately. The model was validated previously (Park et al., 2021).
We placed marker clusters on the upper arms, forearms, thighs, shanks, and the back of heels.
107
Marker positions were calibrated during a five-second standing trial at the beginning of each
trial. We removed all joint markers after the calibration.

We post-processed the kinematic and kinetic data in Visual3D (C-Motion, Rockville,
MD, USA) and Matlab 2020b (Mathworks, USA) to compute variables of interest. Marker
positions and ground reaction forces were low-pass filtered by 4
th
order Butterworth filters with
cutoff frequencies of 6 Hz and 20 Hz, respectively. We selected the type of filter and cut-off
frequency based on previous literature (Kurz et al., 2012; Reisman et al., 2009; Winter, 2009).
We defined foot strike as the point when the vertical ground reaction force became greater than
150N and foot off as the point when vertical ground reaction force became less than 150N. The
timing of perturbations relative to foot-strike was re-examined post-hoc. We removed the
perturbations that occurred more than ~150 ms after the foot-strike.

We created a 13-segment, whole-body model in Visual3D and calculated the angular
momentum of each segment about the body’s center of mass. The model included the following
segments: head, thorax, pelvis, upper arms, forearms, thighs, shanks, and feet. The limb
segments’ mass was modeled based on anthropometric tables (Dempster, 1955), and segment
geometry was modeled based on the description in Hanavan (Hanavan, 1964). Whole-body
angular momentum (L) in the sagittal plane was then computed about the mediolateral axis to the
right through the body CoM as the sum of all segmental angular momenta which were composed
of segmental rotation about the body’s CoM and rotation of each segment about its own CoM
(Silverman and Neptune, 2011). L was normalized by the combination of participant’s mass (M),
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the participant’s height of COM (H), and gravity constant (g) to make L dimensionless and
reduce between subject variability (Eqn. 1)(Martelli et al., 2013).
𝐿𝐿 =
∑ [ 𝑚𝑚 𝑖𝑖 � 𝑟𝑟
� � ⃗
𝐶𝐶𝐶𝐶 − 𝑖𝑖 𝑖𝑖 × 𝑣𝑣 ⃗
𝐶𝐶𝐶𝐶 − 𝑖𝑖 𝑖𝑖 � + 𝐼𝐼 𝑖𝑖 𝜔𝜔 𝑖𝑖 ]
𝑖𝑖 𝑋𝑋 𝑔𝑔 1
2
𝑀𝑀 3
2

(1)

Here, m is segmental mass, r is the distance from segment to the body COM, I is the
segmental moment of inertia, 𝜔𝜔 is segmental angular velocity, and the index i corresponds to
individual limb segments. In the sagittal plane, negative values of angular momentum
represented forward rotation, while positive values represented backward rotation. In addition,
integrated whole-body angular momentum (Lint) was computed as the area under the curve of the
WBAM trajectory for to quantify the degree to which the body rotates about its center of mass
across a step cycle. We categorized the pre-perturbation steps as the last two steps before the
perturbation occurred (Pre-PTB 1 - 2), perturbation steps (PTB) as the step during which the
perturbation was applied, and recovery steps (Recovery 1-4) as the four steps that followed the
perturbation.

In addition to whole-body angular momentum, we used angular impulse to quantify the
mechanical consequences of the reactive control strategies on whole-body dynamics.

We defined the effect of the GRF from each limb on the change in whole-body dynamics
using measures of angular impulse as described in Eqn.2-5 (Figure 22). The earliest strategy that
people could employ to begin recovering from losses of balance during the perturbation step is to
modulate the GRF of the stance limb to reduce the forward momentum about CoM. Thus, we
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first computed the forward pitch impulse during the stance phase (∆LStance) to capture the effect
of the stance limb ground reaction force on whole-body dynamics (Eqn.2).

∆LStance = ∫ r ⃗
𝑠𝑠 ×F
� ⃗
𝑠𝑠 dt
𝐹𝐹 𝐵𝐵 𝐶𝐶𝐵𝐵 𝑇𝑇
(2)
Here, r ⃗
𝑠𝑠 represents the displacement from body CoM to the center of pressure of the
stance limb. F
� ⃗
𝑠𝑠 represents the ground reaction force in the sagittal plane at the stance limb.
Forward pitch impulse is computed by integrating the cross product of the displacement and
ground reaction force during the stance phase defined as 80% of the average time from
midstance to the subsequent foot strike during Pre-PTB steps. Midstance is defined as the
midpoint in time of each step. We use the same time duration across all step types to remove the
effect of time on computing angular impulses. Index s corresponds to the stance leg.
Then, we computed the collision angular impulse to quantify the effect of leading limb
foot placement on whole-body dynamics and push-off angular impulse to quantify the effect of
trailing limb GRF on whole-body dynamics during the collision phase when both limbs were on
the ground (Eqn.3-4).

∆LCol = ∫ r ⃗
𝐿𝐿 ×F
� ⃗
𝐿𝐿 dt
𝐹𝐹 𝐵𝐵 + ∆ 𝑖𝑖 𝐹𝐹 𝐵𝐵
(3)
Here, r ⃗
𝐿𝐿 represents the displacement from body CoM to the center of pressure of the
leading limb. F
� ⃗
𝐿𝐿 represents the ground reaction force in the sagittal plane at the leading limb.
Collision angular impulse is computed by integrating the cross product of the displacement and
ground reaction force during the collision phase ( ∆ 𝐿𝐿 ). The collision phase is defined as 80% of
the average double support time during Pre-PTB steps (Adamczyk and Kuo, 2009). Index L
corresponds to the leading leg.

∆LPush = ∫ r ⃗
𝑇𝑇 ×F
� ⃗
𝑇𝑇 dt
𝐹𝐹 𝐵𝐵 + ∆ 𝑖𝑖 𝐹𝐹 𝐵𝐵
(4)
110
Similarly, r ⃗
𝑇𝑇 represents the displacement from body CoM to the center of pressure of the
trailing limb. F
� ⃗
𝑇𝑇 represents the ground reaction force in the sagittal plane at the trailing limb.
Push-off angular impulse is also computed during the collision phase. Index T corresponds to the
trailing leg.
Lastly, the net angular impulse (∆LNet) during the collision phase is the sum of collision
and push-off angular impulse (Eqn.5).
∆LNet = ∆LCol + ∆LPush (5)
All angular impulse measures was normalized by the same normalization factors as the
whole-body angular momentum. We excluded steps for which people crossed over to the other
belt.

Figure 22: Diagram of computed angular impulse about the body CoM by the leading and
trailing leg during the perturbation (PTB) step and the first recovery step (R1). The PTB ∆LStance
is computed during the phase from midstance of the PTB step until the foot strike of the R1 step.
R1 ∆Lcol and R1 ∆Lpush are computed during the collision phase of the R1 steps. R1 ∆LNet is
111
computed as the sum of R1 ∆Lpush and R1 ∆Lcol .FS: Foot strike; FO: Foot-off. The arrows (+/-)
indicate the backward and forward moments by the GRF about CoM, respectively.

All statistical analyses were performed in Matlab 2020b (Mathworks). For people post-
stroke, if the non-paretic leg was perturbed, the R1, and R3 steps were paretic steps, and Pre-
PTB step, PTP step, R2, and R4 steps were non-paretic steps and vice versa for the paretic
perturbations.
We used a two-sample t-test with unequal variances to test whether there were significant
differences in any participant characteristics between control and post-stroke participants. We
used a two-sample t-test with unequal variances to test whether there were significant differences
in Lint, ∆Lstance, ∆Lcol, ∆Lpush , and ∆LNet during pre-perturbation steps between control and stroke
group (paretic and non-paretic steps). We also used a two-sample t-test with unequal variances to
determine if the deviation of these variables from pre-perturbation values differed between
control participants and stroke participants following paretic and non-paretic perturbations.
These t-test results were adjusted for multiple comparisons using Bonferroni corrections. We
also used the Lilliefors Test to test the residual normality.
We used linear mixed-effect models for stroke participants and control participants,
respectively, to assess if any of the dependent variable Lint, ∆Lstance, ∆Lcol, ∆Lpush , and ∆LNet
following perturbations differed from those measured during pre-perturbation steps. The
independent variables included Step Type (Pre-PTB 1-2, PTB, Recovery 1-3), side of
perturbation (Leg) (paretic and non-paretic side), and the interaction between Step Type and Leg
to determine if changes in any of the dependent variables from pre-perturbation steps differed
between sides. The reference level was set to be Pre-PTB 1. For control participants, we did not
find that any of the variables differed between sides. Thus, we combined values across limbs for
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the remainder of the analysis and the independent variable only included Step Type (Pre-PTB 1-
2, PTB, Recovery 1-3). We included a random intercept for each model to account for
unmodeled sources of between-subject variability. We also determined the number of recovery
steps needed for participants to restore balance by identifying when Lint returned to values
measured before the perturbations. For angular impulses that we computed to quantify the
mechanical consequences of the reactive control strategies on whole-body dynamics, we focused
on the angular impulse during the perturbation and first recovery steps as we anticipated that
reactive stabilization strategies should be the most evident during these two steps than during the
rest of the recovery steps.
Lastly, we computed Pearson correlation coefficients to test for relationships between the
reactive stabilization strategies such as changes in PTB ∆Lstance,, R1 ∆Lcol, and R1 ∆Lpush and
each clinical balance assessment (BBS, FGA, ABC, FM). Significance was set at α = 0.05.
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Figure 23: Whole-body angular momentum in the sagittal plane and ground reaction force for
one representative control participant (A) and a stroke participant during a paretic perturbation
(B) and non-paretic perturbation (C) for both a pre-perturbation stride and a perturbation
stride. Each stride began at foot strike. The gray traces indicate the time series data for a pre-
perturbation stride while the black or colored traces indicate a perturbation stride. Negative
values of angular momentum represent forward rotation while positive values represent
backward rotation. Ground reaction force (% body weight) in the vertical and anterior-posterior
directions for the perturbed and the contralateral limb when perturbations occurred on the
dominant side for the control participant (A), or on the paretic (B) or non-paretic sides (C) for
the stroke participant. For the control participant, black lines indicated the perturbed side and
the dashed lines indicated the contralateral side. For the stroke participant, pink and blue lines
represent the paretic leg and non-paretic leg during the perturbation stride, respectively. Gray
dashed lines represent non-paretic leg and paretic leg during the pre-perturbation stride,
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respectively. Black dashed vertical lines correspond to the time of foot strike. Gray shaded
vertical box corresponds to the double support phase from the time of foot strike to the
contralateral foot-off. Pre-PTB: pre-perturbation stride; PTB: perturbation stride.

The acceleration of the belts caused consistent, immediate increases in forward angular
momentum and triggered multi-step balance recovery responses for both control participants and
people post-stroke. During the perturbation step, angular momentum became more negative as
the body rotated forward. To compensate for the perturbation, participants then generated
positive angular momentum and initiated backward rotation (Figure 23). Compared to control
participants, people post-stroke tended to have a larger range of whole-body angular momentum
during both pre-perturbation steps and perturbation steps.
First, we examined the effects of perturbations on whole-body angular momentum and
determined whether there was a difference between perturbations on the dominant side and non-
dominant side in integrated angular momentum across the step cycle for control participants
(Figure 24). We found no significant main effects for Leg (p = 0.84) or interaction between Leg
and Step Type (p = 0.84). We found a significant main effect of Step Type (F(1,144) = 34.9, p <
0.0001). Control participants significantly increased their forward rotation, indicated by a more
negative Lint, during the perturbation steps relative to pre-perturbation steps (t(144) = -4.82, p <
0.0001). They then restored their angular momentum during the first recovery steps (R1) as
indicated by a more positive Lint, (t(144) = 4.7, p < 0.0001). Control participants restored whole-
body angular momentum to that during pre-perturbation steps by the second step (R2) following
the treadmill perturbations (p = 0.86).
Similarly, we computed the integrated angular momentum across the step cycle to
quantify the effects of perturbations on whole-body rotation for stroke participants (Figure 24).
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We found a significant main effect for Step Type (F(5,444) = 64.7, p <0.001) and a significant
interaction between Leg and Step Type (F(5, 444) = 3.0, p = 0.011). However, we did not find a
significant main effect for Leg (F(1, 444) = 2.12, p = 0.14). We found that Lint was significantly
different from the pre-perturbation steps during the perturbation (PTB) step (t(444) = -6.96, p <
0.001), recovery (R) steps R1 (t(444) = 9.4, p < 0.001), and R2 (t(444) = -2.02, p = 0.04). There
was no significant difference between 𝐿𝐿 𝑖𝑖 𝑖𝑖𝑖𝑖
during the third recovery steps R3 (t(444) = -1.05, p =
0.29). Thus, participants generally were able to restore their angular momentum to pre-
perturbation by the third recovery step. Additionally, changes in Lint from the pre-perturbation
steps during the paretic perturbation steps were significantly more negative compared with the
non-paretic perturbation steps (t(444) = -2.3, p = 0.021), indicating that people post-stroke fell
forward more when the perturbation occurred during paretic stance. This was consistent across
32 out of 38 participants.
Lastly, we compared whether changes in Lint across the step cycle differed between
control participants and stroke participants (paretic and non-paretic perturbations) at each step.
We found that the increase in integrated angular momentum was higher during perturbation steps
in stroke participants following paretic perturbations than that for control participants
(Bonferroni corrected p = 0.018), but there was no difference in increase in integrated angular
momentum between non-paretic perturbation steps and that for control participants (p = 0.56).
During the first recovery step, there was no difference in the increase in integrated angular
momentum between control participants and stroke participants regardless of whether the
perturbation occurred on the paretic or non-paretic steps (all p>0.05).
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Figure 24: Median integrated angular momentum in the sagittal plane over the step cycle
relative to the corresponding pre-perturbation steps (ΔLint) for all participants. Steps alternated
between paretic and non-paretic for stroke participants. PTB: Perturbation; R: Recovery. The
asterisks on top of the boxplots indicate whether the difference in Lint from pre-perturbation steps
was significantly different from zero (*p < 0.05) and the # indicated that the ΔLint was different
between groups. Note that for people post-stroke, if the non-paretic leg was perturbed, the R1
steps were paretic steps, and Pre-PTB step and PTB steps were non-paretic steps and vice versa
for the paretic perturbations.

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Figure 25: Time series trajectories of the PTB ∆LStance, R1∆Lcol, R1∆LPush and the corresponding
trajectories during the pre-perturbation steps for the representative control participant, one
paretic perturbation, and one non-paretic perturbation for one representative stroke participant.
The left column shows the PTB ∆LStance (A), R1 ∆Lcol (D), R1 ∆LPush (G) for the representative
control participant (black) and the corresponding pre-perturbation trajectories (gray). The
middle column shows the PTB ∆LStance (B), R1∆Lcol (E), R1 ∆LPush (H) for the representative
stroke participant during the paretic perturbation (darker red) and the corresponding pre-
perturbation trajectories for comparison (lighter red). The right column shows the PTB ∆LStance
(C), R1∆Lcol (F), R1 ∆LPush (I) for the representative stroke participant during the non-paretic
perturbation (darker blue) and the corresponding pre-perturbation trajectories for comparison
(lighter blue). For the middle and right column, the solid line indicates the non-paretic leg and
the dashed line indicates the paretic leg. FS: Foot strike, FO: Foot-off, MST: Midstance.
Vertical lines indicated gait events and the solid vertical line indicated the ipsilateral limb while
the dashed vertical indicated the contralateral limb.
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During the pre-perturbation steps, the GRF by the stance limb from midstance to the
subsequent foot strike produced a forward moment about the CoM and therefore, forward pitch
impulse (∆LStance) was negative during this phase of the gait cycle (Figure 25A, Figure 26A). The
earliest strategy that people could employ to begin recovering from losses of balance during the
perturbation step is to modulate the GRF of the stance limb to reduce the forward momentum
about CoM. Indeed, control participants produced less of a forward pitch impulse following the
treadmill-induced perturbations from pre-perturbation steps (t(48) = 3.7, p = 0.0005), indicating
that they began to arrest the forward loss of balance during the perturbation steps (Figure 26B).
For stroke participants, the Pre-PTB forward pitch impulse was also negative from
midstance to the subsequent foot strike for both paretic and non-paretic steps (Figure 25B & C,
Figure 26A). However, stroke participants only decreased the PTB forward pitch impulse during
the non-paretic perturbations (t(444) = -2.8 p = 0.005) but not during paretic perturbations
(t(444) = 0.4, p = 0.67). Thirty out of 38 participants showed greater decrease in PTB forward
pitch impulse during non-paretic perturbations than paretic perturbations but we did not find
statistically significant difference between sides (t(444) = 1.7, p = 0.088). There was no
significant difference in the increase in PTB forward pitch impulse in control participants and the
change observed during paretic (Bonferroni corrected p = 0.99) or non-paretic perturbations
(Bonferroni corrected p = 0.78, Figure 26B).
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Figure 26: Median forward pitch impulse ∆Lstance during pre-perturbations and changes in
∆Lstance following perturbations across control and stroke participants. Median ∆Lstance (A)
during pre-perturbation steps across control participants (N = 13) and stroke participants
(N=38) during paretic and non-paretic steps. Gray: Control, Pink: Paretic step, Blue: Non-
paretic step. (B) Median changes in ∆Lstance during the PTB steps compared to measured during
pre-perturbation. The asterisks (*) on top of the boxplots indicated whether the difference in
these variables during each step from pre-perturbation steps was significantly different from
zero (**p<0.001). Gray: Control, Pink: Paretic perturbation, Blue: Non-paretic perturbation.
Solid box: Non-paretic leg, Dash box: Paretic leg.

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Figure 27: Median ∆Lcol, ∆Lpush, and ∆LNet about CoM during pre-perturbations and changes of
those variables following perturbations across participants. The left column shows median ∆Lcol
(A), ∆Lpush (C), ∆LNet (D) during pre-perturbation steps across control participants (N = 13) and
stroke participants (N=38) during paretic and non-paretic step. Gray: Control, Pink: Paretic
step, Blue: Non-paretic step. The right column panel shows median changes in ∆Lcol (B), ∆Lpush
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(D), ∆LNet (E) during the PTB steps and R1 steps compared to those measured during pre-
perturbation. The asterisks (*) on top of the boxplots indicated whether the difference in these
variables during each step from pre-perturbation steps was significant (*p<0.05, **p<0.001,
***p<0.0001). The hashes (#) and the black lines indicated whether the comparison between
two groups were significantly different (#p<0.05, ##p<0.001). Gray: Control, Pink: Paretic
perturbation, Blue: Non-paretic perturbation. Solid box: Non-paretic leg, Dash box: Paretic leg.
Note that for people post-stroke, if the non-paretic leg was perturbed, the R1 steps were paretic
steps, and Pre-PTB step and PTB steps were non-paretic steps and vice versa for the paretic
perturbations.

During pre-perturbation steps, the GRF generated by the leading leg of control
participants produced a backward angular impulse about the CoM during the collision phase
(Figure 25D, Figure 27A). The treadmill-induced perturbations led to a significant decrease in
collision backward impulse during the perturbation steps compared to pre-perturbation steps
(Figure 27B, t(48) = -4.6, p < 0.0001) and thus participants tended to fall forward following the
perturbations. During the collision phase of the subsequent recovery step, participants generated
more backward impulse to help arrest the forward fall., and this was evidenced by a more
positive collision angular impulse for control participants (t(48) = 5.7, p < 0.0001).
For stroke participants, Pre-PTB collision angular impulse generated by the non-paretic
leg was significantly larger than that generated by the paretic leg (t(444) = 5.1, p < 0.0001) and
higher than that of controls (t(49) = -2.6, Bonferroni corrected p = 0.033, Figure 25E & F, Figure
27A). Similar to controls, PTB collision angular impulse significantly decreased from the pre-
perturbation steps during paretic perturbations (t(444) = -2.2, p = 0.027) and non-paretic
perturbations steps (t(444) = -2.3, p = 0.02). During the first recovery steps when the
contralateral limb touched the ground, R1 collision backward impulse increased significantly
from pre-perturbation steps following paretic perturbations (t(444) = 3.5, p = 0.0006) but not
following non-paretic perturbations (t(444) = 1.8, p = 0.075, Figure 25E & F, Figure 27D).
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Lastly, there was no difference between the increase in R1 collision backward impulse during the
first recovery steps between stroke and control participants (All Bonferroni corrected p>0.05).

During the pre-perturbation steps, the GRF by the trailing limb generated a forward
moment about the body’s CoM and thus the Pre-PTB push-off angular impulse was negative
(Figure 25G, Figure 27C). In controls, there was no change in PTB push-off angular impulse
during the perturbation steps (t(48) = 0.17, p = 0.86), but they decreased their forward R1 push-
off angular impulse during the first recovery step (t(48) = 5.1, p<0.0001, Figure 27D). This
indicates that the trailing limb assists with recovery from a forward loss of balance (Figure 27D).
For stroke participants during pre-perturbation steps, the magnitude of Pre-PTB push-off
angular impulse generated by the paretic trailing leg was higher than that generated by the non-
paretic leg (t(444) = 7.0, p < 0.0001) and by the control participants (t(49) = 2.7, Bonferroni
corrected p = 0.028) (Figure 25H & I and Figure 27C). Similar to control participants, there were
no changes in PTB push-off angular impulse from pre-perturbation steps for either paretic
(t(444) = 0.50, p = 0.62) or non-paretic perturbations (t(444) = 0.41, p = 0.68). During the first
recovery steps, forward R1 push-off angular impulse did not change from pre-perturbation steps
during the non-paretic trailing limb push-off following non-paretic perturbations (t(444) = 1.1, p
= 0.27). However, forward R1 push-off angular impulse became more negative following paretic
perturbations during the paretic push-off from pre-perturbation steps (t(444) = -2.0, p = 0.047).
Stroke participants had a greater reduction in forward R1 push-off angular impulse by the non-
paretic limb following non-paretic perturbations than following paretic perturbations (t(444) =
2.2, p = 0.029). In addition, control participants reduced forward R1 push-off angular impulse
more than stroke participants following paretic perturbations (t(49) = 3.7, Bonferroni corrected p
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= 0.0015) but not following non-paretic perturbations (t(49) = 1.5, Bonferroni corrected p =
0.45).

For control participants during the pre-perturbation steps, Pre-PTB net angular impulse
during the collision phase was positive, indicating that the GRF by the leading and trailing limb
produced a increase in backward angular momentum during this collision phase (Figure 27E).
During perturbation steps, control participants decreased the PTB net angular impulse from the
pre-perturbation steps (t(48) = -4.7, p < 0.0001) consistent with the increase in forward
momentum caused by the perturbation. During the first recovery steps, control participants
significantly increased R1 net angular impulse (t(48) = 12.1, p<0.0001), indicating that overall
more backward angular impulse was generated during this phase to arrest the forward fall of the
body (Figure 27F).
For stroke participants during pre-perturbation steps, the Pre-PTB net angular impulse
∆LNet during the collision phase was significantly more negative during non-paretic steps
compared to that during paretic steps (t(444) = -3.8, p = 0.0001) (Figure 27E), indicating that
people post-stroke arrested less forward momentum of the body during the step-to-step transition
from paretic to non-paretic steps compared to during the non-paretic to paretic transition. During
the perturbation steps, stroke participants decreased the PTB net angular impulse following both
paretic (t(444) = -2.3, p = 0.018) and non-paretic perturbations (t(444) = -2.1, p = 0.034). During
the first recovery steps, the R1 net angular impulse increased from pre-perturbation steps
following non-paretic perturbations (t(444) = 3.6, p = 0.0003) but not following paretic
perturbations (t(444) = 1.64, p = 0.1) (Figure 27F). This suggests that people post-stroke did not
arrest the forward falls during the collision phase when the perturbations occurred at the paretic
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side. Lastly, control participants showed a greater increase in R1 net angular impulse from pre-
perturbation steps than stroke participants following paretic perturbations (t(49) = 3.2,
Bonferroni corrected p = 0.007) but not non-paretic perturbations (t(49) = 1.9, Bonferroni
corrected p = 0.07).

We assessed whether reactive stabilization strategies such as changes in PTB forward
pitch impulse, R1 collision angular impulse, and R1 push-off angular impulse from pre-
perturbation steps are associated with clinical assessment of balance and motor impairment
(BBS, FGA, ABC, FM) for stroke participants. We found significant correlations between
paretic R1 push-off angular impulse and scores on clinical assessments of balance and motor
impairment. Reduction in R1 push-off angular impulse following the paretic perturbations
relative to pre-perturbation steps were positively correlated with FM (R
2
= 0.31, p = 0.0002) and
FGA (R
2
= 0.23, p = 0.002, Figure 28). This indicated that participants who scored poorer on
clinical assessments of balance and had greater motor impairment had less capability to reduce
the forward push-off angular impulse using the paretic limb.

Figure 28 Associations between deviation in R1∆LPush in the sagittal plane and clinical
assessments. Deviation of R1 ∆LPush from pre-perturbation steps was positively associated with
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(A) Fugl-Meyer score and (B) the Functional Gait Assessment only following perturbations
occurred at the paretic limb. FM: Fugl-Meyer; FGA: Functional Gait Assessment.

The primary objective for this study was to determine whether reactive control strategies
characterized by whole-body angular momentum and angular impulse differed in people post-
stroke compared to age-matched controls in responses to sudden treadmill accelerations.
Following paretic perturbations, people post-stroke tended to fall forward more compared to
control participants and compared to when perturbations occurred during non-paretic stance. To
recover from these perturbations, control participants decreased the stance limb forward angular
impulse during the perturbed steps and increased the collision angular impulse and decreased
push-off angular impulse during the collision phase of the first recovery step. In contrast,
following paretic perturbations, people post-stroke did not decrease the forward angular impulse
by the stance limb during the perturbed steps as control participants did. People post-stroke also
did not increase the collision angular impulse with the non-paretic leg or reduce the push-off
angular impulse using their paretic leg during the first recovery steps. People post-stroke reduced
their push-off angular impulse during the first recovery step more following the non-paretic
perturbations compared to paretic perturbations. To summarize, people post-stroke primarily
relied on their non-paretic limb to restore balance rather than coordinate both limbs to recover
from the forward fall as non-disabled participants.
People post-stroke needed more recovery steps to restore balance than control
participants did. Studies investigating postural control have used the number of steps to quantify
the ability for people to maintain balance, and the use of multiple recovery steps is indicative of a
higher fall risk (Hilliard et al., 2008; Maki and McIlroy, 2006). For example, older adults,
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particularly those with a fall history, had a higher tendency to adopt multiple steps following
waist pull when standing compared to young adults (Mille et al., 2013). Additionally, people
post-stroke needed more steps to restore balance following stance perturbations compared to age-
matched controls (Martinez et al., 2019). Our results extend these observations to perturbations
during walking by showing that compared with age-matched controls, people post-stroke needed
one more recovery step following the treadmill-induced, slip-like perturbations to restore
balance.
The increase in whole-body angular momentum from pre-perturbation steps in the
sagittal plane following paretic perturbations was higher than non-paretic perturbations,
indicating that people tended to fall more forward during paretic perturbations than non-paretic
perturbations. The increase in the integrated whole-body angular momentum during the paretic
perturbation step was about ~1.5 times higher than that at the non-paretic side. The increase in
whole-body angular momentum from pre-perturbation steps following paretic perturbations was
also significantly higher than that for control participants, indicating that people post-stroke have
impaired regulation of whole-body dynamics following paretic perturbations, which is in line
with the previous study that reported higher instability following paretic perturbations (Kajrolkar
and Bhatt, 2016)
Several post-stroke changes in the lower extremity may explain our observation that
shows difference in angular impulses during non-paretic and paretic stance phase of the
perturbation steps. For example, perturbations of the paretic leg could lead to greater forward
angular momentum than during non-paretic side perturbations if participants were unable to
support body weight and maintain balance on the paretic leg. Additionally, people post-stroke
might have delayed reactions to paretic perturbations due to sensory transmission or processing
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deficits, which might contribute to the increased forward fall during paretic perturbations
(Sharafi et al., 2016). We found that control participants responded to the perturbations by
modulating the stance limb ground reaction force toward the end of the perturbed steps to reduce
forward angular impulse. However, our post-stroke participants did not reduce this impulse when
the paretic limb was perturbed. This could be because the paretic perturbation steps were on
average 145ms shorter than the non-paretic perturbation steps in our study. As a result, there
might not be sufficient time for stroke participants to reduce the angular impulse during the
stance phase during the paretic perturbation step. Future research should also determine if the
paretic limb has longer muscle activation latencies during perturbation responses.
People post-stroke had a reduced ability to increase collision angular impulse using the
paretic limb. People with no sensorimotor deficits control their ground reaction force vectors so
that the vectors intersect right above the CoM throughout the gait cycle (Gruben and Boehm,
2012; Maus et al., 2010). This control strategy was also evident when control participants
stepped down on a camouflaged curb and fell forward as a method to help people to regulate the
body rotational behavior as the ground reaction force vectors at the leading limb in the sagittal
plane continued to direct above the CoM to generate backward moment about CoM (Vielemeyer
et al., 2019). However, this stabilization strategy of using ground reaction forces to control body
dynamics may not be feasible in people post-stroke (Boehm and Gruben, 2016). Generating
sufficient collision angular impulse to redirect forward fall needs both regulation of ground
reaction force and the foot placement location. Although a previous study has reported that
people post-stroke could increase their step length following a forward fall to characterize stroke
participants’ ability to restore balance (Haarman et al., 2017), increasing step length alone does
not directly affect the change in collision angular impulse. In fact, we found that there was no
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association between the increase in collision angular impulse during the first recovery steps and
the increase in step length or distance between foot placement and CoM (all p>0.05). For some
stroke participants, increasing paretic ground reaction force during the collision phase may
generate high impact loading at the paretic limb and potentially cause knee collapse due to the
weakness at the knee extensors. Thus, choosing not increasing collision angular impulse may be
a protective mechanism for these people post-stroke to avoid injury.
Additionally, during the recovery step, people post-stroke had reduced capacity to
modulate their paretic push-off angular impulse following the forward losses of balance in
response to paretic perturbations. At the beginning of the first recovery step, control participants
reduced the push-off forward angular impulse, which could reduce the forward loss of balance
caused by the sudden belt speed increase. One way to reduce the push-off forward angular
impulse is by modulating the propulsive force in the anterior direction. The ability to modulate
propulsive force requires the coordination of the hip flexor, knee extensor, and the ankle
plantarflexor moment of the trailing limb (Debelle et al., 2020; Pijnappels et al., 2005).
Therefore, abnormal coordination patterns post-stroke may prevent people post-stroke from
generating proper force at the trailing limb and redirect the ground reaction force vectors to
reduce the overall push-off angular impulse at the paretic limb (Finley et al., 2008; Sánchez et
al.).

Our study has a few limitations. This study is limited to mild to moderate chronic post-
stroke survivors with ability to ambulate in the community with using any assistive devices. Due
to the study protocol, we only elicited perturbations to induce forward loss of balance during
walking with same perturbation magnitudes for all participants. It remains to be determined if the
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similar reactive stabilization strategies can be extended to larger perturbations and other
directions. In addition, we have only focused on characterizing the reactive control strategies
using kinematic and kinetic based analysis. Future analysis will focus on measures of muscular
coordination to identify the neuromotor factors that might contribute to the heterogeneous
response to perturbations between sides.

We thank Cathy Broderick, Catherine Yunis, Ryan Novotny, Sungwoo Park for their help
with data collection.

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Chapter 6. Reducing step length asymmetry does not improve reactive control
of balance during walking for people post-stroke

Chang Liu, Sungwoo Park, Natalia Sánchez, Julie K. Tilson, Sara J. Mulroy, and James M Finley
Reactive control of balance is impaired for people post-stroke due to their physiological
impairments at their paretic lower extremity. People post-stroke often walk with
spatiotemporally asymmetric gait pattern and reducing such asymmetries is a common objective
of rehabilitation. People post-stroke with larger spatiotemporal asymmetries have impaired
control of whole-body dynamics during walking in the sagittal plane which is evident by a higher
magnitude of whole-body angular momentum. While this may suggest that walking with
asymmetrical gait pattern could impair balance, we did not know whether restoring a more
symmetrical walking pattern would improve reactive control of balance. In this study, we
performed clinical assessments and used unexpected treadmill accelerations to induce forward
falls during walking in a sample of 38 people post-stroke. We computed integrated whole-body
angular momentum to characterize changes in whole-body configuration during multi-step
responses to perturbations. We then used a biofeedback paradigm that allowed 18 of these
individuals to reduce their step length asymmetry and induced forward fall during walking as
well. We also measured the resulting changes in whole-body angular momentum when
participants walked more symmetrically. We first determined the concurrent validity of whole-
body angular momentum during reactive control of balance during walking. We found that the
increase in integraetd angular momentum during perturbation steps from pre-perturbation steps
in the sagittal plane was negatively correlated clinical measures of balance and motor
impairment. However, our results showed that their integrated whole-body angular momentum in
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the sagittal plane did not change following the perturbations when participants walked more
symmetrically. This indicate that reactive control of balance may not be sensitive to changes in
step length asymmetry for people post-stroke.

The ability to use reactive control strategies to maintain balance when encountering
external perturbations is important for people post-stroke, as about 70% of people post-stroke fall
at least once after discharge from the hospital (Sackley et al., 2008; Watanabe, 2005). Preventing
falls requires a combination of proactive control (Patla, 1993), characterized by body
configuration (Bhatt et al., 2005; Cham and Redfern, 2002) and the setting of feedback gains
(Shinya et al., 2016), and reactive control strategies to maintain anti-gravity support and restore
the center of mass (CoM) within the base support (Tang et al., 1998). However, sensory
impairments (Semrau et al., 2013; Tyson et al., 2008), delayed muscle activation to external
perturbations (Kirker et al., 2000; Marigold et al., 2004), weakness (Olney et al., 1996), and
abnormal coordination (Higginson et al., 2006) may impair both proactive and reactive control of
dynamic balance post-stroke and contribute to an increased risk of falls.
One established metric used to characterize dynamic balance during walking is whole-
body angular momentum. This measure reflects the net contribution of all body segments to the
body’s rotation about a given axis, commonly taken to project through the body’s CoM (Herr
and Popovic, 2008; Liu et al., 2018; Martelli et al., 2013). Whole-body angular momentum is
highly regulated within a small range for non-disabled people during unperturbed walking in the
sagittal plane as the peak-to-peak range of angular momentum is much smaller than the angular
momentum of single segments due to the segmental angular momenta cancellation between sides
(Herr and Popovic, 2008; Popovic et al., 2004). Whole-body angular momentum increases
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sharply during sudden perturbations and the subsequent recovery reactions from that measured
during unperturbed walking (Liu et al., 2018; Martelli et al., 2013) and this increase in
momentum captures changes in body posture that might lead to a fall if not arrested.
Additionally, higher values of whole-body angular momentum for people post-stroke during
unperturbed walking are associated with worse performances in clinical assessments of balance
and thus, measures of angular momentum serve as an indicator of impaired balance control (Nott
et al., 2014; Park et al., 2021).
Given the asymmetries in sensorimotor deficits following stroke, reactive control of
dynamic balance during walking may differ based on the side of perturbation (Buurke et al.,
2020; Kajrolkar and Bhatt, 2016). For instance, if a perturbation occurs during the paretic stance
and the paretic leg is too weak to support the body, the non-paretic swing leg may not have
sufficient time to swing forward and arrest the body’s falling toward the ground. Thus, paretic
perturbations led to greater instability at the touch down of the recovery limbthan non-paretic
perturbations (Kajrolkar and Bhatt, 2016). Conversely, if a perturbation occurs on the non-
paretic side, the paretic leg may have difficulty initiating a successful stepping response to help
restore balance due to deficits in propulsion (Allen et al., 2014; Chen et al., 2005; Lauzière et al.,
2015) and decreased hip flexor activity (Rybar et al., 2014) in the paretic leg compared with their
non-paretic side. As a result, people post-stroke demonstrate shorter paretic compensatory step
lengths than non-paretic step lengths for stance perturbations when they got pulled in the anterior
direction and needed more steps to recover when stepping with paretic limbs (Martinez et al.,
2019).
An important question for clinical research is whether restoring a more symmetrical
walking pattern would improve balance during walking for people post-stroke. Reducing step
133
length asymmetries is a common objective of post-stroke rehabilitation (Patterson et al., 2015;
Reisman et al., 2013). Step length asymmetries were also shown to be negatively correlated with
scores on the Berg Balance Scale, suggesting that step length asymmetries may contribute to the
high fall risk in these individuals (Lewek et al., 2014). The presence of such asymmetries
reduces the interlimb angular momentum cancellation in the sagittal plane when contralateral
limbs move in anti-phase so that the range of whole-body angular momentum in people post-
stroke increases (Honda et al., 2019). While asymmetries may increase sagittal plane angular
momentum, restoring a more symmetric walking pattern did not reduce angular momentum
during walking for people post-stroke (Park et al., 2021). Instead, people post-stroke increased
the forward whole-body angular momentum when walking more symmetrically (Park et al.,
2021), suggesting that deviation from self-selected asymmetries may compromise their balance
during walking. However, another important aspect of balance control is whether people post-
stroke could use reactive stabilization strategies to recover from unexpected external
perturbations and it has yet to be determined whether reducing step length asymmetry would
affect reactive control of balance.
Reactive balance control may still benefit from reducing step length asymmetry. If people
post-stroke can advance their limb more anterior to the body CoM to increase the shorter step
lengths and reduce asymmetry, such strategy could reduce the moment arm of the posteriorly-
directed ground reaction force about the body CoM in the sagittal plane. In this case, if a slip-like
perturbation induced by the sudden treadmill acceleration occurs, the change in ground reaction
force in the posterior direction could generate smaller changes in moments at CoM than if taking
shorter steps. Thus, people could fall less forward during the perturbation steps when walking
more symmetrically. On the other hand, if people post-stroke need to increase the step length of
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the contralateral limb to the perturbed limb to reduce asymmetries, this could potentially increase
the step lengths of the recovery step. Increase of the recovery step length could increase the
moment arm of the leading limb ground reaction force to generate more backward moments
about CoM during the recovery steps following perturbations. Thus, reducing asymmetries may
improve reactive balance control during walking but it has yet to be determined.
Here, we aimed to determine the following: (1) the concurrent validity of whole-body
angular momentum as a quantitative measure for reactive control of balance for people post-
stroke during walking, and (2) whether reducing step length asymmetry would improve reactive
control of stability by decreasing angular momentum during the perturbation steps and the
subsequent recovery steps. We addressed these aims by using treadmill accelerations to elicit
slip-like perturbations while participants were provided with biofeedback of their step lengths.
We hypothesized that (1) people who had higher whole-body angular momentum during the
perturbation step would have lower scores on clinical assessments of balance and sensorimotor
impairment. We also hypothesized that (2) reductions in step length asymmetry would be
associated with reductions in angular momentum during the perturbation steps and the
subsequent recovery step. Overall, our findings could contribute to a better understanding of how
voluntarily restoring a more symmetrical gait pattern affects reactive control of balance during
walking for people post-stroke.

We included thirty-eight stroke survivors (Table 1) in this study. Participants were
recruited from the IRB-approved, USC Registry for Aging and Rehabilitation, the USC Physical
Therapy Associates Clinic, and from Rancho Los Amigos National Rehabilitation Center.
Inclusion criteria were the following: 1) a unilateral brain lesion 2) paresis confined to one side,
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and 3) ability to walk on the treadmill for five minutes without holding on to any support. Use of
ankle-foot orthosis was permitted during the experiment. Five out of thirty-eight participants
wore an ankle-foot orthosis. All procedures conformed to the principles set forth in the
Declaration of Helsinki and were approved by the University of Southern California’s
Institutional Review Board.
Table 5 – Participant demographics and clinical assessments (N=38).
Metric Mean (SD) [range]
Female / male 14 / 24
Left / right hemiparetic 15 / 23
Age (years old) 60 (11) [32-78]
Months after stroke 83 (55) [10-394]
History of falls in the past year (Yes / No) 17 / 21
Activity-specific Balance Confidence Scale 77 (13) [53-100]
Berg Balance Scale 51 (6) [27-55]
Functional Gait Assessment 21 (6) [6-29]
10-Meter Walk Test (m s
-1
) 0.8 (0.3) [0.2-1.3]
Falls Efficacy Scale 29 (12) [16-64]
Fugl-Meyer lower extremity 26 (5) [15-34]
Self-selected walking speed (m s
-1
) 0.6 (0.2) [0.2-1]
Step length asymmetry 0.0 (0.11) [-0.19 0.42]

The complete protocol consisted of a set of clinical assessments and walking trials on the
treadmill (Figure 29A). Before the walking trials, we evaluated motor impairment using the
lower extremity portion of the Fugl-Meyer Assessment (FM) (Fugl-Meyer et al. 1975), static
balance using Berg Balance Scale (BBS) (Berg et al. 1992), static and dynamic balance during
locomotion using the Functional Gait Assessment (FGA)(Leddy et al. 2011), and over-ground
walking speed using the 10-meter walking test. Participants also completed questionnaires about
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self-confidence in their balance using Activity Based Confidence Scale (ABC) (Powell and
Myers 1995). Higher scores on these assessments indicated better balance control. Lastly, we
included a Fall History Questionnaire to identify the participants who experienced at least one
fall within the past year.
After completing the clinical evaluations, we instructed participants to walk on the dual-
belt treadmill (Bertec, Columbus, OH, USA). A harness was provided to prevent the participants
from falling but no body weight support was provided. First, the participants walked on the
treadmill to familiarize themselves with the experimental set-up. We determined their self-
selected speed on a treadmill using the following method (Park et al., 2021). We started from
70% of the speed obtained from a 10-meter walking test and adjusted their walking speed by
0.05m/s increments or decrements until the participants verbally indicated that they achieved
their preferred walking speed. For the next trial, the participants walked for three minutes at their
self-selected speed to determine their baseline step lengths. Then the participants completed a
familiarization trial with at least two perturbations on each side. After that, we introduced the
visual feedback to the participants via a monitor placed in front of the treadmill and instructed
them to practice matching their step lengths with the vertical bars which represented their
targeted step length. The visual feedback also showed their ankle position during the swing
phase of the leg.
Participants then performed a total of six trials with three different feedback conditions
and treadmill perturbations: without visual feedback (BASE), feedback with baseline asymmetry
(BASE+FBK), and feedback with symmetrical step length (SYM+FBK). Each trial was three to
four minutes in duration and all the trials were randomized in order. During the trials with visual
feedback, participants were instructed to either match their natural step lengths measured in the
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baseline walking trial (BASE+FBK) or adopt steps of equal length by increasing their shorter
step length to match with their longer step length (SYM+FBK). We provided real-time visual
feedback of their step lengths via a monitor display preprogrammed by Python code written in
Vizard 5 (WorldViz, Santa Barbara, CA) in accordance with previous literature (Park et al.,
2021; Sánchez and Finley, 2018). We also provided points to encourage the participants to match
their steps with the top of the bars at each side at foot strike. The points were calculated using the
following equation (Eqn. 1):
Points = 10 − 10 ∗ |1 −
Step length
Target step length
| (1)
We verbally encouraged the participants to obtain 10 points at each step.
During each trial, we applied five accelerations to the treadmill belts on the paretic and
non-paretic sides at foot-strike using ground reaction forces record by the force plates underneath
the treadmill. Treadmill accelerations were triggered at random intervals within 18 to 24 steps
after the previous perturbation to allow participants to reestablish their walking patterns. Each
perturbation was characterized by a trapezoidal speed profile in which the speed increased by 0.2
m/s at an acceleration of 3 m/s
2
, was held for 0.7 s and then decelerated back to the self-selected
speed during the swing phase of the perturbed leg. Between each trial, participants had breaks of
at least three minutes to minimize fatigue. Participants did not hold on to handrails while walking
on the treadmill.
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Figure 29: A) Experimental protocol. Participants first completed a series of clinical
assessments and familiarization with the treadmill. Then, participants completed trials with
perturbations under three conditions: without visual feedback (BASE), with baseline asymmetry
feedback (BASE+FBK), and symmetry feedback (SYM+FBK). These trials were randomized. For
the visual feedback trials, we provided online visualization of the ankle markers during the swing
phase as a black point. The red lines represented the target range of the step lengths which was
equal to three standard deviations of their baseline step length. We also provided scores on the
top left or right corner of the display to encourage the participants to achieve the target step
lengths.

We used a ten-camera Qualisys motion capture system (Qualisys AB, Gothenburg,
Sweden) to record 3D marker kinematics at 100 Hz and ground reaction forces at 1000 Hz. We
placed a set of 14mm spherical markers on anatomical landmarks to create a 12-segment, full-
body model, including head, trunk/pelvis, left and right upper arms, forearms, thighs, shanks, and
feet (Havens et al., 2018; Song et al., 2012). Since participants wore a harness over their pelvis,
we removed the reflective markers from the pelvis and assumed the pelvis segment to be rigidly
139
connected to the trunk. We have previously validated this 12-segment model with data from five
non-disabled subjects that included the necessary markers to track the pelvis (Park et al., 2021).
We placed marker clusters on the upper arms, forearms, thighs, shanks, and the back of heels. At
the beginning of each trial, marker positions were calibrated during a five-second standing trial.
We removed all joint markers after the calibration. Meanwhile, body weight support system
recorded the weight leaning on the harness.

We post-processed the kinematic and kinetic data in Visual3D (C-Motion, Rockville,
MD, USA) and Matlab 2017b (Mathworks, USA) to compute variables of interest. Marker
positions and ground reaction forces were low-pass filtered by 4
th
order Butterworth filters with
cutoff frequencies of 6 Hz and 20 Hz, respectively (Kurz et al., 2012; Reisman et al., 2009;
Winter, 2009). The timing of perturbations relative to foot-strike was examined post-hoc. We
removed the perturbations that occurred ~150ms after the foot-strike and perturbations that
began deceleration before toe-off of the perturbed leg during post-hoc analysis (Buurke et al.,
2020). We also removed perturbations if 30% of body weight was supported by the safety
harness (Patel & Bhatt, 2018; Salot et at. 2016) or participants held on harness to restore balance.
On average, we excluded 4 (SD 3) perturbations per person that did not fit into our criteria,
leaving 19 (SD 5) perturbations per person per condition.
Foot strike was computed as the point when vertical ground reaction forces reached
150N. Step lengths were defined as the fore-aft distance between heel markers at the leading
limbs’ foot strike. Step length asymmetry (SLA) was computed as the normalized difference in
step lengths based on Equation 2.
140
𝛥𝛥𝐿𝐿 𝑆𝑆 =
𝛥𝛥𝐿𝐿
𝑖𝑖𝐵𝐵 𝑖𝑖𝑝𝑝 𝑏𝑏 𝑟𝑟𝑅𝑅𝑖𝑖 𝑖𝑖 𝑛𝑛 − 𝛥𝛥𝐿𝐿
𝑝𝑝 𝑏𝑏𝑟𝑟𝑅𝑅𝑖𝑖 𝑖𝑖 𝑛𝑛 𝛥𝛥𝐿𝐿
𝑖𝑖𝐵𝐵 𝑖𝑖𝑝𝑝 𝑏𝑏 𝑟𝑟𝑅𝑅𝑖𝑖 𝑖𝑖 𝑛𝑛 + 𝛥𝛥𝐿𝐿
𝑝𝑝 𝑏𝑏𝑟𝑟𝑅𝑅𝑖𝑖 𝑖𝑖 𝑛𝑛 (2)
Here, 𝛥𝛥𝐿𝐿
𝑝𝑝 𝑏𝑏𝑟𝑟𝑅𝑅𝑖𝑖 𝑖𝑖 𝑛𝑛 was computed at paretic foot-strike and 𝛥𝛥𝐿𝐿
𝑖𝑖𝐵𝐵 𝑖𝑖𝑝𝑝 𝑏𝑏 𝑟𝑟𝑅𝑅𝑖𝑖 𝑖𝑖 𝑛𝑛 was computed at
non-paretic foot-strike. Thus, a positive SLA indicates that people take longer non-paretic steps
whereas a negative SLA indicates that people take longer paretic steps. We calculated the
achieved SLA as the mean SLA of the three strides before each perturbation. We used this
achieved SLA instead of target SLA as the independent variable in our statistical analyses.

We computed whole-body angular momentum in the sagittal plane to quantify dynamic
balance. Whole-body angular momentum is the sum of all segmental angular momenta and is
composed of segmental rotation about the body’s center of mass and rotation of each segment
about its own center of mass (Herr and Popovic, 2008). Whole-body angular momentum was
normalized by a combination of the participant’s mass (M), the participant’s height of COM (H),
and the gravity constant (g). (Eqn. 3).
𝐿𝐿 =
∑ [ 𝑚𝑚 𝑖𝑖 � 𝑟𝑟
� � ⃗
𝐶𝐶𝐶𝐶 − 𝑖𝑖 𝑖𝑖 × 𝑣𝑣 ⃗
𝐶𝐶𝐶𝐶 − 𝑖𝑖 𝑖𝑖 � + 𝐼𝐼 𝑖𝑖 𝜔𝜔 𝑖𝑖 ]
𝑖𝑖 𝑋𝑋 𝑔𝑔 1
2
𝑀𝑀 3
2

(3)
Here, m is segmental mass, r is the distance from segment to the body COM, I is the
segmental moment of inertia, ω is segmental angular velocity, and the index i corresponds to
individual limb segments. We defined angular momentum in the sagittal plane such that negative
angular momentum represented forward rotation while positive values represented backward
rotation. Additionally, we used integrated angular momentum (𝐿𝐿 𝑖𝑖 𝑖𝑖𝑖𝑖
) to quantify the amount of
body rotation during each step. 𝐿𝐿 𝑖𝑖 𝑖𝑖𝑖𝑖
can capture the overall body rotation during the perturbation
response and the subsequent recovery steps (Liu et al., 2018; Potocanac et al., 2014; Vielemeyer
et al., 2019). Our current analysis only focused on the fore-aft direction because the perturbations
141
were elicited in this direction and changes in angular momentum was minimal in either
mediolateral or transverse plane

All statistical analyses were performed in Matlab 2017b (Mathworks). We categorized
the pre-perturbation steps as the last two steps before the perturbation occurred (Pre-PTB 1 - 2),
perturbation steps (PTB) as the step during which the perturbation was applied, and recovery
steps (Recovery 1-4) as the four steps that followed the perturbation. If the non-paretic leg was
perturbed, the Recovery 1, and Recovery 3 steps were paretic steps, and Pre-PTB step,
Perturbation step, Recovery 2, and Recovery 4 steps were non-paretic steps and vice versa for
the paretic perturbations.

We computed Spearman correlation coefficients to test for relationships between whole-
boday angular momentum during the perturbation steps and each clinical balance assessment
(BBS, FGA, ABC, FM) from all participants because variables were not normally distributed
tested using Lilliefors test.

To determine if changes in step length asymmetry would affect angular momentum , we
included a subset of participants whose minimal magnitude of baseline step length asymmetry
was 0.05 (Little et al., 2019) and a total of 18 participants were included (11 participants took
longer paretic step and seven participant took shorter paretic step). We first tested whether SLA
differed across BASE, BASE+FBK, SYM+FBK trials using mixed-effect models. The random
142
effects were included to account for the individual differences between subjects. The reference
level was set to be BASE.
Then we fit an LME model for each Step Type to test whether changes in step length
asymmetry would affect changes in 𝐿𝐿 𝑖𝑖 𝑖𝑖𝑖𝑖
during SYM+FBK conditions from BASE+FBK. The
independent variables included 1) changes in the magnitude of step length asymmetry 𝛥𝛥 |SLA|
from BASE+FBK condition, 2) side of perturbation (Leg), and 3) the interaction between Leg
and 𝛥𝛥 |SLA| indicated how perturbations occurring on either the paretic or non-paretic leg affect
the association between 𝛥𝛥 |SLA| and 𝐿𝐿 𝑖𝑖 𝑖𝑖𝑖𝑖
. We defined Leg as a categorical variable with a value
of 0 for non-paretic perturbations and a value of 1 for paretic perturbations. We included a
random intercept to account for individual differences in 𝐿𝐿 𝑖𝑖 𝑖𝑖𝑖𝑖
. Normality was checked using the
Lilliefors Test. Normally distributed data were expressed as mean ± standard deviation of the
corresponding mean. Non-normally distributed data were expressed as median with interquartile
range (IQR), [25% IQR, 75% IQR]. Significance was set at p<0.05 level.

Measures of whole-body angular momentum varied systematically across steps (Figure
30). During pre-perturbation walking (Figure 30, grey line), whole-body angular momentum in
the sagittal plane was negative at the foot strike, capturing the fall of the body during the later
portion of the step cycle. Angular momentum then became more positive as participants rotated
backward during the initial swing phase and again fell forward at the next foot strike. Treadmill
accelerations led to immediate effects on whole-body angular momentum and triggered multi-
step balance recovery responses (Figure 30, black line). During the perturbation step, angular
momentum became more negative as the body fell forward. Participants generated more positive
143
angular momentum and initiated backward rotation during the subsequent recovery step in the
sagittal plane to compensate for the forward fall. Deviations in body rotation in the frontal and
transverse plane from that measured during unperturbed walking were not prominent.

Figure 30 An example of time series whole-body angular momentum in the sagittal plane for a
pre-perturbation and perturbation stride when perturbations occurred on the non-paretic (A) or
paretic side (B). The gray traces indicated the time series data for a pre-perturbation stride
while the black traces indicate a perturbation stride. Each stride began at foot strike. Negative
values of angular momentum represent forward rotation while positive values represent
backward rotation. Black dashed vertical lines correspond to the time of foot strike for the first
recovery step. Pre-PTB: pre-perturbation stride; PTB: perturbation stride.

We first assessed whether measures of angular momentum during the perturbation step
was associated with clinical balance assessments. We found significant correlations between 𝐿𝐿 𝑖𝑖 𝑖𝑖𝑖𝑖

and scores on clinical assessments of balance and motor impairment. Changes in 𝐿𝐿 𝑖𝑖 𝑖𝑖𝑖𝑖
during
paretic perturbation steps relative to pre-perturbation steps were positively correlated with BBS
(R
2
=0.25, p=0.001), FGA (R
2
=0.22, p=0.003), ABC (R
2
= 0.13, p = 0.02), and FM (R
2
= 0.14, p
= 0.02) (Figure 31). This indicated that participants who scored poorer on clinical assessments of
balance and had greater motor impairment fell forward more during paretic perturbations. We
did not find a significant association between changes in 𝐿𝐿 𝑖𝑖 𝑖𝑖𝑖𝑖
during the non-paretic perturbation
steps from pre-perturbation with FGA (p =0.4), BBS (p =0.10), ABC (p = 0.16), or FM (p
=0.76).
144

Figure 31: Associations between deviation in integrated whole-body angular momentum during
the perturbation step in the sagittal plane (ΔLint) and clinical balance assessments. ΔLint was
positively associated with (A) Berg Balance Scale, and (B) the Functional Gait Assessment, (C)
the Activity-based Confidence Scale, and (D) Fugl-Meyer score only when perturbations
occurred at the paretic limb. BBS: Berg Balance Scale, FGA: Functional Gait Assessment, ABC:
Activity-based Confidence Scale, FM: Fugl-Meyer.

Most participants successfully modified step length asymmetry using visual feedback.
Performance from one representative participant showed that during the BASE+FBK, step length
asymmetry was maintained at the same level as in BASE (Figure 32A). During the SYM+FBK,
the participant reduced step length asymmetry when provided with feedback to take steps with
equal lengths. |SLA| did not differ between BASE and BASE+FBK (t(51) = 0.72, p = 0.47), but
participants reduced SLA magnitude during SYM+FBK (Median [25th interquartile, 75th
interquartile]: 0.087 [0.027, 0.18]) compared to both BASE (t (51) = -4.0, p < 0.001) and
BASE+FBK (t(51) = -3.3, p = 0.0017) (Figure 32B). The results confirmed that people post-
stroke could reduce their step length asymmetry using the visual feedback.
145

Figure 32: A) Raw step length asymmetry data for one representative participant who took
longer paretic steps. Each data point represents the step length asymmetry for a stride. B)
Magnitude of step length asymmetry for all participants with baseline |SLA| > 0.05 during
BASE, BASE+FBK, and SYM+FBK. *p<0.05.

Lastly, we investigated whether reductions in step length asymmetry would improve
participant’s reactive control of balance in the sagittal plane. We fit a linear mixed-effects model
relating changes in asymmetry to the changes 𝐿𝐿 𝑖𝑖 𝑖𝑖𝑖𝑖
from BASE+FBK at each step (Pre-PTB,
PTB, and R1). However, contrary to our hypothesis, reduction in step length asymmetry was not
associated with reduction angular momentum during pre-perturbation (F(1,32)=1.3, p =0.26),
perturbation (F(1, 32)=0.41, p = 0.52), or the first recovery steps (F(1,32) = 3.5, p = 0.07; Table
6 and Figure 33).

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Table 6: Parameter Values for Model Relating ΔLint during B1, PTB, and R1 steps to
Spatiotemporal Variables.

Pre-PTB step

Estimate Standard Error t-Statistics p value

Leg -0.016 0.20 -0.08 0.94

∆|SLA| 1.29 2.24 0.49 0.73
Leg: ∆|SLA| -3.96 3.46 -1.15 0.26

PTB step

Estimate Standard Error t-Statistics p value

Leg -0.25 0.33 -0.78 0.44

∆|SLA| 2.12 3.3 0.64 0.52
Leg: ∆|SLA| -10.5 5.3 -1.97 0.058

R1 step

Estimate Standard Error t-Statistics p value

Leg -0.46 0.36 -1.27 0.21

∆|SLA| 6.87 3.67 1.87 0.07
Leg: ∆|SLA| -4.21 5.92 -0.7 0.48

Figure 33: (A) Changes in the 𝐿𝐿 𝑖𝑖 𝑖𝑖𝑖𝑖
in the sagittal plane with changes in SLA magnitude relative
to BASE+FBK during perturbation step when perturbations occurred at the paretic side during
the Pre-PTB steps (A), PTB steps (B), and R1 steps (C). Pink: Paretic perturbations, blue: non-
paretic perturbations, filled: participants who took longer non-paretic steps, unfilled:
participants who took longer paretic steps.
Δ 𝐿𝐿 𝑖𝑖 𝑖𝑖𝑖𝑖
~ Leg ∗ Δ|SLA| + (1|Subject)
ΔL
in t
~ Leg ∗ Δ|SLA| + (1|Subject)

ΔL
in t
~ Leg ∗ Δ|SLA| + (1|Subject)
147

We aimed to understand how manipulation of step length asymmetry magnitude
influence whole-body angular momentum during reactive responses to gait perturbations in
people post-stroke. We used visual feedback to reduce step length asymmetry by increasing step
lengths on the naturally shorter side. We first demonstrated that changes in whole-body angular
momentum during the perturbation steps relative to unperturbed walking were correlated with
scores of multiple clinical assessments. Contrary to our hypothesis, we did not find a significant
relationship between reductions in SLA and reductions in integrated angular momentum during
perturbation or recovery steps.

Participants’ increase in angular momentum in the sagittal plane during the perturbation
step were associated with clinical assessments of balance and motor impairment. We expected
that participants with smaller increase in the integrated angular momentum during the
perturbation would score higher on clinical assessments, indicating that people with better
balance control and less motor impairment would have smaller losses of balance in response to
the perturbation. While this is indeed what we observed, these relationships were only significant
during paretic perturbation steps. We did not find a significant association between clinical
assessements and changes in integrated angular momentum during non-paretic perturbations.
This is likely because control of whole-body angular momentum was similar across participants
when the non-paretic limb was perturbed. Nevertheless, our results were consistent with previous
observations demonstrating that better scores in clinical balance measures were associated with
smaller peak angular momentum in the sagittal plane during unperturbed walking (Park et al. in
148
review). Overall, demonstrate that reactive control of whole-body angular momentum can serve
as a quantitative assessment of balance performance in people post-stroke.

Contrary to our hypothesis, the angular momentum in the sagittal plane did not change
when people adopted a more symmetrical walking pattern. We expected that the integrated
angular momentum which represented the net body rotation during the perturbation steps and the
recovery steps would decrease when walking with more symmetrical step lengths. However, we
found that people post-stroke did not reduce their whole-body rotation during perturbation
response when walking more symmetrically, indicating that improving step length asymmetry
did not reduce body rotation following perturbations. This is likely because the strategy people
post-stroke used to reduce asymmetry might not be consistent due to their heterogeneity in the
level of motor impairment and the direction of step length asymmetry and magnitude. For
example, if the participants chose to place their leading limb farther away from CoM to increase
the shorter step length and reduce step length asymmetry, the effect of perturbation at the foot
strike may generate smaller moments at CoM and thus generate less body rotation at the
perturbation step. However, they would need to increase the propulsion before lifting the limb on
the side where they attempted to increase the step length. Although increasing the push-off force
would help to swing the leg forward, such attempt may lead to an increase in CoM velocity
(Zelik and Adamczyk, 2016) and thus higher forward angular momentum. On the other hand, if
people only increased the stance duration for the trailing limb to elongate their shorter step, they
would shift their CoM more anteriorly relative to the trailing limb and thus retain a higher CoM
velocity when lengthening the shorter steps and induce a higher forward momentum in these
participants. Future studies may investigate whether the strategies that people post-stroke use to
149
reduce step length asymmetry would affect changes in angular momentum following
perturbations.
Although a previous study found that people post-stroke increased their peak forward
angular momentum in the sagittal plane when they walked more symmetrically using visual
feedback, indicating that reductions of step length asymmetry may impair dynamic balance (Park
et al., 2021), we did not find a similar effect of reducing asymmetry on reactive control of
balance. This could because the reactive responses following treadmill perturbations may be
modulated by the neural pathways that generate stereotypical and invariant responses
(Aprigliano et al., 2016, 2017; Liu et al., 2018). A previous study demonstrated that the reactive
control of balance remained invariant across different levels of step length asymmetries for
heathy young people without sensorimotor deficits although the coordination patterns changed
during the perturbation reponses (Liu et al., 2018; Liu and Finley, 2020). Instructing people post-
stroke to walk more symmetrically by increasing the step lengths may not change the
neuromuscular response to the perturbations such as the recruitment sequence and timing of the
muscles. Thus, the reactive stabilization strategies such as modulating the push-off force in the
anterior direction to reduce the forward angular momentum may not be affected by the reduction
in asymmetries (Debelle et al., 2020; Pijnappels et al., 2004).

One of our study's limitations is that participants may have adopted a more cautious gait
when anticipating the perturbations. However, we found no significant differences in the step
width during unperturbed baseline walking and those during pre-perturbation steps for either
paretic (p=0.5) or non-paretic steps (p=0.6). It is possible that adopting a new gait pattern that
deviates from normal walking altered people’s perception of risk, and thus their gait became
150
more cautious. Future studies may assess how people perceive their ability to maintain balance
when instructed to walk differently from their normal gait. Another limitation in this study is
that we provided visual information about the desired and actual step lengths during each step
throughout the symmetry feedback trials, including the perturbation and recovery steps. Since
participants were instructed to achieve the target step lengths for as many steps as possible,
participants may have used this feedback during perturbation recovery to return to their pre-
perturbation walking patterns faster than they otherwise would without visual feedback.
Although there is possibility that participants’ reactive response may be influenced by the visual
feedback, we did not observed difference in measures of momentum between condition with the
feedback and without the visual feedback.

We would like to thank Aram Kim for her help with manuscript editing.

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Chapter 7. Evidence of a nonlinear mapping between body state and foot
placement during perturbed walking

Author: Chang Liu, James M. Finley
Maintaining balance during walking requires the neuromotor system to respond
appropriately to both internally generated and external perturbations. Control of foot placement
is an important stabilization strategy for compensating these perturbations. Previous studies have
characterized the mapping between the foot placement and the body’s state during unperturbed
steady-state walking and found that a linear mapping is sufficient to describe most of the
variance in foot placement. However, it is uncertain whether mappings derived from unperturbed
walking can be extended to explain foot placement in response to perturbations as control of foot
placement during walking may become nonlinear with larger disturbances. To derive such a
mapping to capture foot placement control in response to perturbations during walking, we
applied unexpected changes in belt speed of different magnitudes and directions at foot strike in
thirteen non-disabled participants walking on a dual-belt treadmill at their self-selected speed.
We found that the mapping derived using natural variability in foot placement during steady-
state walking could not explain patterns of foot placement following perturbations, suggesting
that such mapping should be used only with cautious to infer how people control their foot
placement in response to perturbations. Instead, a nonlinear mapping of the body’s state that
counted for differences in the response to forward versus backward perturbations best explained
the variance in foot placement. These results suggest that control of foot placement during losses
of balance is different from the control strategies used during unperturbed walking. Deriving this
nonlinear mapping would help inform foot placement strategies during multidirectional
152
perturbations and help design controllers to generate human-like behavior in response to
perturbations. Future studies could also explore how these individual differences in foot
placement control strategies are associated with balance control capabilities in populations at
high fall risk, which can extend our understanding of how the healthy and impaired nervous
system regulates dynamic balance during walking.

Bipedal locomotion is inherently unstable due to the small base of support, long single-
limb support times, and sensorimotor transmission delays (Winter, 1995; Woollacott and Tang,
1997). Both internal factors such as improper foot placement and incorrect weight shifting
(Robinovitch et al., 2013) and external factors such as unexpected slips or trips could easily lead
to a fall. Our central nervous system needs to integrate sensory feedback from multiple sources
to generate balance corrective responses during walking by, for example, adjusting foot
placement following both external perturbations and internally generated
disturbances(Fitzpatrick et al., 1999; Maeda et al., 2017; O’Connor and Kuo, 2009).
Control of foot placement is one important stabilization strategy during walking and
balance control(Redfern and Schumann, 1994; Vlutters et al., 2016). Adjustment in foot
placement location could adjust the center of pressure location which is the point of application
of ground reaction force and the magnitude of ground reaction force. For example, to recover
from a perturbation in the anterior-posterior direction, one simple way to correct for this
perturbation is to place the foot away from the body center of mass (CoM) so that the leading leg
force has a greater fore-aft component to arrest excessive body rotation and restore balance.
Thus, modulating foot placement from step-to-step is an important strategy for humans to
maintain balance.
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Step-to-step balance corrective strategies can be derived by relating foot placement to the
body’s state using a data-driven approach. This has been done for both non-disabled young
participants (Joshi and Srinivasan, 2019; Rankin et al., 2014; Roden-Reynolds et al., 2015; Wang
and Srinivasan, 2014) and populations who are at risk for falls such as older adults and people
post-stroke (Arvin et al., 2018; Dean and Kautz, 2015). Given a nominal CoM trajectory for non-
disabled people during steady walking on the treadmill, the deviation of CoM state from this
trajectory relates to the deviation in the next foot placement through a linear mapping (Wang and
Srinivasan, 2014). These experimentally derived foot placement mappings can explain ~80% of
the variance in foot placement in the mediolateral direction and ~30% of the variance in the
anteroposterior direction at midstance (Wang and Srinivasan, 2014). The high predictive power,
especially in the mediolateral direction, may indicate that our central nervous system uses
information about the body’s state to actively control the next foot placement during unperturbed
walking. However, few studies have investigated foot placement control strategies during large
external perturbations to determine whether the control strategies inferred from unperturbed
walking generalize to larger losses of balance (Joshi and Srinivasan, 2019; Vlutters et al., 2016)
and whether the linearity assumption of the foot placement mapping during unperturbed walking
still holds when in response to perturbations. A nonlinear controller may indicate how people
regain stability during larger perturbations and whether people have different responses to
multidirectional perturbations.
The primary goal of this study was to determine whether the foot placement mapping
derived from unperturbed walking could explain the variance in foot placement when external
perturbations occurred for non-disabled participants. We hypothesized that the foot placement
mapping derived from unperturbed walking could not capture the control of foot placement in
154
response to perturbations. We also hypothesized that a non-linear mapping would be required to
characterize people’s foot placement during responses to unexpected changes in treadmill speed.
Overall, this study may extend our understanding of how healthy people control their foot
placement to maintain balance during walking and may inform the design of controllers to
generate human-like walking behavior in response to perturbations.

A total of 13 non-disabled individuals were recruited for this study (Participant
characteristics were reported in Chapter 5). The study was approved by the Institutional Review
Board at the University of Southern California, and all participants provided informed consent
before participating. All aspects of the study conformed to the principles described in the
Declaration of Helsinki.

The experiment protocol was the same as in Chapter 2.

We used the same data acquisition methods as in Chapter 2.

We post-processed the kinematic and kinetic data in Visual3D (C-Motion, Rockville,
MD, USA) and Matlab 2020b (Mathworks, USA) to compute variables of interest. Marker
positions and ground reaction forces were lowpass filtered by 4th order Butterworth filters with
cutoff frequencies of 6 Hz and 20 Hz, respectively, based on previous literature (Kurz et al.,
2012; Reisman et al., 2009; Winter, 2009). Foot strike was defined at the time point when the
155
ground reaction reached 80N. We also examined the timing of perturbations relative to foot
strike post-hoc to remove the perturbations that occurred ~150ms after the foot-strike (Buurke et
al., 2020).

Figure 34: Diagram of the simplified model describing the CoM motion (S) and foot positions.
CoM state included the CoM position and velocity in the fore-aft and mediolateral direction.
Blue: swing leg, Red: stance leg). CoM position and the position of the swing foot were
referenced to the stance foot. The gray dashed trajectory represents the nominal (average) CoM
trajectory. The gray solid trajectory represents one measured trajectory. 𝛥𝛥𝛥𝛥 and 𝛥𝛥𝛥𝛥 described
the step-to-step fluctuation of the CoM state and foot placement.

To assess the relationship between the CoM state and foot placement, we followed
previous work (Seethapathi and Srinivasan, 2019; Wang and Srinivasan, 2014) (Figure 34).
Assuming small perturbations during unperturbed walking, fit a linear model relating the
deviation from the mean CoM state and the deviation from the mean foot position (Wang and
Srinivasan, 2014). The full state of our simplified model of the walker included CoM position
and velocity and the positions of the centers of mass of the left and right feet. We defined the
anteroposterior direction to be the x-axis, the mediolateral direction to be the y-axis, and vertical
to be the z-axis. CoM state S = [𝑋𝑋𝑋𝑋 𝑋𝑋 𝐴𝐴𝐴𝐴 ( 𝜏𝜏 ), 𝑋𝑋𝑋𝑋 𝑋𝑋 𝐶𝐶𝐿𝐿 ( 𝜏𝜏 ), 𝑋𝑋𝑋𝑋 𝑋𝑋 ̇ 𝐴𝐴𝐴𝐴 ( 𝜏𝜏 ), 𝑋𝑋𝑋𝑋 𝑋𝑋 ̇ 𝐶𝐶𝐿𝐿 ( 𝜏𝜏 )]
𝑇𝑇 and the
position of the next foot-strike 𝛥𝛥 = [Foot
AP
, Foot
M L
]
𝑇𝑇 were defined relative to the position of
the current stance foot. Velocity variables were computed by differentiating the position
156
variables. We normalized position variables using the height (H) of the participant and velocity
variables using � gH where g is the gravity constant. Each step cycle was divided into 100 time
intervals ( τ).
We defined the nominal trajectories of the CoM (S*) and foot-strike positions (Q*) at
each τ as the average values of these quantities during unperturbed walking. Step-to-step
fluctuations about the nominal trajectory allowed us to examine the relationship between the
deviation in foot positions ΔQ = Q − Q
∗
and deviation of the CoM state ΔS = S − S
∗
. Such
relationship could be approximated by a linear mapping using a Jacobian matrix at each time
interval ( τ) of the step cycle (Eqn.1-5).
[Q
k + 1
− Q
∗
] ≈ 𝐉𝐉 [S
k
− S
∗
] (1)
ΔQ
k + 1
= Δ[Foot
AP
, Foot
M L
]
𝑘𝑘 + 1
𝑇𝑇 = � Foot
x
, Foot
y
�
𝑘𝑘 + 1
𝑇𝑇 − � Foot
x
∗
, Foot
y
∗
�
𝑇𝑇
(2)

ΔS
k
= Δ � 𝑋𝑋𝑋𝑋 𝑋𝑋 𝐴𝐴𝐴𝐴 ( 𝜏𝜏 ), 𝑋𝑋𝑋𝑋 𝑋𝑋 𝐶𝐶𝐿𝐿 ( 𝜏𝜏 ), 𝑋𝑋𝑋𝑋 𝑋𝑋 ̇ 𝐴𝐴𝐴𝐴 ( 𝜏𝜏 ), 𝑋𝑋𝑋𝑋 𝑋𝑋 ̇ 𝐶𝐶𝐿𝐿 ( 𝜏𝜏 ) �
𝑘𝑘 𝑇𝑇 = � 𝑋𝑋𝑋𝑋 𝑋𝑋 𝐴𝐴𝐴𝐴 ( 𝜏𝜏 ), 𝑋𝑋𝑋𝑋 𝑋𝑋 𝐶𝐶𝐿𝐿 ( 𝜏𝜏 ), 𝑋𝑋𝑋𝑋 𝑋𝑋 ̇ 𝐴𝐴𝐴𝐴 ( 𝜏𝜏 ), 𝑋𝑋𝑋𝑋 𝑋𝑋 ̇ 𝐶𝐶𝐿𝐿 ( 𝜏𝜏 ) �
𝑘𝑘 𝑇𝑇 − � 𝑋𝑋𝑋𝑋 𝑋𝑋 𝐴𝐴𝐴𝐴 ( 𝜏𝜏 )
∗
, 𝑋𝑋𝑋𝑋 𝑋𝑋 𝐶𝐶𝐿𝐿 ( 𝜏𝜏 )
∗
, 𝑋𝑋𝑋𝑋 𝑋𝑋 ̇ 𝐴𝐴𝐴𝐴 ( 𝜏𝜏 )
∗
, 𝑋𝑋𝑋𝑋 𝑋𝑋 ̇ 𝐶𝐶𝐿𝐿 ( 𝜏𝜏 )
∗
�
𝑇𝑇
(3)
Δ[Foot
AP
, Foot
M L
]
𝑇𝑇 𝑘𝑘 + 1
= 𝑱𝑱 (𝜏𝜏 ) Δ � 𝑋𝑋𝑋𝑋 𝑋𝑋 𝐴𝐴𝐴𝐴 ( 𝜏𝜏 ), 𝑋𝑋𝑋𝑋 𝑋𝑋 𝐶𝐶𝐿𝐿 ( 𝜏𝜏 ), 𝑋𝑋𝑋𝑋 𝑋𝑋 ̇ 𝐴𝐴𝐴𝐴 ( 𝜏𝜏 ), 𝑋𝑋𝑋𝑋 𝑋𝑋 ̇ 𝐶𝐶𝐿𝐿 ( 𝜏𝜏 ) �
𝑘𝑘 𝑇𝑇

(4)

𝐉𝐉 =
⎣
⎢
⎢
⎢
⎡
𝜕𝜕 𝐹𝐹𝑋𝑋 𝑋𝑋 𝐿𝐿 𝐴𝐴𝐴𝐴 𝜕𝜕 𝑋𝑋𝑋𝑋 𝑋𝑋 𝐴𝐴𝐴𝐴 𝜕𝜕 𝐹𝐹𝑋𝑋 𝑋𝑋 𝐿𝐿 𝐴𝐴𝐴𝐴 𝜕𝜕 𝑋𝑋𝑋𝑋 𝑋𝑋 𝐶𝐶𝐿𝐿 𝜕𝜕 𝐹𝐹𝑋𝑋 𝑋𝑋 𝐿𝐿 𝐴𝐴𝐴𝐴 𝜕𝜕 𝑋𝑋𝑋𝑋 𝑋𝑋 𝐴𝐴𝐴𝐴 ̇ 𝜕𝜕 𝐹𝐹𝑋𝑋 𝑋𝑋 𝐿𝐿 𝐴𝐴𝐴𝐴 𝜕𝜕 𝑋𝑋𝑋𝑋 𝑋𝑋 𝐶𝐶𝐿𝐿 ̇ 𝜕𝜕 𝐹𝐹𝑋𝑋 𝑋𝑋 𝐿𝐿 𝐶𝐶𝐿𝐿 𝜕𝜕 𝑋𝑋𝑋𝑋 𝑋𝑋 𝐴𝐴𝐴𝐴 𝜕𝜕 𝐹𝐹𝑋𝑋 𝑋𝑋 𝐿𝐿 𝐶𝐶𝐿𝐿 𝜕𝜕 𝑋𝑋𝑋𝑋 𝑋𝑋 𝐶𝐶𝐿𝐿 𝜕𝜕 𝐹𝐹𝑋𝑋 𝑋𝑋 𝐿𝐿 𝐶𝐶𝐿𝐿 𝜕𝜕 𝑋𝑋𝑋𝑋 𝑋𝑋 𝐴𝐴𝐴𝐴 ̇ 𝜕𝜕 𝐹𝐹𝑋𝑋 𝑋𝑋 𝐿𝐿 𝐶𝐶𝐿𝐿 𝜕𝜕 𝑋𝑋𝑋𝑋 𝑋𝑋 𝐶𝐶𝐿𝐿 ̇ ⎦
⎥
⎥
⎥
⎤

(5)
157
Here, k is the step number. J is a 2x4 Jacobian matrix that can be estimated by the least
square fit. * corresponds to the nominal trajectory of CoM state or foot-strike position. We
computed the Jacobian matrix at each time interval 𝜏𝜏 to determine the relationship between CoM
state and the subsequent foot placement over the step cycle. We assumed left-right symmetry so
that the foot positions and the CoM state were mirrored about the sagittal plane (Ankaralı et al.,
2015; Joshi and Srinivasan, 2019).

Our objective was to determine whether the foot placement mapping between
unperturbed and perturbed gait differed. We first used the Jacobian matrix derived from
unperturbed walking steps to predict foot placement during perturbed steps. We also derive foot
placement mappings using both the perturbed steps and an equal number of unperturbed steps at
the end of the unperturbed walking trial. Both a linear mapping (LR, Eqn.6) and a nonlinear
mapping such as piecewise linear regression (PLR, Eqn.7) that account for the directional
differences in responses to increases and reductions in belt speed were fit to the data and the
better model was selected based on Akaike information criterion to control for model complexity
(AIC)(Akaike, 1981). A lower AIC value indicated better model fit.
Linear mapping (LR):
ΔQ
𝑇𝑇 𝑘𝑘 + 1
= 𝑱𝑱 ΔS
k
𝑇𝑇
(6)
Piecewise linear regression (PLR):
ΔQ
𝑇𝑇 𝑘𝑘 + 1
= �
𝑱𝑱 1
ΔS
k
𝑇𝑇
𝐿𝐿 𝑖𝑖 Δ 𝑋𝑋𝑋𝑋 𝑋𝑋 ̇ 𝐴𝐴𝐴𝐴 > 0
𝑱𝑱 2
ΔS
k
𝑇𝑇 𝐿𝐿 𝑖𝑖 Δ𝑋𝑋𝑋𝑋 𝑋𝑋 ̇ 𝐴𝐴𝐴𝐴 < 0
(7)
158
We used mixed-effects models to account for the variability across participants for all
foot placement mapping models. We included random slope for each predictor in the models.
Additionally, we calculated the coefficient of determination (R
2
) to capture how well the
variance in foot placement could be explained based on the body’s state. We derived the model
and compared the model fit and coefficients at midstance which was defined as 50% ( τ = 50) of
the step cycle. We chose midstance beacuse the mapping between foot placement and CoM state
was well characterized at this time point in previous studies and there was sufficient time
remained to allow for changes in foot placement by the swing limb (Joshi and Srinivasan, 2019;
Kim and Collins, 2017; Wang and Srinivasan, 2014). We used AIC to determine which model
best fits the foot placement data while controlling model complexity. AIC was computed for
each individual fit for both the linear mapping model and the piecewise linear regression model.
We also used paired sample t-test to compare whether the regression coefficients from perturbed
walking were different from those derived from unperturbed walking at midstance. Significance
will be set at p<0.05.

During unperturbed walking, we found that the foot position in the mediolateral direction
was associated with the CoM position and velocity in the mediolateral direction at midstance:
ΔFoot
M L
≈ 1.6ΔCoM
M L
+ 1.13Δ 𝑋𝑋𝑋𝑋 𝑋𝑋 ̇ M L
( 𝑅𝑅 2
= 0.60). In response to a rightward CoM
deviation at midstance, the person will step toward the right side. Similarly, we found that the
foot position in the anteroposterior direction with associated with the CoM position and velocity
in both mediolateral and anteroposterior direction: ΔFoot
AP
≈ −0.81ΔCoM
ML
−
0.94Δ𝑋𝑋𝑋𝑋 𝑋𝑋 ̇ M L
+ 0.71ΔCoM
AP
+ 0.77Δ𝑋𝑋𝑋𝑋 𝑋𝑋 ̇ 𝐴𝐴𝐴𝐴 (𝑅𝑅 2
= 0.38) in the anteroposterior direction.
159
Thus, a larger forward speed at midstance was associated with a longer step while a larger lateral
CoM speed at midstance was associated with a shorter step. These results are consistent with
previous literature (Wang and Srinivasan, 2014).

However, this mapping between foot position and CoM state at the midstance during
unperturbed waking did not generalize to foot positions following perturbations (Figure 35B).
We compared the AIC among three models: 1) fit the mapping derived from unperturbed
walking to perturbed steps (UNPTB); 2) apply the linear mapping derived from perturbed
walking to perturbed steps (LR), and (3) applying the piecewise linear regression model to
perturbed steps (PLR) (Figure 35C & D). AIC did not differ across three models in the
mediolateral directions (all p > 0.05). In the anteroposterior direction in which we applied
perturbations, AIC significantly decreased for the LR model and PLR model from UNPTB
model (both p < 0.0001), indicating that the model derived directly using perturbed steps
explained the variance in foot placement better than generalizing the unperturbed mapping to
perturbed steps. We found that the piecewise linear model’s AIC was significantly lower than
that for the linear fit for perturbed steps (t(24) = 2.8, p = 0.0097) in the anteroposterior direction.
Thus, the piecewise linear regression model could best capture the foot placement variance
during perturbed steps. The derived foot placement mapping in the anteroposterior direction had
the following form (Figure 36):
Δ 𝐹𝐹𝑋𝑋𝑋𝑋 𝐿𝐿 𝐴𝐴𝐴𝐴
≈ �
− 0.76ΔCoM
M L
− 0.35Δ 𝑋𝑋 𝑋𝑋𝑋𝑋 ̇ M L
+ 1.27ΔCoM
AP
+ 0.4Δ 𝑋𝑋 𝑋𝑋𝑋𝑋 ̇ AP
𝐿𝐿 𝑖𝑖 Δ 𝑋𝑋 𝑋𝑋𝑋𝑋 ̇ A P
> 0 ( 𝐹𝐹 𝑋𝑋𝑟𝑟 𝐹𝐹 𝐹𝐹 𝑟𝑟 𝐹𝐹 )
− 1.5ΔCoM
M L
+ 1.5Δ 𝑋𝑋 𝑋𝑋𝑋𝑋 ̇ M L
+ 2.3ΔCoM
A P
+ 1.5Δ 𝑋𝑋 𝑋𝑋𝑋𝑋 ̇ A P
𝐿𝐿 𝑖𝑖 Δ 𝑋𝑋𝑋𝑋𝑋𝑋
̇ AP
< 0 ( 𝐵𝐵 𝐹𝐹 𝐵𝐵 𝐵𝐵 𝐹𝐹 𝐹𝐹 𝑟𝑟 𝐹𝐹 )

We also compared the coefficients of the foot placement mapping derived from
unperturbed and perturbed walking during forward and backward perturbations to compare the
sensitivity of foot placement to changes in CoM state (Figure 36). Coefficient estimates for each
160
individual were computed by summing the random effects and the fixed effects from each mixed
effect model. We found that the magnitude of the coefficients for ΔCoM
AP
derived from
backward perturbations were significantly higher than those derived from forward perturbations
(t(12) = -4.1, p = 0.0015) and unperturbed walking (t(12) = -5.5, p = 0.0001). The coefficients
for Δ𝑋𝑋𝑋𝑋 𝑋𝑋 𝐶𝐶𝐿𝐿 ̇ derived from backward perturbations were also significantly higher than those
derived from forward perturbations (t(12) = -3.4 , p = 0.006) and unperturbed walking (t(12) = -
4.9, p = 0.0004) and note that the coefficients for backward perturbations had the opposite signs
to both those coefficients derived from forward perturbations and unperturbed walking. The
coefficients for Δ 𝑋𝑋𝑋𝑋 𝑋𝑋 𝐶𝐶𝐿𝐿 ̇ were also significantly higher derived from forward perturbations than
unperturbed walking (t(12) = -3.5, p = 0.0047). Lastly, the coefficients for Δ𝑋𝑋𝑋𝑋 𝑋𝑋 𝐴𝐴𝐴𝐴 ̇ derived from
backward perturbations were also significantly higher than those derived from forward
perturbations (t(12) = -4.1, p = 0.014) and unperturbed walking (t(12) = -2.78, p = 0.017) and the
coefficients were significantly higher derived from forward perturbations than unperturbed
walking (t(12) = 3.0 , p = 0.012).
161

Figure 35: Scatter plots showing the foot placement during unperturbed walking and following
perturbation responses for a representative participant and goodness of fit for each model. (A)
Left foot placement relative to the right perturbed stance foot during unperturbed steps and
perturbed steps; (B) Actual foot placement v. fitted foot placement in the anteroposterior
direction during perturbed walking at midstance using the mapping derived during unperturbed
walking. Colored dots indicate foot placement following perturbations. Gray dots represent foot
placement during unperturbed walking. (C & D) AIC of deviations in foot placement after
applying the mapping derived at midstance from unperturbed walking to perturbed steps
(UNPTB), applying the linear mapping derived from perturbed walking to perturbed steps (LR),
and applying the piecewise linear regression model to perturbed steps (PLR). (C) Mediolateral
direction. (D). Anteroposterior direction. (*p<0.05, **p<0.001,***p<0.0001). Dots represent
each participant.

162

Figure 36: The estimated coefficients of the partial derivative of the foot position with respect to
CoM state at midstance in the Jacobian matrix. Boxplot shows coefficient estimates across
participants. Gray: estimates during unperturbed walking, Green: estimates from piecewise
linear regression for forward perturbations, Blue: estimates from piecewise linear regression for
backward perturbations. Dots represented individual estimates of coefficients by summing the
random effects and the fixed effects from the mixed effect models (*p<0.05,
**p<0.001,***p<0.0001).

Our study’s primary objective was to determine whether the foot placement mapping
derived from unperturbed walking could explain the variance in foot placement when external
perturbations occurred for non-disabled participants. We found that the mapping derived from
the natural variability of foot placement during steady-state walking could not explain patterns of
foot placement in response to perturbations during walking. Instead, a nonlinear mapping of
CoM state accounted for differences in responses to forward versus backward perturbations best
explained foot placement patterns. Overall, this study suggested that nonlinear behavior emerges
163
when people need to adjust their foot placement in response to relative large deviation in CoM
state. Ultimately, these results may extend our understanding of about the foot placement control
to maintain balance during walking, which could be applied to design controllers to capture
walking dynamics.
The foot placement mapping derived from natural foot placement variability during
unperturbed walking in non-disabled participants was similar to that previously reported for
young adults despite the fact that our population was on average older (Wang and Srinivasan,
2014). Our derived foot placement mappings can explain ~60% of the variance in foot placement
in the mediolateral direction and ~40% of the variance in the anteroposterior direction at
midstance, which is also comparable with the previous study (Wang and Srinivasan, 2014). In
the fore-aft direction, more lateral deviation of CoM position and CoM velocity at midstance was
associated with a shorter step while a more forward deviation of CoM position and CoM velocity
was associated with a longer step. In the mediolateral direction, more lateral deviation of CoM
position and velocity was associated with a more lateral step. In both directions, participants
stepped in the direction of the CoM deviation. Such association between deviation in CoM state
and foot placement could be attributed to passive dynamics of the swing leg and active control of
foot placement to maintain balance. Additionally, as in the non-disabled young population, the
coefficient of determinant at midstance was higher for mediolateral deviations in foot placement
than fore-aft deviations, indicating that people may adopt a tighter control their foot placement in
the mediolateral direction than in the fore-aft direction.
Consistent with our hypothesis, we found that the foot placement control strategy differed
for forward perturbations versus backward perturbations when the mapping was derived at
midstance. For instance, changes in foot placement in anteroposterior direction were more
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sensitive to changes in fore-aft CoM position and velocity at midstance following backward
perturbations than forward perturbations. The discrepancy in foot placement mapping between
forward and backward perturbations may result from people modulating the ankle torque more of
the perturbed limb to shift the center of pressure location forward following forward
perturbations but not as much during backward perturbations during the stance phase since our
perturbations occurred at foot strike (Vlutters et al., 2016). Shifting the center of pressure
location forward by activating the ankle plantarflexors at the stance phase following forward
perturbations could help people to generate backward moment about body CoM to reduce the
forward rotation of the body. As a result, a smaller backward moment needs to be generated
about the body’s CoM at the next foot placement and less foot placement deviation from the
nominal trajectory was needed than following backward perturbations.
It is possible that other stabilization control strategies such as modulating the ankle push-
off also play an important role in maintaining balance(Pijnappels et al., 2005). Kim & Collins
derived a controller that used both foot placement and ankle push-off impulse to stabilize a biped
in the sagittal plane when negotiating through random changes of the ground's height during
walking (Kim and Collins, 2017). Therefore, future studies may investigate how different
balance recovery strategies coordinate together following the deviation in body’s state and
whether such coordination may explain the difference in foot placement mapping following the
forward and backward perturbations.
There could be learning effect during perturbation response. Previous work has reported
that people increase the distance between CoM position relative to the rear base of support and
CoM velocity during repeated exposure to tripping over obstacles (Wang et al., 2012). However,
we did not observe the similar learning effects in our study. We compared the distance from the
165
CoM to the rear edge of the base of support and CoM velocity in the anteroposterior direction
during the first perturbation and last perturbation for each level of treadmill speed change.
However, we did not observe any significant differences between the first and last perturbations
(CoM position to : p = 0.25; CoM velocity: p = 0.20). This discrepancy between our results and
previous work could be because we randomized the perturbation magnitude, side, and direction,
making it harder for participants to make specific adjustments prior to the perturbations.

Our study has a few limitations. Our current study used CoM state as the predictor to
derive the foot placement mapping. However, it is uncertain if CoM state provides the best
predictive value. Other studies have used the swing leg state at the swing initiation (Dean and
Kautz, 2015), the stance leg state (Arvin et al., 2018), or the ankle state (Maus et al., 2015) to
construct predictive models that describe how humans control balance during walking or
running. Future studies should perform a more comprehensive model comparison to determine
the best set of predictors to construct the foot placement mapping between changes in body’s
state and foot placement location.
Although a recent study using an open-loop stable 2D model showed that the
predictability of the ‘next’ foot position using CoM state in the fore-aft at midstance is more than
80% with passive dynamics alone (Patil et al., 2019), the relationship between CoM state and the
subsequent foot positions can arise from a combination of passive dynamics due to the intrinsic
body segment inertia and active control of foot placement. One primary objective of our study
was to derive the foot placement mapping during relatively large perturbations that required
reactive responses to avoid falls. In addition, the previously examined 2D bipedal model did not
take into account the inertial properties of the swing limb or consider control of the torso that
166
helps to maintain an upright posture. Thus, a more complex model with segment inertias (Nozari
and Finley, 2019) needs to be developed to untangle the relative contribution of passive
dynamics and active control to the correlation between body’s state and foot placement and draw
inference about how people use sensory feedback information to generate corrective response.

167
Chapter 8. Discussion and future work
Maintaining balance when faced with internally generated or externally imposed
perturbations can be very challenging during bipedal locomotion. Reactive control strategies are
elicited in response to unexpected perturbations during walking to maintain balance. Overall,
the objective of this dissertation was to characterize the reactive control strategies for healthy
populations and people post-stroke by imposing treadmill-induced perturbations during walking
and identify the contributors to the impaired reactive control of balance in people post-stroke.
We also investigated whether modulating spatiotemporal gait asymmetries would affect people’s
reactive control of balance. One common way to characterize the resulting perturbation recovery
strategies is the measure of whole-body angular momentum, which reflects the contribution of all
body segments to the body’s rotation about a given axis. Our first research question was whether
changing the reference axis of whole-body angular momentum would change our interpretation
of dynamic balance during walking. The second question was to assess whether changing step
length asymmetry would affect people’s reactive balance control. We performed two studies: we
asked whether introducing spatiotemporal gait asymmetries would impair reactive control of
balance for healthy individuals, and if improving spatiotemporal asymmetries would improve
reactive balance in people post-stroke who usually have spatiotemporal asymmetries during
walking. The third objective was to investigate whether reactive control strategies differed in
people post-stroke compared to age-matched controls. Lastly, our final goal was to derive a foot
placement mapping that relates the body’s state and the subsequent foot placement following
perturbations and during unperturbed walking. Our findings will help us understand the
contributors to impaired balance control in people post-stroke and the relationship between
asymmetry and reactive balance of both healthy and people post-stroke. In this way, this work
168
will provide a mechanistic understanding of how the central nervous system implements reactive
control strategies to restore balance for both healthy and people post-stroke and advance the
development of effective interventions that seek to improve mobility and reduce fall risk for
people post-stroke.

We found that in the sagittal plane, referencing whole-body angular momentum to a
mediolateral axis projecting through the body’s center of mass provided complementary
interpretations about how healthy people coordinated the segments to maintain whole-body
angular momentum in response to perturbations when compared to the use of an axis that
projected through the leading edge of base of support. We found that peak backward angular
momentum during the perturbed steps referenced to both axes was positively associated with
perturbation amplitude. In addition, the low-dimensional intersegmental coordination patterns
extracted referenced to both axes had one similar component and one dissimilar component
during perturbation steps. We found that analyzing coordination patterns referenced to an axis
that projected through the edge of the base of support may provide insights about how the upper
body responded to sudden loss of balance while referencing to the CoM axis may help us to
identify the degree of segmental angular momentum cancellation during walking and
perturbation responses. This analysis may provide a better understanding of how people regulate
whole-body angular momentum by coordinating the body segments’ rotation in response to
perturbations.
169

Chapter 5 aimed to assess whether the side of a perturbation would affect the reactive
control of balance in people post-stroke and compared these strategies with control participants.
We found that people post-stroke increased the integrated angular momentum more during
paretic perturbations than they did following non-paretic perturbations and compared to controls.
Control participants modulated the mechanical effect of ground reaction forces on the body
dynamics for both limbs to reduce the forward loss of balance induced by the sudden treadmill
acceleration while stroke participants could not modulate the body dynamics using their paretic
limb at either paretic foot collision or push-off phase. In addition, we found that people post-
stroke decreased their push-off angular impulse to reduce forward body rotation more using their
non-paretic limb than they did with the paretic limb. These post-stroke deficits in reactive
stabilization strategies may contribute to the need for one extra compensatory step following
perturbations to restore balance compared with control participants. These results may provide
implications for designing perturbation-based trainings. These trainings may target both the
weight-bearing capacity at the paretic limb to help prevent the body from falling forward during
the perturbations and improve the ability to modulate force production and force directions at the
paretic side so that less forward momentum would be generated during the late stance phase
following the perturbations.

Chapter 3,4, and 6 aimed to investigate whether modulating step length asymmetry would
affect people’s reactive control of balance during walking for both healthy and people post-
stroke. An important question for clinical researchers is whether there is a causal relationship
between gait asymmetry and the ability to maintain balance. Our results suggested that although
170
walking with asymmetrical gait patterns changed the limb coordination during perturbation
responses, there was no association between step length asymmetry and impaired reactive
control of balance in the absence of neuromotor impairments. These results also indicated that
our nervous system may allow for variability in limb coordination to regulate a higher-order
performance variable such as the whole-body angular momentum.
After studying how the intact neuromotor system modulates coordination between the
upper and lower extremities in response to changes in asymmetry, we extended to investigate
whether reducing spatiotemporal gait asymmetries would improve post-stroke survivors’ reactive
balance control during perturbation responses. We found that reducing asymmetry during a cross-
sectional design study did not immediately reduce people’s reactive control of balance. Our
findings may also inform the design of rehabilitation programs that seek to improve balance
control for other clinical populations with asymmetric walking patterns such as amputees and
people with Parkinson disease who also tend to have impaired ability to correct for perturbations
due to sensorimotor deficits.

Chapter 7 aimed to derive a foot placement mapping between the CoM state and foot
placement location in healthy individuals as control of foot placement is an important balance
control strategy for people to make step-to-step correction. We aimed to determine whether the
foot placement mapping derived from unperturbed walking could capture the patterns of foot
placement observed following external perturbations for healthy participants. We found that the
mapping derived during unperturbed, normal walking could not explain the foot placement
location during perturbation responses. Instead, a nonlinear mapping that accounted for the
difference in strategies during forward and backward perturbation response could better explain
171
the data. These results provided evidence that nonlinear behavior emerge around the nominal or
average trajectories when large perturbations occur, which may provide new insights into the
design of robust controllers to generate human-like walking behavior in response to
perturbations.

People post-stroke tend to have sensory impairments at the lower limb due to the deficits
in the ascending pathway that transport sensory information to the brain (Tyson et al., 2008;
Wutzke et al., 2013). However, despite the prevalence of sensory impairments post-stroke, few
studies have assessed how the reduced sensation, especially at the paretic limb after stroke,
affects the initiation of reactive control strategies which involves using feedback information to
make corrective response. We could use sensory tests to assess the touch thresholds at the foot
sole for people post-stroke. We could investigate if the reduced sensation at the paretic limb is
associated with longer muscle activation latency at the paretic perturbed limb following
perturbations. We would expect that people with more sensory deficits will have longer delays
before making corrective feedback adjustment in response to perturbations. People post-stroke
may not arrest the falls fast enough due to such sensorimotor delays, which could contribute to
the increased risk of falls post-stroke. These results may potentially help develop haptic-based
rehabilitation device to improve sensation and thus improve balance control.
We could also assess the effects of increasing propulsive force on body’s rotational
behavior for people post-stroke. Increasing propulsive force during the push-off phase has been
widely used in gait rehabilitation studies to reduce step length asymmetries and increase walking
speed(Awad et al., 2020). People post-stroke still retain the capacity to increase the paretic
propulsive force (Lewek et al., 2018). Lewek et al. showed that people post-stroke could increase
172
their paretic propulsion impulse by more than 200% when walking on the treadmill against an
impeding force applied at their center-of-mass (Lewek et al., 2018). However, no study has
quantified the effect of increasing propulsive force on dynamic balance control measured by, for
example, whole-body angular momentum. In Chapter 5, we found that the push-off angular
impulse at the paretic side that increased forward angular momentum of the body was higher
than the non-paretic side and control participants during walking. People post-stroke also has less
capability modulating the push-off angular impulse following perturbations. Thus, it is possible
that increasing the paretic propulsive force could lead to increase in forward angular momentum
during the paretic to non-paretic step transition and thus compromise balance. Not using the full
reserve of paretic propulsive force could be an adaptive strategy to maintain dynamic balance for
people post-stroke.
Another future direction would be to determine the foot placement mapping for people
post-stroke during perturbation responses. We would expect that the maximum perturbation that
people post-stroke could recover from without falling would be lower than control participants.
Like non-disabled participants, there would be differences in the strategies in response to
forward and backward perturbations. We can then compare the foot placement mapping during
perturbation response between stroke and healthy participants. Deriving individual-specific foot
placement mapping may provide some insights about individual responses to external
perturbations, such as the sensitivity of modulating foot placement to deviation of a sensed
body’s state. Also, we would expect people post-stroke may need to make more corrections in
foot placement following forward perturbations than control participants. This is because people
post-stroke may have less capability to reduce the forward falls using the perturbed limb in
stance phase. Thus, deriving the foot placement mapping in people post-stroke could provide a
173
better understanding of the control mechanism they use to maintain balance and how the control
mechanism changed post-stroke when compared to non-disabled participants. Also, by applying
different levels of perturbations, we could potentially find the maximum disturbance that the
patient population could recover from, which could capture the nonlinear behavior that would
not be normally observed during the small magnitude of perturbations.
Additionally, in Chapter 7, we derived a simple model of foot placement control during
perturbation response by using the CoM state and foot placement location. Previous studies that
assessed this feedback control during unperturbed walking also inferred a linear relationship
between the next foot placement and a single variable of interest that encapsulates the body’s
state, for example, the pelvis state (Wang and Srinivasan, 2014) or the swing leg state (Rankin et
al., 2014). While studies limit the number of regressors in this linear feedback model to predict
the next foot placement to avoid collinearity and improve interpretability, our model and the
previously reported models did not take into account that people maintain balance by
incorporating multiple sources of sensory information. Our central nervous system, by
integrating sensory feedback information from multiple sensory sources such as vision,
vestibular, and proprioception, generates motor commands to adjust foot placement during
walking in response to external perturbations and internal physiological noise. One particular
important sensory information for balance control is the vestibular system. The information
transmitted by vestibular afferents that carry signals to the vestibular nuclei can be approximated
by measures of head kinematics and provide valuable information about the signals that the
nervous system uses to detect losses of balance. Therefore, future studies could aim to derive a
more physiologically plausible controller while deal with the collinearity issue using advanced
data analytics such as partial least square regression to inform us of the balance corrective
174
control strategies following perturbations for both pathological population and healthy
population.

175
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Appendices

Control of balance during walking is not an easy task for humans. Challenges during
bipedal locomotion arise from our small base of support (BoS) defined as the border of the
standing foot, long single-limb support times, and heavy upper body (Winter, 2009). For
example, when walking in the anteroposterior direction, the body center of mass (CoM)
constantly moves out of the edge BoS during the single support phase, which poses a challenge
to maintain dynamic balance. Preventing falls requires both proactive control (Patla, 1993) and
reactive control strategies to maintain anti-gravity support and restore the center of mass (CoM)
within the base support (Tang et al., 1998). While reactive control strategies involve the use of
feedback information about the body’s state to make corrective responses, proactive control of
balance refers to the balance control mechanisms used prior to a perturbation (Patla, 1993).
One common proactive adjustment is to change spatiotemporal gait patterns to prepare
for the known upcoming perturbations (Tang et al., 1998). Adoption of proactive control
strategies to maintain balance depend on prior experiences (Heiden et al., 2006), anticipation of
perturbations (Pater et al., 2015), and the metabolic/energetic cost of the proactive strategies
(Donelan et al., 2004). People tend to adopt proactive strategies to reduce the risk of falls when
they expect to encounter slips or trips during walking. For example, people shorten their step
lengths in anticipation of a backward loss of balance either when they are informed of a slippery
surface ahead (Cham and Redfern, 2002) or a slip that would be induced by a forward translation
of the movable platform during walking (Espy et al., 2010). People also shift their CoM more
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anteriorly at foot strike and increase the distance between the projected CoM and the posterior
edge of BoS when anticipating a backward loss of balance by the forward translation of the
platform (Bhatt et al., 2006). Similarly, during repeated obstacle-tripping perturbations causing
forward rotation of the body, people increase the margin between the anterior edge of the BoS
and CoM when anticipating the obstacle that would obstruct the swing leg after the initial
exposure to the obstacle tripping (Wang et al., 2012). Other strategies, such as increasing step
width, may also be beneficial for balance recovery (Tang et al., 1998). When people are
informed about an upcoming slippery surface before they step, they tend to step on the surface
with an enlarged mediolateral distance between CoM and foot placement (Marigold et al., 2003).
Overall, people tend to adopt proactive control strategies to increase the margin between the
edge of the base of support and CoM state when facing impending perturbations. However, it
remains to be determined whether such proactive strategies adapt over time as changing
spatiotemporal gait features from unperturbed walking can be energetic costly and
uncomfortable.
The use of proactive control strategies may be direction and timing specific.
Perturbations in the sagittal plane could elicit both forward and backward rotation of the body. It
is uncertain what proactive strategies people would prioritize to overcome perturbations of
unknown directions. If people prioritize preparing for forward falls, they would likely increase
the forward safety margin so that the backward safety margin would decrease and vice versa.
Thus, preparing for perturbations for one direction would likely impair the other. Additionally,
healthy individuals may only make proactive locomotor adjustment if they expect to be
perturbed at the next step (Major et al., 2018; Shinya et al., 2016). When walking on a treadmill,
given a set of randomly directed perturbations with random magnitudes, it is possible that
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participants cannot predict the timing or direction of the perturbations and thus not increase
stability margin in either direction for the upcoming perturbations.
In addition to the measures that assess the relationship between the BoS and CoM
dynamics, whole-body angular momentum can characterize the overall control of whole-body
dynamics when people are anticipating perturbations. Whole-body angular momentum about the
CoM is highly regulated during normal human locomotion (Herr and Popovic, 2008). The peak-
to-peak range of whole-body angular momentum is much smaller than the angular momentum of
single segments due to momentum cancellation between the limbs (Herr and Popovic, 2008),
which indicates that segment rotation about the CoM is coordinated in a way that whole-body
angular momentum remains small. When facing unexpected trips or slips, Whole-body angular
momentum dramatically increases and is then restored to that of unperturbed walking over the
next few steps (Liu et al., 2018; Martelli et al., 2013). People commonly use peak-to-peak range
of whole-body angular momentum to describe the regulation of body rotation as the deviation
from the equilibrium position may pose challenges to balance control. Populations with known
balance deficits such as stroke survivors and unilateral amputees show an increased range of
angular momentum during normal walking (Honda et al., 2019; Nott et al., 2014; Vistamehr et
al., 2016). These findings suggest that regulating angular momentum is essential for balance
control, but whether people would reduce whole-body angular momentum as a proactive
adjustment to improve balance remains to be seen.
The purpose of this study was to characterize how healthy people learn to use proactive
adjustments during treadmill-induced trips and slips. We hypothesized that people would
increase their step width, decrease step length, and increase the portion of double support phase
compared to unperturbed walking during the initial phase of the repeated perturbation trials.
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Since changing spatiotemporal gait patterns from unperturbed walking can be energetic costly
and uncomfortable, when people become familiar with the repeated trips or slips, their gait
patterns may return to those measured during unperturbed walking. We also hypothesized that
people would not change their forward and backward stability margin because the direction of
the perturbations was unpredictable, but they would increase their stability margin compared to
unperturbed walking in the mediolateral direction to help stabilize the body. Additionally, we
hypothesized that participants would decrease their range of whole-body angular momentum in
the fore-aft and mediolateral direction during the initial phase of perturbation trials to tightly
control their body rotation when anticipating perturbations.

A total of 11 healthy young individuals (5M, 27±3yrs old, 67.9±17.2kg, self-selected
walking speed = 1.0m/s±0.1m/s) with no musculoskeletal or gait impairments participated in this
study. Lower limb dominance was determined by asking participants which leg they would use
to kick a ball. All participants were right dominant. The study was approved by the Institutional
Review Board at the University of Southern California, and all participants provided informed
consent before participating. All aspects of the study conformed to the principles described in the
Declaration of Helsinki.

Experiment protocol is the same as that reported in Chapter 2.

A ten-camera motion capture system (Qualisys AB, Gothenburg, Sweden) recorded 3-D
marker kinematics at 100 Hz and ground reaction forces at 1000 Hz. We placed a set of 14 mm
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spherical markers on anatomical landmarks to create a 13-segment, full-body model (Havens et
al., 2018; Song et al., 2012). We placed marker clusters on the upper arms, forearms, thighs,
shanks, and the back of heels. Marker positions were calibrated during a five-second standing
trial at the beginning of each trial. We removed all joint markers after the calibration.

We post-processed the kinematic and kinetic data in Visual3D (C-Motion, Rockville,
MD, USA) and Matlab 2020b (Mathworks, USA) to compute variables of interest. Marker
positions and ground reaction forces were lowpass filtered by 4
th
order Butterworth filters with
cutoff frequencies of 6 Hz and 20 Hz, respectively. We selected the type of filter and cut-off
frequency based on previous literature (Kurz et al., 2012; Reisman et al., 2009; Winter, 2009).
We defined foot strike as the point when the vertical ground reaction force became greater than
80N and foot off as the point when vertical ground reaction force became less than 80N. Step
length was defined as the fore-aft distance between the heel marker of the leading foot and the
trailing foot at foot-strike as step width was defined as the mediolateral distance between the 5th
metatarsal marker positions of the two feet at foot strike. Step time was defined as the time
interval between each foot strike. Double support phase (%) was defined by the percentage of
time when both feet were on the ground during each step. Additionally, we defined the pre-
perturbation steps to be all steps excluding the perturbed steps and four recovery steps after each
perturbation. We defined the unperturbed phase as the last 100 steps during unperturbed
(baseline) walking (No PTB). We defined the initial perturbation phase as the first 100 pre-
perturbation steps during the perturbation trials (INIT) and the final perturbation phase as the last
100 pre-perturbation steps during the perturbation trials (FINAL).
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Margin of stability in the anteroposterior direction (AP MoS) and the mediolateral
direction (ML MoS) were calculated for the dominant and non-dominant leg independently as
the distance between the edge of BoS and the extrapolated center of mass (XCoM) based on
previous literatures (Hof, 2008) (Eqns. 1).
𝑋𝑋𝑋𝑋 𝛥𝛥 = 𝐵𝐵 𝑋𝑋 𝛥𝛥 − 𝑋𝑋𝑋𝑋𝑋𝑋 𝑋𝑋 (1)
We used CoM position and velocity to compute the extrapolated CoM (Eqn. 2).
𝑋𝑋𝑋𝑋𝑋𝑋 𝑋𝑋 = 𝑋𝑋𝑋𝑋 𝑋𝑋 + 𝑋𝑋𝑋𝑋 𝑋𝑋 ̇ ∗ 𝜔𝜔 0
− 1
, 𝜔𝜔 0
= � 𝑔𝑔 /𝑙𝑙
(2)

Here, l was leg length, g was the gravity constant equal to 9.81. CoM velocity was
computed by differentiating CoM position with respect to time. AP CoM velocity was computed
by adding the heel marker velocity of the leading stance limb during each step. We defined the
edge of BoS in the mediolateral direction using the 5th metatarsal marker position, edge of BoS
in the AP direction using 1st distal phalanx marker position to calculate the forward MoS and the
heel marker to calculate the backward MoS (Figure 37).
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Figure 37: Diagram of MoS in the fore-aft direction for the forward and backward MoS. We
calculated the forward MoS and the backward MoS using the toe and heel markers, respectively
at foot strike.

We calculated the forward MoS (FWD MoS) and backward MoS (BWD MoS), at each
foot strike. Positive MoS indicated that the XCoM is posterior to the edge of BoS. For the frontal
plane, the mediolateral MoS during the stance phase (ML MoS) was calculated as the minimum
value during each step. This indicated the minimum distance between XCoM and the lateral
boundary of BoS defined by the 5th metatarsal marker. Positive MoS indicated that the XCoM
was medial to the edge of the BoS while negative MoS indicated that XCoM was lateral to the
BoS.

We created a 13-segment, whole-body model in Visual3D and calculated the angular
momentum of each segment about the body’s center of mass. Segmental angular momenta
captured how the rotational and translational behavior of each body segment changed in response
to the treadmill perturbations. The model included the following segments: head, thorax, pelvis,
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upper arms, forearms, thighs, shanks, and feet. The limb segments’ mass was modeled based on
anthropometric tables (Dempster 1955), and segment geometry was modeled based on the
description in Hanavan (Hanavan 1964). Whole-body angular momentum (L) was then computed
as the sum of all segmental angular momenta which were composed of segmental rotation about
the body’s center of mass and rotation of each segment about its own center of mass (Silverman
and Neptune 2011). Then, L was normalized by the participant’s mass (M), treadmill velocity
(V), and the participant’s height (H) (Eq.3).
𝐿𝐿 � ⃗
=
Σ
i
[ 𝑚𝑚 𝑖𝑖 � 𝑟𝑟
� � ⃗
𝐶𝐶𝐶𝐶 − 𝑖𝑖 𝑖𝑖 × 𝑣𝑣 ⃗
𝐶𝐶𝐶𝐶 − 𝑖𝑖 𝑖𝑖 � + 𝐼𝐼 𝑖𝑖 𝜔𝜔 𝑖𝑖 ]
𝑋𝑋 𝑀𝑀𝑀𝑀
(3)

Here, m is segmental mass, r is the distance from segment to the body COM, I is the
segmental moment of inertia, 𝜔𝜔 is segmental angular velocity, and the index i corresponds to
individual limb segments. In the sagittal plane, negative values of angular momentum
represented forward rotation, while positive values represented backward rotation. In the frontal
plane, we specified rotation toward the right side as positive and rotation toward the left side as
negative.

All statistical analyses were performed in Matlab R2020b (Mathworks, Natick, MA,
USA). Repeated measures analysis of variance (RM-ANOVA) was used to determine if the
outcome variables differed between Condition (unperturbed walking, initial phase of the
perturbed walking, and the final phase of the perturbed walking), Leg (Dominant/Non-dominant)
and if there was an interaction between Condition and Leg. Since peak-to-peak range of whole-
body angular momentum was calculated over a stride, we only included Condition as the within
subject predictor for this outcome measure. Normality assumptions were checked using Lilliefors
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Test and violations of sphericity were assessed using the Mauchly’s test. If the assumption of
sphericity was violated, results were corrected using a Greenhouse-Geisser correction. Post hoc
comparisons used the Tukey-Kramer correction for multiple comparisons. Significance value
was set as α = 0.05.

Participants showed changes in spatiotemporal gait patterns during the INIT phase
compared to the unperturbed baseline walking. Participants tended to decrease the step length,
increase step width, and increase the double support phase during the INIT phase compared to
the unperturbed walking (Figure 38). These patterns became more similar to those measured
during unperturbed walking during FINAL phase.

Figure 38 Spatiotemporal patterns throughout unperturbed trial and perturbation trials for one
representative participant. (A) Step length, (B) Step width, and (C) double support phase. Blue
dots: unperturbed baseline trial, gray dots: perturbed trials, red cross: the first perturbation.
Vertical lines indicate No PTB, INIT, FINAL intervals and each intervals included 100 steps.

Participants changed their spatiotemporal gait patterns when anticipating the treadmill
perturbations (Figure 39). We did not find any significant effects of Leg or interaction between
210
Leg and Condition on any of the spatiotemporal gait patterns. There were significant effects of
Condition on step lengths(F(2,20) = 4.48, p = 0.02), step width(F(2,20) = 22.5, p = 0.003), and
double support phase(F(2,20) = 12.85, p = 0.0003). Participants decreased their step lengths
during the INIT phase of perturbation trials compared with unperturbed walking (p = 0.01,
Figure 39A). Participants also increased their step width during the INIT phase of perturbation
trials compared to unperturbed walking (p < 0.001) but not during the FINAL phase (p = 0.20,
Figure 39B). Also, participants increased the portion of the double support phase (p = 0.006)
during each step during the INIT phase (Figure 39C). Although participants reduced their double
support phase toward the end of the perturbation trials, the double support phase during the
FINAL phase was still higher than that during the unperturbed walking (p = 0.04).

Figure 39 Average (A) step length, (B) step width, and (C) double support phase for all
participants (N = 11) during unperturbed baseline walking (No PTB), initial phase, and final
phase of perturbed walking.
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Figure 40: Example gait cycle from one participant showing XCoM position, edge of base of
support (BoS) position, and MoS in AP (A) and ML (B) direction during unperturbed baseline
walking. Shaded area: double support phase from initial contact to the contralateral limb’s toe
off. Solid line: right step. Dashed line: left step. Note that we calculate FWD MoS and BWD MoS
at each foot strike by using the MoSToe of the leading limb and MoSHeel of the trailing limb,
respectively.
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We did not find any significant effects of Leg or the interaction between Leg and
Condition on the FWD MoS, BWD MoS, or ML MoS. There were significant effects of
Condition on ML MoS (F(2,20) = 14.35, p = 0.0001) and FWD MoS (F(2,20) = 3.57, p = 0.047)
but not BK MoS (F(2,20) = 1.5, p = 0.24) (Figure 41). ML MoS increased during the INIT
phase compared to unperturbed walking (p < 0.001) but decreased during the FINAL phase
compared to the INIT phase (p = 0.049). Although there was significant effect of Condition on
FWD MoS, FWD MoS was not significantly different across conditions after multiple
comparison corrections.

Figure 41: Average (A) forward MoS, (B) backward MoS, and (C) ML MoS for all participants
(N = 11) during unperturbed baseline walking (No PTB), initial phase, and final phase of
perturbed walking.

Whole-body angular momentum varied throughout the gait cycle and captured the body’s
rotational behavior of falling forward and then rotating backward. Whole-body angular
momentum in the sagittal plane was most negative during the transition from the swing phase to
the stance when the peak forward momentum occurs (Figure 42A). Then participants increased
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the angular momentum to positive until mid-swing and the angular momentum becomes negative
again during the late stance. The peak backward momentum occurred during the mid-swing
which corresponded to ~25% and ~75% of the gait cycle. Whole-body angular momentum in the
frontal plane was most positive near each right foot strike and then reached its peak negative
value near left foot strike (Figure 42B).
Participants increased their peak-to-peak angular momentum in both the sagittal plane
and ML plane during FINAL phase(Figure 42C, D). We found significant effects of Condition
on peak-to-peak whole-body angular momentum in the sagittal plane (F(2,20) = 7.99, p =
0.0028) and the ML plane (F(2,20) = 11.05, p = 0.0006). In the sagittal plane, the range of
whole-body angular momentum increased during the FINAL phase compared to both INIT (p =
0.046) and the unperturbed trial (p = 0.027). Similarly, in the frontal plane, the range of whole-
body angular momentum increased during the FINAL phase compared to both INIT (p = 0.002)
and the unperturbed baseline trial (p = 0.016).
214

Figure 42: The average angular momentum (L) across the gait cycle for one participant during
unperturbed walking (No PTB), initial phase of perturbed walking, and final phase. (A) Angular
momentum in the sagittal plane. (B) Angular momentum in the frontal plane. Average peak-to-
peak range of whole-body angular momentum in the sagittal plane (C) and frontal plane (D) for
all participants (N = 11) during No PTB, initial phase, and final phase of perturbed walking.

The objective of this study was to quantify how healthy people adapt their proactive
control strategies during repeated treadmill-induced trips or slips. We hypothesized that people
would use proactive strategies to change their spatiotemporal gait features to improve stability
during the initial phase of the repeated perturbation trials while these features would return to
that measured during the unperturbed walking trial toward the end of the perturbation trials.
During the initial perturbation phase, we found that people increased their step width, decreased
their step time, and increased the portion of double support time within the step. Contrary to our
hypothesis, we did not find changes in forward and backward MoS during perturbed trials
215
compared to unperturbed walking. Also, the range of whole-body angular momentum did not
reduce during the initial phase of perturbation trials. Instead, the peak-to-peak range of whole-
body angular momentum increased during the final phase of perturbation trials.

Our findings demonstrated that participants made proactive adjustments in
spatiotemporal gait features during walking in a destabilizing environment. Although the
treadmill-induced perturbations were in the anteroposterior direction, participants tended to
increase the step width to increase the lateral edge of BoS in preparation for the upcoming
perturbations. Decrease in step lengths and step time are in line with previous literatures that
applied continuous perturbations during walking (Hak et al., 2012). Additionally, participants
increased the portion of double-support phase within each step as the proactive adjustments
when anticipating for perturbations. Such a strategy could increase stability by increasing the
portion to time for both legs to shift the CoM position and increase the time for people to
modulate the trailing limb ankle moment to reduce the effects of perturbations (Reimann et al.,
2017).
Consistent with our hypothesis, changes in spatiotemporal features observed during the
early perturbation period returned toward baseline during the final phase of perturbed walking.
Previous studies that assessed proactive adjustment during prolonged perturbed walking did not
find significant differences in proactive adjustment during pre-perturbation steps such as step
width (Major et al., 2018), step length, or cadence compared with unperturbed walking
(Madehkhaksar et al., 2018). Our study has several differences. First, our study excluded the
perturbation and recovery steps from our analysis since our focus was on pre-perturbation
proactive adjustments. Second, we analyzed the initial perturbation phase and final phase of the
216
perturbed trials separately as we hypothesized that people may only make proactive adjustment
at the beginning of the perturbed trials before they fully adapted to the treadmill perturbations.
Instead, previous studies averaged the spatiotemporal features throughout the entire perturbed
walking trial (Madehkhaksar et al., 2018; Major et al., 2018).
Several factors could potentially explain the adaptation of proactive adjustments of
spatiotemporal features. Deviations from normal gait patterns can increase energetic cost or
metabolic cost of transport. For example, increasing the step widths and decreasing the step time
to increase cadence would both increase the metabolic cost of transport (Donelan et al., 2001).
After prolonged exposure to perturbations, people may prioritize reducing metabolic cost by
returning to the baseline gait patterns rather than choosing to walk with spatiotemporal gait
patterns that would heighten the metabolic cost but improve balance. Thus, our study may
indicate that people change their spatiotemporal gait features to prioritize stability mostly during
the initial phase of the perturbed walking.

Although treadmill-elicited perturbations were elicited in the anteroposterior direction at
random foot strikes, participants only increased their MoS in the mediolateral direction during
the initial phase of the perturbed trials. This may indicate that participants actively control
stability in the frontal plane regardless of the perturbation directions. Consistent with this idea,
our study demonstrated that neither forward nor backward MoS changed during either initial or
the final phase of the perturbed trials. There were two potential explanations for our observation.
Our results of MoS in the anteroposterior direction (the direction of perturbations) were
consistent with the previously proposed ‘constant offset control’ of MoS where people tend to
maintain a constant MoS in destabilizing environment (MacLellan and Patla, 2006; Rosenblatt
217
and Grabiner, 2010). Alternatively, since increasing MoS in one direction (for example, forward)
would decrease MoS in the other direction (for example, backward), participants may not
prioritize preparation for perturbations of any specific direction and thus leave the MoS in the
anteroposterior direction constant.
Changes in spatiotemporal features during walking can spontaneously lead to changes in
MoS. Based on the theoretical model, the MoS is proportional to the impulse required to move
the XCoM outside of the base of support so that a corrective step becomes necessary to maintain
balance. Increasing the MoS indicated that a larger perturbation was required to for participants
to use a stepping response to maintain balance and thus more stable. Hak et al. found that
increase cadence would increase ML MoS and shortening step length would increase backward
MoS (Hak et al., 2012). Also, decreasing single support time or increasing double support time
would decrease the time for the CoM to fall before foot placement to reduce the sway of CoM.
Mediolateral MoS would thus be increased by widening the steps and by increasing the double
support time (Buurke et al., 2019).

Whole-body angular momentum and the margin of stability can describe different aspects
of locomotion stability. While MoS describes a strategy that participants use in response to
perturbations (Wu et al., 2020), whole-body angular momentum can describe the whole-body
rotational dynamics throughout the stride cycle. Contrary to our hypothesis, the range of whole-
body angular momentum during the initial phase of perturbed trials did not change compared to
unperturbed walking but the range of whole-body angular momentum during the final phase of
the perturbations increased. A potential explanation for the discrepancy between our hypothesis
and the observed results is that people adapted their ground reaction force to prepare for the
218
perturbations. Alterations in the ground reaction force loading may contribute to the increased
peak-to-peak range of angular momentum at the final phase of the perturbed trial. Participants
may take more cautious steps by adopting flat-foot landing strategy to reduce the rate of reaction
force loading and peak braking and vertical ground reaction force during foot strike (Cham and
Redfern, 2002; Marigold and Patla, 2002; Shinya et al., 2016). Consequently, participants had to
reduce the overall propulsion impulse during the push-off phase as the braking and propulsion
impulse needs to be equal for the CoM position to be stationary on the treadmill (Marigold and
Patla, 2002). Although increased in range of peak-to-peak angular momentum may increase the
energetic cost as more effort was needed to reach the equilibrium point during walking, it is
possible that such increase in energetic cost may not be large enough to be detected by our
central nervous system so that we do not decrease the range of angular momentum after long
exposure to perturbations. Alternatively, control of whole-body angular momentum in a small
range may require more effort to regulate the body movement. Thus, as participants became
fatigued walking on the treadmill, they could not regulate whole-body angular momentum within
a small range as well as during the initial phase of walking. Future study should test whether
people increase their range of peak-to-peak angular momentum after prolonged unperturbed
walking.

One major limitation of the study is that although our study aimed to randomize the steps
in between each perturbation, the perturbation magnitudes, and directions to make the
perturbations unpredictable, participants may be able to figure out the minimal steps in between
each perturbation. In this case, participants may have learned the timing of the perturbation
consciously or subconsciously so that they would only need to make specific adjustments a few
219
steps before each perturbation. Another limitation is that our study used a split-belt treadmill and
we instructed participants to walk with one foot on one belt if possible, which may change
people’s natural gait patterns such as increasing step width.

To summarize, our study found that healthy young adults make proactive adjustments in
spatiotemporal gait patterns and MoS mostly during the initial phase of perturbed walking and
these adjustments tended to return to those measured during unperturbed walking over prolonged
exposure to treadmill perturbations. Our study may provide implications for clinical populations
with sensorimotor deficits such as amputees and people post-stroke who rely on proactive
control strategies to maintain locomotion stability. For example, people post-stroke usually
adopted gait patterns with widened steps and shorter step lengths. These changes in gait patterns
could be an adaptive change to cope with increased gait stability post-stroke and help them to
maintain balance despite the higher energetic cost associated with these changes in gait patterns.

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225

Falls are the leading cause of injury for people post-stroke. Internal factors such as
improper foot placement largely contribute to the increased risk of falls (Robinovitch et al.,
2013). People post-stroke experience almost double the rate of falls as non-disabled individuals
of similar age. Among people post-stroke, up to 70% of them fall at least once every year
(Sackley et al., 2008; Watanabe, 2005), and most of the falls are related to walking (Weerdesteyn
et al., 2008). The high fall risk reduces the ease and confidence for people post-stroke walk
safely in everyday life and may eventually result in decreased physical capacity.
Control of foot placement is one important stabilization strategy during walking and
balance control(Bruijn and Van Dieën, 2018). The foot placement variability observed during
walking may reflect how humans respond to internally generated perturbations as our central
nervous system integrates sensory information to adjust commands to muscles of the lower
extremity (Fitzpatrick et al., 1999; Maeda et al., 2017; O’Connor and Kuo, 2009). For example,
one simple way to recover from a forward loss of balance is to place the foot further ahead of the
body’s center of mass (CoM) so that the leading leg force has a greater fore-aft component to
arrest excessive body rotation and restore balance. Thus, modulating foot placement from step-
to-step is an important strategy for humans to maintain balance.
Step-to-step balance corrective strategies can be derived by relating foot placement to the
body’s state using a data-driven approach. This has been done for both non-disabled young
participants (Joshi and Srinivasan, 2019; Rankin et al., 2014; Roden-Reynolds et al., 2015; Wang
and Srinivasan, 2014) and populations who are at risk for falls such as older adults and people
226
post-stroke (Arvin et al., 2018; Dean and Kautz, 2015; Stimpson et al., 2019). Given a nominal
trajectory, the deviation of CoM state from this trajectory explains the deviation of the next foot
placement location through a linear mapping for non-disabled people during steady walking on
the treadmill (Wang and Srinivasan, 2014). These experimentally derived foot placement
mappings can explain ~80% of the variance in foot placement in the mediolateral direction and
~30% of the variance in the anteroposterior direction at the midstance in non-disabled
individuals (Wang and Srinivasan, 2014). The high predictive power, especially in the
mediolateral direction, may indicate that our central nervous system uses information about the
body’s state to actively control the next foot placement during unperturbed walking.
However, the foot placement mapping derived from populations at risk of falls differed
from that for non-disabled young adults (Arvin et al., 2018). For example, older adults showed a
reduced coefficient of determination in the foot placement mapping between the CoM state and
the subsequent foot placement compared with young adults, indicating older adults may have
less tight control their foot placement in the mediolateral direction (Arvin et al., 2018). In
addition, the coefficients of the foot placement mapping that relate the body’s state to foot
positions at each step were also different for people post-stroke from that for a group of age-
matched controls (Dean and Kautz, 2015). For people post-stroke at higher fall risk, they had a
weak association between their paretic mediolateral foot placement location and the step-to-step
changes in the swing leg velocity or swing leg muscle activity at the initiation of the swing phase
(Dean and Kautz, 2015). These results may suggest that people post-stroke may have a reduced
ability to make changes in the foot placement to correct for changes in the body’s state, which
would potentially contribute to high fall risk. However, it has yet to be determined whether
227
people post-stroke use different foot placement control strategies in fore-aft direction compared
to non-disabled individuals during walking.
The primary goal of this study was to determine the mapping between the body’s state
and foot placement in people with stroke and non-disabled age-matched adults during steady-
state walking in both anteroposterior and mediolateral directions. We compared the foot
placement mapping derived from unperturbed walking for people post-stroke and controls. We
hypothesized that compared with non-disabled age-matched adults, the coefficient of
determination of the mapping would be lower in people post-stroke due to their reduced ability
to adjust in foot placement. We also hypothesized that the coefficient of determination would be
lower when explaining variance in paretic foot placement than for the non-paretic limb. Lastly,
we hypothesized that the coefficient of determination would be positively correlated clinical
assessments of balance and motor impairment. Overall, these results may help us understand
how post-stroke motor impairments impact dynamic balance control strategies during walking.
These results may eventually help us identify the underlying mechanisms that contribute to
increased instability and heightened fall risk in people-stroke.

A total of 13 non-disabled age-matched individuals and 38 stroke participants were
recruited for this study (Participant characteristics were reported in Chapter 5). We based our
sample size calculation on a set of pilot data (n = 4) collected with non-disabled young adults
and (n=38) with stroke survivors. The sample size n = 18 was computed using GLIMMSE based
on the preliminary coefficient of determination of the derived foot placement mapping during
unperturbed walking of our pilot study with power at 0.8 and Type I error at 0.05. The observed
228
effect size and standard deviation of the difference in R
2
for this pilot data was 0.13 and 0.14,
respectively. Participants self-reported their dominant limb when asked which leg they would
use to kick a ball. The study was approved by the Institutional Review Board at the University of
Southern California, and all participants provided informed consent before participating. All
aspects of the study conformed to the principles described in the Declaration of Helsinki.

The complete protocol consisted of a set of clinical assessments and walking trials on the
treadmill. For stroke participants, before the walking trials, we evaluated motor impairment
using the lower extremity portion of the Fugl-Meyer Assessment (FM) (Fugl-Meyer et al., 1975),
static balance using Berg Balance Scale (BBS) (Berg et al., 1992), static and dynamic balance
during locomotion tasks using the Functional Gait Assessment (FGA)(Leddy et al., 2011), and
over-ground walking speed using the 10-meter walking test. Participants also completed
questionnaires about balance confidence using the Activity-Specific Balance Confidence Scale
(ABC) (Powell and Myers, 1995). Higher scores on all these assessments indicated better
balance control or higher balance confidence. After completing the clinical evaluations, we
instructed stroke participants to walk on the dual-belt treadmill (Bertec, Columbus, OH, USA). A
harness was provided to prevent the participants from falling but no body weight support was
provided. First, the participants walked on the treadmill to familiarize themselves with the
experimental set-up. We used the following method to identify participants’ preferred walking
speed on a treadmill (Park et al. In Review). We started from 70% of the speed obtained from a
10-meter walking test and adjusted their walking speed by 0.05m/s increments or decrements
until the participants verbally indicated that they achieved their preferred walking speed. For the
next trial, the participants walked for three minutes at their self-selected speed.
229
Control participants also completed a set of clinical assessments including ABC, FES,
BBS, and over-ground 10-meter walking test. We instructed the participants to walk at matched
speeds with a stroke participant of similar age.

We used the same data acquisition methods as in Chapter 7.

Data processing was same as in Chapter 7.
For stroke participants, we could not assume the mapping to be symmetric between left
and right side due to their gait asymmetries. Thus, we computed a different Jacobian matrix
(J
NP
) for non-paretic foot placement and vice versa (J
P
) (Eqn.1). Such relationship could be
approximated at each time interval ( τ) of the step cycle.
Δ 𝛥𝛥 𝑇𝑇 𝑘𝑘 + 1
= �
𝑱𝑱 𝑵𝑵𝑵𝑵
ΔS
𝑘𝑘 𝑇𝑇
𝑱𝑱 𝑵𝑵 ΔS
𝑘𝑘 𝑇𝑇
(1)
Here, k is the step number. J is a 2x4 Jacobian matrix that can be estimated by the least
square fit. ΔQ
k + 1
= Δ[Foot
A P
, Foot
ML
]
𝑘𝑘 + 1
𝑇𝑇 and ΔS
k
= Δ � 𝑋𝑋𝑋𝑋 𝑋𝑋 𝐴𝐴𝐴𝐴
( 𝜏𝜏 ), 𝑋𝑋𝑋𝑋 𝑋𝑋 𝐶𝐶𝐿𝐿 ( 𝜏𝜏 ), 𝑋𝑋 𝑋𝑋𝑋𝑋 ̇ 𝐴𝐴𝐴𝐴
( 𝜏𝜏 ), 𝑋𝑋 𝑋𝑋𝑋𝑋 ̇ 𝐶𝐶𝐿𝐿 ( 𝜏𝜏 ) �
𝑘𝑘 𝑇𝑇 . Chapter
7 includes more details about how the mapping was derived.

We developed mixed-effects models for control participants and stroke participants,
respectively, to account for the individual variability. We included a random slope for each
predictor in the models. In these models, the independent variables included the deviation of
CoM state from the nominal trajectory, and the dependent variables included the deviation of
foot placement from the nominal foot placement. We calculated the coefficient of determination,
𝑅𝑅 2
, for each participant to measure the goodness of fit. We used an unequal variance t-test to
230
compare the coefficients in the Jacobian matrix and coefficient of determination for age-matched
controls and people post-stroke to test whether people post-stroke and age-matched people used
different foot placement mapping during unperturbed walking at midstance. We chose midstance
as the mapping between foot placement and CoM state was well characterized at this time point
in previous studies and there was sufficient time remained to allow for changes in foot placement
by the swing limb (Joshi and Srinivasan, 2019; Kim and Collins, 2017; Wang and Srinivasan,
2014). We also used a linear regression model to assess the relationship between the coefficient
of determination computed for the relationship between foot placement and CoM state and motor
impairment and balance metrics for people post-stroke. Since this coefficient of determination
may be associated with self-selected walking speed, we included self-selected walking speed as a
covariate in the linear model to account for this confounding factor, which allowed us to assess
the association between the foot placement controller and balance independent of walking speed.
Significance level is set at p<0.05.

The average coefficient of determination R
2
across all participants increased as
progressing during the step cycle for people post-stroke and control participants in both
mediolateral and fore-aft direction (Figure 43A). In the mediolateral direction, coefficient of
determination was similar for paretic and non-paretic foot placement and similar for people post-
stroke and non-disabled participants. In contrast, coefficient of determination was consistently
higher for non-paretic foot placement (Figure 43A, Blue) than paretic foot placement (Figure
43A, Red) in the anteroposterior direction. Coefficient of determination was also higher for non-
paretic foot placement than non-disabled participants (Figure 43A, Black). Coefficient of
determination at midstance in the anteroposterior direction was higher during paretic step
231
compared to non-paretic step (t(37) = 4.9, p < 0.001) and higher than control participants (t(49) =
2.15, p = 0.037, Figure 43B).
We also found that the coefficients of the foot placement mapping derived at midstance
for people post-stroke walking was different from control participants. Control participants
generally had larger regression coefficients for the velocity-related predictors compared to stroke
participants. In the mediolateral direction, coefficients for ΔCoM
̇ M L
of control participants were
higher than stroke participant (Non-paretic foot placement v. Control: t(49) = -11.8 , p < 0.001 ;
Paretic foot placement v. Control: t(49) = -14.0, p < 0.001). In the anteroposterior direction, both
coefficients for ΔCoM
̇ M L
(Non-paretic foot placement v. Control: t(49) = 6.8, p < 0.001; Paretic
foot placement v. Control: t(49) = 6.7, p < 0.001) and for ΔCoM
AP
̇ (Non-paretic foot placement
v. Control: t(49) = -11.8 , p < 0.001 ; Paretic foot placement v. Control: t(49) = -14.0, p < 0.001)
were higher in control participants than stroke participants for both paretic and non-paretic foot
placement. Additionally, the coefficient for ΔCoM
AP
was higher during non-paretic foot
placement for stroke participants compared to that during paretic foot placement (t(49) = 5.6, p <
0.001) and control participants (t(49) = 3.2, p = 0.002, Figure 43C) at midstance (See
Supplementary Figure 1 for the average coefficients across step cycle for all stroke participants
at paretic foot placement and non-paretic foot placement).
232

Figure 43: The coefficient of determination and coefficients from the foot placement mapping for
control participants and stroke participants during both paretic and non-paretic steps. (A)
Fraction of foot placement variance (R
2
) explained by the CoM state throughout the step cycle
during unperturbed walking for control participants (Gray), post-stroke participants during non-
paretic steps (Red) and paretic steps (Blue) in mediolateral and anteroposterior direction.
Dashed vertical line: Midstance of the gait cycle. Darker lines: mean value across participants.
Shaded area: standard deviation across participants in each group; (B) Fraction of foot
placement variance explained by CoM state at midstance. (C) Coefficients of the foot placement
mapping in the mediolateral direction and anteroposterior direction. Bar height: mean values
across participants. Error bars: standard deviation (*p<0.05).

We used a linear model to assess whether there is an association between the coefficient
of determination and clinical assessment scores for people post-stroke when self-selected
walking speed was included as a covariate. We found that the coefficient of determination

of the
foot placement mapping during the non-paretic step in the mediolateral direction was positively
233
associated with scores on Functional Gait Assessments (FGA) (P = .01). We did not find any
other association between coefficient of determination

of the foot placement mapping and any of
other clinical assessment scores in the mediolateral direction or the anteroposterior direction at
midstance.
Table 7 Statistical results of the association between coefficient of determination

of the foot
placement mapping and the clinical assessment scores for people post-stroke
BBS FM FGA
Paretic step
AP
P = .91 P = .74 P = .19
ML P = .17 P = .88 P = .14
Non-paretic step AP P = .08 P = .65 P = .38
ML P = .15 P = .78 P = .01

Our study’s primary objective was to derive and compare foot placement mappings from
unperturbed walking for both people post-stroke and control participants. This foot placement
mapping at midstance had higher coefficient of determination in the anteroposterior direction at
non-paretic foot placement than paretic foot placement. The coefficient of determination was
also higher at non-paretic foot placement than that of control participants while there was no
difference in coefficient of determination in the mediolateral direction. We also found that foot
placement strategies in people post-stroke were less sensitive to deviations in CoM velocity for
both the paretic and non-paretic side and in both the anteroposterior and mediolateral direction
compared to non-disabled participants. These results indicated that people post-stroke adopt a
different foot placement control strategy in response to the variation in CoM state, especially
when using the paretic limb. Our study may potentially provide a mechanistic understanding of
the cause of increased gait instability in people post-stroke.
234
The coefficient of determination explaining the step to step deviation in foot placement
using the deviation in CoM state during unperturbed walking was higher using the non-paretic
limb for the next foot placement than using the paretic limb in the fore-aft direction. This result
is consistent with a previous study that reports higher coefficient of determination using the non-
paretic limb foot placement than the paretic limb in the mediolateral direction (Stimpson et al.,
2019). These results suggest that people post-stroke may have impairments in the ability to
modulate the paretic foot placement due to impairments such as reduced hip flexor activity
(Rybar et al., 2014) or the abnormal joint torque coupling (Cruz and Dhaher, 2008; Finley et al.,
2008; Sánchez et al.; Sulzer et al., 2010).
However, contrary to our hypothesis, the coefficient of determination of the derived foot
placement mapping at midstance in non-disabled participants was lower than for the non-paretic
limb of post-stroke participants in the anteroposterior direction. Therefore, higher coefficient of
determination

using the non-paretic limb foot placement could emerge from the need to correct
for the higher deviation of CoM state at the paretic stance phase than at the non-paretic stance
phase. Using corrective response that requires adjusting non-paretic foot placement may become
particularly important to avoid falls when the deviation of CoM state during paretic stance phase
was higher. On the contrary, there is less urgency to modulate foot placement to actively correct
for deviation in CoM during the non-paretic stance phase in the anteroposterior direction.
Moreover, the regression coefficients that we derived differed for people post-stroke and
non-disabled, age-matched participants. Particularly, the magnitudes of velocity-related
regression coefficients were much smaller for stroke participants than non-disabled participants,
although both populations walked with similar speed. Such a discrepancy in the sensitivity of
foot placement to CoM velocity between groups may be attributed to the disrupted sensory
235
feedback post-stroke (Tyson et al., 2008). Future studies should examine whether the
insensitivity to deviations in CoM velocity may contribute to the increased risk of falls post-
stroke.
Although the current analysis explored the natural variability in foot placement, this
analysis may not extend to capture how people control foot placement in response to
perturbations such as slips or trips. In Chapter 7, we found that the foot placement mapping
during unperturbed walking could not extend to explain the behavior during perturbation
responses in non-disabled individuals. Therefore, future studies should focus on deriving the foot
placement control strategies during perturbation responses for people post-stroke, which would
provide more insight into the individual response to perturbations and may potentially provide a
better understanding of how people post-stroke may use the information of the body’s state to
make corrective responses given their sensorimotor deficits.

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Supplementary Figure 1: Sensitivity analysis for Jacobian coefficients during perturbed, self-
selected walking for stroke participants (N = 38) during step cycle (Red: Non-paretic step; Blue:
Paretic step). Dark line: mean value across participants. Shaded area: standard deviation of
Jacobian coefficients across participants.

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Asset Metadata

Creator Liu, Chang (author)

Core Title Understanding reactive balance control strategies in non-disabled and post-stroke gait

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Biomedical Engineering

Degree Conferral Date 2021-08

Publication Date 07/28/2021

Defense Date 04/26/2021

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

Tag Balance,gait,OAI-PMH Harvest,reactive control,stroke

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Finley, James M. (committee chair), Leech, Kristan (committee member), McNitt-Gray, Jill L. (committee member), Valero-Cuevas, Francisco (committee member)

Creator Email liu.maggie.chang@gmail.com,liuchang@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-oUC15657561

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