Transients, Variability, Stability and Energy in Human Locomotion

Dissertation

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

Nidhi Seethapathi, B.S., M.S.

Graduate Program in Mechanical Engineering

The Ohio State University

2018

Dissertation Committee:

Dr. Manoj Srinivasan, Advisor Dr. Alison Sheets-Singer Dr. Giorgio Rizzoni Dr. Martin Golubitsky Dr. Rama K. Yedavalli c Copyright by

Nidhi Seethapathi

2018 Abstract

Most research in human locomotion is limited to steady-state, constant speed and symmetric locomotion behaviors. However, walking and running in everyday life requires us to adapt our locomotion strategies to intrinsic noise-like transients, external environmental irregularities and more general practical demands on moving from place-to-place. Here, we investigate such locomotion behaviors in three broadly related studies: (I) the dynamics of walking with changing speeds, (II) control strategies for running stably in the presence of noise-like deviations, and (III) the dynamics of asymmetric walking.

In part I, we measured the metabolic cost of walking with changing speeds by having subjects speed up and slow down on a constant speed treadmill. We ﬁnd that the metabolic cost of changing speeds when walking is signiﬁcant and can constitute

8 to 20% of our daily walking energy budget. We explain the incremental cost with simple mathematical models, correlating the cost with increased mechanical work.

Moreover, this increased metabolic cost predicts lower preferred walking speeds of humans walking short distances, which we conﬁrmed by further experiments on non- amputee and amputee subjects.

In part II, we reveal the control strategies hidden within the step-to-step variability in steady-state human running data, adopted by humans in order to run stably in the presence of unavoidable and intrinsically generated sensorimotor errors. The

ii stabilizing control is largely implemented when the leg is on the ground and is well- predicted by deviations in the center-of-mass states during the previous ﬂight phase.

We show that humans use almost-deadbeat control of horizontal velocity deviations; killing about 70-100% of horizontal velocity deviations within one step by appropri- ately modulating the placement of the foot and the force applied on the ground by the leg. Further, deviations in the center-of-mass motion predicts the swing foot placement before the swing foot itself. On implementing these strategies on two simple computational biped models, we ﬁnd that the models can withstand deviations up to ten times larger than the step-to-step variability from which they were inferred.

This suggests that the control strategies humans use for small intrinsic errors can be extended to deal with larger external perturbations.

In part III, we predict the steady-state adaptation behaviors in humans constrained to walk asymmetrically. The constraints are in the form of asymmetric masses and asymmetric belt speeds. We conduct metabolic energy optimization of simple biped models, making changes in the dynamics of the model corresponding to each type of asymmetry. We ﬁnd that the optimization-based predictive models agree with trends in the stance time asymmetry observed in experiments for the asymmetric mass and asymmetric belt speed conditions. For split-belt walking, the optimization-based model predicts that humans will adapt to taking longer steps on the fast belt and shorter steps on the slow belt. This qualitative prediction made by the model matches with behavior that has been observed by other researchers over many days of adaptation to the new paradigm.

iii This dissertation is dedicated to the memory of my Bombay Paati (maternal grandmother), who would have surely become a scientist given her curious mind,

diligent work ethic and eye for analytical detail. If only she had been given a

fraction of the opportunities and freedom I have.

iv Acknowledgments

I am immensely grateful to Manoj Srinivasan for his strong intellectual mentor- ship and his kind support of all my pursuits. He has been extremely patient, non- judgemental and encouraging whilst my attention and interests happily meandered across a range of research topics. I greatly cherish our many sciencey conversations, which have been a crucial ingredient in driving me to pursue meaningful and sound science. Secondly, I would like to thank Schlumberger Foundation Faculty for the

Future for giving me access to a forum of passionate and awe-inspiring female scien- tists all over the world. I would also like to thank both the Schlumberger Foundation and NSF for their ﬁnancial support throughout my PhD. I thank Dr. Anil Jain and

SDMH hospital in Jaipur, India for being generous with their time and for the fruitful collaboration in obtaining data on amputees in India. My research experience during graduate school has greatly benefited from attending a number scientific conferences and meetings and I thank everyone who made these meetings stimulating. A special mention for the CoSMo 2017 summer school organizers and attendees; this experience greatly helped me clarify my future scientific directions. Also, I would like to thank the vibrant biomechanics and neuroscience Twitterverse for keeping me clued into the latest research news. I would like to thank all the members of my doctoral committee for their support, helpful comments and for serving on the committee.

v I would like to thank my friends and labmates for the pleasant times we spent together. I thank Bharat Hegde for being a warm and caring friend through the ups and downs of graduate school. I thank Karthika Shanmugam and Roopa Comandur for the fun travels together. I thank the Association for India’s Development Colum- bus Chapter for giving me the opportunity to volunteer and make an impact, however small, outside of my research. I thank Gopal Iyer for being a caring and supportive cousin brother and wish him much success on his own academic ventures.

Finally, I thank my parents Lakshmi Iyer and Seethapathi Vaidyanathan for their undying support and for believing in me ﬁercely. Thanks Appa for always making me laugh and having a few points of wisdom for every occasion. Thanks Amma for always being my loudest cheerleader and for giving me the freedom to pursue whatever I choose.

vi Vita

2012 ...... B.S. Mechanical Engineering Veermata Jijabai Technological Insti- tute 2012-present ...... Doctoral Fellow in Mechanical Engi- neering, The Ohio State University.

Publications

Research Publications

N. Seethapathi and M. Srinivasan “The metabolic cost of changing walking speeds is signiﬁcant, implies lower optimal speeds for shorter distances, and increases daily energy estimates.” Biology letters, 11.9 (2015): 20150486.

Fields of Study

Major Field: Mechanical Engineering

vii Table of Contents

Page

Abstract ...... ii

Dedication ...... iv

Acknowledgments ...... v

Vita ...... vii

List of Figures ...... xiii

1. Introduction ...... 1

1.1 Evidence for energy optimality ...... 2 1.2 Predictive models of human locomotion ...... 3 1.2.1 Simple models of human locomotion ...... 3 1.2.2 Complex models of human locomotion ...... 3 1.2.3 Objective functions for optimization-based predictions . . . 5 1.3 Acceleration and deceleration in walking ...... 5 1.4 Stability and control of running ...... 6 1.5 Steady-state adaptations during asymmetric walking ...... 8 1.5.1 Experimental work on asymmetric walking ...... 9 1.5.2 Asymmetric walking on a split-belt treadmill ...... 10 1.5.3 Modeling changes speciﬁc to asymmetric gait ...... 11 1.5.4 Application to rehabilitation and prostheses ...... 12 1.6 Overground walking rehabilitation measures for asymmetric walking 13 1.7 Objectives ...... 14

2. The Metabolic Cost of Changing Speeds and its Eﬀect on Preferred Walk- ing Speeds ...... 18

2.1 Summary ...... 18

viii 2.2 Materials and methods ...... 19 2.2.1 Experiment: Metabolic cost of oscillating-speed walking . . 19 2.2.2 Experiment: Preferred walking speed ...... 20 2.2.3 Model: Kinetic energy ﬂuctuations ...... 20 2.2.4 Model: Inverted pendulum walking ...... 23 2.3 Results ...... 23 2.3.1 Metabolic rate of oscillating-speed walking ...... 23 2.3.2 Model predictions ...... 26 2.3.3 Daily energy budget for starting and stopping ...... 27 2.3.4 Optimal and preferred walking speeds are lower for shorter distances ...... 27 2.3.5 Daily energy budget for starting and stopping ...... 29 2.3.6 Comparison with a previous study ...... 30 2.4 Discussion ...... 30

3. Preferred Walking Speeds for Unilateral Amputees Wearing A Passive Prosthetic Leg ...... 33

3.1 Summary ...... 33 3.2 Experimental methods ...... 35 3.2.1 Walking for short distances ...... 35 3.2.2 Walking in circles ...... 37 3.3 Computational methods ...... 37 3.4 Results ...... 38 3.4.1 Preferred walking speed for above-knee unilateral amputees depends on distance walked ...... 38 3.4.2 The rate of decreases of preferred walking speed with distance walked is lower for unilateral above-knee amputees compared to non-amputees ...... 39 3.4.3 Increased metabolic cost of constant speed walking predicts ﬂatter speed-distance relationship for above-knee amputees . 39 3.4.4 Distance-dependence of preferred walking speeds for below- knee unilateral amputees is inconclusive ...... 42 3.4.5 Preferred walking speeds for unilateral amputees walking in circles depends on the circle radii ...... 42 3.4.6 Preferred walking speeds for unilateral amputees walking in circles is independent of turning direction ...... 45 3.4.7 Preferred walking speeds are proportional to degree of amputation ...... 45 3.5 Discussion ...... 45

ix 4. Control Strategies Inferred from Variability in Human Running . . . . . 50

4.1 Summary ...... 50 4.2 Materials and methods ...... 51 4.2.1 Notation and terminology ...... 51 4.2.2 Calculating input state variables during flight apex . . . . . 52 4.2.3 Calculating the output control variables during stance . . . 52 4.2.4 Linear regressions between the outputs and the inputs . . . 54 4.3 Results ...... 54 4.3.1 Center of mass variability is phase-dependent and direction- ally uncoupled ...... 55 4.3.2 Apex-to-apex maps show fast decay of perturbations and stable periodic motion ...... 55 4.3.3 Ground reaction force variability is needed to control center of mass motion ...... 58 4.3.4 Ground reaction impulses can be controlled independently in different directions ...... 58 4.3.5 Force impulses correct velocity deviations in about about one step ...... 59 4.3.6 Horizontal impulses are modulated by changing forces, not duration ...... 60 4.3.7 Within-step vertical impulse modulations correct the vertical position deviations ...... 61 4.3.8 GRF control is phase-dependent ...... 63 4.3.9 Step in the direction of the deviation ...... 64 4.3.10 Landing leg length is changed in response to vertical deviations 66 4.3.11 Horizontal forces are modulated by force direction and vertical by force magnitude ...... 66 4.3.12 Leg force is mostly along leg direction ...... 67 4.3.13 Is the GRF component modulated by changing leg force direction or magnitude? ...... 67 4.3.14 Approximate left-right symmetry and possible asymmetry in the control ...... 68 4.3.15 Swing foot control ...... 68 4.3.16 Station-keeping and speed do not affect control ...... 70 4.3.17 Continuous control is no better than apex-based control . . 70 4.3.18 Attitude control: GRFs and foot placement do not directly respond to trunk tilts ...... 71 4.3.19 Foot center of pressure modulation has systematic modulation, but is not a big contributor to center of mass control . 71 4.4 Discussion ...... 72

x 5. Experimentally Inferred Running Controllers Implemented on Simple Biped Models ...... 76

5.1 Summary ...... 76 5.2 Implementing the data-derived control on simple biped models . . . 76 5.3 Human-derived controller stabilizes a running model ...... 81

6. Predicting Steady-state Motion Adaptations for Asymmetric Walking . . 92

6.1 Summary ...... 92 6.2 Experimental methods ...... 93 6.2.1 Experiment design ...... 93 6.2.2 Equipment and data-collection ...... 94 6.3 Computational methods ...... 94 6.3.1 Optimization of a simple model ...... 94 6.3.2 Modiﬁcations for modeling the asymmetry ...... 96 6.3.3 Formulation of the optimization problem ...... 97 6.3.4 Optimization convergence and uniqueness ...... 98 6.3.5 Comparison with the simple model ...... 98 6.4 Experimental observations on steady-state walking with asymmetric weights ...... 99 6.5 Simple model predictions for walking with asymmetric leg masses and lengths ...... 100 6.6 Simple model predictions for split-belt walking ...... 102 6.6.1 Optimal stance time asymmetry for split-belt walking predicts the observed trends ...... 102 6.6.2 Optimal step length asymmetry for split-belt walking predicts the observed trends ...... 104 6.6.3 The model predicts that observed metabolic cost could go lower with more time for adaptation ...... 104 6.6.4 Longer step length on fast belt is accompanied by lesser net work done on the fast belt ...... 106 6.7 Discussion ...... 109

7. Conclusions ...... 112

Appendices 116

A. Optimization-based Predictive Model of Walking with Changing Speeds . 116

xi A.1 Deriving step-to-step transition work for changing speeds ...... 116 A.2 Optimal multi-step inverted pendulum gaits satisfying experimental protocol ...... 120

Bibliography ...... 123

xii List of Figures

Figure Page

1.1 Point-mass model with telescoping legs. The ﬁgures shows a point-mass telescoping leg model of walking with a ﬁxed massless stance leg and a swing leg with a point mass at the foot. a) Model for symmetric walking. b) Model for walking with asymmetric leg masses. c) Model for walking with asymmetric leg lengths...... 4

2.1 Experimental setup and kinetic energy model for treadmill walking with changing speeds. a) Subject walking with oscillating speeds on a constant speed treadmill, walking faster and slower than the belt, moving between two prescribed positions. A longitudinal bungee cord (never to be made taut) constraints the rear-most position and a bungee cord perpendicular to sagittal plane (not shown, never to be touched) constrains the forward-most position. b) Sacral marker fore-aft velocities, original and smoothed...... 21

2.2 Experiment and model to study preferred short-distance walking speeds. a) Measuring preferred walking speeds as a function of bout distance D; subjects start and stop at rest. b) An idealized bout:

human travels the whole distance D at single speed vopt, starting and stopping instantaneously...... 22

2.3 Optimization of simple biped with step-to-step transition. a) A nine-step periodic inverted pendulum walking motion, with five steps faster and four steps slower than average. Initial and final stance-leg direction are shown for each step (red and blue). b) Details of one step-to-step transition is shown; downward velocity at the end of one step is redirected by push-off and heel-strike impulses...... 24

xiii 2.4 Metabolic cost of changing speeds is significant. Difference ˙ ∆Eexpt between oscillating-speed and constant speed walking metabolic rates for six oscillating-speed trials (P1 to P6): three (Tfwd,Tbck) combinations and two average speeds. Box-plot shows median (red bar), 25-75th percentile (box), and 10-90th percentile (whiskers); p-values use one-sided t-tests for the alternative hypothesis that metabolic rate differences are from a distribution with greater-than-zero mean. . . . 25

2.5 Model predictions of energy increase. a) The experimentally ˙ measured increase in metabolic energy cost ∆Eexpt compared with a) ˙ ˙ kinetic energy model ∆Eke and b) inverted pendulum model ∆Eip; we show experimental and model means (black ﬁlled circles), the best- ﬁt line (red, solid), and all subjects’ trials (scatter plot, gray dots); no scatter plot for inverted pendulum model as it produces only one prediction per trial...... 26

2.6 Preferred walking speed decreases with distance walked. Distance- dependence of the model-based energy-optimal walking speeds (blue, solid) and experimentally measured preferred speeds (red and black error bars). Ranges of model-based energy-optimal speeds within 1% (blue line, thin) and 2% (blue band) of optimal energy cost are shown. We show whole bout “average” speeds (red) and “steady-state” speeds over middle 1.42 m (black, thick), indistinguishable from over middle 0.75 m (gray, thin). Average preferred speeds for 0.5-14 m trials were signiﬁcantly lower than that for the 89 m trial (paired t-test, p < 0.01); similarly, the “steady-state” speeds for 2-6 m were signiﬁcantly lower than that for 89 m (paired t-test, p < 0.04)...... 28

3.1 Overground walking experiment setup. We measured the preferred walking speed of walking for unilateral amputees wearing a passive prosthetic leg in two conditions: a) walking a range of short distances, starting and stopping each bout at rest and b) walking in circles of diﬀerent radii, both clockwise and anti-clockwise...... 36

3.2 Decrease in preferred walking speed with distance walked for above-knee amputees. a) Above-knee amputees showed a greater decrease in preferred walking speed for short distances speed compared to able-bodied individuals. The error bars represent the standard deviations. b) The rate of change in preferred walking speed with distance is steeper for able-bodied individuals than for the unilateral amputees. 40

xiv 3.3 Minimization of total metabolic cost predicts short-distance walking speed. The total cost of the walking a short distance includes a term due to constant speed cost and a changing-speed cost. We ﬁnd that minimizing this total cost predicts the observed trends in changing preferred walking speed with distance for both amputees and non-amputees...... 41

3.4 Decrease in preferred walking speed with distance walked for below-knee amputees. The below-knee amputees also show a significant decrease in preferred walking speed for short distances compared to able-bodied individuals. However, the change in speed with distance does not show a consistent trend like that for above-knee amputees. . 43

3.5 Preferred walking speeds for circle walking. The preferred walking speed for all the unilateral amputees showed a decrease with radius of the circle walked. The unilateral amputees did not show a signiﬁcant diﬀerence in preferred walking speed when walking with the prosthesis- leg a) inside and b) outside the circle...... 44

4.1 Idealized running control. A schematic of a running trajectory, perturbed sideways during ﬂight. The runner can recover back to steady- state by changing the ground reaction force — say, by altering the foot placement and the leg force magnitude...... 53

4.2 Step-to-step variability during running. a) Variability in the center of mass velocity at ﬂight apex. b) Variability in the fore-aft and sideways impulses due to the GRFs. c) Variability in foot position relative to torso at the beginning of stance phase. In panels a, b and c each dot corresponds to a separate step and the scatter plot is for all subjects with each subject’s mean value subtracted, so that only variability about the mean is shown for 500 randomly-chosen steps. Red dots denote right steps and blue dots denote left steps. d) Mean GRFs in three directions (blue line) and one standard deviation around the mean (yellow band) for a right stance phase; left stance phase GRFs is similar in fore-aft and vertical directions, but the sideways GRF is negative of that for the right stance. Green text indicates standard deviation values in all panels. GRFs are in fraction of body weight. . 56

xv 4.3 Variability in states as a function of phase. Standard deviations of a) sideways CoM velocityx ˙, b) fore-aft CoM velocityy ˙, and c) vertical position z. The standard deviation of the sideways and fore-aft velocities are minimum near mid-stance and maximum during flight. The vertical position standard deviation is maximum during flight and minimum during stance, a little before mid-stance. We show how much of this standard deviation is explained by the CoM state alone or CoM and swing foot state at the immediately preceding flight apex. . . . . 57

4.4 Vertical position control by differential impulse control. Us- ing a unimodal vertical GRF, we find that the way to lower vertical position over a step is to move the peak force to the right. This is equivalent to increasing the vertical impulse on the second half of the step and decreasing the vertical impulse over the first half. Conversely, to increase the vertical position over a step, we find that the peak force needs to be moved to the left...... 62

4.5 Phase-dependent control of GRFs. We show how the GRF components respond to perturbations at the previous ﬂight apex, as estimated by our phase-dependent GRF model. a) Sideways GRF response to a (rightward) sideways velocity perturbation. b) Fore-aft GRF response to a forward velocity perturbation. In both cases, the change in GRF from nominal (shown by the arrow) is obtained as a product of the perturbation size and the sensitivity of the GRF to the perturbation. 65

4.6 Swing foot control before foot placement. The fraction of a) sideways and b) fore-aft foot placement variance at beginning of stance predicted by the center of mass (CoM) state or swing foot state during the previous one step (ﬂight and stance). The solid and dashed lines represent right and left foot placements respectively...... 69

5.1 Minimal mathematical biped models used for implementing control. Two simple biped models were simulated: a telescoping leg model with direct force control and a kneed biped with activation control of the muscle at the knee...... 78

xvi 5.2 Details of the model implementations. a) The muscle in the second model is a classic Hill-type muscle [1], composed of an active contractile element, a series elastic element (tendon), and a parallel

elastic element. The force FCE in the active contractile element for the muscle model depends on the muscle length `m through a force-length ˙ relationship ψ`, on the muscle length rate `m through a force-velocity relation ψv, and the activation a, so that FCE = aFisoψvψ`, where Fiso is the maximum isometric force in the muscle [1]. b) The control input for both models is represented as a fourier combination of two −1 −1 sine waves of frequencies Tstance and 2Tstance. For the ﬁrst model, the force is represented by this combination and for the second model, the muscle activation...... 80

5.3 Comparing the simple biped models and human running data. Both models ﬁt the experimental GRFs and feedback gains reasonably well, despite not having been made to match them explicitly: a) Mean GRFs over stance in three directions. GRFs are reported as a fraction of body weight. b) Phase-dependent feedback gains describing the sensitivity of the sideways GRF to sideways velocity perturbation, fore- aft GRF to a fore-aft velocity perturbation, and the vertical GRF to vertical position perturbation at the previous ﬂight apex. For standard deviations of the experimental curves, see Figures 4.2 and 4.5. . . . . 83

5.4 Stability of controlled simple biped runner in response to a forward or vertical push. We illustrate the stability of the running model by showing how large perturbations at ﬂight apex decay. a) Decay of a forward velocity perturbation. b) Sagittal view of running, recovering from an upward position perturbation. On the ﬁrst step, the leg touch-down angle is steeper than the touch-down angle during unperturbed running and the contact time is shorter. All quantities are non-dimensional...... 84

5.5 Stability of controlled simple biped runner in response to a sideways push. a) Decay of a sideways velocity perturbation. b) Decay of a sideways velocity perturbation. c) Top view center of mass trajectory, showing the unperturbed running motion and a running motion that recovers from a rightward velocity perturbation. The rightward perturbation elicits a rightward foot placement, compared to the nominal foot placement during unperturbed running. All quantities are non-dimensional...... 85

xvii 5.6 Basins of attraction. The basin of attraction for steady running: that is, the set of all perturbations at ﬂight apex from which the runner can recover back to steady-state. Each of the basins correspond to recovery or non-recovery from perturbations added at the ﬂight apex before a right stance. The basin for perturbations at an apex preced-

ing a left stance is identical, except for being mirror image for ∆x ˙ a. The three panels in the ﬁgure show basins of attraction for combined perturbations in: a) sideways velocity - forward velocity, b) sideways velocity - vertical position and d) forward velocity - vertical position. 87

5.7 Work loop and net leg work. Work loops for the muscle-driven biped model without perturbations and with perturbations in the a) fore-aft velocity, b) sideways velocity, and c) vertical position. The work loop plots force versus leg length and the signed area included in it is the net work performed by the leg...... 88

5.8 Running stably with noise. Multiple steps of the biped model running in the presence of noisy foot placement and muscle activations (green and pink). Periodic nominal motion in the absence of noise (black)...... 90

5.9 Model-generated state variability at flight. Deviations in flight apex state, GRF impulse, and foot placement from nominal for a 1000 steps (500 left, 500 right), showing behavior analogous to Figure 4.2. This variability is not explicitly specified, but instead emerges from the interaction between the motor noise and the controlled dynamics. 91

6.1 Experimental setup and sample experimental data. a) Lab setup with motion capture cameras from Vicon and an instrumented Bertec split-belt treadmill. b) Sample ground reaction force captured by the force plates in the treadmill. c) Mass was added to the leg using ankle weights. d) Sample capture from post-processed marker data with seven body segments and four markers on each segment. . . 95

6.2 Simple model predictions for walking with asymmetric leg mass. This ﬁgure shows the results from optimizing the simple model for mass asymmetries. a) When a mass is added to the swing leg, the stance time for the other leg increases quadratically. b) Preferred walking speed decreases gradually as the mass of either leg is increased. 101

xviii 6.3 Simple model predictions for walking with asymmetric leg length. This ﬁgure shows the results from optimizing the simple model for leg length asymmetries. When the length of one leg is increased, the longer leg shortens to the eﬀective length of the shorter leg. . . . 102

6.4 Optimization of a simple model predicts stance time asymmetry for split-belt walking. The slow belt speed for the model is held constant at 0.5 m/s and the fast belt speed is swept through the speeds shown on the x-axis in the ﬁgure. We ﬁnd that the stance time is shorter on the fast belt and longer on the slow belt. This prediction is in agreement with past experimental observations...... 103

6.5 Optimization of a simple model predicts step length asymmetry. The optimal gait of the simple model predicts that people will take longer steps on the fast belt and shorter steps on the slow belt. . 105

6.6 Optimal metabolic cost of walking on a split belt treadmill. The optimal cost predicted by the model for walking on a slow-fast split-belt paradigm is in between the optimal cost for the corresponding slow-slow and fast-fast case and lower than the mean-mean condition. 107

6.7 Net leg work prediction for walking on a split-belt treadmill. The model predicts that the net leg work is lesser for the leg on the fast belt compared to that on the slow belt. This prediction is similar to recent experimental observations made by [2]...... 108

6.8 Complex multibody models of human walking with muscles. The ﬁgure shows a more complex 7-link model of walking and the ways in which it could be modiﬁed to model asymmetric gait. a) This model has sixteen muscles on both legs. The muscles are shown only on one leg, the other leg has identical muscles. Asymmetries are introduced in this model by asymmetrically changing b) muscle properties, c) adding masses or d) varying segment lengths...... 111

xix A.1 Step-to-step transition to change speed. a) The walking motion is assumed to be inverted pendulum-like with the transitions from one inverted pendulum step to the next accomplished using push-off and heel-strike impulses. Overlaid is the ‘hodograph’ (a depiction of velocity changes) during the step-to-step transition, when push-off happens entirely before heel-strike. b) Details of the velocity changes during step-to-step transition, with push-off before heel-strike and slowing down. c) Analogous to panel-c, except the walking speeds up during the transition. d) Velocity changes and impulses when the heel-strike precedes push-off entirely...... 118

xx Chapter 1: Introduction

Although there has been a lot of research on analyzing various aspects of human locomotion, most such work focuses on constant speed, steady-state and symmetric movements. However, in real life, people are constantly changing their movement dynamics and their control strategies in response to intrinsic noise-like deviations, external perturbations and other constraints on movement. Moreover, people with walking disorders often walk asymmetrically and studies that are limited to symmetric walking are not representative of their daily lives. Here we try to analyze and understand the strategies and behaviors that people use to adapt their locomotion in the presence of such realistic demands. To understand such adaptive locomotion behavior, we infer control strategies from noise-like perturbations in steady-state movement and implement them on simple biped models to make predictions about transient responses to larger perturbations.

An important goal of biomechanics and motor control research is not just to analyze but also to predict human locomotion in various distinct situations. If we build such broadly predictive models, we could then use them for testing and improving proposed rehabilitation therapies, physiotherapeutic measures and prosthetic limbs.

However, most current models only predict very simple movements without making

1 signiﬁcant changes to the model and few models can predict adaptive locomotion behaviors of the type we study here.

Metabolic energy optimization has been shown to predict aspects of steady-state walking behaviors in habitual and even in some novel tasks. In some other tasks, people find that stability is prioritized over energy optimality. There could be other competing costs like minimizing joint pain and minimizing error as well, in addition to minimizing energy and being stable. Here, we use optimization of the metabolic cost of movement to predict human locomotion behavior in different situations. In the rest of this chapter, we first discuss this key concept of energy optimality further.

We then provide introductory background and describe prior work on each of the broadly related topics being discussed in this thesis: (I) the mechanics of changing speeds and walking short distances, (II) the stability and control of steady human running despite noise-like deviations, and (III) energy optimality while walking with asymmetries. Given that this detailed introductory material is provided in this ﬁrst chapter, each individual chapter simply begins with a summary and directly transitions into the Methods and Results. We ask the reader to refer back to this chapter for this background material.

1.1 Evidence for energy optimality

There is considerable experimental and modeling-based evidence for the predictive ability of metabolic cost for human walking. Experiments have shown that people walk at speeds [3], step lengths [4], step widths [5] and step frequencies [6] that roughly minimize metabolic energy. Additionally, various simple models [7, 8, 9] and a few complex models [10, 11] have been able to predict some aspects of human

2 locomotion using energy optimality. Here, to predict multiple walking phenomena, we will use the metabolic cost of walking as an initial guess for the objective function being minimized.

1.2 Predictive models of human locomotion

1.2.1 Simple models of human locomotion

Qualitative aspects of symmetric human locomotion have been predicted by fairly simple models. A point mass model with a telescoping leg (Figure 1.1), when opti- mized for energy consumption per unit distance, predicted that walking is optimal at low speeds and running is optimal at high speeds, as is seen in people [7]. Here, we use simple models to study both walking and running. We distinguish a walk from a run, typically, by the presence or absence of a ﬂight phase. If a ﬂight phase is present, we conclude that the model, and by extension, the person, is running. Adding slight complexity to the model in the form of a knee, muscles, a spring etc. leads to more detailed qualitative predictions for the ground reaction forces and the center of mass trajectory [9]. Simple models of human walking can be a good starting point to explore the hypothesis of energy optimality and to determine the modeling details necessary to capture important features of the movement, resulting in meaningful and accurate predictions. Here, in all the chapters detailed below, we use such simple models to make novel predictions about qualitative aspects of human locomotion.

1.2.2 Complex models of human locomotion

Quantitative and detailed predictions of kinematics, ground reaction forces, muscle forces and activations may warrant more complex models. Indeed, models of varying degrees of complexity have been used to predict aspects of human locomotion. Some

3 a) Symmetric b) Asymmetric leg mass c) Asymmetric leg length

stance leg Swing added length Telescoping leg leg

Stance leg swing foot swing foot mass added mass

Figure 1.1: Point-mass model with telescoping legs. The ﬁgures shows a point- mass telescoping leg model of walking with a ﬁxed massless stance leg and a swing leg with a point mass at the foot. a) Model for symmetric walking. b) Model for walking with asymmetric leg masses. c) Model for walking with asymmetric leg lengths.

models simulate walking for the whole musculoskeletal complex [10], others assume key segments of the body as rigid links actuated by just the most signiﬁcant muscles

[11], and still others assume joint torques instead of muscle forces as the source of actuation [12]. Often these models use nonlinear constrained optimization to make predictions. Symmetric walking at different speeds, stance phase knee flexion [11], walking in low gravity [13] and walking with osteoarthritis [14] are just some walking tasks that have been explored by such detailed models. However, none of the models have been able to make qualitative or quantitative predictions for more than one walking task without substantially changing the modeling details or without at- tempting to track experimental data in some capacity [15, 16]. Here, we focus only on simple model-based qualitative predictions for different locomotion tasks and do not use such complex models. However, the broad applicability of our simple models to

4 diﬀerent types of locomotion can be extended to such complex models in the pursuit of quantitative predictions.

1.2.3 Objective functions for optimization-based predictions

Metabolic cost has been modeled in a number of ways such as, the work done by the muscles taking into account muscle eﬃciencies [17, 18] or as a sum of some positive exponent of individual muscle forces, stresses, activations or joint torques

[16, 11, 12]. Other plausible candidate objective functions are fatigue, pain, robustness to perturbations and robustness to uncertainty about the body states. For instance, a very high power of muscle activation has been used to model fatigue and has been shown be able to predict stance phase knee ﬂexion [11]. Here, we limit our study of the role of metabolic energy cost and stability in predicting movement behavior.

1.3 Acceleration and deceleration in walking

Walking in daily human life involves and requires changing speeds. Human activity measurements suggest that a majority of the walking over a day is performed in short bouts [19], starting and ending at rest. In Chapter 2, to better understand such naturalistic behavior, we measure the metabolic cost of changing walking speeds.

Although there is extensive literature on the energy cost of steady-state (constant speed) walking e.g., [20, 21], the cost of changing walking speeds has not been measured without non-inertial treadmill speed changes or step-frequency control (e.g.,

[22], as discussed later). Measuring the metabolic cost of changing speeds lets us quantify the contributions of acceleration-deceleration costs versus ‘constant speed’ costs to daily human walking energy expenditure. The cost of changing speeds may also have signiﬁcant behavioral implications. For instance, humans seem to move in

5 a manner that minimizes metabolic energy costs, preferring nearly energetically optimal speeds [21]; the cost of changing speeds, if signiﬁcant, may lower such optimal walking speeds. Here, we show that the cost of changing speeds is (statistically) sig- niﬁcant and an appreciable fraction of typical energy budget. Further, in Chapter 2 and 3, we predict (using a mathematical model) that the metabolic cost of changing speeds reduces the optimal walking speeds for short distances for both amputee and non-amputee individuals; we then measured and found that these individuals indeed prefer lower speeds for shorter distances.

1.4 Stability and control of running

The models described so far are normative and physics-based. As a complement to such physics-based analyses, we also considered data-derived modeling in this thesis, speciﬁcally of stability and control in human running.

Constant speed human running is not exactly periodic. Body motion during running varies from step to step. This step-to-step variability could be due to muscle force noise, sensory noise, or external perturbations [23, 24, 25]. To run without falling, the body’s ‘running controller’ must prevent these perturbations from growing too large. We do not yet have a complete characterization of this controller. That is, we do not yet understand how humans run without falling down. In Chapters 4 and

5, we derive a controller from variability in human running data and implement the controlled on simple biped models of running.

Many mathematical models have been proposed to describe running control, but these models are not derived completely from experimental data on how humans control running. Most such models simply assume that the human leg behaves like a

6 linear spring or variants thereof [26, 27, 28, 29, 30, 31]. However, it is clear from simple arguments that this linear spring assumption cannot, fundamentally, be consistent with how running is controlled [32]. Consider a large push to the runner, so that the runner’s kinetic energy is increased or decreased. If the runner had to recover her balance and return back to normal running, she must have some mechanism to change her mechanical energy back to normal. A perfectly springy leg cannot eﬀect such changes in energy, because perfect springs conserve energy. Legs must perform positive and negative work to actively stabilize typical deviations from steady-state running. Another issue with spring-mass-like models are based on ﬁtting to the average center of mass motion during running [26, 33, 28], not deviations from the average. But understanding human running stability requires understanding how deviations from the average running motion are corrected and controlled.

One way of characterizing the running controller is to apply perturbations during running and examine how the body recovers from the perturbations [34, 35, 36]. In- stead of such external perturbations, here, we use the naturally occurring step-to-step variability [37, 32] to characterize the controller. Previous attempts at examining such variability for controller information focused only on walking [38, 37] or assumed only variants of the spring-mass model [32], which we demonstrate to be an unnecessary assumption. These attempts did not focus on a mechanistic understanding of how running is stabilized. Here, we obtain a mechanistic understanding of human running stability by characterizing how the leg forces and foot placement are modulated in response to deviations from normal running motion.

A human-derived controller such as the one proposed here could inform monitoring devices to quantify running stability or fall likelihood [39]. Also, implementing such

7 controllers into robotic prostheses and exoskeletons [40, 41] will allow the human body to interact more ‘naturally’ with the device, rather than having to compensate for an unnatural controller. Numerous running robots have demonstrated stable periodic running, using a variety of control schemes [42, 43, 44]. But these robots fall short of human performance and versatility. Understanding human running may lead to better running robots.

1.5 Steady-state adaptations during asymmetric walking

Asymmetric walking is commonly seen in people with injuries, movement disorders, or asymmetric limb amputations. Often, such asymmetry is viewed as unde- sirable and rehabilitation and prosthesis-design efforts aim to decrease asymmetry in human walking. However, there is not sufficient evidence that such rehabilitation efforts are effective in the long term. An experimentally-verified predictive model of human locomotion, for both symmetric and asymmetric walking, may inform such rehabilitation efforts. While energy optimality during human locomotion has been used as a predictive hypothesis [10, 12, 11, 45], this optimization hypothesis has not yet been demonstrated to predict a broad range of walking tasks without modifying the objective function. Our objective is to use optimization to predict aspects of symmetric as well as asymmetric walking with a single model and compare to experiment.

Walking around in the world might seem like a highly pre-programmed activity that humans perform without much conscious modiﬁcation. However, our body and our environment are constantly challenging us to modify various aspects of our walking like speed, step length, step time, forces etc. These challenges may be internal in

8 the form of sensorimotor and neural noise [38, 37, 23] or external like uneven terrain

[46] and obstacles from our non-uniform environment [47, 48]. If these challenges are

ones we have faced before, our body may have a fast reaction to it and very quickly

modify the gait to meet the challenge. However, when our movements are faced with

a novel challenge, we explore our options and settle on a particular solution on a

longer timescale [49, 50]. With repeated exposure to this novel challenge, perhaps,

our reaction time decreases and we will have a go-to stereotypical response to it. Un-

derstanding the mechanisms behind such adapted walking behaviors is important to

understand human locomotion in able-bodied as well as individuals with movement

pathologies. Here, we aim to build a predictive model of the adaptations in walking

behavior that people exhibit when subject to novel asymmetric constraints.

1.5.1 Experimental work on asymmetric walking

Multiple studies have found signiﬁcant spatio-temporal asymmetries in the loco-

motion of individuals with movement disorders and asymmetric amputations [51, 52,

53]. In these studies, gait parameters like stance time, kinematics, ground reaction

forces, leg lengths and leg range of motions were experimentally measured. A sym-

metry index is typically used to quantify asymmetry of one or more of the above

gait parameters [54]. If a gait parameter θ is measured on the left and right leg, a symmetry index SI is typically deﬁned as

θ − θ SI = L R (1.1) θL + θR

. Such asymmetries have also been simulated in healthy subject populations in a

number of ways: asymmetrically loaded ankle weights [55], artiﬁcially lengthening

one leg [56], diﬀerent belt speeds on a split-belt treadmill [57] and adding a knee

9 brace [58] to one leg. Here, we aim to simulate such able-bodied asymmetric gaits, so as to make predictions about gait for both the symmetric and asymmetric cases.

1.5.2 Asymmetric walking on a split-belt treadmill

Most walking and running movements in able-bodied individuals are roughly symmetric [59]. Given this, when people are constrained to walk in an asymmetric way, it becomes a novel movement that they need to learn or adapt to. There is a sizeable body of research studying how people adapt their walking on a split-belt treadmill

[57, 60, 61]. A split-belt treadmill has two belts, one for each leg, and the speed of each belt can be controlled independent of the other. Past research has looked at the biomechanical parameters and muscle activations during the adaptation and de-adaptation phases of asymmetric walking on a split-belt treadmill [62]. The adaptation phase typically refers to subjects learning to walk in the asymmetric-belt conditions. In contrast, the de-adaptation phase refers to the storage of some learning eﬀects that need to be unlearned when subjects walk on symmetric belts after having walked asymmetrically.

For left-right speed differences ranging from 0.5 to 1.5 m/s, people adapt to a new steady-state movement within 10 to 20 stride cycles [57]. Many studies find consistent trends in the adaptation of various biomechanical parameters such as stance time %, total stride time, step length, joint angles and muscle EMGs [57]. They find that the stance time of the ‘fast leg’ (leg constrained on faster belt) is a smaller fraction of the total stride time compared to the symmetric case; the swing time is correspondingly a bigger fraction of the stride time compared to the symmetric case. Similarly, the stance time of the ‘slow leg’ (leg constrained on the slower belt) is a larger fraction

10 of the stride time and its swing time a smaller fraction, compared to the symmetric case. There is an increased activity of the agonist muscle (gastrocnemius) on the fast leg and a decreased activity of the antagonist muscle (tibialis anterior) on the slow leg. There is, however, no change in the activity of the antagonist muscles on the fast leg and the agonist muscles on the slow leg.

There has been research theorizing the neural (spinal or cerebral) origins of the adaptation behaviors seen for humans and animals walking on a split-belt treadmill

[63]. Initially, most theories of human adaption to split-belt walking pertained to theories of spinal coordination because past work in spinal cats were able to show similar adaptation behaviors. It has been suggested that proprioceptive feedback from the fast leg is primarily used for modulating the muscle activations. And that the slow leg is controlled by the spine, after processing the proprioceptive input from the fast side. It was found that spinal stretch reflexes probably do not play a role in such inter-limb coordination because numbing the afferent neurons (type Ia) necessary for stretch-reflex did not change muscle response for an inter-limb perturbation task when standing. Moreover, the spinal stretch reflex is thought to be suppressed during gait. Later work in [57] investigates the role of supra-spinal and cerebral processes in split-belt locomotor adaptation behaviors.

1.5.3 Modeling changes speciﬁc to asymmetric gait

Models of human locomotion are hybrid, with a diﬀerent governing diﬀerential equation depending on the phase of locomotion; the stance, swing and double support phase are modeled distinctly. Often, the work done or the metabolic cost associated with each of these phases are also modeled distinctly [17, 64, 65]. Depending on the

11 way that the asymmetry is introduced, it will affect one or more of the phases of walking. Swing phase of locomotion has been shown to account for 10 to 30% of human walking cost and to directly depend on the mass of the leg [65, 66, 67]. So when mass is added to one leg, the swing cost model becomes significant in predicting the asymmetric locomotion. Similarly, asymmetry in leg length has been shown to increase joint loading for the shorter leg during the stance phase [56], making the stance phase modeling a more significant predictor of walking with unequal leg lengths. Here, we will explore asymmetries that affect different aspects of walking, so as to isolate the effect on different terms in the objective function.

1.5.4 Application to rehabilitation and prostheses

An accurate model of asymmetric human locomotion can help better understand and rehabilitate pathological asymmetric gait. The most common method in analyzing pathological gait is inverse dynamics, where muscle forces are inferred from the observed motion and ground reaction forces [68, 69, 70]. However, such inverse dynamics-based methods cannot be used to predict the results of a rehabilitation technique or the performance of a prosthetic leg because they rely on experiments.

Some ad hoc interventions have attempted to decrease the amount of asymmetry in joint loading in patients with osteoarthritis with the help of a split belt treadmill or some form of inverse dynamics-based biofeedback [51, 57, 60]. While these studies have induced temporary symmetry in their subjects, their inherent asymmetries appear to resurface over time [61]. Such temporal learning eﬀects on symmetry and asymmetry can be predicted using optimization [49]. Here, we attempt to build an experimentally-veriﬁed model of human locomotion which can make such predictions.

12 1.6 Overground walking rehabilitation measures for asymmetric walking

Preferred overground walking speed is a commonly used metric to test a patient’s improvement after being ﬁt with a new prosthetic leg or after undergoing physiotherapy [71, 72]. Often, the preferred speed used to gauge patient progress is measured for walking in a straight line over a relatively short distance [73]. In the past, we have shown that even for non-amputee subjects, the preferred walking speed for walking in a straight line is distance-dependent and can be explained by the cost of speeding up and slowing down [74]. The higher the cost of speeding up and slowing down, the more likely it is to aﬀect the preferred walking speed at short distances. Since it is known that amputees walking with passive prosthetic legs have a higher metabolic cost [75] compared to non-amputees, it would not be surprising if they had a higher cost of speeding up and slowing down as well. This would imply that their preferred walking speeds over short distances would be even less representative than is the case for non-amputee people. Also, it may mean that it is not appropriate to compare the walking speed of an amputee over a short distance to that of a non- amputee person over the same short distance. Here, we measure the walking speeds of amputees wearing prosthetic legs over a range of short to long distances. Using simple optimization-based biped models of walking, we also estimate how their cost of speeding up and slowing down may compare to that of non-amputee individuals.

13 be higher than that for non-amputee people. So, it may be meaningful to use circle- walking performance as an additional measure of the eﬀectiveness of a prosthetic leg or a physiotherapeutic intervention. Although such curved walking interventions have been used for other disabled populations [77, 78], there less literature on the subject for amputee-walking. Here we measure the walking behavior of unilateral amputees wearing the Jaipur Foot passive prosthetic leg as they walk in circles of diﬀerent radii.

In the past, metabolic energy optimality has been used to make predictions for a number of aspects of overground and treadmill walking behaviors for non-amputee people [79, 21, 5]. However, much less work has been done towards making such predictions for a disabled population [49, 80]. It could be that for such a population a consideration like stability may take precedence over metabolic energy optimality.

Here, we use simple models to make predictions about overground walking behavior and compare it to experiments to see how much is explained by energy optimality alone.

The experiments and models in this study may be used to develop the most relevant tests for evaluating prosthetic function in an amputee population. Speciﬁcally, this study may have implications for the distance and paths for subjects to walk during the prosthesis-ﬁtting to reveal insights into their needs for the device to easily useable.

1.7 Objectives

Part I: Energetics of walking with changing speeds

Hypothesis Ia is that the metabolic cost of changing speeds is signiﬁcant. Hy- pothesis Ib is that the metabolic cost of changing speeds can be explained by the

14 change in kinetic energy and the cost of transitioning from one step to the next.

Hypothesis Ic is that the metabolic cost of changing speeds will predict aspects of walking behavior in humans where this cost dominates. We test the above hypotheses by carrying out the following tasks:

1. Measure the metabolic cost of changing speeds. By using expired gas

VO2 measurements, we measure the metabolic cost of changing speeds in human

walking. We constrain the velocity ﬂuctuations and acceleration-deceleration by

asking subjects to walk back and forth on a constant speed treadmill.

2. Compare the metabolic cost of changing speeds to change in kinetic

energy. We measure the kinetic energy ﬂuctuations as the walks with changing

speeds and compare this change in kinetic energy to the observed metabolic cost.

3. Predict the metabolic cost of changing speeds using optimization of

a simple inverted pendulum model. We conduct a multistep optimization

of a simple inverted pendulum model with step-to-step transitions for pushoﬀ

and heelstrike work.

4. Measure and predict preferred walking speeds over short distances.

Measure the preferred walking speed over a range of short distances, where

acceleration-deceleration is a signiﬁcant portion of the total movement bout.

Compare the measured behavior to predictions based on the observed and mod-

eled metabolic cost of changing speeds.

15 Part II: Stability and control from variability in running

Hypothesis IIa is that the step-to-step variability in human running contains information about running stability and control. Hypothesis IIb is that the devia-

tions in the center of mass motion predict the control (deviations) of the the leg during

the stance phase. Hypothesis IIc is that control inferred from experiment will be

able to stabilize a simple model of a biped runner. We test the above hypotheses by

carrying out the following tasks:

1. Inferring a running controller from variability. Conduct experiments to

measure variability in body states and ground reaction forces during constant

speed running. Derive linear models between variability in input state and

output control variables.

2. Predictions with a simple biped model. Implement the inferred controller

on a simple biped model and observe its responses to perturbations.

Part III: Energy optimization to predict asymmetric walking

Hypothesis IIIa is that qualitative aspects of symmetric and asymmetric walking

can be predicted using the optimization of an energy-like objective function with a

simple model. Hypothesis IIIb is that qualitative aspects of overground asymmetric

walking behavior in able-bodied and amputee individuals can be predicted using the

energy-optimization. We test the above hypotheses by carrying out the following

tasks:

1. Experiments on walking with asymmetric masses. Perform experiments

on subjects walking with asymmetric weights and record ground reaction force

16 and motion data. For split-belt walking, we compare model predictions to data

already collected in previous studies.

2. Predictions with a simple biped model. Perform numerical optimization

with a simple point-mass-telescoping-leg model of a human with asymmetric

mass and leg speed conditions to predict optimal motion and walking speed that

minimizes metabolic energy. Then, we compare the simple model predictions

for asymmetric walking to trends in the corresponding experiments.

17 Chapter 2: The Metabolic Cost of Changing Speeds and its Eﬀect on Preferred Walking Speeds

2.1 Summary

Humans do not walk at exactly constant speed, except perhaps on a treadmill.

Normal walking entails starting, stopping, and changing speeds. In this chapter, we measure the metabolic energy cost of human locomotion when changing walking speed. Subjects walked with oscillating speeds on a constant speed treadmill, alternating between walking faster and slower than the treadmill belt by moving back and forth in the laboratory frame. Metabolic rate of walking with changing speeds was statistically significantly higher than walking at constant speed. The metabolic cost increase was correlated with kinetic energy changes while ignoring within-step velocity fluctuations. Extrapolating our empirical measurements to typical daily walking- bout distribution data, we estimate that the cost of changing speeds may account for about 10% of daily walking energy budget. Further, because of the cost of starting and stopping, we predicted that the optimal walking speed would be lower for shorter distances. We measured human preferred walking speeds for different walking distances, obtaining lower walking speeds for shorter distances.

18 2.2 Materials and methods

2.2.1 Experiment: Metabolic cost of oscillating-speed walking

Subjects (N=16, 12 males and 4 females, 23.25 ± 2.1 years, height 177.08 ± 7.4 cm, mass 75.99 ± 12.94 kg, mean ± s.d.) performed both “steady” (constant speed) and “oscillating-speed” walking trials. Oscillating walking speeds were achieved on a constant speed treadmill by alternately walking faster and slower than the belt

(Figure 2.1a). Two distinct audible tones of durations Tfwd and Tbck alternated in a loop indicating whether the subjects should move towards the treadmill front or rear.

We used three (Tfwd,Tbck) combinations, (1.9, 1.9), (2.8, 2.8), and (1.9, 2.8) seconds, obtaining different speed fluctuations. We instructed subjects to walk between fixed positions on the treadmill (0.48 m apart, Figure 2.1a) giving mean excursion length

L = 0.41 ± 0.08 m. The subjects obeyed the imposed back-and-forth time period constraints: average periods diﬀered from prescribed periods by 0.97% ± 0.24%.

While humans do not usually walk with oscillating speeds, this protocol was designed to isolate the cost of changing speed.

Oscillating-speed trials were at one or both constant treadmill speeds 1.12 m/s and 1.56 m/s (equal to average speeds): 10 subjects at both speeds, 4 subjects at 1.12 m/s only and 2 at 1.56 m/s only, with random speed order. Steady walking trials were performed at speeds ranging from 0.89 to 1.78 m/s, including 1.12 and 1.56 m/s.

Metabolic rate per unit mass (W/kg) was estimated using respirometry (Oxycon ˙ ˙ ˙ ˙ ˙ Mobile), approximated as E = 16.58 VO2 + 4.51 VCO2 (V in ml/kg/s), denoted Esteady ˙ and Eosc for steady and oscillating-speed trials respectively. Trials lasted 7 min: 4 min to reach metabolic steady-state and 3 min to estimate average metabolic rate.

19 The speed oscillation periods (3.8-5.6 s) are much smaller than typical metabolic

time-constants (30 s), so our metabolic steady-state is nominally constant. A sacral

marker’s motion was measured with marker-based motion capture.

2.2.2 Experiment: Preferred walking speed

Subjects (N=10) were asked to walk ten distances (D = 0.5, 1, 2, 4, 6, 8, 10, 12, 14, 89

m) at a comfortable speed, starting and ending at rest (Figure 2.2a). These subjects

were diﬀerent from those that performed the changing speeds experiment above with

metabolic energy measurements. We had three trials per distance, all trials in random

order, but performed 0.5-1 m trials separately.

2.2.3 Model: Kinetic energy ﬂuctuations

In this model, we attribute the metabolic cost increase for oscillating-speed walk-

ing over steady walking to fore-aft kinetic energy ﬂuctuations beyond what happens

within each step in constant speed walking. Figure 2.1b shows fore-aft velocity vx(t)

of the sacral marker for oscillating-speed and steady walking, approximating center of

mass motion. Smoothing vx(t) with an averaging window equal to step period gives v¯x(t), removing within-step speed ﬂuctuations. The mass-normalized metabolic cost increase for oscillating-speed walking over steady walking due to the kinetic energy

2 2 −1 −1 ﬂuctuations for each cycle is modeled as ∆Eke = (¯vmax − v¯min)(ηpos + ηneg ) /2,

wherev ¯max andv ¯min are maximum and minimum smoothed fore-aft speeds for that

cycle and ηpos = 0.25 and ηneg = 1.2 are typical positive and negative work eﬃcien- ˙ cies [81]. Model-predicted metabolic rate increase ∆Eke for each oscillating-speed

trial was the median ∆Eke/Tperiod over all cycles.

20 a) Subjects walk with changing speeds, moving between these positions in lab frame

Treadmill belt at constant speed Bungee cord prescribes maximum excursion

Treadmill Treadmill rear front

b) Sample experimental velocity !uctuations

Oscillating-speed walking Steady walking Moving Moving to rear to front Natural within-step speed !uctuations in 2.5 normal walking 2

fore-aft 1.5 speed (m/s) 1 t t 01 2 30 1 2 3 Un-smoothed Smoothed Treadmill speed

Figure 2.1: Experimental setup and kinetic energy model for treadmill walking with changing speeds. a) Subject walking with oscillating speeds on a constant speed treadmill, walking faster and slower than the belt, moving between two prescribed positions. A longitudinal bungee cord (never to be made taut) constraints the rear-most position and a bungee cord perpendicular to sagittal plane (not shown, never to be touched) constrains the forward-most position. b) Sacral marker fore-aft velocities, original and smoothed. 21 a) Preferred walking speeds for a nite distance

Bout distance D

START STOP

b) Idealized walking bout

speed walk at constant speed v opt

START time t STOP instantaneous instantaneous acceleration deceleration

to v opt to rest

Figure 2.2: Experiment and model to study preferred short-distance walking speeds. a) Measuring preferred walking speeds as a function of bout distance D; subjects start and stop at rest. b) An idealized bout: human travels the whole distance D at single speed vopt, starting and stopping instantaneously.

22 2.2.4 Model: Inverted pendulum walking

We consider inverted pendulum walking of a point-mass biped, for which the total

walking metabolic cost is the sum of: (1) a step-to-step transition cost (described

below), and (2) a leg-swing cost [82]. Using numerical optimization, we found the

multi-step-periodic inverted pendulum walking motion (e.g., Figure 2.3a), satisfying

our oscillating-speed experimental constraints and minimizing this metabolic cost.

Here, we derived and used expressions for step-to-step transition cost for non-constant

speed walking, generalizing previous constant speed expressions. This step-to-step

cost accounts for the push-oﬀ and heel-strike work redirecting the center of mass

velocity during the step-to-step transition (Figure 2.3b) and depend on leg-angles ˙ and center of mass velocities. The model prediction ∆Eip is the diﬀerence between

the optimal oscillating-speed and constant speed costs at the same average speed.

A metabolic cost term proportional to the integral of leg forces contributed almost

equally to oscillating-speed and constant speed walking costs and did not contribute

to their diﬀerence. See Appendix A for details.

2.3 Results

2.3.1 Metabolic rate of oscillating-speed walking

Metabolic rates for all six oscillating-speed trials (P1 to P6, Figure 2.4) were sig-

niﬁcantly higher than the corresponding steady-state costs. Metabolic rate increment ˙ over constant speed walking ∆Eexpt was signiﬁcantly greater than zero for all trials

(one-sample t-test, all p-values < 2×10−3). Oscillating-speed trials with higher speed

ﬂuctuations had higher metabolic rates with one exception (P1>P3>P2, P4>P6 and

P4>P5, all p < 0.02).

23 a) Inverted pendulum walking, alternating fast and slow

One complete period Center FastSlow Fast (again) of Mass trajectory

Push-o! Heel-strike

b) Mechanics of the step-to-step transition

vafter Heel-strike impulse

vbefore Push-o! vafter : velocity impulse just after heel-strike

vbefore : velocity Trailing Leading just before leg leg push-o!

Figure 2.3: Optimization of simple biped with step-to-step transition. a) A nine-step periodic inverted pendulum walking motion, with five steps faster and four steps slower than average. Initial and final stance-leg direction are shown for each step (red and blue). b) Details of one step-to-step transition is shown; downward velocity at the end of one step is redirected by push-off and heel-strike impulses.

24 Incremental cost of unsteady walking over steady walking for 6 unsteady trials: P1 to P6

Average Speed Average Speed

vavg = 1.12 m/s vavg = 1.56 m/s

Prescribed (T fwd , T bck ) in seconds 1.2 P1 P5 (1.9,1.9) (2.8,2.8) P6 (1.9,2.8) P3 P2 (1.9,2.8) (W/kg) 0.6 (2.8,2.8) expt DE

P4 (1.9,1.9) 0

p-values = 1x10 -4 3x10-3 1x10-4 1x10 -6 1x10 -3 4x10 -5

Figure 2.4: Metabolic cost of changing speeds is significant. Difference ˙ ∆Eexpt between oscillating-speed and constant speed walking metabolic rates for six oscillating-speed trials (P1 to P6): three (Tfwd,Tbck) combinations and two average speeds. Box-plot shows median (red bar), 25-75th percentile (box), and 10-90th percentile (whiskers); p-values use one-sided t-tests for the alternative hypothesis that metabolic rate differences are from a distribution with greater-than-zero mean.

25 a) Kinetic energy model b) Inverted pendulum model

1.5 1.0 0.8 1.0 0.6 slope slope 0.79 0.67 0.4 0.5 0.2 0 0 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1.0

−0.4

Figure 2.5: Model predictions of energy increase. a) The experimentally mea- ˙ sured increase in metabolic energy cost ∆Eexpt compared with a) kinetic energy model ˙ ˙ ∆Eke and b) inverted pendulum model ∆Eip; we show experimental and model means (black ﬁlled circles), the best-ﬁt line (red, solid), and all subjects’ trials (scatter plot, gray dots); no scatter plot for inverted pendulum model as it produces only one prediction per trial.

2.3.2 Model predictions

˙ Both the kinetic energy ﬂuctuation model ∆Eke and the inverted pendulum model ˙ ˙ ∆Eip were correlated with measured metabolic rate increments ∆Eexpt (Figure 2.5). ˙ The kinetic energy model and experimental costs are best-ﬁt by the line: ∆Eexpt =

˙ 2 λke (∆Eke)−0.04, with λke = 0.67 whether we use trial means (R = 0.96, 95% C.I. of

2 λke = ±0.19) or all data (R = 0.24, 95% C.I. of λke = ±0.28). Similarly, the inverted ˙ ˙ pendulum model and experimental costs are best-ﬁt by ∆Eexpt = λip (∆Eip) + 0.05

2 with λip = 0.79 (R = 0.88, 95% C.I. of λip = ±0.40).

26 2.3.3 Daily energy budget for starting and stopping

Humans mostly walk in short bouts [19]. For simplicity, we idealize a bout of

distance D and average speed v as instantaneously accelerating from rest to speed v, walking at constant speed v, and stopping instantaneously at time D/v (Figure 2.2b).

The total metabolic energy per unit mass Ebout(D, v) for this idealized bout has two

components: (1) a starting and stopping cost, extrapolating from the kinetic energy

−1 −1 2 model, λke(ηpos + ηneg)v /2, and (2) a cost for steady-state walking at speed v, given

˙ 2 −1 2 by Esteady = a + bv with a = 2.22 W/kg and b = 1.15 W/kg/(ms ) [20], so that

−1 −1 2 2 Ebout = λke(ηpos + ηneg)v /2 + (a + bv )D/v. Applying this model to data in [19] with p vopt = a/b = 1.39 m/s and the 95% C.I of λke suggests that starting-stopping costs are 4-8% of daily walking energy expenditure (Appendix A); this cost fraction (4-8%) may apply primarily to the subject population of [19], adults working in oﬃces, but could be estimated for other populations given their walking behavior.

2.3.4 Optimal and preferred walking speeds are lower for shorter distances

For the idealized bout of distance D (Figure 2.6), the energy-optimal walking speed vopt that minimizes Ebout(D, v) is given by the implicit function:

3 −1 −1 2 λkevopt ηneg + ηpos /(a − bvopt) = D. (2.1)

This metabolically-optimal speed increases with distance D, approaching vopt = pa/b for large distances (Figure 2.6).

As predicted by the distance-dependence of optimal walking speeds, preferred hu-

man walking speeds in our experiment, both “average” and “steady-state” speeds,

increased with distance (Figure 2.6). “Average” preferred speed is the average over

27 Preferred walking speeds (expt.) versus energy-optimal walking speeds (model)

Speeds, m/s 2 Long- Experiment (preferred speeds) distance “Steady state” Average walking 1.6 walking speed speed over whole distance 1.2 Theoretical Predictions (optimal speeds) 0.8

Optimal walking speed 0.4 Within 1% of optimum energy 0 Within 2% of optimum energy 0 2 4 6 8 10 12 14 89 Distance walked (m)

Figure 2.6: Preferred walking speed decreases with distance walked. Distance-dependence of the model-based energy-optimal walking speeds (blue, solid) and experimentally measured preferred speeds (red and black error bars). Ranges of model-based energy-optimal speeds within 1% (blue line, thin) and 2% (blue band) of optimal energy cost are shown. We show whole bout “average” speeds (red) and “steady-state” speeds over middle 1.42 m (black, thick), indistinguishable from over middle 0.75 m (gray, thin). Average preferred speeds for 0.5-14 m trials were sig- niﬁcantly lower than that for the 89 m trial (paired t-test, p < 0.01); similarly, the “steady-state” speeds for 2-6 m were signiﬁcantly lower than that for 89 m (paired t-test, p < 0.04).

28 the whole bout; a proxy for the “steady-state” preferred speed is the average over

the bout’s middle 0.75 m (indistinguishable from averaging the middle 1.4 m). Model-

predicted optimal speeds have a 0.96 correlation coeﬃcient (Pearson’s) with exper-

imental steady-state preferred speeds, which were within 1-2% optimal cost. Our

subjects could accelerate to higher average or steady-state speeds, but they preferred

not to. Therefore, the time taken to accelerate-decelerate cannot explain lower speeds

for shorter distances.

2.3.5 Daily energy budget for starting and stopping

Subjects in [19] performed a majority of the walking over a day in short bouts;

they walked in 43914 bouts and took a total of 1717730 steps. Assuming a typical

step length of 0.6 m [83], the subjects walked a total distance of 1030638 m. As-

suming the subjects walked the whole distance at a constant speed of 1.4 m/s, we

can predict a total constant-speed energy expenditure to be 2262600 J, based on a

˙ 2 parabolic relationship given by Esteady = a + bv with a = 2.22 W/kg and b = 1.15

W/kg/(ms−1)2 [20]. But, such a cost would ignore the cost of accelerating from and

to rest at the start and the end of the bout. We can approximate this daily unsteady

cost for the 43914 bouts to be 137380 J by extrapolating our results from the kinetic

−1 −1 2 energy-based model with the unsteady cost for one bout given by, λke(ηpos +ηneg)v /2, where λke = 0.67. The ratio of the cost of changing speed to cost of walking is thus found to be 0.06, in other words, the unsteady cost of walking per day is 6% of the steady cost on average. Using the 95% C.I. for λke gives us 4-8% as reported in the main manuscript. This approximate calculation shows that the cost of changing speeds is a signiﬁcant fraction of the energy humans consume in daily walking.

29 2.3.6 Comparison with a previous study

Our oscillating speed protocols had speed fluctuations between ±0.13 and ±0.27 m/s. As noted in the main manuscript, one previous article [84] attempted to measure the cost of changing speeds, with greater speed fluctuations (±0.15 to ±0.56 m/s) and higher kinetic energy fluctuations per unit time than our study. The rate of kinetic energy fluctuations for both experiments can be compared by comparing v∆v/T , for a fluctuation between speeds v − ∆v and v + ∆v in T seconds. In [84] v∆v/T ranges between to 0.0226 and 0.127 m2s−3 and in our protocol, v∆v/T ranges between 0.026 and 0.1108 m2s−3. Thus, the kinetic energy fluctuation rates were similar in the two studies. Nevertheless, the study [84] found significant increase only for their highest speed fluctuation but not lower. As noted in the main manuscript, this study [84] required walking on oscillating-speed treadmill belts or controlling step durations in overground walking (derived from oscillating-speed treadmill). An oscillating-speed treadmill, being a non-inertial reference frame, can perform mechanical work on the subject, and is not mechanically equivalent to oscillating-speed walking overground.

Further, controlling step durations [84] will produce incorrect speed ﬂuctuations that do not obey the speed-step-duration relation for directly controlling walking speed, as established by Bertram and Ruina [85] in the case of steady walking.

2.4 Discussion

We have shown that oscillating-speed walking costs more than constant speed walking. These cost-increments are correlated with kinetic energy ﬂuctuation and inverted pendulum model-predictions; inverted pendulum model-predictions were closer

30 to experimental values (regression slope closer to 1), perhaps because the kinetic en-

ergy model ignores walking mechanics. The cost of changing speeds implies lower

energy-optimal speeds for shorter distances, reﬂected in our preferred speed experi-

ments here and previous amputee data [86].

Preferred walking speeds are used to quantify mobility and rehabilitation [87], so

bout distances should be chosen to avoid artiﬁcially lowering speeds. Using the cost

of changing speeds may improve daily activity tracking, energy balance estimations

for obesity, and metabolic estimations during sports (e.g., soccer [88]).

A previous experiment [22] considered walking with greater speed ﬂuctuations

(±0.15 to ±0.56 m/s) than our study (±0.13 to ±0.27 m/s) and similar kinetic energy

fluctuations (Appendix A), but found significant cost increase over steady walking only for their highest speed fluctuation. However, this previous study [22] required walking on oscillating-speed treadmill-belt or controlling step durations in overground walking

(derived from oscillating-speed treadmill trials). An oscillating-speed treadmill, being a non-inertial frame (in contrast to a constant speed treadmill), can perform mechanical work, and is not mechanically equivalent to overground oscillating-speed walking.

Thus, the metabolic cost measured in such a set up would underestimate the cost of changing speeds. In fact, the results in [22] do underestimate the metabolic cost of changing speeds when walking compared to what we find here. Further, prescribing step durations to control overground speed fluctuations is different from prescribing speed fluctuations directly [85].

Future work could involve overground experiments, detailed biped and metabolic cost models (including muscle force and history dependence), using diﬀerent speed

ﬂuctuations, and measuring metabolic cost while subjects alternate between walking,

31 stopping and starting (being directly applicable to walking bouts, relying less on extrapolation).

32 Chapter 3: Preferred Walking Speeds for Unilateral Amputees Wearing A Passive Prosthetic Leg

3.1 Summary

Preferred overground walking speed is a commonly used metric to test a patient’s improvement after being ﬁt with a new prosthetic leg or after undergoing physiotherapy or rehabilitation [71, 72]. Often, the preferred speed used to gauge patient progress is measured for walking in a straight line over a relatively short distance [73].

Even for non-amputee subjects, the preferred walking speed for walking in a straight line is distance-dependent. Humans walk slower for short distances. This distance dependence of speed can be explained by the cost of speeding up and slowing down

[74]. The higher the cost of speeding up and slowing down, the more likely it is to aﬀect the preferred walking speed at short distances.

Since it is known that amputees walking with passive prosthetic legs have a higher metabolic cost [75] compared to non-amputees, it would not be surprising if they had a higher cost of speeding up and slowing down as well. This would imply that their preferred walking speeds over short distances would be even less representative than is the case for non-amputee people. Also, it may mean that it is not appropriate to

33 compare the walking speed of an amputee over a short distance to that of a non- amputee person over the same short distance. Here, we measure the walking speeds of amputees wearing prosthetic legs over a range of short to long distances. Using simple optimization-based biped models of walking, we also estimate how their cost of speeding up and slowing down may compare to that of non-amputee individuals.

In daily life, people often need to walk along curves in addition to walking in a straight line. In the past, it has been shown that walking along curves costs more energy than walking in a straight line and the energy cost depends on the radius of the curve [89]. For amputees, the increase in cost with turn radius when walking may be higher than that for non-amputee people. So, it may be meaningful to use circle- walking performance as an additional measure of the eﬀectiveness of a prosthetic leg or a physiotherapeutic intervention. Although such curved walking interventions have been used for other disabled populations [77, 78, 90], there less literature on the subject for amputee-walking. Here we measure the walking behavior of unilateral amputees wearing the Jaipur Foot passive prosthetic leg as they walk in circles of diﬀerent radii.

Metabolic energy optimality has been used to make predictions for a number or aspects of overground and treadmill walking behaviors for non-amputee people [79,

21, 5]. However, there is less work making such predictions for a disabled population

[49, 80, 91]. It could be that for such a population a consideration like stability may take precedence over metabolic energy optimality. Here, we use simple models to make predictions about overground walking behavior and compare it to experiments to see how much is explained by energy optimality alone.

34 3.2 Experimental methods

The experimental protocol was approved by the Ohio State University Institution

Review Board. The overground walking behavior of unilateral amputees wearing

the Jaipur Foot prosthetic leg was recorded and analyzed for two types of walking

experiments: (i) walking for a range of distances in a straight line and (ii) walking

along circles of diﬀerent radii. 12 subjects who are unilateral amputees (11 males, 1

female) performed randomized overground walking trials with informed verbal consent

for a total of 40 minutes each. Their walking was recorded using simple video capture

on a laptop. A stopwatch was also used to manually record the times to complete each

walking trial. Out of 12 subjects 7 were above-knee and 4 were below-knee amputees.

The subjects had weight 65.75±12.6 kg (with prosthesis and shoes), height 1.67±0.09 meters and age 39.67 ± 15.31 years (mean ± standard deviation). All of them wore the above knee or below-knee Jaipur Foot prosthetic leg which was manufactured and

ﬁt in SDMH hospital in Jaipur.

3.2.1 Walking for short distances

Firstly, subjects were instructed to walk in a straight-line for five different distances in random order: 4 m, 6 m, 8 m, 10 m and 23 m (Figure 3.1a). There were four trials for each distance and the subject was asked to “walk the way they usually walk”. Also, they were asked to start and end each trial standing so they had to speed up from and slow down to rest. The walking speeds for different distances and radii were measured manually from the recorded videos and were verified with the stopwatch data recorded during the experiment. The video-based and the stopwatch- based measures agreed with each other to within 0.5 seconds.

35 a) Straight line walking experimental protocol

prosthesis intact leg leg

distance walked (d)

d = 4 m, 6 m, 8 m, 10 m or 24 m

b) Circle walking experimental protocol

circle radius (r)

r = 1 m, 2 m or 3 m

Figure 3.1: Overground walking experiment setup. We measured the preferred walking speed of walking for unilateral amputees wearing a passive prosthetic leg in two conditions: a) walking a range of short distances, starting and stopping each bout at rest and b) walking in circles of diﬀerent radii, both clockwise and anti-clockwise.

36 3.2.2 Walking in circles

Subjects were then asked to walk in circles of three diﬀerent radii: 1 m, 2 m and

3 m (Figure 3.1b). They were asked to walk 5, 4 and 3 laps for each of the circles respectively, so that their average tangential speed can be approximated. Unlike the straight line walking trials, we were not interested in capturing the eﬀects of speeding up from and slowing down to rest for circle walking. We believe that the large number or laps will help remove any such eﬀects. The direction of walking along the circle was also considered. To do this, each radius of circle-walking was repeated twice: once with the prosthetic leg inside the perimeter and once with the prosthetic leg outside the perimeter of the circle. The average speed was obtained by averaging over all laps for each trial. All the trials, with the combination of circle radius and walking direction, were randomized.

3.3 Computational methods

We compare the experimentally-observed preferred walking speed results to the energy-optimal walking speed predicted by a simple model. This model is similar to equation 2.1 in Chapter 2 and in [74]. For simplicity, we assume that people walking a distance D start from rest (speciﬁed in experiment), then instantaneously speed up to some speed vopt, continue at that speed for the whole distance and then instantaneously come to rest again. Thus, the total cost of walking the distance includes the cost of accelerating from rest to speed vopt at the start, walking at constant speed vopt and then decelerating to rest at the end of the walking bout.

For above-knee amputees, the metabolic cost of walking at a constant speed has been measured at diﬀerent speeds and provides a u-shaped relationship between energy cost

37 and speed of walking, similar to that for able-bodied subjects. In the past, we have found the metabolic cost of changing speeds to be 0.67 KE for able-bodied subjects, where KE is the change in kinetic energy. For the optimal speed calculation here, we

first use the same regression equation for the cost of changing speeds. However, we also modify the slope of the equation to see if some other slope better explains the observed behavior (see Results). After obtaining the cost, we find the walking speed that minimizes the cost for each distance. We also find the speeds that are within 1% of the minimum energy cost, because the energy curves are usually flat and a small change in speed near the minimum energy usually results in an even smaller energy change.

3.4 Results

We measure the preferred walking speeds for unilateral amputees wearing passive prostheses as a function of distance walked in a straight line and as a function of diﬀerent radii when walking in circles. We use optimization of a simple biped model of unilateral amputee walking to predict the energy-optimal walking speeds and compare to experimentally-observed walking behavior.

3.4.1 Preferred walking speed for above-knee unilateral amputees depends on distance walked

The unilateral above-knee amputees we considered here show a decrease in preferred walking speed for short distances. The mean preferred walking speeds for the short distances (4 m, 6 m, 8 m and 10 m), pooled across all subjects, were signiﬁ- cantly lower that the preferred walking speed for the long-distance trial (23 m) with p < 0.01 for a left-tailed paired t-test. The decrease in preferred speed, compared

38 to the long-distance speed, is higher for shorter distances and ranges from a 7% to a

13% decrease in speed, depending on distance walked. The percentage decreases in preferred walking speeds with distance for unilateral above-knee amputees are shown in Figure 3.2a. The preferred speed, pooled across all seven subjects, for the 8 m and

10 m walks are not signiﬁcantly diﬀerent from one another (p = 0.37).

3.4.2 The rate of decreases of preferred walking speed with distance walked is lower for unilateral above-knee amputees compared to non-amputees

Although we find that the preferred walking speed for unilateral above-knee amputees decreases with distance walked, the rate of decrease is flatter compared to that for able-bodied individuals. As shown in Figure 3.2b, the slope of the best-fit line for the short-distance trials for able-bodied subjects is three times steeper than that for the above-knee amputee subjects. This means that the above-knee amputees slowed down by a much smaller percentage for short-distance walking trials, compared to able-bodied subjects. However, they still slowed down by a statistically significant amount.

3.4.3 Increased metabolic cost of constant speed walking predicts ﬂatter speed-distance relationship for above-knee amputees

Our simple model of the distance-speed relationship for above-knee amputees (see methods) predicts a flatter distance-dependence of optimal walking speed, when we take into account the increased constant speed metabolic cost of walking previously measured in experiments. Increasing the cost of changing speeds does not affect the slope in a similar manner. Thus, we find it likely that the above-knee amputees do

39 a) Preferred walking speeds decrease with distance walked. able-bodied above-knee amputee 40

compared to long-distance walk (%) Decrease in preferred walking speed 0 4 m walk 6 m walk 8 m walk 10 m walk Distance walked

b) Rate of decrease in speed is "atter for the amputee subjects. For able-bodied subjects experiment datapoint model datapoint 1.2 experiment !t model !t

0.8

For above-knee amputee subjects experiment datapoint 0.4 model datapoint Preferred (m/s) walking speed experiment !t model !t

0 4 6 8 10 Distance walked (m)

Figure 3.2: Decrease in preferred walking speed with distance walked for above-knee amputees. a) Above-knee amputees showed a greater decrease in preferred walking speed for short distances speed compared to able-bodied individuals. The error bars represent the standard deviations. b) The rate of change in preferred walking speed with distance is steeper for able-bodied individuals than for the unilateral amputees. 40 upper bound of within 1% optimum upper bound of within 1% optimum optimum able-bodied walking speed optimum above-knee amputee walking speed 2 lower bound of within 1% optimum lower bound of within 1% optimum preferred able-bodied walking speed preferred above-knee amputee walking speed

1.6

1.2

0.8 Walking speed (m/s) speed Walking

0.4

0 0 2 4 6 8 10 24 89 Distance walked (m)

Figure 3.3: Minimization of total metabolic cost predicts short-distance walking speed. The total cost of the walking a short distance includes a term due to constant speed cost and a changing-speed cost. We ﬁnd that minimizing this total cost predicts the observed trends in changing preferred walking speed with distance for both amputees and non-amputees.

not slow down as much for short distances because, the cost of moving at a reduced speed for the whole distance outweighs the cost of changing speeds from rest. As seen in Figure 3.3, the model prediction for optimal walking speed, with the bounds that are within 2% of optimum, predict the observed preferred walking speeds of the above-knee amputees.

41 3.4.4 Distance-dependence of preferred walking speeds for below-knee unilateral amputees is inconclusive

With the small initial sample set (4 subjects) of unilateral below-knee amputees, we find that they show less dependence of preferred walking speed on distance walked, compared to the above-knee amputees. The walking speeds for the short distances, pooled over all below-knee subjects, were not significantly different from each other for a left-tailed paired t-test. However, the results were variable and the sample is deemed too small to be conclusive. We found, as have others in the past, that the preferred walking speed for the below-knee unilateral amputees were higher than those for the above-knee unilateral amputees [92]. Figure 3.4 shows these preferred walking speed results for unilateral below-knee amputees.

3.4.5 Preferred walking speeds for unilateral amputees walking in circles depends on the circle radii

We ﬁnd that, irrespective of type of amputation, all the unilateral amputees that we observed slowed down when walking along circles of smaller radii (Figure 3.5). We suspect that, similar to able-bodied individuals, this is because the cost of turning the body about a vertical axis is higher, when walking along a circle of smaller radius.

The slope of decrease in preferred walking speed with radius of circle walked, is quite similar for the able-bodied and amputee populations. The slope is 0.1 for able-bodied individuals and 0.08 for both the amputee populations pooled together. Since these slopes are similar, we conjecture that the mechanism for the decreases in speeds must also be analogous.

42 45 able-bodied

below-knee amputee

compared to long-distance to (%) walk compared 5 Decrease in preferred walking speed in preferred Decrease

-5 4 m walk 6 m walk 8 m walk 10 m walk

Distance walked

Figure 3.4: Decrease in preferred walking speed with distance walked for below-knee amputees. The below-knee amputees also show a signiﬁcant decrease in preferred walking speed for short distances compared to able-bodied individuals. However, the change in speed with distance does not show a consistent trend like that for above-knee amputees.

43 a) Preferred walking speeds with prosthesis leg inside circle

10 compared to long-distance to walking compared (%)

Percentage decrease in preferred walking speed in preferred decrease Percentage 0 1 m radius 2 m radius 3 m radius Radius of circle walked able-bodied all unilateral amputees

b) Preferred walking speeds with prosthesis leg outside circle

10 compared to long-distance to walking compared (%)

Percentage decrease in preferred walking speed in preferred decrease Percentage 0 1 m radius 2 m radius 3 m radius Radius of circle walked

Figure 3.5: Preferred walking speeds for circle walking. The preferred walking speed for all the unilateral amputees showed a decrease with radius of the circle walked. The unilateral amputees did not show a signiﬁcant diﬀerence in preferred walking speed when walking with the prosthesis-leg a) inside and b) outside the circle. 44 3.4.6 Preferred walking speeds for unilateral amputees walking in circles is independent of turning direction

We had subjects walk both clockwise and anti-clockwise along circles drawn on the ground. We did this so as to check for any effects due to having the prosthesis-leg as the pivot, as opposed to the intact leg as the pivot. Interestingly, we did not find significant differences between the two conditions for any of the unilateral amputee populations considered here p > 0.05. To the eye, the above-knee amputees seem to walk slightly faster when the prosthetic leg was outside the circle. However, this effect of turning direction on preferred circle-walking speed was not significant.

3.4.7 Preferred walking speeds are proportional to degree of amputation

For all the cases above i.e., straight-line walking and circle-walking, we found that the preferred walking speed for the above-knee amputees are the lowest and that for the able-bodied individuals are the highest. The preferred walking speeds for the below-knee amputees are higher than above-knee and lower than able-bodied preferred speeds. Thus, it can be said that the preferred walking speeds depend on the degree of amputation, being lower for higher degree of amputation. This result for straight-line long-distance walking over long distance echoes other work in the past.

However, the result for short-distance and circle-walking is new, even if unsurprising.

3.5 Discussion

Preferred walking speed is very often used as a measure of progress in walking rehabilitation for various disabled populations like people with neuromuscular disorders and amputees. An implicit assumption made when relying on such measures is that

45 higher speeds means more improvement. However, past theoretical and experimental work on able-bodied subjects shows that the speed at which people choose to move depends on the constraints of the motion itself like the distance walked and the cur- vature of the motion. So, depending on the situation, people may sometimes move at a lower speed than physically possible to satisfy some other objective, like minimizing energy. Here, we ﬁnd that these observations extend to disabled populations as well and may have signiﬁcant implications for the usage of preferred walking speeds as a measure of performance. These implications are detailed in the paragraphs below.

Past research has found that able-bodied individuals slow down when walking short distances. Here, we find that above-knee amputees also slow down when walking short distances. This implies that the distance over which the preferred walking speed is measured and interpreted during rehabilitation may systematically overestimate or underestimate the progress that the patient has made. Majority of studies measuring the progress of people with walking disabilities post-rehabilitation report preferred walking speeds over very short distances (4m or 6m). In order to circumvent these distance-effects, we suggest measuring the speed over a few distances, not just one or two. Also, when comparing the preferred speed values for amputees to the able-bodied values, we suggest comparing to the values for the same distance walked. Because, we find here that the difference in walking speed between able-bodied and above-knee amputees over short distances is lesser than the corresponding difference over longer distances.

Everyday walking consists of not just straight line walking but also turning. Pre- vious work [76] has found that able-bodied individuals slow down when walking in circles of smaller radii. Here, we ﬁnd that unilateral amputees also slow down when

46 taking sharp turns (circles of smaller radii) similar to able-bodied individuals. We also

ﬁnd that the corresponding walking speeds are lower for the amputees for all radii, compared to those for able-bodied individuals. For all these reasons, we propose that circle-walking lends itself as an additional useful measure of walking performance during rehabilitation.

In the past, people have predicted aspects of walking behavior such as walking speed, walking step width and walking step frequency using energy minimization as a hypothesis. However, most of these studies look at constant speed straight line walking on treadmills. Moreover, a majority of these studies attempt to predict able- bodied walking behavior. Here, we provide evidence that energy-minimization can predict aspects of non-steady overground walking behavior in a disabled population.

Speciﬁcally, we have found that, minimizing a combined cost of changing speeds and constant speed walking explains the short-distance walking behavior in above-knee unilateral amputees. Thus, we have added to the body of evidence that metabolic cost of walking is predictive of overground walking behavior for amputees.

We have studied the preferred walking speeds of unilateral amputees wearing a passive prosthetic leg (Jaipur Foot). The decision to select a population wearing passive prostheses was intentional and was motivated by the fact that preferred walking speeds as a performance measure is even more common in developing countries where low-cost passive prostheses such as Jaipur Foot are used. Thus, we feel that our conclusions regarding the qualiﬁcation of preferred walking speeds as a performance measure would be more relevant in hospitals in the developing world, where access to other measures of performance, such as a gait lab with motion capture and

47 force plates, may be limited. Also, we believe that the short-distance and circle- walking measures proposed here are easy to implement even in environment where the resources available are limited.

One limitation of this study is the relatively small sample size of subjects. The experiments were conducted in a hospital in India in a short span of time and not all the patients had the time or resources to participate in the study. Despite the small sample size, the qualitative results for the above-knee amputees are robust (low p-values).

The results for the below-knee amputees, however, are variable and inconclusive. A follow-up study will be conducted to investigate the below-knee population with a larger sample size. Another potential limitation is that we did not limit our amputee sample by years since amputation and age. However, we believe that this limitation is offset by the fact that we focus here on the change in preferred speeds of each subject under different conditions relative to his/her own long-distance straight-line walking speed. So, the individual differences in terms of years of being accustomed to wearing the prosthesis will not affect our results. We compare the able-bodied walking speeds of a diverse group of American graduate students to those of Indian amputees. It would be more tenable to measure the able-bodied speeds also for the same population (Indian able-bodied adults). However, past studies of variations in walking speed due to location or country are much lesser than the differences between able-bodied and amputee populations that we observe here. So, we believe that our fundamental findings will not change even with the more appropriate Indian able-bodied population.

In the future, we would like to study the preferred walking speeds of amputees wearing low-cost prostheses in real-life conditions by using wearable devices. Such

48 work would help understand what kind of motions are most common among these amputees and would allow us to understand the priorities for rehabilitation measures and prosthesis design. Another follow-up study is to measure the constant speed, changing-speed and turning metabolic costs for amputees and use those measured metabolic costs in our predictive models.

49 Chapter 4: Control Strategies Inferred from Variability in Human Running

4.1 Summary

Although running mechanics has been studied extensively, we do not yet have a complete characterization of how humans run without falling down. Sensory noise, muscle force noise, and other small environmental variations perturb the running motion away from perfect periodicity, even in the absence of external perturbations.

Thus, in human running, each step is diﬀerent from every other step. In this chapter, we obtain a controller that stabilizes running by mining this step-to-step variability in human running data. We infer linear models describing how humans change their leg forces, foot placement, stance duration and leg length in response to deviations of the center of mass from the nominal periodic motion. We found humans continuously modulate their leg force so as to oppose a velocity deviation, correcting the deviation mostly within a step, and use foot placement to modulate their leg force direction.

Such human-derived control strategies for running could be used to design better robotic exoskeletons and prostheses, understand movement disorders in runners, and inform running robots that may match human running performance.

50 4.2 Materials and methods

The methods involved performing human subject experiments, obtaining linear

models to characterize the control, and performing dynamic simulations using the

inferred controller. The protocols were approved by the Ohio State University IRB

and subjects provided informed consent. Eight subjects, three female and ﬁve male

(age 25.0 ± 5.3 years, weight 66.8 ± 7 kg, height 1.8 ± 0.14 m, leg length 1.05 ±

0.08 m, mean ± s.d.) ran on a split-belt treadmill for about 3.5 minutes on average at three constant speeds: 2.5, 2.7 and 2.9 m/s, presented in random order. Subjects wore a loose safety harness that did not constrain their motion. Three-dimensional

GRFs were recorded by two six-axis load cells in the treadmill, one under each belt

(Bertec Inc., 1000 Hz). Body segment motion was measured using marker-based motion capture (Vicon T20, 100 Hz) with four markers on each foot and on the torso.

4.2.1 Notation and terminology

We measured body motion and ground reaction forces (GRFs) of humans running on a treadmill at three speeds: 2.5, 2.7 and 2.9 m/s (see Materials and Methods). The

results we present are for the highest speed. Figure 4.1 shows the coordinate system

and sign convention: x is sideways, y is fore-aft, and z is vertical. Each running

step consists of a ﬂight phase, with neither foot on the ground, and a stance phase,

with one foot on the ground. A ﬂight apex is deﬁned as when the center of mass

height z is maximum. The center of mass position and velocity at ﬂight apex are

denoted by (xa, ya, za) and (x ˙ a, y˙a, z˙a) respectively—withz ˙a = 0 by deﬁnition and thus not usually considered as an explanatory variable. Unless otherwise speciﬁed, all quantities and results are non-dimensionalized using body mass m, acceleration

51 due to gravity g, and leg length `max. For instance, forces are normalized by mg, √ p positions and distances by `max, speeds by g`max, time by `max/g, impulses by √ m g`max, etc.

4.2.2 Calculating input state variables during ﬂight apex

We deﬁne ﬂight apex as when the center of mass velocity reaches its peak height

(z ˙ = 0). The input to the running controller is the center of mass state at ﬂight

apex. The body center of mass motion during ﬂight is nearly parabolic, ignoring air

resistance. The center of mass velocities are obtained by integrating the center of

mass accelerations, equal to the mass-normalized net force on the body. To obtain

the integration constants, we assume that the mean velocity and acceleration are zero,

because the person does not translate appreciably in the lab frame over a trial. To

remove the slow integration drift in the center of mass velocity, we used a high-pass

ﬁlter with a frequency cut-oﬀ equal to an eighth of mean step frequency. Changing

this filter cut-off to a twentieth of the step frequency does not affect this article’s

conclusions. We ignored air drag here, as including it changes the velocities by less

than 10−5 ms−1, much smaller than the step-to-step variability. We use a sacral marker as a proxy for the center of mass position [93]. Unless otherwise specified, we use the flight apex state (x ˙ a, y˙a, za) as inputs in our linear models. The vertical velocityz ˙a at flight apex is zero by definition and hence not included as an input.

The sideways and fore-aft positions are also usually not included.

4.2.3 Calculating the output control variables during stance

We assume that the following variables are used to control the runner: GRFs, foot placement, stance duration, and the landing leg length. Stance phases are deﬁned

52 A conceptual model of running control

Stance phase (when leg pushes on ground) Flight phase (parabolic trajectory under gravity)

some relevant states at !ight apex

!ight apex unperturbed average trajectory perturbed trajectory

ground reaction

z (vertical) forces change

D irec foot position change tion y (f of ys) orw tra wa ard vel ide ) x (s

Figure 4.1: Idealized running control. A schematic of a running trajectory, perturbed sideways during ﬂight. The runner can recover back to steady-state by changing the ground reaction force — say, by altering the foot placement and the leg force magnitude.

53 as when Fz > 30 N. The corresponding stance duration is Ts. The GRF impulses

(Px,Py,Pz) are obtained by integrating the GRF over the stance phase. To approxi-

mate how the GRFs changes with the stance phase fraction φ, we compute the binned

averages of the GRF in each of n = 20 bins of size Ts/n.

4.2.4 Linear regressions between the outputs and the inputs

We compute the mean values of the inputs over all steps in each trial and obtain

deviations from these means (∆x ˙ a, ∆y ˙a, ∆za) = ∆Ba. Similarly, we compute the deviations of the output variables ∆F (φ), ∆P , ∆Ts, and ∆(xf − xs, yf − ys). We use least squares regression to obtain linear models between the inputs and the outputs.

This gives the Jacobian matrix for each output as ∂Ts/∂Ba, ∂Fs(φ)/∂Ba, and so on,

to quantify the sensitivity of the outputs to the inputs. We interpret the elements of

these matrices as feedback gains relating an output control variable to the input state

deviations. The results presented are based on pooled deviations over all subjects.

The outputs considered here are the ground reaction forces, the foot placement, the

leg length and the foot center-of-pressure.

4.3 Results

In this section, we ﬁrst describe the step-to-step variability in the motion and

forces during human running. We then show how this variability, which superﬁcially

appears random, contains information about how running is controlled.

54 4.3.1 Center of mass variability is phase-dependent and di- rectionally uncoupled

The center of mass speeds (x ˙ a, y˙a) in the sideways and fore-aft directions have low correlations (Pearson’s |r| < 0.1) and have similar standard deviations (Figure 4.2a).

On the other hand, the center of mass position has very different variabilities in the three directions: specifically, a standard deviation of 6 cm in the fore-aft (ya), 2 cm in the sideways (xa), and 5.9 mm in the vertical (za) direction. The center of mass state variability changes systematically throughout a step, being higher during flight and lower during stance (Figure 4.3). Later, we interpret this systematic variation as reflecting the effect of control and the addition motor noise through leg forces during stance.

4.3.2 Apex-to-apex maps show fast decay of perturbations and stable periodic motion

We computed the ‘apex-to-apex maps’ i.e., linear models from the center of mass state at one ﬂight apex to the next. The right-to-left map from the state SR =

(x ˙ a, y˙a, za) at an apex preceding a right stance to the state SL at the next apex preceding left stance is:

∆SL = KRL ∆SR (4.1)

where the best-ﬁt KRL is

"−0.05 −0.02 +0.31# KRL = −0.08 +0.27 −0.15 +0.02 +0.06 +0.46

with p < 0.05 for all elements except KRL(1, 1) and KRL(1, 2). The low value

of KRL(1, 1), not signiﬁcantly diﬀerent from zero, suggests that a purely sideways

55 eito ausi l aes Rsaei rcino oyweight. body of standard fraction the indicates in text but are Green directions, GRFs stance. vertical panels. right stance and the all right fore-aft for in a values in that (blue for deviation of similar directions band) negative is three (yellow is GRFs in GRF mean GRFs sideways phase the Mean stance around d) left deviation denote steps. phase; the standard dots left and one Red denote step and only steps. dots separate that line) randomly-chosen blue so a 500 subtracted, and for to value steps shown mean corresponds of right is subject’s dot beginning mean each each the the with at c about subjects torso variability impulses and all to sideways for b relative is and a, plot position fore-aft panels scatter foot the In in in Variability phase. Variability c) stance b) GRFs. the apex. to ﬂight due at velocity mass of 4.2: Figure Average sideways GRF fore-aft speed (m/s) variability a) Flight apex state -0.2 0.2 stance fraction (fstance fraction sideways speed(m/s) 0 -0.2

(Body wt.) 0.2 -0.2 0.2 0 1 0 0.033 m/s d) Ground forces: reaction mean and standard deviation tpt-tpvraiiydrn running. during variability Step-to-step 0 0.034 0.5

0.029 m/s stance )

fore-aft impulse (m/s) b) Leg impulse variability -0.2

Average fore-aft GRF 0.2 stance fraction (fstance fraction sideways impulse(m/s) 0 (Body wt.) -0.2 -0.4 0.4 0 0.049 m/s 1 0 0.025 0.2 0 56 0.5 0.018 0.035 m/s stance ) c) Foot placement variability )Vraiiyi h center the in Variability a) Average vertical GRF fore-aft foot stance fraction (fstance fraction placement (m)

(Body wt.) -0.06 0.06 0 1 2 3 0 0.12 0 00 0.06 0 -0.06 0.075 placement (m) sideways foot 0.5 0.014 m 1 stance

) 0.011 m a) Variability in sideways velocity as a function of phase 0.04 Total standard deviation

Explained 0.02 Explained by CoM + by CoM state swing foot 0 0 100 fraction of stance (%)

b) Variability in fore-aft velocity as a function of phase Total standard deviation 0.03

0.02

0.01 Explained by CoM state 0 0fraction of stance (%) 100

c) Variability in vertical position as a function of phase 0.01 Total standard deviation 0.005 Explained by CoM state 0 0fraction of stance (%) 100

Figure 4.3: Variability in states as a function of phase. Standard deviations of a) sideways CoM velocityx ˙, b) fore-aft CoM velocityy ˙, and c) vertical position z. The standard deviation of the sideways and fore-aft velocities are minimum near mid-stance and maximum during flight. The vertical position standard deviation is maximum during flight and minimum during stance, a little before mid-stance. We show how much of this standard deviation is explained by the CoM state alone or CoM and swing foot state at the immediately preceding flight apex.

57 velocity perturbation gets corrected over one step. The value KRL(2, 2) suggests that

73% of a forward velocity deviation is corrected in one step. The left-to-right matrix

KLR is similar to KRL, except for the sign changes due to mirror-symmetry. In other words, the control for the left leg is identical to that for the right leg except for a half period phase shift and a mirror reﬂection about the sagittal plane. The matrix product of KLR and KRL (also known as Jacobian of the Poincare map [94]) quantify how apex state deviations grow or decay over one stride (two steps). The eigenvalues of these matrices are all less than one in absolute value, indicating a stable periodic motion.

4.3.3 Ground reaction force variability is needed to control center of mass motion

Humans need to ensure that this step-to-step motion variability does not grow without bound. From Newton’s second law, it follows that the only way to control the center of mass motion is to modulate the total ground reaction components during stance phase. Thus, as expected, the ground reaction force (GRF) components over the stance phase are variable (Figure 4.2d). The variability of these GRF components

(Fx,Fy,Fz) depend on the time fraction φ of stance phase, being higher during mid- stance and lower during the beginning and end of stance.

4.3.4 Ground reaction impulses can be controlled independently in diﬀerent directions

The eﬀect of forces on velocity over a given period is captured by the integral of force, namely, the impulse. Indeed, the mass-normalized GRF impulses over one step

(Px,Py), that is, the time-integrals of the GRFs over stance phase, give us the change

58 in velocity over one step. The variability of these stance impulses in the sideways and fore-aft direction are largely uncorrelated (Pearson’s |r| < 0.1, Figure 4.2b). This lack of correlation suggests that the fore-aft and sideways impulses do not constrain each other and they can be modulated mostly independently.

The GRF impulses over one step (Px,Py,Pz) quantify the eﬀect of the ground reaction forces over one step. The F = ma equation does not directly constrain how quickly the deviations are corrected. But it tells us how big the force changes need to be for the velocities to be corrected in a certain amount of time.

4.3.5 Force impulses correct velocity deviations in about about one step

The sideways and fore-aft ground reaction impulses are well-predicted by the center of mass state (x ˙ a, y˙a, za) at the previous ﬂight apex. A given velocity deviation in the sideways direction is corrected almost entirely in a step, whereas about 70% of a fore-aft velocity deviation is corrected in a step. Further, there is not much coupling between the fore-aft and sideways directions. The sideways impulse depends on sideways velocity, but not much on fore-aft deviations. The fore-aft impulse depends on the fore-aft velocity, but not much on sideways deviations. Both sideways and fore-aft impulses depend on vertical position deviations. The sideways GRF impulse

Px is predicted by the previous apex state via the linear model:

2 Left: ∆Px = −1.03∆x ˙ a + 0.05∆y ˙a − 0.29∆za with R = 0.57, and (4.2) 2 Right: ∆Px = −1.07∆x ˙ a − 0.08∆y ˙a + 0.31∆za with R = 0.55.

These models have a simple interpretation. The ∆x ˙ a coeﬃcient of about -1 in these models (that is, ∆Px ≈ −∆x ˙ a when ∆za = 0) implies that sideways velocity deviations are completely corrected, on average, within approximately one step. This

59 correction could have been done over many steps (as would be implied by a coeﬃcient of -0.5, say), but humans seem to exhibit a dead-beat controller for sideways velocity deviations. Of course, this single-step correction is not perfect; R2 < 0.6 suggests that the system over-corrects or under-corrects deviations for any given step. The fore-aft GRF impulse Py is predicted by the previous apex state via the linear model:

2 Left: ∆Py = 0.05∆x ˙ a − 0.72∆y ˙a − 0.19∆za with R = 0.33 and (4.3) 2 Right: ∆Py = −0.08 ∆x ˙ a − 0.72 ∆y ˙a − 0.19∆za with R = 0.34.

These regressions suggest that about 72% of a forward velocity deviation is corrected in a single step, on average. All coeﬃcients in these equations are signiﬁcant with p < 10−4.

4.3.6 Horizontal impulses are modulated by changing forces, not duration

The impulse can be changed by changing the average GRF component over stance or the stance duration Ts. We ﬁnd that about 99% of the variance in the sideways and fore-aft impulses is due to changes in the average ground reaction forces in those directions, and only about 1% of the variance is due to stance time variation. Thus, we next consider how the ground reaction forces are changed. In contrast to the horizontal impulses, the vertical impulse has considerable covariance with both the total force magnitude and the total stance duration. The percentages sum to greater than 100% because the force magnitude and the total stance duration themselves are anti-correlated.

However, we ﬁnd that the stance duration is largely independent ofx ˙ a andy ˙a,

−4 given by: ∆Ts = −1.4 ∆za (p < 10 ). Thus, sideways and fore-aft velocity deviations

60 are corrected by changing the GRFs, rather than the stance duration. We also see

that an upward position perturbation at apex results in a shorter stance phase.

4.3.7 Within-step vertical impulse modulations correct the vertical position deviations

The control of vertical position is qualitatively diﬀerent from that of control in

the fore-aft and sideways directions, as we cannot use net vertical impulse for vertical

position control due to the impulse-momentum considerations below. A ﬂight apex

occurs when the center of mass vertical velocity is zero. So, the net vertical impulse

between two consecutive ﬂight apexes is also zero (as it equals the change in vertical

momentum, according to the impulse-momentum equation). Therefore, changing the

net vertical impulse over a stance phase will not accomplish any meaningful control

in the vertical direction. However, we will show that by diﬀerentially modulating the

vertical impulse within one stance phase, we can change the vertical position (za)

from one ﬂight apex to the next, without changing the net impulse (Figure 4.4).

To show this most simply, consider inﬁnitesimal ﬂight phases and a stance phase

from t = 0 to t = Tstep. The total impulse Pz due to the vertical ground reaction

R Tstep R Tstep force Fz(t) equals that due to gravity given by, Pz = 0 Fz(t) dt = 0 mg dt =

mgTstep. For a triangular stance force (Figure 4.4) with peak force Fpeak at tpeak, we

get Fpeak = 2mg. Then, the change in vertical position z(Tstep) − z(0) over a step is

given by: g z(T ) − z(0) = (T − t )2 − t2 . (4.4) step 6 step peak peak

If the step was symmetric about mid-stance (tpeak = Tstep/2), there is no vertical position change over a step (z(Tstep) = z(0)). The ﬂight apex vertical position on

the next step z(Tstep) can be changed by changing tpeak relative to Tstep/2 (Figure

61 Vertical position control: a simple model

F (t) Shift peak later z F peak to decrease vertical position

z(T step )

nominal GRF

0 tpeak Tstep

z(t) CoM trajectory vertical position

z(T step ) decreases

0 Tstep

Figure 4.4: Vertical position control by differential impulse control. Using a unimodal vertical GRF, we find that the way to lower vertical position over a step is to move the peak force to the right. This is equivalent to increasing the vertical impulse on the second half of the step and decreasing the vertical impulse over the first half. Conversely, to increase the vertical position over a step, we find that the peak force needs to be moved to the left.

62 4.4). For example, if z(0) at one ﬂight phase was greater than its nominal value

and the runner wishes to reduce it, this simple model predicts that the runner will

decrease the ﬁrst-half impulse and increase the second-half impulse; doing this is

equivalent to delaying tpeak relative to Tstep/2. This prediction is in agreement with

the following experimentally-derived linear relations for the ﬁrst half vertical impulse

∆P |Tstep/2 (from t = 0 to T /2) and the second half vertical impulse ∆P |Tstep z 0 step z Tstep/2

(from t = Tstep/2 to Tstep):

Left: ∆P |Tstep/2 = −2.5∆z and ∆P |Tstep = +2.5∆z , with R2 = 0.35 (4.5) z 0 a z Tstep/2 a

Right: ∆P |Tstep/2 = −2.3∆z and ∆P |Tstep = +2.3∆z , with R2 = 0.30 (4.6) z 0 a z Tstep/2 a

In addition to the vertical impulse, the landing leg length is also modulated in response to vertical ﬂight apex deviations. Regressing the leg length ` at the beginning

of stance with the ﬂight apex state, we found that this landing leg length is mostly a

function of the vertical position at ﬂight apex:

−4 2 ∆`landing = 0.3∆za with p < 10 ,R = 0.25. (4.7)

Thus, a downward position deviation at ﬂight apex would result in landing with a

shorter leg length than nominal (e.g., via ﬂexed knee or ankle).

4.3.8 GRF control is phase-dependent

We have shown that GRF impulses are modulated every step to control the center

of mass motion. Next, we characterize how these impulse modulations are achieved

by modulating GRFs as a function of stance fraction φ. For each φ, we obtained

63 linear models for the deviations of force components ∆Fx(φ) and ∆Fy(φ) of the form:

∆Fx(φ) = J1(φ)∆x ˙ a + J2(φ)∆y ˙a + J3(φ)∆za and (4.8)

∆Fy(φ) = J4(φ)∆x ˙ a + J5(φ)∆y ˙a + J6(φ)∆za (4.9)

where Ji(φ) are a phase-dependent coeﬃcient. This linear model shows that Fx is decreased over the whole step to correct an increased sideways velocity (Figure 4.5a).

Thus, in response to an increased fore-aft speed, the fore-aft GRF is modulated so that there is a net negative force on the body over the next step (Figure 4.5b).

Signiﬁcantly, Fy is changed much more during the deceleration phase (φ < 0.5) than during the acceleration phase (φ > 0.5).

4.3.9 Step in the direction of the deviation

Placing the foot relative to the body allows a runner to modulate the leg force direction and thus the GRF components. The variability in the horizontal foot position (xf , yf ) relative to the center of mass position (xs, ys) at the beginning of stance is shown in Figure 4.2c. The modulation of this relative foot position is predicted by the ﬂight apex state as follows:

2 ∆(xf − xs) = 0.92∆x ˙ a ∓ 0.25∆za with R = 0.55 and (4.10) 2 ∆(yf − ys) = 0.37∆y ˙a − 0.8∆za with R = 0.45. where the two signs on the za-coeﬃcient correspond to the left and right stances respectively. These equations have the following simple interpretation. A sideways velocity perturbation to the body results in the foot being placed further along the direction of the perturbation: a rightward perturbation results in a more rightward step. Analogously, a forward velocity perturbation results in the foot being placed further forward relative to the body. There is no signiﬁcant coupling between sideways

64 a) Sideways GRF response to sideways perturbation average perturbation sideways GRF modulated + = sideways GRF size sensitivity sideways GRF 0.2 0 0.2 0.1 (right stance) (right stance) -1 0.1 0 + = (Body wt.) 0 -1

(ms ) (Body wt.) -0.1 -2 -0.1 -0.2 -3 -0.2 0 1 0 1 0 1

stance fraction (f stance ) stance fraction (f stance) stance fraction (f stance)

b) Fore-aft GRF response to fore-aft perturbation average perturbation fore-aft GRF modulated + = fore-aft GRF size sensitivity fore-aft GRF 0.4 0 0.4 = 0.2 -1 0.2 0

(Body wt.) 0 + -2 (Body wt.) -0.2 (ms -1 ) -0.2 -0.4 -3 -0.4 0 1 0 1 0 1

stance fraction (f stance ) stance fraction (f stance) stance fraction (f stance)

Figure 4.5: Phase-dependent control of GRFs. We show how the GRF components respond to perturbations at the previous ﬂight apex, as estimated by our phase-dependent GRF model. a) Sideways GRF response to a (rightward) sideways velocity perturbation. b) Fore-aft GRF response to a forward velocity perturbation. In both cases, the change in GRF from nominal (shown by the arrow) is obtained as a product of the perturbation size and the sensitivity of the GRF to the perturbation.

65 and fore-aft directions. Foot placement modulations also depend on vertical position

deviations, so that an upward position deviation results in landing with a steeper leg

angle.

4.3.10 Landing leg length is changed in response to vertical deviations

Regressing the leg length at the beginning of stance with the ﬂight apex state, we

found that this landing leg length is mostly a function of the vertical position at apex:

−4 ∆` = 0.3∆za (p < 10 ). Thus, a downward position deviation at ﬂight apex results in landing with a shorter leg length than nominal (e.g., via ﬂexed knee or ankle).

4.3.11 Horizontal forces are modulated by force direction and vertical by force magnitude

The force components in the three salient directions can be changed either by changing the direction of the force or changing the magnitude of the force or both.

We ﬁnd that in the horizontal directions (fore-aft and sideways), most of the force component modulation is achieved by changing the force direction, rather than force magnitude. Speciﬁcally, say the force magnitude is F and the force direction unit

vector is [ux uy uz]. Then, in the sideways direction, the sideways force component

is given by Fx = F ux and changes in the sideways force are given, to ﬁrst order, by

∆Fx = ∆F ux + F ∆ux. The second term, F ∆ux, due to changes in direction explains

97 ± 1.9% of sideways force changes ∆Fx over all subjects and trials. Similarly, in the fore-aft direction, the term F ∆uy due to changes in direction explains 83.2 ± 5% of the fore-aft direction changes. In contrast, in the vertical direction, the term due to magnitude changes ∆F uz, explains 99.5 ± 0.4%.

66 4.3.12 Leg force is mostly along leg direction

The leg force direction is largely aligned with the leg direction (from foot to torso),

so that the dot product between their corresponding unit vectors is very close to 1 (on

average 0.99 ± 0.01). This alignment of direction is used in the modeling of leg force direction as being exactly along the leg in numerous mathematical models of running and human locomotion. The leg force being roughly along the leg is an inevitable consequence of small leg mass compare to whole body and small hip torques during running. However, if the runner had much larger hip torques, the leg force could have a substantial component perpendicular to the leg. This result is of importance because, in the next chapter, we use a simple model with a point mass and telescoping legs to model the running motion. The fact that the leg force is along the direction of the leg gives some validity to the use such a point-mass model.

4.3.13 Is the GRF component modulated by changing leg force direction or magnitude?

Changes to GRF component in the sideways direction are mainly achieved by

making changes to the foot placement relative to the hip at stance start and less so

by changing the leg force magnitude. In the fore-aft direction, during the initial phase

of stance (about 30% into the stance), the leg force magnitude and foot placement

are both modulated to a similar extent to bring about a change in the GRF compo-

nent. However, as stance progresses, a majority of the contribution to GRF fore-aft

component comes from the foot placement and not the leg force. Overall, we ﬁnd

that foot-placement is the major variable that contributes to control. Foot placement

67 is eﬀective in redirecting the GRF components because it can be modulated independently in the sideways and fore-aft directions, whereas changing the leg force directly changes components in all directions.

4.3.14 Approximate left-right symmetry and possible asymmetry in the control

The running control gains have bilateral (left-right) symmetry. The gains that map center of mass state to foot placement or impulses are not significantly different for the left and right stances, except for a sign changes, when applicable. Specifically, the gains that couple sideways direction and either fore-aft or vertical directions have mirror-symmetry. That is, these gains for the left stance are the negatives of corresponding gains for the right stance. Other gains, for instance, those that couple sideways and sideways or fore-aft and fore-aft are the same for left and right, without any such sign changes. This mirror symmetry in running control likely follows from the approximate mirror symmetry in body physiology about the sagittal plane, but is in contrast to the substantially different roles that the two hands play during manipulation [95] or even asymmetric legged tasks such as kicking [96].

4.3.15 Swing foot control

We have found that the foot placement deviations at the beginning of stance are well explained by center of mass state at ﬂight apex. When are these foot placement deviations achieved? One possibility is that these deviations are achieved early on during the swing phase and this deviation is preserved until the next foot placement.

However, this does not appear to be the case. At the beginning of ﬂight phase, the CoM state is a vastly better predictor of the foot placement than the swing

68 a) Predicting sideways foot placement using CoM and swing foot states

Rapid increase Stance Flight in predictive

2 ability during 1 !ight

CoM state is CoM state’s more predictive predictive ability 0.5 early remains !at during !ight

Swing foot state does not predict future foot

Sideways Foot Placement R 0 0 50 100 placement during Fraction of one step (%) early !ight

Predicting future foot placement Predicting future foot placement using just the swing foot state using just the CoM state b) Predicting fore-aft foot placement using CoM and swing foot states

Stance Flight

2 1

CoM state is more predictive 0.5 early

Fore-aft Foot Placement R 0 0 50 100 Fraction of one step (%)

Figure 4.6: Swing foot control before foot placement. The fraction of a) sideways and b) fore-aft foot placement variance at beginning of stance predicted by the center of mass (CoM) state or swing foot state during the previous one step (ﬂight and stance). The solid and dashed lines represent right and left foot placements respectively. 69 foot itself (see Figure 4.6). In particular, less that 10% foot placement deviation

is predicted by the swing foot deviation at the beginning of ﬂight phase, and the

rest of the swing foot deviation appears to be achieved during the ﬂight phase. The

explanatory power of the CoM remains ﬂat during ﬂight—this could be due to one of

two reasons: presumably because CoM state follows a parabolic path during ﬂight and

thus accumulating no new variation. On the other hand, we see that the explanatory

power of the swing foot rises rapidly from less than 10% to a 100% when it becomes the

next stance phase. During this brief period, information from CoM state is transferred

to the foot, presumably via some mixture of feedback control or feedforward dynamics.

4.3.16 Station-keeping and speed do not aﬀect control

In the linear models above, adding the sideways and fore-aft apex body position to

the explanatory variables improves the R2 values by only 1±0.1%. Thus, our runners did not prioritize controlling their position relative to the treadmill (station-keeping).

Further, the regression coeﬃcients did not vary signiﬁcantly across the three running speeds (p > 0.05, paired t-tests).

4.3.17 Continuous control is no better than apex-based control

As an alternative to control based on the flight apex state, we considered a ‘continuous control’ model. Specifically, we obtained linear models for the GRFs based on the current center of mass state during stance (x, ˙ y,˙ z). These linear models did not differ significantly in the fraction of GRF variance explained, compared to the apex-based control model (paired t-test p = 0.94).

70 4.3.18 Attitude control: GRFs and foot placement do not directly respond to trunk tilts

Running requires control of other degrees of freedom, not just the center of mass.

For instance, torso attitude (orientation) may be controlled and could be accomplished using hip moments [97]. Nevertheless, including the torso attitude and angular velocity did not substantially improve explained foot placement or the ground reaction force variance, over and above the variance explained by center of mass state: R2 increase is less than 2% in sideways and 10% in fore-aft direction. Even though the correlation between the deviations fore-aft GRF and deviations in trunk angles at

flight apex are weak, the coefficient relating the sagittal plane trunk angle deviation to the fore-aft GRF is negative and significant with p < 10−5. That is, if the trunk is deviated forward in the sagittal plane, the forward GRF increases. The weak dependence of GRFs on orientation state may suggest that while humans use hip moments to control trunk attitude, they have enough muscles in the legs that the ground reaction forces may respond primarily to center of mass deviations and need not change much in response to orientation deviations.

4.3.19 Foot center of pressure modulation has systematic modulation, but is not a big contributor to center of mass control

Humans do not have a point foot; they have an extended foot and have the ability to control the center of pressure without physically lifting and lowering the foot. Such center of pressure modulations are thought to be useful in the control of standing and walking [98, 99]. Here, we ﬁnd that there is only a small degree of within-step center of pressure modulation over and above what is accomplished with the foot or the

71 mean center of pressure. The mean center of pressure or the foot position during stance explains over 96% of the center of pressure variation during most of the stance phase (except the initial and ﬁnal tenth of a stance phase). The within-step center of pressure modulation relative to the foot has a small standard deviation of about 5 mm in the sideways direction and 13 mm in the fore-aft direction. On average, less than

10% of this variance is explained by past center of mass state in both the sideways direction and the fore-aft direction. Our subject pool contained both supinate and pronate runners, who had the center of pressure progress on the outside or inside of the foot, respectively.

One way to potentially modulate the center of pressure in addition to foot placement is to use foot orientation during stance (foot yaw). Foot yaw has previously been found to be a useful control variable in simple models of walking and was found to have a significant but weak correlation with sideways foot placement [100]. Here, for running, we found that the foot orientation is similarly significantly, but weakly explained, by past center of mass state. We find that foot depends on sideways deviations in the same manner as sideways foot placement – when the right foot is moved outward, the foot is also rotated clockwise, so that the center of pressure may be moved outward.

4.4 Discussion

We have mined the variability in human running data to show how humans modulate various control variables to run stably. We then used this data-derived controller on a biped model, demonstrating robustness to perturbations.

72 We have shown that humans use foot placement or leg angle control in a manner that they step in the direction of the perturbation, thereby directing the leg force so as to oppose the perturbation. This result provides an empirical basis for ad hoc assumptions about leg angle control made in previous running models [31, 101, 102].

Our inferred foot placement controller is similar to the classic Raibert controller

[42] in that the foot placement opposes velocity deviations with no sideways-fore-aft coupling, but diﬀers in that it has a dependence on vertical position perturbations.

Humans use similar foot placement control in walking, stepping in the direction of the perturbation [38, 99]. We find that the gain relating sideways foot placement and sideways velocity deviation is about 2.5 times greater than the gain relating fore- aft foot placement and fore-aft velocity deviation; a similar factor of 3 was found in walking [38], perhaps reflecting the greater sideways instability of a biped without foot placement control [31]. We also find that the recovery from a sideways perturbation is faster than from a fore-aft perturbation, perhaps reflecting lower control authority and a greater fall propensity due to a smaller basin of attraction in the sideways direction. While station-keeping was not prioritized over a single step, it may occur on a slower time-scale with a multi-step controller, not considered here.

We ﬁnd that that when a runner starts at a higher-than-normal height at ﬂight apex, or equivalently, encounters a step-down, the runner lands with a steeper leg angle and a shorter contact time. Such behavior has been observed in humans and bipedal running birds running with large unforeseen or visible step-downs [35, 103, 97].

Conversely, step-ups decrease touch-down angle, as predicted [104]. This behavior has been attributed to swing leg retraction just before foot contact [101], but our foot placement controller captures this phenomenon even without explicit leg swing

73 dynamics. These comparisons between experiment with large perturbations (5-20 cm) and our experiments based on tiny deviations (s.d. 5 mm) indicates that humans may use qualitatively similar control strategies for large and small perturbations. Our analysis may ignore some long-term trends in the control due to ﬁltering.

Our results suggest that purely spring-like leg behavior, often assumed in running models, cannot fully explain running stability. An energy-conservative spring cannot add or remove mechanical energy from the system [31, 28] and cannot explain the

GRF modulations seen in our data (Figure 4.5). Spring-like legs also rule out running on an incline. Thus, a force-generating actuator-like leg, such as used here, is needed to explain human running control. A previous article [32] fit running data to variants of the spring-mass model, allowing the spring stiffness and spring length to change during stance. In contrast, here, we have shown that a model without any spring-like assumptions can capture both the average running motion and the human running controller. Also, direct control of leg force is more parsimonious than the simultaneous control of two variables, namely, spring stiffness and length. We have shown that humans modulate GRF continuously over the whole stance phase for control, not captured by [32], who assume a sudden energy input at mid-stance for simplicity.

The experimental methods used here are simple, easily reproduced, and do not need any mechanism to apply perturbations. Thus, these methods are suitable for analyzing diﬀerences in diﬀerent populations like athletes and non-athletes, the young and the elderly, male and female runners, and adults with and without movement disorders.

While we have focused on control of stance based on the previous ﬂight apex, the center of mass state at the end of the previous stance has the same state information

74 and would allow us to infer equivalent controllers. Our flight phase durations are considerably longer than the typical short latencies in reflex or feedback loops [105], suggesting feasibility of feedback based on flight phase information. We have obtained a running controller for a simple model. Because humans have extended feet, non- point-mass upper bodies and legs with masses, the point-mass model may not capture all aspects of the running data [28, 106]. Future work will involve obtaining controllers for more complex biped models, which, for instance, might do feedback control of not just the center of mass state, but the states of individual body segments.

75 Chapter 5: Experimentally Inferred Running Controllers Implemented on Simple Biped Models

5.1 Summary

In this chapter, we implement the experimentally-derived control strategies from the previous chapter onto simulations of a simple biped model, showing that it runs stably and recovers from perturbations that are an order of magnitude larger than the natural step-to-step variability. The model predicts that humans sensitivity to perturbations are not equal in all directions. The model also naturally discovers behaviors previously observed in experiment: for instance, that humans modulate the leg angle of attack and stance durations to recover from step-up and step-down perturbations. Such human-derived control strategies for running could be used to design better robotic exoskeletons and prostheses, understand movement disorders in runners, and inform running robots that may match human running performance.

5.2 Implementing the data-derived control on simple biped models

We consider two simple models of running, similar in spirit to previous models in terms of simplicity [26, 33, 28, 107], but with controller details now inferred from our

76 experimentally-obtained linear models. Both biped models have point-mass upper

body and massless legs [107, 7], that can change eﬀective leg length like humans do

by changing leg joint angles. During ﬂight phase, the point-mass body undergoes

parabolic free ﬂight. The legs can apply forces on the upper body during stance

phase. The equations of motion are as follows:

F (x − x ) x¨ = f (5.1) m ` F (y − y ) y¨ = f (5.2) m ` F (z − z ) z¨ = f − g (5.3) m `

where F is the leg force coming from the muscle or from the telescoping leg and

xf , yf and zf are the sideways, forward and vertical positions of the foot. The variables x, y and z correspond to the position of the center of mass and ` is the

eﬀective instantaneous leg length of the model. The two models, dubbed “direct force

control model” and “muscle control model” diﬀer in how the leg force is produced

and controlled (Figure 5.1). For the muscle control model, we use a Hill muscle model

with force-length and force-velocity relations. In the “direct force control model”, the

object of control is, as the name suggests, is the leg force during stance phase. In the

“muscle control model”, the object of control is the muscle activation.

Both models have two terms in their control: (1) a feedforward or ‘nominal’

term, that depends only on the average or desired periodic motion and (2) feedback

modiﬁcation of the control in response to state deviations at ﬂight phase. Both the

nominal running motion and the feedback controller for this biped model are obtained

from the running data. We obtain the nominal running motion by making it match

properties of the steady-state running trajectory. We model the nominal leg force or

77 Two simple biped models

Telescoping leg Kneed biped with biped with direct control of force control muscle activation z (vertical) Direction of

y travel (f ys) orw wa ard ide ) x (s

Figure 5.1: Minimal mathematical biped models used for implementing control. Two simple biped models were simulated: a telescoping leg model with direct force control and a kneed biped with activation control of the muscle at the knee.

78 −1 −1 muscle activation as a two-term sine series with frequencies (2Ts) and Ts (Figure 5.2b). We parameterize the running motion using stance duration, ﬂight duration, 2D foot placement, 3D initial conditions for stance, and the coeﬃcients of the two-term sine series describing the leg force. We solve for these variables to obtain a periodic running motion that matches the forward speed, step period, step width, and peak leg force from experimental data. The runner leaves the ground when it reaches the maximum leg length, but the nominal leg length at landing is assumed to be shorter

(95%) than the maximum leg length, as seen in running data [108]. We enforce that left and right stances are mirror symmetric. Unlike most classical running models, our model’s nominal periodic motion has non-zero step width and a stance phase that is asymmetric about mid-stance. This asymmetric stance is due to unequal landing and take-oﬀ leg lengths. For details about the muscle model see Figure 5.2a.

The foot placement control is given by the linear model in equation. The leg force controller, based on apex body state, for the direction force control model has feedback control gains that are diﬀerent for the ﬁrst and the second half of stance.

The muscle activation controller muscle control model has feedback control gains that simply modify the two Fourier coeﬃcients. These control gains were derived so that the linear map from one apex to the next is the same for the model and the data

(equation). When the controller attempts to make leg force negative, we set the leg force to zero. The foot placement and leg force feedback control are activated only when the apex state deviates from nominal.

To obtain a running simulation over many steps, we break up each step into three phases: flight from apex to beginning of stance, the stance phase, and flight from the end of stance to flight apex. The control actions for the next stance are chosen

79 a) Hill-type muscle model b) Two-term Fourier

Series Contractile Two individual elasticity element Fourier components (tendon) F muscle Fmuscle 1

0 Parallel elasticity -1 0

fstance= t/Tstance

Muscle force-velocity Muscle force-length Sum of Fourier terms relation relation

Short- 1 -ening Parallel 1 elasticity Lengthening 0 -10 1 0 1 2 0 Muscle length rate 0 1 Muscle length fstance= t/Tstance

Figure 5.2: Details of the model implementations. a) The muscle in the second model is a classic Hill-type muscle [1], composed of an active contractile element, a series elastic element (tendon), and a parallel elastic element. The force FCE in the active contractile element for the muscle model depends on the muscle length ˙ `m through a force-length relationship ψ`, on the muscle length rate `m through a force-velocity relation ψv, and the activation a, so that FCE = aFisoψvψ`, where Fiso is the maximum isometric force in the muscle [1]. b) The control input for both models −1 is represented as a fourier combination of two sine waves of frequencies Tstance and −1 2Tstance. For the ﬁrst model, the force is represented by this combination and for the second model, the muscle activation.

80 at flight apex. The flight apex is usually whenz ˙ = 0; we declare the end of stance to be flight apex ifz ˙ < 0 when stance ends. The end of flight and the beginning of stance is determined as when the distance between the body and the target foot position is equal to the landing leg length. The leg length at landing is also controlled based on apex state, based on the linear model in the Results. At flight apex, if the distance to the next foot position is less than the target landing leg length, the runner immediately goes into stance.

We computed the basin of attraction by simulating for a dense 200 × 200 grid of perturbed initial conditions at ﬂight apex, perturbing two state variables at a time.

The basin reported is the set of perturbed initial conditions that did not result in the runner’s body falling to the ﬂoor in 20 steps. We may thus slightly overestimate the true size of the basin of attraction. The initial conditions in the reported basin converged to the nominal running motion.

5.3 Human-derived controller stabilizes a running model

The experimentally-derived control strategies described above are suﬃcient to control the running dynamics of a simple mathematical model of a biped. We consider a biped with point-mass upper body and massless telescoping legs capable of generating arbitrary force proﬁles (unlike a spring). We considered two versions of this biped model (Figure 5.1a), one with direct control of the leg force and another that produces leg forces via Hill-type muscles (Figure 5.2a).

The models’ ground reaction forces are similar to experimental data despite not explicitly matching the curves (Figure 5.3a). The phase-dependent ground reaction force feedback gains for the models are qualitatively similar to the gains inferred from

81 experiment (Figure 5.3b), again, despite not explicitly ﬁtting these gains. This shows

that these simple models can not only capture the average motion during running,

but also how the runner responds to deviations from the average motion.

The model’s running motion is not stable without the controller: an arbitrarily

small perturbation makes it diverge from the original running motion. With the foot

placement and leg force controller turned on, the running motion is asymptotically

stable. Figure 5.4a-b shows the model recovering from fore-aft and vertical pertur-

bations at ﬂight apex. Figure 5.5a-b shows the model recovering from a sideways

perturbations at ﬂight apex. It is a mathematical theorem that a stable periodic

motion that can reject perturbations at one phase (say, ﬂight apex) can reject per-

turbations at any phase [94]. So, it follows that our model rejects perturbations at

any phase.

The inputs to the feedback controller (x ˙ a, y˙a, za) do not include the absolute side-

ways and fore-aft position (xa, ya) of the runner. Therefore, the controller does not correct position perturbations (station-keeping). A sideways or fore-aft push to the model results in convergence to the nominal running motion, except for a sideways or fore-aft position oﬀset.

Basin of attraction of the controlled model is large

The controlled running model recovers from substantial perturbations from nominal, at least 10σ in each direction, where σ is the standard deviation of the ﬂight apex

states, a measure of the variability in our running data. Figure 5.6 shows the basin

of attraction of the steady running motion: that is, the set of all perturbations at

ﬂight apex from which the biped can recover. This basin of attraction is asymmetric

82 a) Nominal GRFs: Model vs Data

0.2 0.4 3 (right stance) 0.2 2 0 0 1 (Body Wt.) (Body

(Body Wt.) (Body -0.2 (Body Wt.) (Body -0.2 -0.4 0 00.5 1 00.5 1 00.5 1 Average Vertical GRF Vertical Average Average fore-aft GRF Average Average sideways GRF sideways Average stance fraction (fstance ) stance fraction (fstance) stance fraction (fstance)

Force control Data Muscle control model model b) Feedback gains for GRF changes: Model vs Data 0 1 15 10 -1 0 5 -2 -1 0 -3 -2 00.5 1 00.5 1 -5 00.5 1 stance fraction (fstance ) stance fraction (fstance) stance fraction (fstance)

Figure 5.3: Comparing the simple biped models and human running data. Both models ﬁt the experimental GRFs and feedback gains reasonably well, despite not having been made to match them explicitly: a) Mean GRFs over stance in three directions. GRFs are reported as a fraction of body weight. b) Phase-dependent feedback gains describing the sensitivity of the sideways GRF to sideways velocity perturbation, fore-aft GRF to a fore-aft velocity perturbation, and the vertical GRF to vertical position perturbation at the previous ﬂight apex. For standard deviations of the experimental curves, see Figures 4.2 and 4.5.

83 Simulated bipedal runner can run stably and recover from large pushes a) Forward speed decay after a forward push

1.0

0.9

y dot (forward) 0 5 10 time

!ight phase trajectory leg directions at beginning and end of stance stance phase trajectory foot positions

b) Recovery from an upward push results in landing steeper

vertical perturbation (equivalent to a step down)

perturbed steady state 1 normal more steps z (vertical) 0 steeper landing foot y (fore-aft) angle than nominal position

Figure 5.4: Stability of controlled simple biped runner in response to a forward or vertical push. We illustrate the stability of the running model by showing how large perturbations at ﬂight apex decay. a) Decay of a forward velocity perturbation. b) Sagittal view of running, recovering from an upward position perturbation. On the ﬁrst step, the leg touch-down angle is steeper than the touch-down angle during unperturbed running and the contact time is shorter. All quantities are non-dimensional.

84 a) Sideways speed decay after a sideways push

0.1

x dot (sideways) x 0 5 10 time

!ight phase trajectory leg directions at beginning and end of stance stance phase trajectory foot positions

c) Top view: Decay of a sideways push

foot position -0.2 Unperturbed periodic motion 0

x (sideways) 0.2 Forward direction

-0.2 Recovery after sideways perturbation

x (sideways) 0.2 0 foot 2 y (forward) 3 rightward position velocity moves perturbed rightward here

Figure 5.5: Stability of controlled simple biped runner in response to a sideways push. a) Decay of a sideways velocity perturbation. b) Decay of a sideways velocity perturbation. c) Top view center of mass trajectory, showing the unperturbed running motion and a running motion that recovers from a rightward velocity perturbation. The rightward perturbation elicits a rightward foot placement, compared to the nominal foot placement during unperturbed running. All quantities are non-dimensional.

85 about the origin. For instance, the biped recovers from a larger rightward than a leftward velocity perturbation just before the right stance (Figure 5.6a-b). It recovers from a larger range of perturbations in the fore-aft direction, compared to the sideways direction (Figure 5.6a). It recovers from larger upward perturbations than downward perturbations at ﬂight apex (Figure 5.6b-c). These upward or downward perturbations can be interpreted as the ﬂoor moving down or up, respectively. Thus, the model is much more robust to step-downs than step-ups.

Running stably cannot be purely passive and involves active leg work

The classic spring-mass model of running is an energy-conservative system and cannot produce net positive or negative work. Here, in contrast, the derived controller can and does net leg work to reject perturbations. Figure 5.7 shows the leg work- loop for the unperturbed run (net zero work) and when positive perturbations are applied to sideways and fore-aft velocities, and vertical positions. All such positive perturbations result in net negative work on the ﬁrst step after the perturbation, reﬂected in the work-loops with net negative area within them. Such net leg work during recovery is inconsistent with a passive spring-like leg.

Explaining variability: Muscle-driven running model does not fall despite noise

To simulate the step-to-step variability in real human running, we added ‘noise’ to our foot placement and leg forces (for the direct force control model) or muscle activations (for the muscle-driven model). This noise is meant to model the phenomenon that desired muscle forces tend to deviate from actual muscle forces due to motor

86 Basin of attraction for the running model with respect to !ight apex perturbations

a) sideways-forward basin b) sideways-vertical basin

0.4 0.4 0.6 0.4

0 0 0.2 0.2

0.1 0.2 0.2 0.2 -0.4 -0.4

-0.4 0 0.4 -0.40 0.4

c) forward-vertical basin

0.4

0.4 0 0.2

-0.4 0.1 0.4

-0.40 0.4

Figure 5.6: Basins of attraction. The basin of attraction for steady running: that is, the set of all perturbations at flight apex from which the runner can recover back to steady-state. Each of the basins correspond to recovery or non-recovery from perturbations added at the flight apex before a right stance. The basin for perturbations at an apex preceding a left stance is identical, except for being mirror image for ∆x ˙ a. The three panels in the figure show basins of attraction for combined perturbations in: a) sideways velocity - forward velocity, b) sideways velocity - vertical position and d) forward velocity - vertical position.

87 Non-zero leg work modulations are needed to control perturbations

a) Work loop for forward velocity perturbation

3 forward velocity perturbation = 0.05 2

leg force 1 unperturbed average motion 0 0.85 0.9 0.95 1 leg length

b) Work loop for sideways velocity perturbation

3 sideways velocity perturbation = 0.05 2

leg force 1 unperturbed average motion 0 0.850.9 0.95 1 leg length

c) Work loop for forward velocity perturbation

3 vertical position perturbation = 0.05 2

leg force 1 unperturbed average motion 0 0.850.9 0.95 1 leg length

Figure 5.7: Work loop and net leg work. Work loops for the muscle-driven biped model without perturbations and with perturbations in the a) fore-aft velocity, b) sideways velocity, and c) vertical position. The work loop plots force versus leg length and the signed area included in it is the net work performed by the leg. 88 noise [23]. We use the unexplained variance in the foot placement and leg forces from experimental regressions as a simple model of this intrinsic noise. We re-simulated the two point-mass running models for hundreds of steps, in the presence of these noisy foot placements and leg forces or activations. We find that while the direct leg force control model falls down, the runner with muscles does not fall down for hundreds of steps despite the noise. The stable motion of the center of mass in the presence of noise-like perturbations is shown in Fig 5.8. The variability in the center of mass state at flight apex for the model (Fig 5.9) as a result of the simulated noisy control is qualitatively similar to the variability found in experiment. The model is also able to run without falling despite vertical position perturbations at flight apex, which are mathematically equivalent to uneven terrain. Thus, even though the model was derived using data on horizontal ground, it is capable of running robustly on uneven terrain. The muscle-driven model is robust to motor noise presumably because of the stabilizing properties of force-length and force-velocity relations [109, 110].

89 Runner model does not fall despite noisy actuation and shows step-to-step variability

Running Running Stance phases Flight phases with without noise noise 0.04 0.98 0.98 0.02 0.94 0.94 0 0.90 0.90

(sideways velocity) (sideways -0.02 0.86 (forward velocity) (forward

0.86 (vertical position) 0 5 10 15 0 5 10 15 05 10 15 time t time t time t

Figure 5.8: Running stably with noise. Multiple steps of the biped model running in the presence of noisy foot placement and muscle activations (green and pink). Periodic nominal motion in the absence of noise (black).

90 Model step-to-step variability under noisy actuation

0.3 0.3

0 0 fore-aft (m/s) velocity -0.3 fore-aft impulse (m/s) Δ Δ Δ -0.3

-0.3 0 0.3 -0.3 0 0.3 Δ sideways velocity (m/s) Δ sideways impulse (m/s)

0.08

-0.08 -0.08 0 0.08

fore-aft foot placement (m) fore-aft placement foot Δ sideways foot placement (m) Δ

Figure 5.9: Model-generated state variability at flight. Deviations in flight apex state, GRF impulse, and foot placement from nominal for a 1000 steps (500 left, 500 right), showing behavior analogous to Figure 4.2. This variability is not explicitly specified, but instead emerges from the interaction between the motor noise and the controlled dynamics.

91 Chapter 6: Predicting Steady-state Motion Adaptations for Asymmetric Walking

6.1 Summary

In this chapter, we study three types of asymmetric walking behaviors: (i) asymmetric leg masses, (ii) asymmetric leg lengths and (iii) asymmetric leg speeds (on a split-belt treadmill). We use energy optimization of a simple point mass model with a telescoping leg to predict the asymmetric gaits. We ﬁnd that the point-mass model predicts increased stance time for the unloaded leg and unchanged stance time for the loaded leg. The model also predicts a decrease in optimal walking speed for the asymmetric mass condition compared to walking without any weights. Experiments on human subjects walking with asymmetric masses show increased stance times for the unloaded leg, as predicted by the model. However, a decrease in stance time for the loaded leg is seen in the experiments but not captured in the simple model. A signiﬁcant decrease in preferred overground speed is seen for heavier (10 lb and 20 lb) weights, as qualitatively predicted by the model. For the asymmetric leg length condition, the simple model predicts a shortened-leg gait where the longer leg shortens to match the length of the shorter leg. For the split belt condition, the model predicts a decreased stance time and a longer step length for the fast leg and the

92 reverse for the slow leg. These predictions match past experimental observations in split-belt walking experiments by other researchers. Moreover, the model’s prediction based on energy-optimality is also reﬂected in experiments, which ﬁnd a decrease in total metabolic cost and less net mechanical work in the fast leg, as people adapt to the asymmetric gait on the split-belt treadmill.

6.2 Experimental methods

6.2.1 Experiment design

We perform human subject experiments to test predictions from the simple models. Four subjects participated with informed consent in treadmill and overground walking trials wearing asymmetric weights. We have IRB approval to conduct all the experiments described below. Before the experiment, measurements of key anatomi- cal landmarks were made through static motion captures of the subjects or through direct measurements. Photographs were taken in a few standard perspectives with a calibrated background, so that further body measurements can be inferred later, if needed. The following treadmill walking trials were conducted on a split-belt treadmill (Figure 6.1a, separate treadmill belt for each leg) with built-in force plates, at two to three constant speeds:

1. No mass on either leg

2. No mass on one leg and a mass between 5 to 20 lb on the other

3. Mass on both legs between 5 to 20 lb

We also performed overground walking trials to measure the preferred walking speed over a suﬃciently long distance, for the various symmetric and asymmetric load.

93 Ground reaction force data and motion capture data were collected for all the above

trials.

6.2.2 Equipment and data-collection

The motion of the various segments of the body were tracked using an eight-

camera marker-based motion capture system (Vicon T20, 500 Hz at full 2 megapixel

resolution, about 0.2 mm marker error). The motion of a given segment will be

approximated by using a redundant marker set on each salient body segment (hip,

thighs, shanks and feet). A Bertec split-belt treadmill with built-in force plates was

used for all the treadmill trials. The subjects wore a safety harness on their torso,

suspended from an eye-nut on the ceiling. These trials were ﬁve minutes long to

ensure that the VO2 dynamics go to steady-state.

6.3 Computational methods

6.3.1 Optimization of a simple model

We consider a two-dimensional point mass biped model with a massless telescoping

leg identical to the model described in [7] and shown in Figure 1.1. The equations of

motion for this simple model are given by:

F (y − y ) y¨ = f (6.1) m ` F (z − z ) z¨ = f − g. (6.2) m ` Here, F is an arbitrary force applied by the telescoping leg on the point mass, y and z are the are the fore-aft and vertical position of the two dimensional model and yf and zf are the fore-aft and vertical positions of the stance foot. Also, ` is

the instantaneous leg length during stance. We consider two steps of walking and

94 a) Experimental setup

Cameras

Treadmill

b) Sample force plate data d) Sample motion capture data

900

600 Left Right leg 300 leg Body segment

0 Markers 0 1 2 Time (in sec) (4 per segment) c) Ankle weight and markers Vertical ground reaction force (in N) reaction force ground Vertical Ankle Weights

Re!ective marker

Figure 6.1: Experimental setup and sample experimental data. a) Lab setup with motion capture cameras from Vicon and an instrumented Bertec split-belt treadmill. b) Sample ground reaction force captured by the force plates in the treadmill. c) Mass was added to the leg using ankle weights. d) Sample capture from post- processed marker data with seven body segments and four markers on each segment.

95 parameterize each step by the body initial state, piecewise-linear leg forces, step du-

rations (stance durations) and foot position. We then determine the optimal values

of these variables that minimize a metabolic cost-like objective function subject to

various linear and nonlinear constraints. Speciﬁcally, continuity across a step, peri-

odicity over two steps, and leg length constraints are imposed, in additional to simple

bounds on leg forces. When ﬁnding the optimal stance times, the forward speed is

also constrained. For this simple model, the objective function is the sum of two

terms: a stance phase cost (due to the leg on the ground) and a leg swing cost. The

stance phase cost is related to the work done by the telescoping leg when it exerts a

force F while elongating at the rate `˙ [7]. The swing cost is modeled as the kinetic

energy expended in instantaneously increasing the foot speed to vf at the beginning

of stance and instantaneously decreasing it to 0 at the end of stance [66]. The positive

and negative muscle eﬃciencies, η+ = 0.25 and η− = 1.2, are obtained from [111] to

ﬁnd the metabolic-equivalent work done. The objective functions used for the stance

Wst and swing Wsw phases are given by,

Z 1 1 1 1 1 W = [F `˙]+ + [F `˙]− dt W = + m v2 (6.3) st η+ η− sw η+ η− 2 f f two steps

˙ + ˙ − [F `] and [F `] represent the positive and negative parts of the power P . Also, mf

is the mass of the swing foot and so, the cost of swinging the leg is modeled as the

cost to take the foot mass through a certain distance in a certain time.

6.3.2 Modiﬁcations for modeling the asymmetry

We introduce the following types of asymmetry in the model: (i) adding a mass to

the swing leg (ii) increasing the length of the stance leg or (iii) changing the speeds at

which the two feet move relative to the body. In the model, asymmetry in leg mass

96 is imposed by increasing the the mass of the swing foot in the objective function for

the swing cost in equation 6.3. Asymmetry in leg length is imposed by increasing the

maximum inequality bound for the leg length during stance for one of the legs.

For the split-belt model, asymmetry is imposed in the velocity constraint, the leg

swing cost and the equations of motion. For split-belt walking, the average velocity

of the person walking on the treadmill is taken to be 0, implemented as a nonlinear

inequality constraint. During stance, the feet move backward at unequal speeds

corresponding the the speed at which the two belts are moving. The leg swing cost

from equation 6.3 is changed for the split-belt model by changing the forward velocity

of the swing foot vf depending on whether the fast or slow leg is in stance. For the

vfasttfast fast leg, vf = , where vfast is the speed of the fast belt and tfast and tslow are the tslow stance times on the fast and slow belts respectively. The expression for the swing foot speed for the slow leg is analogously obtained. Finally, in the equation of motion 6.1, the stance foot’s fore-aft position yf takes into account backward movement of the stance foot by subtracting vfastt or vslowt from the foot position yf at the beginning

of the stance phase.

6.3.3 Formulation of the optimization problem

A reasonably general trajectory optimization problem is of the following form. T Minimize a functional J where J = R h(x(t), u(t), t) dt such that x(t) and u(t) are 0 related by some ordinary diﬀerential equations (e.g. rigid body equations of motion and muscle dynamics) that can be written in the form: A(x(t), t)x ˙ = B(x(t), u(t), t), where x(t) ∈ R2n is the vector of state variables (body segment positions and velocities) at time t, u(t) ∈ Rm is the value of the control at time t (e.g., muscle forces or

97 muscle activations). In addition there may be conditions and constraints on the state x(t) (e.g., range of motion inequality constraints) and the control u(t) (e.g., muscle force bounds, including muscle force positivity), including boundary conditions of various kinds (e.g. periodicity constraints).

6.3.4 Optimization convergence and uniqueness

Techniques for such optimization problems based on necessary conditions of optimality such as the Pontryagin minimum principle (a generalization of the classical calculus of variations) are generally difficult to apply to such problems, both because of their hybrid nature, as well as due to the difficulty in solving the resulting two-point boundary value problems. So instead, the infinite-dimensional continuous- time problem is converted to a finite dimensional optimization problem by discretiz- ing the muscle forces as piecewise linear, with continuity being enforced at the grid points. Then, we directly solve the finite dimensional optimization problem using standard nonlinear programming software SNOPT [112], an implementation of Se- quential Quadratic Programming. This method has been demonstrated to be robust and reliable for smooth large-scale problems of the kind we wish to solve [91]. If the numerical optimization converges to more than one local minimum when started from multiple initial seeds, we will pick the local minimum with the lowest cost.

6.3.5 Comparison with the simple model

The model predictions detailed above will be compared both qualitatively and quantitatively to the experimental results. The results from the models and experiments will be compared in their non-dimensional forms. We will compare experimental data with the optimal step length, step durations, and the walking speeds

98 predicted by the simple model. The leg force and the leg length as a function of

time during gait are more idealized for the simple model, so no quantitative match

is expected. But a qualitative comparison of these simple model predictions will be

made to the similar measures from experiments, to see if the simple model is able to

explain the trends in the experimental results.

6.4 Experimental observations on steady-state walking with asymmetric weights

We performed some pilot experiments on four subjects. The subjects walked with

ankle weights ranging from 2 to 20 lb on one or both legs, on the split-belt treadmill

while the motion and ground reaction forces were captured. The ground reaction

force data was used to calculate the stance time for each leg, deﬁned as the average

time for which each leg remains on the ground in one stride. The regressed relation

between non-dimensional left and right stance times (t` and tr) and masses added to

the left and right legs (m` and mr) is given by,

t` = −1.5m` + 0.6mr and tr = 0.5m` − 1.3mr. (6.4)

The difference in coefficients for the left and right legs are likely due to the small sample size. We performed overground walking experiments in a corridor over a sufficiently long distance of 30 m, to measure the preferred walking speeds for the same four subjects who performed the treadmill trials. The regression between overground preferred walking speeds (vpref ) and normalized added mass for both legs, is given by:

vpref = −0.6m` − 0.4mr + 0.4 (6.5)

This indicates that the preferred speed of the subjects decreased as the mass of the swing leg was increased.

99 6.5 Simple model predictions for walking with asymmetric leg masses and lengths

We performed optimization over two steps for the point mass telescoping leg model for leg mass and leg length asymmetries. The asymmetric mass condition results in an increased swing cost on one side. This increased swing cost results in an increased swing time for the loaded leg and a correspondingly increased stance time for the unloaded leg (see Figure 6.2a), as has been seen in experiments [113]. The stance time for the unloaded leg increased roughly quadratically with an increase in the mass added to the swing leg. This model’s prediction agreed with our experimental

ﬁndings that stance time decreases for the loaded leg. However, the experiment also shows a corresponding increase in stance time for the unloaded leg in comparison to the symmetric walking case, which was not seen in the model. An optimization to

ﬁnd the preferred walking speed to cover a certain distance, predicts a decrease in optimal walking speed with mass added to the swing leg (Figure 6.2b), as seen in experiment. Although the simple model appears to capture many of the trends seen in the experiments, it appears to over-predict the changes due to asymmetry.

For the asymmetric leg length condition, a shortened-leg gait, such that the longer leg shortens to match the length of the shorter one, was optimal and is shown in Figure

6.3. This might just be due to the fact that this model has no actuator at the knee and so, is unable to capture the increased cost for shortening of the leg. These asymmetric leg length predictions can be tested with future experiments.

100 a) Simple model asymmetric leg mass results (one leg loaded) 2.8 1.4 2.6 1.2 Optimal results Center of mass trajectory 1 2.4

0.8 2.2 Quadratic !t to optimal 0.6 results 2 0.4 1.8 0.2 Stance time for unloaded leg unloaded for time Stance 0 1.6 -0.4 -0.20 0.2 0.4 0.6 0.8 1 1.2 0 0.05 0.1 0.15 Stance phase Stance phase Mass fraction added to swing of lighter leg of heavier leg

b) Simple model optimal asymmetric walking speed results

Contour plot for optimal walking speed (non-dimensional) 0.15

0.14

0.1

0.15

0.05 0.16

mass added to right foot right to added mass 0.17 Δ 0.18 0.19 0.21 0 00.05 0.1 0.15 Δ mass added to left foot

Figure 6.2: Simple model predictions for walking with asymmetric leg mass. This ﬁgure shows the results from optimizing the simple model for mass asymmetries. a) When a mass is added to the swing leg, the stance time for the other leg increases quadratically. b) Preferred walking speed decreases gradually as the mass of either leg is increased.

101 Simple model asymmetric leg length results Center of mass trajectory Contour plot of stance times (non-dimensional) 0.7 0.6 0.2 2.0 0.5 1.9 0.1 0.4 1.8 0 0.3 1.7 Longer leg 0.2 matches shorter -0.1 1.6

Shorter leg’s length Right Leg

0.1 leg’s length during Δ -0.2 1.5 stance phase stance 0 1.4 -0.3 -0.2-0.10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -0.3 -0.2 -0.10 0.1 0.2 Δ Left Leg length

Figure 6.3: Simple model predictions for walking with asymmetric leg length. This ﬁgure shows the results from optimizing the simple model for leg length asymmetries. When the length of one leg is increased, the longer leg shortens to the eﬀective length of the shorter leg.

6.6 Simple model predictions for split-belt walking

6.6.1 Optimal stance time asymmetry for split-belt walking predicts the observed trends

The model’s energy-optimal split-belt gait predicts walking with a shorter stance time on the fast belt and a longer stance time on the slow belt (Figure 6.4). This prediction agrees with what has been observed in experiments in the past [61, 63].

Further, the model predicts that the degree of stance time asymmetry increases as the diﬀerence in the belt speeds increases. For a stance time tf on the fast belt and ts on the slow belt, stance time symmetry is deﬁned as follows:

t − t STS = s f . (6.6) ts + tf

The model makes a unique (fast, slow) step time prediction for each belt speed combination and the relationship between the belt speed combination and the stance time

102 2

Slow leg stance time Fast leg stance time 1.5 Total stride time Stance Time

0.5 0.50.6 0.7 0.8 0.91 1.1 1.2 1.3 1.4 1.5 Fast Leg Speed

Figure 6.4: Optimization of a simple model predicts stance time asymmetry for split-belt walking. The slow belt speed for the model is held constant at 0.5 m/s and the fast belt speed is swept through the speeds shown on the x-axis in the ﬁgure. We ﬁnd that the stance time is shorter on the fast belt and longer on the slow belt. This prediction is in agreement with past experimental observations.

combination is non-linear. However, the stance time on the slow belt is always greater than that on the fast belt. Most papers only explore a single (1.5,0.5) m/s belt speed combination [61, 63]. Because of this, the predictions made here about how the step time symmetry changes with belt speed combinations could be tested with further experiments.

103 6.6.2 Optimal step length asymmetry for split-belt walking predicts the observed trends

Our model predicts walking with a longer step length on the fast belt and a shorter step length on the slow belt (Figure 6.5). This prediction is in contrast to past experimental observations that suggested that people tend towards symmetric step lengths after walking on the split-belt treadmill for some time. However, more recent work with much longer time for adaptation and repeated trials over many days suggest that people indeed take longer steps on the fast belt [2], as we predict here.

As in previous papers, for a step length `f on the fast belt and `s on the slow belt, step length symmetry is deﬁned as follows:

` − ` SLS = f s . (6.7) `s + `f

In contrast to the case for stance time symmetry, the model predicts that the of step length symmetry is positive and plateaus oﬀ after an initial increase, as the diﬀerence in the belt speeds increases. Again, the predictions made here about how the step length symmetry changes with belt speed combinations can be tested with further experiments.

6.6.3 The model predicts that observed metabolic cost could go lower with more time for adaptation

A previous experimental split-belt study found that the metabolic cost of walking decreases through the adaptation phase of split-belt walking [49]. Since we use energy- optimality as the driving hypothesis to make predictions for our model, we can make predictions about the optimal metabolic cost of the model and compare it to the observed metabolic cost at the end of the adaptation phase. As shown in Figure

104 0.35

0.3

0.25 Slow leg Fast leg 0.2 Step Length

0.15

0.1

0.05 0.50.6 0.7 0.8 0.91 1.1 1.2 1.3 1.4 1.5 Fast Leg Speed (Slow leg is maintained constant at 0.5 m/s)

Figure 6.5: Optimization of a simple model predicts step length asymmetry. The optimal gait of the simple model predicts that people will take longer steps on the fast belt and shorter steps on the slow belt.

105 6.6, we ﬁnd that the optimal metabolic cost for the (fast-slow) split belt case is between the optimal metabolic cost for the (slow-slow) and the (fast-fast) tied belt cases, as observed previously in experiment. Moreover, if vf is the fast belt speed, vs is the slow belt speed and the average of both is vavg, the model predicts that the optimal metabolic cost of walking at tied belt speeds (vavg,vavg) is higher than the metabolic cost of walking at the split belt speeds. This prediction disagrees with the experimental observation of metabolic cost at the end of adaptation in previous work. However, similar to the step length prediction, more time for adaptation may decrease the metabolic cost further, thus agreeing with our prediction.

6.6.4 Longer step length on fast belt is accompanied by lesser net work done on the fast belt

As mentioned above, the model’s optimal solution produces longer step length on the fast belt compared to the slow belt. Also, as a result of this, the net mechanical work done on the fast belt is less than that on the slow belt and the two are proportional (see Figure 6.7). This decrease in net mechanical work is one simple hypothesis that could explain the positive asymmetry predicted by the model. The decreased net mechanical work agrees with recent simple experimental calculations of net mechanical work performed in [2]. This result is also intuitive as, the fast belt helps you pull your leg back during stance and you may potentially need to resist this by performing more negative work. Because of this, the net work would be lower on the fast belt compared to the slow belt. In the past, it has been found that decreasing step length asymmetry, as shown by the subjects during the adaptation phase, is correlated with a decrease in metabolic cost. Our model also ﬁnds a similar relationship between step length asymmetry and metabolic cost.

106 0.08 fast-fast condition

0.07

0.06

0.05

0.04

0.03 mean-mean condition Optimal Metabolic Cost Optimal

0.02

0.01 slow-slow condition

0 0.5 0.6 0.7 0.8 0.91 1.1 1.2 1.3 1.4 1.5 Fast Leg Speed (Slow leg is maintained constant at 0.5 m/s)

Figure 6.6: Optimal metabolic cost of walking on a split belt treadmill. The optimal cost predicted by the model for walking on a slow-fast split-belt paradigm is in between the optimal cost for the corresponding slow-slow and fast-fast case and lower than the mean-mean condition.

107 -3 10 2 Slow leg Fast leg 0

-2

-4

-6 Individual stance net work

-8

-10 0.5 0.6 0.7 0.8 0.91 1.1 1.2 1.3 1.4 1.5 Fast Leg Speed (Slow leg is maintained constant at 0.5 m/s)

Figure 6.7: Net leg work prediction for walking on a split-belt treadmill. The model predicts that the net leg work is lesser for the leg on the fast belt compared to that on the slow belt. This prediction is similar to recent experimental observations made by [2].

108 Optimization of the simple biped model, described above makes prediction about various aspects of the ‘adapted’ split-belt walking behavior. Predictions made about stance time, step length, metabolic energy cost and mechanical work are compared to previous experimental observations. Some of the predictions made by the model can be tested by further, more detailed, experiments.

6.7 Discussion

Here, we have predicted trends in asymmetric walking adaptations for diﬀerent types of asymmetries without modifying the model (except to incorporate the changed dynamics). This modeling approach is in contrast to the more complex models that have been used for asymmetric walking in the past [114, 115]. Moreover, we make these predictions for diﬀerent types of asymmetries without changing the overarching hypothesis of the model, unlike past models which only make predictions for one type of asymmetry and with changes to the hypothesis (for eg. minimizing joint forces versus minimizing metabolic energy) [14, 115].

While energy-optimization based models have been used to explain common daily walking behaviors like walking at a constant speed [17, 7] and walking with changing speeds [74], much less work has been done to build predictive models of gaits that people converge to after some learning and exploration, in novel locomotion situations.

For split-belt walking, it has been found that people minimize metabolic energy cost as they adapt their gait to the paradigm [49]. Similar evidence for energy optimization has been found when people learn to explore a modiﬁed metabolic cost landscape [50].

Also, people have found that mechanical work, which is closely linked to metabolic energy cost, is correlated with some behaviors in split-belt walking adaptation [2].

109 In summary, we have used energy-optimality to explain the asymmetric steady-state behaviors that people adapt to in a novel situation.

Here we have used only simple models and have successfully predicted some trends in asymmetric walking. This lends some support for the use of the energy-like objective functions for future implementation on more complex models. Thus, future work might involve repeating the energy optimal walking predictions using more complex models such as shown in Figure 6.8.

110 a) Symmetric b) Asymmetric muscle properties

c) Asymmetric leg d) Asymmetric leg mass length

Figure 6.8: Complex multibody models of human walking with muscles. The ﬁgure shows a more complex 7-link model of walking and the ways in which it could be modiﬁed to model asymmetric gait. a) This model has sixteen muscles on both legs. The muscles are shown only on one leg, the other leg has identical muscles. Asymmetries are introduced in this model by asymmetrically changing b) muscle properties, c) adding masses or d) varying segment lengths. 111 Chapter 7: Conclusions

We have studied diﬀerent aspects of adaptive behaviors in human locomotion; non-constant speed, transient and asymmetric behaviors in human walking and running. Overall, we have found that the minimization of metabolic energy cost is a good qualitative predictor of trends in observed walking behaviors for steady-state, periodic movements (like constant speed walking and walking with oscillating speeds) or learned movements (like split-belt walking). In Chapters 2 and 3, we found that minimization of the total metabolic cost i.e., sum of constant speed and changing-speed cost, of a short walking bout predicts the speed at which people choose to walk the bout. For very short bouts, where the metabolic cost of changing speeds dominates the total cost, people choose to walk at a lower speed as predicted by the energy- minimization model. In Chapter 6, we found that energy-minimization predicts experimentally observed trends in stance time and step length choices for walking with asymmetric weights and asymmetric foot speeds. Moreover, energy minimization predicts the overground walking behavior in both able-bodied and amputee individuals.

In the case of asymmetric foot speeds, computational energy-minimization of a simple model predicts the behavior that people tend towards after a long adaptation over many days. This result suggests that, when adapting to a novel paradigm, people minimized metabolic energy cost over a longer timescale.

112 We have studied the control strategies that are hidden within the step-to-step transient variabilities in human movement. In Chapters 4 and 5, we found that even very small deviations (less that 0.1% of the mean) in the motion and forces of a person running contain information about stability and control. We discovered low- dimensional control hidden in the high-dimensional variability present in the running data. We found that the center-of-mass motion is a good predictor of the control behaviors that people show, perhaps indicating that the center-of-mass deviations are used for feedback or feedforward-based control in human movement. On mining the variability in the running data, we discovered a decoupled control of deviations in sideways and fore-aft directions. We implemented this experimentally-inferred controller on simple biped models and were able to stabilize the models for perturbations up to ten times larger than the variability from which the controller itself was derived.

This result points to the fact that the control strategies inferred from small intrinsic noise-like deviations may be capable of predicting responses that people have to larger external perturbations.

We have shown that simple models can be useful to make predictions about the movement behaviors that people adapt to in diﬀerent situations as well as to gain insights into the mechanisms underlying those behaviors. In Chapter 2, we were able to explain the metabolic cost of changing speeds by using very simple kinetic energy- based and step-to-step transition cost-based models of human walking. Moreover, by minimizing a simple kinetic energy-based formulation of metabolic cost, we were able to explain the overground walking behavior choices of both able-bodied (Chapter 2) and amputee individuals (Chapter 3). In fact, for the amputee subjects, we were able to use a simple model to understand whether the cost of constant speed or the

113 cost of changing speed largely inﬂuences overground walking behavior. In Chapter

5, we found that a very simple model with just one muscle added at the knee is able to remain stable to continuous noise-like perturbations that a muscle-less model is unable to withstand. This result, although derived from a very simple model, suggests that the intrinsic dynamics of a muscle might be crucial to withstanding continuous noise-like perturbations.

Many of the methods used here and the insights found have implications for gait rehabilitation therapies. In Chapter 4, we are able to infer insights into stability and control without applying any external perturbations. Such a non-invasive measure could be potent in its use to study and rehabilitate the gait stability in people with movement disorders and injuries. Moreover, once validated in a lab setting, such variability-based control can be used to predict falls in elderly individuals by appropriate coding of wearable devices. In Chapter 6, we are able to predict trends in asymmetric walking behavior using simple models. On further validation and improvement, such models can be used to test the effectiveness of a proposed rehabilitation therapy before it is implemented. Specifically, we have made predictions about the effectiveness of a split-belt treadmill walking paradigm, which has been used to rehabilitate stroke patients. The fact that these predictions accurately predict the effect on people’s gait after many days of training suggest that such models can be used to predict the way people will respond to novel rehab environments.

All of our results can be extended to the understanding of movement behaviors in other animals and in robots. In Chapters 4 and 5, some of the control behaviors that we ﬁnd in humans reﬂect past results in running birds. Also, the strategies that people use to run without falling down are similar to some strategies that roboticists

114 have used to prevent their running or hopping robots from falling. Our human-derived controller may be appropriate for implementing on a robotic exoskeleton in order to obtain more seamless interactions with its human wearer. In Chapter 6, we have used optimization to predict how people adapt their behavior when the dynamics of their movement are altered. This type of modeling will be very beneﬁcial for robots, where the robot may also need to adapt to changing dynamics as it moves around in the world (due to some wear and tear, say).

In the future, we would like to extend the work done here with more complex models and a wider variety of experiments as described in more detail in the respective chapters. As an extension to Chapter 2, we could further measure the metabolic cost of changing speeds for a wider range of movement speeds (including running).

This will help us predict a wide variety of human behavior where such acceleration- deceleration dominates without any extrapolation. In extension to Chapters 4 and

5, we could use modern machine learning methods to infer a more exhaustive (but perhaps, not much more insightful) controller based on some unsupervised learning methods. Also, we could implement the simple controller derived here on more complex models of human running to see how the range of stability changes as the model’s complexity changes. In other words, how well can a simple controller stabilized models of increasing degrees of complexity? Finally, we can extend the work done in

Chapter 6 by using more terms in the objective functions such as stability and joint forces or by using more complex dynamics (many muscles and segments).

115 Appendix A: Optimization-based Predictive Model of Walking with Changing Speeds

A.1 Deriving step-to-step transition work for changing speeds

The mechanical work done to redirect the velocity of the center of mass from downward to upward when transitioning from one step to the next is thought to be a major determinant of the metabolic cost of walking at a constant speed [116, 117, 118].

Mathematical expressions for this step-to-step transition cost have previously been derived for such steady state walking, with the simplifying assumption that humans walk with an inverted pendulum gait [119, 116, 117, 118]. When a person walks at continuously varying speeds, as in our experiments, the step-to-step transition will include a change in both the magnitude and direction of the COM velocity. Here, we derive an expression for the work done in the step-to-step transition when walking at changing (non-constant) speeds, thereby generalizing previous work [119, 116, 117,

118]. We allow that the length of the leg to change during the step-to-step transition

(unequal θbefore and θafter) while having constant leg length during the stance phase.

Push-oﬀ before heel-strike. Figure A.1 describes the transition from one inverted pendulum to the next using push-oﬀ and heel-strike impulses; panels a-c describe

116 situations in which the push-oﬀ happens entirely before heel-strike, which we consider

ﬁrst. In particular, Figure A.1a-b shows a ﬁnite reduction in speed being accomplished

during the step-to-step transition, with push-oﬀ before heel-strike. We focus on Figure −→ A.1b in the following derivation. Here, vector OA with magnitude OA = Vbefore and

making angle θbefore with horizontal, is the body velocity just before push-oﬀ at the −→ end of one inverted pendulum phase. Vector OC, with magnitude OC = Vafter and

making angle θafter, is the body velocity just after heel-strike at the beginning of the

next inverted pendulum phase. −→ −→ A push-oﬀ impulse is applied along the trailing leg to change velocity OA to OB, −→ along AB. Then, a heel-strike impulse is applied along the leading leg to change −→ −→ −→ velocity OB to OC, along BC. The push-oﬀ positive work Wpos is the kinetic energy

−→ −→ 1 2 1 2 change from OA and OB given by 2 mOB − 2 mOA and the heel-strike negative work

−→ −→ 1 2 1 2 is the kinetic energy change from OB to OC given by Wneg given by 2 mOC − 2 mOB , which simplify to:

1 1 W = m (AB)2 and W = m (BC)2 respectively. (A.1) pos 2 neg 2

As opposed to the steady walking situation [119, 116, 117, 118], when changing speeds,

the step-to-step push-oﬀ positive work Wpos and the step-to-step heel-strike negative

work Wneg will be unequal.

First, we note that angle that in triangle EBD, ∠DEB = π/2 − θafter, ∠BDE =

π/2 − θbefore, and ∠EBD = θafter + θbefore. We use the geometric relations that AB =

AD-BD, BC = EB+CE, AD = OA tan θbefore, CE = OC tan θafter,

OA OC BD cos θ EB cos θ ED = − , = after , and = before cos θbefore cos θafter ED sin (θafter + θbefore) ED sin (θafter + θbefore) (A.2)

117 a) Hodograph for unsteady walking -- speed decrease (push-o! before heel-strike) velocity just after Velocity change due to heel-strike heel-strike impulse along leading leg Velocity change due to velocity push-o! impulse q q along trailing leg b a just before push-o!

Step to step transition with push-o! before heel-strike (details of panel-a) b) Slowing down C c) Speeding up C

B O qa E D

qb O qa D E

B qb A q q b a A qb qa

d) Step to step transition with heel-strike before push-o! Q Slowing down C G

O qa F

qb P A

qb qa

Figure A.1: Step-to-step transition to change speed. a) The walking motion is assumed to be inverted pendulum-like with the transitions from one inverted pendulum step to the next accomplished using push-off and heel-strike impulses. Overlaid is the ‘hodograph’ (a depiction of velocity changes) during the step-to-step transition, when push-off happens entirely before heel-strike. b) Details of the velocity changes during step-to-step transition, with push-off before heel-strike and slowing down. c) Analogous to panel-c, except the walking speeds up during the transition. d) Velocity changes and impulses when the heel-strike precedes push-off entirely.

118 in Eq. A.1 to obtain:

2 1 cos θafter Vbefore Vafter Wpos = m Vbefore tan θbefore − − and 2 sin (θafter + θbefore) cos θbefore cos θafter (A.3) 2 1 cos θbefore Vbefore Vafter Wneg = m Vaftertan θafter + − , (A.4) 2 sin (θafter + θbefore) cos θbefore cos θafter

with OA = Vbefore and OC = Vafter. We obtain exactly the same expressions for speeding up during step-to-step transition as shown in figure A.1c (though the figure looks superficially different, a couple of negative signs cancel, thereby giving the same answers). Note that the main qualitative difference between panels b and c of figure −→ A.1 is that the velocity OB is above or below the horizontal.

An implicit assumption in the above derivation is that the push-oﬀ and heel-strike impulses do not require tensional leg forces (the leg cannot pull on the ground) and are in the directions shown. This requirement is satisﬁed when the ratio of the two speeds obey the following condition:

Vafter 1 cos (θafter + θbefore) ≤ ≤ . Vbefore cos (θafter + θbefore)

When Vafter/Vbefore = cos (θafter + θbefore), the necessary push-oﬀ impulse becomes zero and when Vbefore/Vafter = cos (θafter + θbefore), the necessary heel-strike impulse becomes zero.

Heel-strike before push-off. When the heel-strike impulse precedes the push- off impulse, the negative work Wneg by the heel-strike and the positive work by the push-off Wpos are given by the respective kinetic energy changes:

1 1 W = m (OA2 − OG2) and W = m (OC2 − OG2), (A.5) neg 2 pos 2

119 2 2 2 where OA = Vbefore, OC = Vafter, OG = OQ + QG , OQ = OA cos (θafter + θbefore), angle ∠QGC = θafter + θbefore, QG = CG cos (θafter + θbefore), and CG = AB in ﬁgure A.1b. i.e.,

cos θafter Vbefore Vafter CG = AB = Vbefore tan θbefore − − sin (θafter + θbefore) cos θbefore cos θafter using the derivation for push-off before heel-strike; the final expression is easily obtained by substituting these relations into equation A.5. As has been shown before for steady state walking [119, 117], we find that a transition with heel-strike before push- off requires more metabolic cost than push-off before heel-strike even when changing speeds with the simplifying assumption that θbefore = θafter.

A.2 Optimal multi-step inverted pendulum gaits satisfying experimental protocol

We use a metabolic cost that is a sum of two terms: (1) the step-to-step transition cost and (2) a swing cost.

Step-to-step transition cost. The step-to-step transition cost Es2s is a weighted sum of the push-oﬀ work Wpos and heel-strike work Wneg, summed over all steps, and scaled by the approximate eﬃciencies of positive and negative work respectively:

X −1 −1 Es2s = ηposWpos + ηnegWneg steps using the equations A.3 and A.4 for the work expressions.

Swing cost. The step-to-step transition cost does not account for the work required to swing the legs. We use a simple model of the metabolic cost required to swing the legs forward [120] equal to

3 Eswing = µDstride/Tstep

120 where Tstep is the step duration, Dstride is the distance travelled by the swing foot during the step (distance between previous and successive foot contact points), and the proportionality constant µ = 0.06 when all other quantities are non-dimensional, chosen so as to best ﬁt steady walking metabolic costs [120].

Representing a multi-step inverted pendulum walking motion. Each step

of the inverted pendulum walking motion was represented using ﬁve variables: the ˙ initial leg angle θ0, the initial (post-heel-strike) angular velocity θ0, step duration

Tstep, the constant leg-length over the step `leg, and the foot-ground contact position

in the forward direction xcontact. Nonlinear equality constraints make sure that the

body position at the end of one step is equal to that at the beginning of the next

step.

Numerical optimization. We used numerical optimization to determine the

multi-step walking motion that satisﬁes the oscillating-speed experimental protocol

and minimizes the model metabolic cost as described above. The biped model al-

ternates between a higher speed vavg + L/Tfwd and a lower speed vavg − L/Tbck, each lasting a few steps, so that the net average speed is vavg, and the forward and backward movement in lab frame have periods equal to Tfwd and Tbck. The number of steps for the forward and backward movements are chosen based on the number of steady walking steps in the durations Tfwd and Tbck. Other constraints included an upper bound on the leg length (< `max) and a periodicity constraint on the body height over one period of back and forth walking. The optimization problem was solved in MATLAB using the optimization software SNOPT, which employs the sequen- tial quadratic programming technique [121]. At each average speed, we also computed

121 the optimal constant-speed inverted pendulum walking gait (repeating calculations in [118, 82]), so as to subtract from the optimal oscillating-speed walking cost.

Leg force cost. We repeated the calculations above with a cost for leg force, proportional to the integral of the leg force, with a proportionality constant as in

[122]. We found that this leg force cost did not change our overall predictions for the diﬀerence between oscillating-speed and constant-speed metabolic costs, as both these costs increase by the almost same amount due to the leg force cost. This result can be explained intuitively as follows: because the legs make relatively small angle with the vertical, as explained in [82], the average leg force is approximately equal to the average vertical force, which has to be equal to the total body weight for periodic motion – be it constant-speed walking or oscillating-speed walking.

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