Simulating Human-Prosthesis Interaction and Informing Robotic Prosthesis Design Using Metabolic Optimization

Simulating human-prosthesis interaction and informing robotic prosthesis design using metabolic optimization

Dissertation

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

Matthew L. Handford, B.S., M.S.

Graduate Program in Mechanical Engineering

The Ohio State University

2018

Dissertation Committee:

Dr. Manoj Srinivasan, Advisor Dr. Steve Collins Dr. Kiran D’Souza Dr. Rob Siston c Copyright by

Matthew L. Handford

2018 Abstract

Robotic lower limb prostheses can improve the quality of life for amputees. Devel- opment of such devices, currently dominated by long prototyping periods, could be sped up by predictive simulations. In contrast to some amputee simulations, which track experimentally determined non-amputee walking kinematics, we can instead explicitly model the human-prosthesis interaction to produce a prediction of the user’s walking kinematics. To accomplish this, we use large-scale trajectory optimization on a muscle-driven multi-body model of an amputee with a robotic prosthesis to obtain metabolic energy-minimizing walking gaits. While this computational framework can be applied to a wide range of passive or biomechatronic prosthetic, exoskeletal, and assistive devices, here, we focus on unilateral ankle-foot prostheses. We use this optimization to determine optimized prosthesis controllers by minimizing a weighted sum of human metabolic and prosthesis costs and develop Pareto optimal curves between human metabolic and prosthesis cost with various prostheses masses and at various speeds. We also use this optimization to obtain trends in the energetics and kinematics for various net prosthesis work rates produced by given prosthesis feedback controllers. We ﬁnd that the net metabolic rate has a roughly quadratic relationship with the net prosthesis work rate. This simulation predicts that metabolic rate could be reduced below that of a non-amputee, although such gaits are highly asymmetric and not seen in experiments with amputees. Walking simulations with bilateral

ii symmetry in kinematics or ground reaction forces have higher metabolic rates than asymmetric gaits, suggesting a potential reason for asymmetries in amputee walking.

Our findings suggest that a computational framework such as one presented here could augment the experimental approaches to prosthesis design iterations, although quantitatively accurate predictions of experiments from simulation remains an open problem. We run a series of optimizations to examine additional objective functions, which may improve the prediction. These objective functions include mechanical muscle costs and socket interaction costs. Finally, we consider a simple point-mass model of a unilateral amputee, finding that the point-mass models make broad qualitative predictions similar to those of the complex model: as the prosthesis produces more net work, the metabolic cost to the person is reduced and the bilateral asymmetry of the gait increases; favoring the affected side.

iii Acknowledgments

This work was supported in part by NSF CMMI grants 1300655 and 1254842 and informed extensively by the work of Dr. Manoj Srinivasan and collaborative conversations with and ongoing prosthesis research from Steve Collins, Josh Caputo,

Roberto Quesada, and others. I would like to thank my advisor Dr. Manoj Srinivasan for his guidance and support over the years as I have worked on this research. I appreciate the knowledge, experience, and advice he shared with me through our many conversations. I would like to thank Dr. Steve Collins, Dr. Kiran D’Souza, and

Dr. Rob Siston for serving on my committee and for encouraging me throughout my time at Ohio State. I would also like to thank my lab-mates and colleagues for all of their insight and conversations, which pushed my project to greater heights. Finally,

I would like to thank my wife Larissa, my family, and my friends for their love and support as I spent long hours researching in the lab.

iv Vita

2012 ...... B.S. Mechanical Engineering, The Ohio State University. 2016 ...... M.S. Mechanical Engineering, The Ohio State University. 2013-present ...... Graduate Research Associate, The Ohio State University.

Publications

Research Publications

ML. Handford and M. Srinivasan “Energy-optimal human walking with feedback- controlled robotic prostheses: a computational study”. IEEE TSNR, doi:10.1109/ tnsre.2018.2858204, Sept. 2018.

ML. Handford and M. Srinivasan “Robotic lower limb prosthesis design through simultaneous computer optimizations of human and prosthesis costs”. Scientiﬁc Reports, doi:10.1038/srep19983, Feb. 2016.

ML. Handford and M. Srinivasan “Sideways walking: preferred is slow, slow is optimal, and optimal is expensive”. Biology Letters, doi: 10.1098/rsbl.2013.1006, Jan. 2014.

Fields of Study

Major Field: Mechanical Engineering

Specializations: Biomechanics, Energetics, Dynamic Simulation, and Optimiza- tion

v Table of Contents

Page

Abstract ...... ii

Acknowledgments ...... iv

Vita...... v

List of Tables ...... x

List of Figures ...... xi

1. Introduction ...... 1

1.1 Introduction to prosthesis design and simulation ...... 1 1.2 Literature review ...... 2 1.2.1 Current passive prostheses ...... 2 1.2.2 Reduced mobility and metabolic efficiency in amputees . . . 3 1.2.3 Active prostheses ...... 4 1.2.4 Effects of active prostheses ...... 6 1.2.5 Prosthesis controller experimentation ...... 6 1.2.6 Energy optimality ...... 7 1.2.7 Simulation through tracking ...... 7 1.3 Thesis objective ...... 8 1.4 Thesis organization ...... 9 1.5 Research significance ...... 11

2. Human and prosthesis simulation and optimization ...... 12

2.1 Human and prosthesis model ...... 12 2.1.1 Simpliﬁcation and parameterization ...... 12 2.1.2 Non-amputee model for comparison ...... 15

vi 2.1.3 Ground contact ...... 15 2.1.4 Ordinary diﬀerential equations throughout gait ...... 16 2.1.5 Collisions ...... 17 2.2 Optimization problem set-up ...... 19 2.2.1 Periodic gait deﬁned through multiple shooting ...... 19 2.2.2 Objective function ...... 20 2.2.3 Linear and nonlinear constraints ...... 25

3. Prosthesis control with time-dependent torques ...... 28

3.1 Optimal human-prosthesis trade-oﬀs ...... 28 3.2 Results ...... 29 3.2.1 Optimizing mostly just the human metabolic cost ...... 29 3.2.2 Comparison with non-amputee gait ...... 29 3.2.3 Optimal trade-oﬀs between human and prosthesis cost. . . . 32 3.2.4 Symmetry is expensive ...... 32 3.2.5 Lighter feet are less expensive...... 34 3.2.6 Greater human cost reduction at higher speeds...... 35 3.2.7 Passive prosthesis can be metabolically expensive ...... 35 3.3 Discussion ...... 36

4. Human and prosthesis optimization with feedback control ...... 43

4.1 State-based prosthesis controllers ...... 43 4.1.1 Variable work feedback control ...... 44 4.1.2 Comparing results of trajectory optimization to experiment 45 4.2 Changes to the optimization setup ...... 47 4.3 Results ...... 48 4.3.1 Increase in prosthesis work rate reduces metabolic rate . . . 48 4.3.2 Simple feedback is worse than optimized control but both are better than SACH foot ...... 48 4.3.3 Zero work prostheses can give near-able-bodied costs . . . . 51 4.3.4 All energy-optimal gaits are asymmetric gaits ...... 51 4.3.5 Symmetry constraints increase cost but promote kinematics closer to experiment ...... 54 4.3.6 Reduced limb mass or limb muscle strength do not aﬀect qualitative features ...... 56 4.4 Discussion ...... 56

5. Improving model predictions through controller and cost function modiﬁ- cation ...... 67

vii 5.1 Methods ...... 67 5.1.1 Controller with nonlinear initial stiffness ...... 68 5.1.2 Cost function sweep ...... 68 5.1.3 Socket interaction cost ...... 71 5.2 Results ...... 73 5.2.1 A stiff cubic controller produces higher metabolic cost regardless of kinematic symmetry constraint ...... 73 5.2.2 Cubic controller changes affected limb kinematics and dynamics 76 5.2.3 Added force rate squared costs has a greater effect on cost than added force squared or work costs ...... 76 5.2.4 Force rate cost causes larger changes in stride kinematics and kinetics than work and force costs ...... 81 5.2.5 Muscle forces change with additional mechanical costs . . . 81 5.2.6 Cost and symmetry relationships to prosthesis work rate for added costs ...... 86 5.2.7 Socket loading costs ...... 86 5.2.8 Socket costs effect on kinematics and dynamics...... 86 5.3 Discussion ...... 92

6. Point mass biped walking with a unilateral prosthesis or exoskeleton . . . 100

6.1 Simple point mass model ...... 100 6.1.1 Past models ...... 100 6.1.2 Our model ...... 101 6.2 Point mass optimization ...... 103 6.2.1 Work-based objective function ...... 103 6.2.2 Constraints ...... 104 6.3 Results ...... 105 6.3.1 Optimization discovers pendular walking for zero or low assistance ...... 105 6.3.2 Increasing assistance increases bilateral asymmetry...... 105 6.3.3 Asymmetric gaits have a lower cost than symmetric . . . . . 110 6.3.4 Trends when velocity is constrained are similar to when it is not ...... 110 6.3.5 Changing stance cost ...... 110 6.4 Discussion ...... 113

7. Contributions and Future Work ...... 119

7.1 Contributions ...... 119 7.2 Future work ...... 120

viii Bibliography ...... 122

ix List of Tables

Table Page

2.1 Biped body segment parameters. The lengths of each body segment define the distance along the segment for the HAT, thigh, and shank segments. The foot and prosthesis dimensions are shown by the length of the foot bed and the height from the foot bed to the ankle (given in the parentheses). For both the unaffected foot and the prosthesis, the heel is located 0.06 m behind the ankle. The center of mass distances are measure from the origin of the segment connected at the proximal joint. The x distance is along the segment while y is perpendicular to the segment. The moment of inertia are about the z axis (perpendicular to sagittal plane), through the center of masses of the respective segments. All properties are set to the values shown by default and are only altered for specified tests...... 14

2.2 Biped muscle parameters. Maximum isometric force Fiso, maximum contractile velocity vmax, tendon stiﬀness ktendon, and moment arm d at each joint for all eight muscle groups. Positive or negative moment arms indicate that a muscle force will create a counter-clockwise or clockwise torque (respectively) to the distal body segment attached to that joint. Moment arms marked as zero indicate that the muscle- tendon unit does not cross that joint and thus provides no moment. These muscle groups are shown graphically in Figure 2.1B. All parameters are from [54,55]...... 15

2.3 Biped joint range of motion bounds. The joint angles are displayed graphically in Figure 2.1d. This table presents the bounds on these angles (in radians). Some bounds are assumed to be larger than anatomical, but such bounds are never active at the optimal solution, as is clear from the optimal kinematics depicted in Chapters 3 and 4. 26

x List of Figures

Figure Page

1.1 Examples of passive lower limb prostheses. This image from [9], shows an standard ESR (Energy Storage and Return) passive prosthesis (a), a SACH (Soft Ankle Cushioned Heel) foot (b), and a cross section of the SACH foot (c)...... 4

1.2 Examples of active lower limb prostheses. Images taken from [17, 20, 21]. a) Ossur¨ Propio Foot [20], b) BiOM iWalk [18, 19, 21], c) universal prosthesis emulator [6,7,17]...... 5

2.1 Biped model. A) The sagittal-plane model of a human with a unilateral ankle-foot prosthesis. It is made of seven rigid bodies (one HAT, two thighs, two shanks, one foot, and one prosthesis). The side with transtibial amputation is labeled as the affected limb and while the other side is labeled the unaffected limb. B) The model assumes the five intact joints are actuated via thirteen uni- and biarticular muscles. The muscles of the unaffected limb include the illiopsoas (Il), glutei (Glut), hamstrings (Ham), rectus (Rec), vasti (Vas), gastrocnemius (Gas), soleus (Sol), and tibialis anterior (TA). The affected limb shares the same musculature as the unaffected limb except the muscles that cross the ankle joint (gastrocnemius, soleus, and tibialis anterior) are replaced with a single torque motor associated with the prosthesis. C) The assumed contact phases of a human walking gait. Each contact phase is defined by which heels and toes are in contact with the ground (e.g. HT0T refers to the heel and toe of the unaffected limb and the toe of the affected limb in contact with the ground)...... 13

xi 2.2 Biped model. A) The configuration variables used to define the position of the biped model. Shown are the horizontal and vertical hip positions, and the absolute angles for each segment relative to the global horizontal axis. Each position variable has an analogous velocity and acceleration variable. B) The Free body diagram of the whole body used for constructing linear momentum balance equations. These free body diagrams include all gravitational and ground reaction forces. Each ground reaction force is set to zero if the associated point is not in contact with the ground. C) Free body diagrams for constructing angular momentum balance equations at each joint. These diagrams are valid for both limbs (affected and unaffected)...... 18

2.3 Structure of the optimization. A) We divide the walking gait into the contact phases defined by the points in contact with the ground. These contact phases are divided into six to eight segments depending on the test. B) Each segment is given a set of initial conditions, which are used for the given ODE over short time segment. Then the state variables at the end of each segment are constrained to be equal to the state variables at the beginning of the next. This way, each short time segment can be optimized with unknowns that are local to that segment, while still producing a continuous walking cycle once the optimization finds a minimum and satisfies the constraints...... 21

2.4 Individual muscle metabolic cost details. The metabolic cost ex- 2 pression, Ci = [ (ai + ai ) + ai φ (¯vi)] Fiso,ivmax,i, adapted from [30,31], has two terms: an activation cost and a muscle-shortening related term φ. The muscle-shortening related term φ, is based on empirical heat and ATPase activity [30] and approximated by φ = 0.1 + 0.9(¯v)+ + 0.2(¯v)− +9(¯v)+(¯v)+, wherev ¯ is the muscle contractile velocity over the maximum contractile velocity, (¯v)+ is the positive part of thev ¯, and (¯v)− is the negative part ofv ¯. This function is used to model the differing metabolic cost between a contracting muscle and an extending 2 muscle. See [31] for more details. The activation cost, (ai + ai ) where = 0.05, is a function that captures the empirical result that about 40% of isometric muscle exertion cost is the activation cost [56]. Be- cause we optimize for a ﬁxed walking speed, we do not include a resting cost to the total metabolic cost. Adding a ﬁxed resting metabolic rate will simply add a constant term to all the metabolic costs and does not change any of the optimal strategies [57]...... 23

xii 2.5 Illustration of optimization in progress. These ﬁgures give an estimate of current optimization progress by displaying the prosthesis toque proﬁle (both with respect to time and prosthesis angle), the biped position at the start of each contact phase, the current constraint violations, and a sample of the past objective function and constraint function values...... 27

3.1 Optimal trade-offs (Pareto curves) between human and prosthesis costs. Results for optima without a symmetry constraint (blue line) and with a symmetry constraint (red line) are shown. Different markers (legend) denote results from different λ’s (0.1-0.95). Optima

from passive prostheses with four non-dimensional stiﬀnesses (K1 = 0.1, K2 = 0.5, K3 = 1.0, K4 = 1.5), as are non-amputee optimum and the non-amputee optimum with muscle replacement strategy (cost-free ankle muscle costs)...... 30

3.2 Comparison with non-amputee data. A) Optimal gait kinematics for amputee walking cycle (with and without a bilateral symmetry constraint) with cost function weighting λ = 0.95 and non-amputee walking cycle. The aﬀected side (or analogous side for non-amputee) is signiﬁed by the blue prosthesis. B) Optimal joint angles for all three conditions over one gait cycle with periods of 2.876 (asymmetric), 2.412 (symmetric), and 2.857 (non-amputee)...... 31

3.3 Prosthesis torque over one stride with gait symmetry unconstrained and constrained. Each curve represents an optimization with different λ (0.1-0.95) as shown in the figure legend. The stride for each gait starts on the contact phase following the prosthesis toe-off (HT00)...... 33

xiii 3.4 Pareto curves for two other cost models. Pareto curves for three different cost functions are shown (solid curves): (1) using Alexander- Minetti metabolic cost for human and smoothed torque-squared cost for prosthesis, as in the main manuscript (Figure 3A), (2) using a scaled muscle-force-squared cost for the human and a scaled torque-squared cost for the prosthesis, (3) using a muscle work-based cost for the human and a motor work cost for the prosthesis. All costs shown are normalized by body weight and leg length. Thus, these Pareto curves show that all these different costs give qualitatively similar trade-offs between human and prosthesis energy costs for the amputee. We also show the optimal costs for a non-amputee walker (long-dashed line) and the “muscle replacement strategy” (short-dashed line) for the three cost functions. The optimal robotic prosthesis actuation reduces amputee energy cost below that of a non-amputee and a non-amputee walker with cost-free muscles crossing an ankle...... 40

3.5 Trade-offs between human and prosthesis cost for five different prosthetic foot inertial parameters. Masses / moment of inertias range from 50% to 150% unaffected foot mass. Lowering prosthetic foot mass decreases metabolic and prosthesis costs for a given cost weighting factor λ...... 41

3.6 Trade-offs between human and prosthesis cost with five different walking speeds. Speeds range from 0.7 m/s to 1.5 m/s. As shown, high prosthesis effort is more important to reducing metabolic cost at higher speeds than at lower speeds...... 42

xiv 4.1 Torque-angle work-loops during stance for three prosthesis controllers. A) Controller 1. Push-oﬀ work is controlled through the

addition of a constant torque ∆τ by increasing the rest angle by ∆αrest while maintaining stiﬀness. B) Controller 2. Push-oﬀ work is controlled

through adjusting the rest angle (by ∆αrest) and stiﬀness to create a continuous torque throughout stance. C) Controller 3. Push-oﬀ work is controlled through maintaining the maximum torque produced during

dorsiflexion over a range of ankle angles ∆αrest, then returns to the linear stiffness used in the initial stance phase. D) Passive controller. The work-loop shown is taken from experimental data while walking with a SACH foot [2]. E) Prosthesis stance phase sequence. Heel strike, toe strike, and braking all belong to the initial stance phase. Initial stance is followed by push-off, which lasts until the toe leaves the ground. The leg then continues into the swing phase. Numbers 1-5 are overlaid on panels A-D, corresponding to the contact phase sequence shown...... 46

4.2 Controller torque-angle-loops for increasing work rate A-C) The torque angle relationship of Controller 1, 2, and 3 over a full gait cycle. The gradient from blue to red indicates the controller with increasing net prosthesis work rate. D-E) The optimized controller and SACH torque-angle-loops. Only a single color is shown as neither have variable work...... 49

4.3 Eﬀects of prosthesis work on metabolic rate. For Controller 1 (c1, blue), Controller 2 (c2, red), and Controller 3 (c3, yellow), the relationship between net metabolic rate and net prosthesis work rate is non-monotonic and has a minimum. A quadratic ﬁt to pooled data ˙ ˙ ˙ 2 from all three controllers (C = 2.4−4.0Wpros +3.2Wpros) is shown. The simulated SACH foot and the optimized prosthesis controller (both shown in green) occupy two extremes of high metabolic rate / low prosthesis work for the SACH foot and low metabolic rate / high prosthesis work for the optimized controller. Metabolic rate versus work rate for human subject experiments shows no systematic trend and has high variability (mean ± standard deviation, shown in black and gray) [8]...... 50

xv 4.4 Bilateral symmetry is correlated with prosthesis work. The bilateral symmetry, as quantified by percent step time difference (the percent of the stride time spent on the unaffected side minus the percent of stride spent on the affected side), is shown for each controller (c1, c2, c3) at various work rates. The percent step time difference decreases, roughly linearly, with increased positive prosthesis work. Thus, the biped spends more time on the affected side as the robotic prosthesis provides more net positive power. In contrast, negative work generally corresponds to less time spent on the affected side. Experimental data [8] (black and gray error bars) and the simulated non-amputee result (pink circle) are shown for comparison. The SACH foot results in a highly asymmetric gait with more time spent on the unaffected limb, whereas the optimized robotic prosthesis results in large asymmetry in the opposite direction (green circles)...... 52

4.5 Joint angles over one stride have bilateral asymmetry for all controllers. The angles (in radians) for the hip, knee, and ankle are shown for each controller at various levels of work. Each group of plots belongs to the same controller and work level. The aﬀected and unaﬀected sides are shown in blue and gray respectively. All plots start at heel strike and display extension as positive...... 53

4.6 Controller 1 with various symmetry constraints. Applying step time, kinematic, or ground reaction force (GRF) symmetry constraints to the optimization increases the metabolic rate over the unconstrained condition. When both kinematic and GRF constraints are enforced, the metabolic rate is much higher than any constraint alone. Exper- imental metabolic data [8] (black and gray error bars) and simulated non-amputee metabolic rate (pink circle and line) are shown for comparison...... 55

4.7 Joint angle comparisons with experiment with and without symmetry constraints. This figure presents the joint angles for selected tests (with similar levels of net prosthesis work) using controller 1 with various step time, kinematic, and/or ground reaction force (GRF) symmetry constraints, overlaid with amputee data [8], depicted as a mean curve and 95% confidence interval band (black solid line and gray band). Positive angle values refer to extension and negative refer to flexion; plantar flexion and dorsiflexion are positive and negative respectively for the ankle and prosthesis. Each stride begins with the heel-strike of the corresponding leg being plotted...... 57

xvi 4.8 Joint torque comparisons with experiment with and without symmetry constraints. This figure presents the joint torques for selected tests (with similar levels of net prosthesis work) using controller 1 with various step time, kinematic, and/or ground reaction force (GRF) symmetry constraints, overlaid with amputee data [8], depicted as a mean curve and 95% confidence interval band (black solid line and gray band). Positive torque values refer to extension and negative refer to flexion; plantar flexion and dorsiflexion are positive and negative respectively for the ankle and prosthesis. Each stride begins with the heel-strike of the corresponding leg being plotted...... 58

4.9 Error between joint torques/angles from optimization and experimental data. Error is measured using the standard deviation of the diﬀerence between experiment and optimization results. Joint torque and angle data is taken from selected tests (with similar levels of net prosthesis work) using controller 1 with various step time, kinematic, and/or ground reaction force (GRF) symmetry constraints. The legend refers to the symmetry constraints applied during each optimization...... 59

4.10 Normalized muscle forces with and without symmetry constraints. Normalized muscle forces are presented over one stride period, as a fraction of maximum isometric force. The curves shown are from optimizations that differ only by the symmetry constraints used (including no symmetry constraints). All optimizations had similar net prosthesis work rate and used prosthesis controller 1. Each stride shown starts from the heel-strike of the leg in which the corresponding muscle is present. Horizontal black bars atop each figure panel are an on/off representation of EMG data taken from experiment [15,71,72,73]. 60

5.1 Simple active controller with cubic stiffness during dorsiflexion and plantar flexion A) The ideal torque loop for the cubic spring controller. Work is added using a constant torque input at the onset of push-off. The numbers 1-5 refer to the stage of the gait cycle. B) Stages of the gait cycle, same as in Chapter 4...... 69

xvii 5.2 Aﬀected shank structure and internal loads. A) The shank of the aﬀected side consists of the residual limb, which resides in the socket, and the pylon, which attaches to the base. B) By making a section cut at the end of the socket, we expose the interfacial loading. This includes the force normal to the cross-sectional area of the cut (and axial to the shank) N, the shear force planar to the surface (perpendicular to the axis of the shank) V , and the bending moment M. We calculate these forces and moment using the ground forces on the prosthesis, the gravitational forces on the prosthesis and pylon, and the accelerations of those bodies...... 72

5.3 Cubic spring controller torque-angle relationships. Net prosthesis work rate ranging from -0.2 and 0.8 W kg−1 is displayed as a gradient from blue to red...... 74

5.4 Eﬀects of net prosthesis work rate on metabolic rate. The relationship between net metabolic rate (W kg−1) and net prosthesis work rate (W kg−1) are shown for controller 1 (c1) and the cubic spring controller (csc) with symmetry unconstrained and constrained (sym c1 and sym csc respectively). The costs for the optimized controller, SACH foot, non-amputee, and experimental data [8] are also displayed for comparison...... 75

5.5 Joint angle comparison between cubic spring controller optimizations and experiment. Joint angles for controller 1 (c1) and the cubic spring controller (csc) as well as the angles for both controllers with symmetry constrained (c1 sym and csc sym respectively). Experimental data [8] is shown in black and gray...... 77

5.6 Joint torque comparison between cubic spring controller optimizations and experiment. Joint torques for controller 1 (c1) and the cubic spring controller (csc) as well as the torques for both controllers with symmetry constrained (c1 sym and csc sym respectively). Experimental data [8] is shown in black and gray...... 78

5.7 Error between optimization and experimental joint torques and angles. The joint torque and angle data is from selected tests (with similar levels of net prosthesis work) using controller 1 with various step time, kinematic, and/or ground reaction force (GRF) symmetry constraints...... 79

xviii 5.8 Cost comparison results of augmented cost function optimizations. The resultant total cost (A) and the net metabolic cost (B) of various optimizations. Each test used equation 5.2 with one lambda set to a value ranging from 0 to 1 and the other two set to zero. All optimizations used a periodic stride with controller 1 set with a ∆τ = 42 N m. A) The total cost of all components summed together

(e.g. Ctotal = Cmet + λFCF. B) The contribution of the metabolic cost alone for each test...... 80

5.9 Joint angles from optimizations with added mechanical costs. The results shown use controller 1 with a torque input of 42 N m and ˙ ˙ ˙ are calculated using Cmet (blue) plus 0.5CF (orange), 0.5CW (yellow), ˙ or 0.1CFR (purple). Joint angles from experiment [8] are shown in grey and black...... 82

5.10 Joint torques from optimizations with added mechanical costs. The results shown use controller 1 with a torque input of 42 N m and ˙ ˙ ˙ are calculated using Cmet (blue) plus 0.5CF (orange), 0.5CW (yellow), ˙ or 0.1CFR (purple). Joint torques from experiment [8] are shown in grey and black...... 83

5.11 Error between optimization and experimental results. The errors in both joint torques and angles are consistently reduced with the addition of a force rate cost. The addition of improves some predictions (e.g. unaffected knee angle) but has no significant effect on other joint angles or joint torques...... 84

5.12 Muscle forces from various cost functions. The muscle forces (scaled by max isometric force) for the metabolic cost are shown in blue. All other lines depict the muscle forces produced by the metabolic cost plus the force squared cost (orange), the work cost (yellow), or the force rate squared cost (purple). The weighting coefficients for each cost function are shown in the figure legend. All muscle force profiles are compared to experimental EMG data [15,71,72,73], shown as black bars...... 85

xix 5.13 Total cost of walking gait at various levels of work. The curves depict the total cost produced by the metabolic cost by itself (blue) and with the additional force squared cost (orange), the work cost (yellow), or the force rate squared cost (purple). The weighting coeﬃcients for each cost function are shown in the ﬁgure. All results are compared to the cost-vs-work relationship found in experiment (black and gray). . 87

5.14 Metabolic cost at various levels of work. The curves depict the metabolic cost produced when the objective function is defined as the metabolic cost by itself (blue) or with the additional force squared cost (orange), work cost (yellow), or force rate squared cost (purple). The weighting coefficients for each cost function are shown in the figure. All results are compared to the cost-vs-work relationship found in experiment (black and gray)...... 88

5.15 Step-time difference with each objective function. The curves depict the bilateral symmetry produced when the objective function is defined as the metabolic cost by itself (blue) or with the additional force squared cost (orange), work cost (yellow), or force rate squared cost (purple). The weighting coefficients for each cost function are shown in the figure. All results are compared to the cost-vs-work relationship found in experiment (black and gray)...... 89

5.16 Net metabolic rate with socket loading costs. Each line represents the net metabolic rate of the optimization using diﬀerent weighted cost functions, which include the net metabolic rate and a cost associated with one of the socket loads (normal force, shear force, or bending moment). An added normal force cost is shown to increase the metabolic cost more than and added shear or moment cost. Experi- mental data is shown in black and gray for comparison [8]...... 90

5.17 Percent step time with socket loading costs. The percent step time difference between the unaffected and affected side is shown for the normal force, shear force, and bending moment cost functions. Added normal force cost produces strides with longer unaffected stance times. 91

xx 5.18 Joint angles with socket loading costs. The curves represent the joint angles found using the standard metabolic cost (blue) or with the addition of the socket normal cost (orange), shear cost (yellow), or moment cost (purple). Each cost function produces similar joint angles for all affected joints and the unaffected hip and ankle. Added socket loading cost did produce larger unaffected knee flexions during swing. 93

5.19 Joint torques with socket loading costs. The curves represent the joint torques found using the standard metabolic cost (blue) or with the addition of the socket normal cost (orange), shear cost (yellow), or moment cost (purple). Joint torques produced by each cost function are not significantly different from the results produced using metabolic cost alone except in minor differences in timing...... 94

5.20 Error between simulated and experimental data with socket loading costs. There is no consistent decrease in error for any one socket loading function for either joint angles or joint torques. . . . . 95

5.21 Muscle forces produced by tests with socket loading costs. Each cost function changes the muscle activations throughout gait. . 96

6.1 Point mass model. A) A point mass representation of a biped capable of pushing oﬀ the ground with one leg at a time. The free body diagram is shown with the gravitational force G and the leg force F. The direction of F is determined based on the mass position (x, y) and

the foot position xfoot on the ground. The foot mass mfoot is used in calculating the cost of swing. B) A periodic stride consisting of two steps, one on the unaﬀected limb (without amputation) and one on the aﬀected limb (with amputation and a prosthesis)...... 102

6.2 Periodic stride. The plots show the mass position through each step of the periodic stride at various levels of prosthesis work assistance 0 ≤ λ ≤ 0.99. The positions are separated by the unaﬀected and aﬀect limb stance phases. The gaits with velocity unconstrained are shown in blue and red while those with velocity constrained are shown in blue and green...... 106

xxi 6.3 Leg length throughout stride. The leg length is limited to the

maximum length Lmax = 1. When the assistance level is low or zero, both legs have a mostly constant leg length. When the assistance level is high (λ ≥ 0.6) there is a noticeable decrease in length during the aﬀected limb stance phase. The gaits with velocity unconstrained are shown in blue and red while those with velocity constrained are shown in blue and green...... 107

6.4 Symmetry changes in the presence of of greater levels of assistance. This plot displays the percent difference of the stance time, stance work, and step length, defined as the difference between the unaffected and affected sides over the stride total, at increasing levels of prosthesis work assistance, λ. As shown by the red and green lines, representing the unconstrained velocity (unc.v) and constrained velocity (con.v) conditions respectively, the stance time percent and the stance work percent increase with increasing assistance. The step length, shown by the dashed red and green lines, has a similar trend except at the highest values of λ...... 108

6.5 Forces over the percent gait cycle. The leg forces for the optimization with velocity unconstrained (blue-red) and velocity constrained (blue-green) are shown. As the level of assistance increases, the affected side forces increase while those on the unaﬀected side decrease. 109

6.6 Asymmetry leads to larger decreases in cost. Both the total work cost per unit distance and the movement cost (total cost minus the resting cost) are shown. In both measurements, the cost decreases as the prosthesis provides more assistance. The dotted lines display the cost of the model with a symmetric gait (found using the kinematic and dynamic results from the zero assistance condition) and applying the prosthesis assistance at the various levels. For the velocity-constrained condition (shown in green), the asymmetric gaits have a lower total and movement cost. For the velocity-unconstrained conditions (shown in red) the total cost of the asymmetric gates is always lower while the movement cost is only lower at high levels of assistance...... 111

6.7 Velocity increases with increased prosthesis assistance when unconstrained The red line displays the speed at all levels of assistance when velocity is unconstrained while the green line displays the ﬁxed velocity of 1 m/s normalized by gravity and leg length...... 112

xxii 6.8 Changes in leg length and force with increasing average power assistance. Gradient from blue to red displays the increase in assistance from λ = 0 to λ = 1. As provided power increases, the magnitude of the leg force throughout gait does not change while the unaﬀected leg length gets systematically shorter...... 114

6.9 Percent diﬀerence in stance work, stance time, and step length with added power. As the prosthesis adds more power to the affected side, the stance work, stance time, and step length asymmetry increases and favors the aﬀected side...... 115

6.10 Metabolic cost with added power assistance. Decrease in metabolic cost is associated with increase assistance similar to previous tests. . . 116

xxiii Chapter 1: Introduction

1.1 Introduction to prosthesis design and simulation

Humans are bipedal animals who rely on their ability to walk and run to navigate their environment. So what happens when a person needs to have a portion of their leg amputated due to vascular problems or traumatic injury? In order to maintain their ability to navigate their world, they could to make use of crutches, a wheel chair, or they could have their limb replaced with a prosthetic device. Fifty years ago, the state of the art for a prosthetic foot consisted of a wooden core, wrapped in a foam foot facsimile, which was designed for some shock absorption [1]. This technology has improved over the years to include carbon fiber energy storage and return (ESR) passive prostheses [2, 3]. However, despite significant advances in prosthesis design, the vast majority of amputees still experience significant mobility issues and much higher energy usage during even common tasks (e.g. walking around the house, up stairs, on uneven grass, etc.) [4, 5]. More recently, many researchers have begun designing robotic prostheses that can provide the amputee with intelligent joint actuation and active push-off that isn’t possible with passive devices. These active prostheses could greatly reduce amputees’ metabolic cost as they walk and help them to navigate more difficult terrain such as large slopes, stairs, or unstable surfaces.

1 However, the current active prostheses seem to be under-performing in their purpose and progress in improving them is slow. Designing and testing new prostheses takes considerable time and eﬀort. Most prostheses must be designed from scratch, at great cost to the designers, and go through months of veriﬁcation. It could take another six months to a year testing the device with non-amputees in a simulator boot before any amputee subjects don the prototype device. This long process can be improved by prosthesis emulators which are capable of mimicking other physical devices [6], but we could further decrease this time through computational simulations.

If we could observe controllers and device designs on a computational model as a ﬁrst pass, we could rapidly iterate those controllers in the span of weeks. We could also make use of optimization to tune the many parameters of a controller where human- in-the-loop tuning on a physical device would be impractical. Here, we present an optimization framework that seeks to predict amputee kinematics and dynamics using existing or novel controllers through principles of energy optimality. The modeling and computational framework is more generally applicable to the design and control of biomechatronic devices such as exoskeletons, prostheses, and assistive devices, which exchange considerable mechanical energy with the human. Such a computer simulation framework could be complementary to hardware emulator-based simulation and iteration of multiple prostheses [6,7,8].

1.2 Literature review

1.2.1 Current passive prostheses

Transtibial amputees, those who have had one or both legs amputated below the knee, are often prescribed ‘passive’ prostheses, which provide some stiﬀness and

2 damping. One of the older prostheses models still in use is the SACH (Soft Ankle

Cushioned Heel) foot [1,9], shown in Figure 1.1. This prosthesis was designed without an ankle joint to reduce maintenance and with a foam damper to absorb the shock of heel strike. While these design choices reduce cost and maintenance of the SACH foot, the prosthesis limits the users’ range of motion and requires the users to exert considerable eﬀort [2]. More advanced passive prostheses attempt to provide a more comfortable gait through energy storage and return (ESR). ESR prostheses (Figure

1.1) include the Seattle foot and the Flex foot [2, 3]. This class of passive prosthesis is designed to store energy in an elastic component, typically a ﬁberglass or carbon

ﬁber leaf spring, and return it to provide positive power later in stance rather than dissipate it through damping. Such ESR prostheses can be designed with a clutch mechanism so that the energy stored in heel strike can be more pointedly applied to the toe during push oﬀ [10]. While there is evidence that a clutched ESR prosthesis could help reduce metabolic cost over what is possible with a prosthesis without a clutch, no commercial device is currently available.

1.2.2 Reduced mobility and metabolic eﬃciency in amputees

Passive prostheses are more often prescribed than robotic prostheses since they are typically lighter weight, more durable, and more aﬀordable. However, the majority of passive prosthesis users experience reduced mobility [4]. In a previous study, when asked to rate their household and community mobility, only about 20% of re- spondents stated that they were able to walk without the use of crutches or other aids and almost 10% stated that they had given up entirely on their prosthesis [5].

Alongside this reduced mobility comes an increase in metabolic cost. Regardless of

3 Figure 1.1: Examples of passive lower limb prostheses. This image from [9], shows an standard ESR (Energy Storage and Return) passive prosthesis (a), a SACH (Soft Ankle Cushioned Heel) foot (b), and a cross section of the SACH foot (c).

the passive prosthesis in use, most transtibial amputees experience a 10-30% increase in there metabolic cost when compared to non-amputees [2, 11, 12, 13]. This reduced performance is partially due to reduced foot control (e.g. users can’t dorsiflex their foot during swing to avoid tripping) and partially due to passive prostheses’ inability to produce net positive work, thus placing that burden on the amputee’s unaffected joints. However, in a few recent studies, there has been evidence that these metabolic inefficiencies may have more to do with physical fitness and training on the prosthesis than the design of the prosthesis alone [14,15].

1.2.3 Active prostheses

To overcome the limitations of passive ESR prostheses, robotic (active) ankle- foot prostheses make use of microcontrollers and electrical motors (or other forms

4 a) b) c)

Figure 1.2: Examples of active lower limb prostheses. Images taken from [17, 20,21]. a) Ossur¨ Propio Foot [20], b) BiOM iWalk [18,19,21], c) universal prosthesis emulator [6,7,17].

of actuation) to assist the user throughout gait. Some robotic prostheses today use very similar designs to their passive counterparts but add active dorsiﬂexion while the prosthesis is not in contact with the ground. Such prostheses include the Ossur¨

Propio Foot, which is capable of adapting the prosthesis ankle angle based on the

current gait condition [16].

The more recent robotic prostheses make use of powerful motors, which can pro-

vide a push-oﬀ torque at the end of stance. Some of the prostheses are even capable

of producing human levels of torque and power during a walking task with relatively

little increase in mass. These include laboratory bound prosthesis emulators which

have oﬀ-board motors (so as to reduce the mass of the device while still allowing for

high performance) [6,7,17] and standalone devices like the BiOM iWalk [18,19] which

are prescribed to patients.

5 1.2.4 Eﬀects of active prostheses

In some lab tests, when users switched from passive to active prostheses, metabolic reductions of about 10% have been observed [18,19]. This makes intuitive sense given that any work done by the prosthesis could replace the work that the user would otherwise have to do themselves. However, the correlation between prosthesis work and metabolic cost may be complicated. When testing the relationship between metabolic cost and prosthesis work rate on non-amputees in simulator boot (a device for disabling ankle motion while attaching a prosthesis in series with an intact limb), there appears to be an almost linear correlation [7]. When performing the same test with amputees, the reduction in metabolic rate was not observed for the average user [8].

1.2.5 Prosthesis controller experimentation

Robotic lower limb prostheses are often controlled through a mixture of feedback and feedforward control. Feedback controllers, which change control inputs based on state current, past, or future state of the system, are often based on prosthesis angle, angular rate, center of pressure, or myoelectric activity (EMG) of neighboring muscles

[6,19,22]. Nominally feedforward controllers, which apply torques as a function of time based on pre-programed behavior, are usually not entirely feedforward. They typically have some state-based or event-based resets, such as restarting the controller at every heel-strike [23,24,25]. The parameters of such controllers are often designed and tuned with verbal feedback from the user or measured kinematics and energetics [19]. Given that these control parameters may be numerous and their eﬀects often coupled, tuning these parameters with human intuition alone may never discover the ‘optimal’ setup.

6 Such controller tuning is sometimes formalized using ‘body-in-the-loop’ optimization or adaptive control frameworks [26,27,28,29].

1.2.6 Energy optimality

Numerous studies suggest that humans move in an approximately energy optimal manner [30, 31, 32]. The natural walking gait is correlated with optimal walking speeds [33], step length or frequency [34], and step width [35]. This energy optimality extends to those using unpracticed motions [36], moving unsteadily [37], and even amputee walking gaits [33, 38, 39, 40, 41]. While amputees may not use an energy optimal gait until after a (possibly long) learning phase [42, 43], metabolic cost can be a useful metric for analyzing the performance of diﬀerence prostheses.

1.2.7 Simulation through tracking

If we know that people are naturally drawn to energy optimal movements, then we can use this fact to design optimizations. We can build a model of a person and determine the kinematic and dynamic strategies that will minimize the amount of energy used by the model for a given task. In many previous prosthesis computational studies, the human-prosthesis models tracked experimental data to inform gait kinematics and used some form of muscle eﬀort optimization to determine actuation [44, 45, 46]. This tracking is done through a weighted cost function which minimizes some combination of energy and kinematic and/or dynamic agreement between the model and experimental data [15,41,45,47,48]. However, simulations with these tracking terms necessarily ignore human adaptation to the prosthesis [49] by assuming that the person using the device will default to a non-amputee gait. Given that non-amputee gait kinematics and dynamics change in the presence of even small

7 changes in mass [50], we should assume that the loss of a limb will likewise alter the gait.

1.3 Thesis objective

The purpose of this thesis is to build a computational simulation of a lower limb amputee, which can be used to make real world predictions about the eﬀects of various prosthesis controllers. We use large-scale numerical optimization to compute energy-optimal walking motions and prosthesis actuations of a human wearing a unilateral robotic prosthesis. We start by computing optimal trade-oﬀs between human metabolic and prosthesis torque costs when the prosthesis is unconstrained by a particular controller. We show how increasing prosthesis mass, using passive prostheses, or forcing left-right symmetry increase human costs for a given walking speed and how increasing speed increases cost. We predict that optimal prosthesis actuation can reduce the amputee metabolic cost far below normal human metabolic cost. The high dimensionality of the prosthesis actuation as a piecewise linear function of time allows us to discover close to the greatest possible metabolic reduction in a mathematical model.

However, such an idealized calculation may only be useful as a benchmark and may not be practical for a real world device since it produces purely time-based, feedforward control for a perfectly periodic gait at a single walking speed. Therefore, we study state-based controllers by constraining the prosthesis actuation to four simple feedback controllers: three active controllers based on ankle ‘torque-angle relationships’ [6,19,51,52,53] and one passive prosthesis based on the SACH foot [2]. Each of the three active controllers is designed with a diﬀerent torque-angle loop and a single

8 parameter, which can be used to vary the amount of work being applied by the prosthesis. We examine the implications of the assumed torque-angle loop shape and test whether increasing net prosthesis work always reduces net metabolic rate. We also determine a relationship between bilateral asymmetry and net metabolic rate using these controllers both with and without bilateral symmetry constraints. We compare our simulation results with experimental data and ﬁnd good qualitative agreement.

With the goal of better understanding the diﬀerences between our model and experimental data, we alter the cost function to include terms involving muscle work, muscle force, and muscle force rate. We also study the eﬀects of added penalty terms to the cost function based on the interfacial forces between the prosthesis socket and the residual limb. We complete the study by exploring a simple point mass model of a unilateral amputee to determine if our predictions from our complex model still hold true.

1.4 Thesis organization

The rest of this thesis is organized as follows.

Chapter 2 provides a description of the human-prosthesis model we use throughout this research. We discuss how this model is deﬁned within the simulation and the simplifying assumptions we use. We then describe the structure of the numerical optimization including the objective functions for various studies and the constraints we apply to maintain physically any physiologically meaningful results.

Chapter 3 describes calculations in which we optimize a weighted combination of human and prosthesis costs. Here, the prosthesis is unconstrained by a speciﬁed controller. We observe how the prosthesis torques change based on whether the

9 human or the prosthesis cost has a higher weight in the objective function. We also study the change in the cost trade-oﬀ when we apply symmetry constraints, change the mass of the prosthesis, or change the speed of the walking gait.

Chapter 4 describes the optimization of human metabolic rate while constraining the prosthesis to actuate via a set of predetermined feedback controllers. These controllers are designed to provide various net prosthesis work rates to alter the amount of assistance from each controller. We then compare the kinematics and energetics produced using these controllers to data found through previous experimental studies.

Chapter 5 describes the results of repeating the previous studies with various modifications. First, we consider a controller with a non-linear spring meant to mimic the stiffness of an intact ankle joint better. Next, we examine the effects of changing the cost function to include different mechanical objective functions including muscle work, force, or force-rate terms. Finally, we study the addition of cost functions associated with the internal loading at the residual limb and socket interface. We compare each of these tests to experimental data to determine if they improve the quantitative predictions of a physical system.

Chapter 6 presents a point mass model of a biped walking with a prosthesis or an exoskeleton. This simple model is used to test if the results obtained from the more complex model are robust to substantial model changes. We consider various levels of assistance to the aﬀected stance leg and observe the eﬀect of prosthesis work on bilateral asymmetry, cost, and leg forces.

Chapter 7 summarizes the results of this study and presents possible avenues for future work.

10 1.5 Research signiﬁcance

This research is signiﬁcant as it presents a methodology for predicting human walking kinematics and dynamics in the presence of a prosthesis. If a simulation is limited by tracking kinematics or ground reaction forces of able-bodied data, as is common, the results of that simulation will only be predictive of users who already walk with those attributes. However, if we can successfully predict a user’s interaction with a prosthesis, even if quantitative agreement is not perfect, we can determine the controller parameters that are most important for producing a healthy gait (one that is not tiring, painful, or damaging to the patients’ long-term health) and improving the patients’ long-term quality of life. Such computer simulations will also provide a tool for rapid prototyping and iteration of controllers as well as a framework for optimizing complex prostheses with minimal human-intuition-based tuning.

11 Chapter 2: Human and prosthesis simulation and optimization

This chapter describes the model and optimization parameters used throughout the study to simulate a person walking with a prosthesis.

2.1 Human and prosthesis model

2.1.1 Simpliﬁcation and parameterization

The human body has more degrees of freedom than may be necessary to simulate a walking gait for our purposes. We simplify the model by constraining motion to the sagittal plane and reducing the number of body segments through combining the head, arms, and torso into a single segment: the HAT segment. We assume that all of the body segments are rigid and connected via revolute joints with constant joint centers. This results in a seven-segment rigid body system, shown in Figure 2.1A.

To simulate a unilateral transtibial amputee, one of the feet is considered to be the prosthetic foot. Typically, a transtibial prosthesis is attached to the residual limb using a socket interface made of a liner, shell, sleeve, and pylon. Here, we assume the residual limb is rigidly attached to the prosthesis and thus forming the ‘prosthesis shank.’ The inertial parameters (mass, moment of inertia, and center of mass) and size parameters of the body segments are taken from past work, [54,55]. While these

12 A) Model of amputee B) Muscles C) Contact phase sequence and prosthesis HT00 0T00 0TH0 0THT

Il Glut Rec 00HT 000T H00T HT0T A!ected limb Ham Vas Una!ected Prosthesis Gas limb ankle motor Robotic TA prosthesis Sol

Figure 2.1: Biped model. A) The sagittal-plane model of a human with a unilateral ankle-foot prosthesis. It is made of seven rigid bodies (one HAT, two thighs, two shanks, one foot, and one prosthesis). The side with transtibial amputation is labeled as the affected limb and while the other side is labeled the unaffected limb. B) The model assumes the five intact joints are actuated via thirteen uni- and biarticular muscles. The muscles of the unaffected limb include the illiopsoas (Il), glutei (Glut), hamstrings (Ham), rectus (Rec), vasti (Vas), gastrocnemius (Gas), soleus (Sol), and tibialis anterior (TA). The affected limb shares the same musculature as the unaffected limb except the muscles that cross the ankle joint (gastrocnemius, soleus, and tibialis anterior) are replaced with a single torque motor associated with the prosthesis. C) The assumed contact phases of a human walking gait. Each contact phase is defined by which heels and toes are in contact with the ground (e.g. HT0T refers to the heel and toe of the unaffected limb and the toe of the affected limb in contact with the ground).

parameters can be diﬀerent between the amputated (aﬀected) limb and the intact

(unaﬀected) limb, for the majority of our testing, we used the parameters deﬁned in

Table 2.1 for both limbs.

To actuate the biped, we use thirteen uni- and biarticular muscle-tendon units with eight crossing the three joints on the unaffected side and five crossing the hip and knee on the affected side, see Figure 2.1B. The muscles that would cross the

13 Segment Length, (m) Mass, (kg) MoI, (kg m2) x CoM, (m) y CoM, (m) HAT 0.6 50.85 3.177 0.3155 0 Thigh 0.4410 7.5 0.1522 0.1910 0 Shank 0.4428 3.49 0.0624 0.1917 0 Foot 0.21(0.07) 1.087 0.0184 0.0351 0.0768 Prosthesis 0.21(0.07) 1.087 0.0184 0.0351 0.0768

Table 2.1: Biped body segment parameters. The lengths of each body segment define the distance along the segment for the HAT, thigh, and shank segments. The foot and prosthesis dimensions are shown by the length of the foot bed and the height from the foot bed to the ankle (given in the parentheses). For both the unaffected foot and the prosthesis, the heel is located 0.06 m behind the ankle. The center of mass distances are measure from the origin of the segment connected at the proximal joint. The x distance is along the segment while y is perpendicular to the segment. The moment of inertia are about the z axis (perpendicular to sagittal plane), through the center of masses of the respective segments. All properties are set to the values shown by default and are only altered for specified tests.

ankle of the aﬀected limb are removed and replaced with a single torque motor to actuate the prosthesis joint. We assume the muscles can produce piece-wise linear forces and that each muscle has a constant moment arm about each joint they cross.

The experimentally derived properties of each muscle, including maximum isometric force, maximum shortening velocity, tendon stiﬀness, and constant moment arms are shown in Table 2.2 [54, 55]. The force bounds for each muscle and the amount that they contribute to the metabolic cost of the simulated biped are scaled by muscle and tendon properties, as well as muscle velocity. As a model simpliﬁcation, we assume that the muscles do not have a length dependence [54,55].

14 F , v , k , d , d , d , Muscle iso max tendon hip knee ankle (N) (m/s) (N/mm) (mm) (mm) (mm) iliopsoas 1500 1.069 264.1 50 0 0 glutei 3000 2.097 477.7 -62 0 0 hamstrings 3000 1.090 224.6 -72 -34 0 rectus 1200 0.8492 75.37 34 50 0 vasti 7000 0.9750 784.8 0 42 0 gastrocnemius 3000 0.5766 178.6 0 -20 -53 soleus 4000 0.5766 408.2 0 0 -53 tibialis anterior 2500 0.8597 197.2 0 0 37

Table 2.2: Biped muscle parameters. Maximum isometric force Fiso, maximum contractile velocity vmax, tendon stiﬀness ktendon, and moment arm d at each joint for all eight muscle groups. Positive or negative moment arms indicate that a muscle force will create a counter-clockwise or clockwise torque (respectively) to the distal body segment attached to that joint. Moment arms marked as zero indicate that the muscle-tendon unit does not cross that joint and thus provides no moment. These muscle groups are shown graphically in Figure 2.1B. All parameters are from [54,55].

2.1.2 Non-amputee model for comparison

As it is useful to compare the results of our amputee model to a non-amputee

model, we also use an intact seven-segment model. This non-amputee model is the

same as the amputee model except that both ankles are actuated via muscle tendon

units, thus the model has 16 uni- and biarticular muscles.

2.1.3 Ground contact

For our model, we have assumed that the foot and prosthesis can only make

contact with the ground at the heel and toe. With this simpliﬁcation we deﬁne

diﬀerent ‘contact phases’ based on which part of each foot is in contact with the

ground. For an average walking gait, we assume the contact phase sequence shown in

Figure 2.1C: The stride begins during swing with the unaﬀected foot completely ﬂat

15 (HT00), then lifts the heel of the unaﬀected leg (0T00) before the heel of the aﬀected

limb strikes the ground (0TH0), followed by the aﬀected toe (0THT). Once the toe of

the unaﬀected side is lifted oﬀ the ground (00HT), this pattern repeats for the swing

of the unaﬀected leg (00HT, 000T, H00T, HT0T) to complete a single periodic stride.

For each of these contact phases, we deﬁne a set of non-linear, ordinary diﬀerential

equations (ODE) in which the biped can produce ground reaction forces (GRF) only

at points that are in contact with the ground.

2.1.4 Ordinary diﬀerential equations throughout gait

We develop the ordinary diﬀerential equations for the human using Newton-Euler

formalism. We specify the human-prosthesis model using nine conﬁguration variables

corresponding to the horizontal and vertical position of the hip and the absolute angle

of each segment:

T q = [xhip, yhip, θHAT, θthigh,u, θthigh,a, θshank,u, θshank,a, θfoot, θpros] , (2.1) where x defines the horizontal position of the hip, y the vertical position of the hip from the ground, and θ the angles of each segment relative to the global horizontal axis, see Figure 2.2A. The subscript u describes segments on the unaffected side and the subscript a describes those on the affected side. We then define the position vectors for important points in the system as a function of q including the joints (e.g. pankle,u(q)), heel/toe locations (e.g. ptoe,u(q)), and center of mass positions of each of the segments (e.g. pg,foot,u(q)). Through differentiating the various center of mass positions pg(q) we produce the velocity vectors vg(q, ˙q) and acceleration vectors ag(q, ˙q, ¨q) for each of the body segments. Using the free body diagrams shown in

Figure 2.2A-C, we write a system of ordinary diﬀerential equations. We start with

16 a linear momentum balance equation based on the whole body free body diagram

(Figure 2.2B) of the form:

X X mag = mg + Fheel,u + Ftoe,u + Fheel,a + Ftoe,a, (2.2) where the sum is over all body segments in the free body diagram, m is the mass of each body segment, ag is the center of mass acceleration vector of each segment, g is the acceleration vector due to gravity, and Fheel/toe is the force vector at the heels and toes on each leg. For contact phases where one or more points are not in contact with the ground, the forces associated with those points are set to zero. When a point is in contact with the ground, the force associated with that location is calculated through constraint equations that set the acceleration of that contact point equal to zero; namely:

aheel/toe = 0. (2.3)

We write the angular momentum balance equations at each joint which includes all segments that are distal from that joint (Figure 2.2C):

X X (Iα + rg × mag) = (τ + rg × mg) + rheel × Fheel + rtoe × Ftoe, (2.4) where I is the moment of inertia of each segment, α is the angular acceleration vector for each segment, rg is the position vector from the current joint to the enter of mass of each segment (e.g. rg = phip − pg,thigh), τ is the joint torque vector (calculated from the muscle forces and moment arms for all unaﬀected joints), and rheel/toe is the position vector from the joint to the heel and toes.

2.1.5 Collisions

When the biped transitions between the various contact phases, the ground reaction forces at the feet change abruptly whenever a new contact is made. To simplify

17 A) Con"guration B) Linear momentum C) Angular momentum balance variables balance

Fhip1

τhip1 α HAT F GHAT knee1 y hip τknee1 x hip F α ankle1 thigh2 G G τhip1 αthigh1 thigh2 thigh1 τhip2 τankle1

αshank2 F hip1 Fhip2 αshank1 G Gshank2 shank1 α α foot2 foot1 Fheel2 Gravitational force G F Joint force / torque foot2 F toe1 toe2 Fheel1 Gfoot1 Ground reaction forces

Figure 2.2: Biped model. A) The configuration variables used to define the position of the biped model. Shown are the horizontal and vertical hip positions, and the absolute angles for each segment relative to the global horizontal axis. Each position variable has an analogous velocity and acceleration variable. B) The Free body diagram of the whole body used for constructing linear momentum balance equations. These free body diagrams include all gravitational and ground reaction forces. Each ground reaction force is set to zero if the associated point is not in contact with the ground. C) Free body diagrams for constructing angular momentum balance equations at each joint. These diagrams are valid for both limbs (affected and unaffected).

18 the collision mechanics, we treat these collisions as being perfectly plastic. That is, when a new point makes contact with the ground, the velocity of that point instan- taneously goes to zero. We then deﬁne a series of collision equations to calculate the change in velocity throughout the biped given this instantaneous deceleration. We do this by writing a linear momentum balance equation for before and after collisions of the form:

X X mvafter = mvbefore + Jheel,u + Jtoe,u + Jheel,a + Jtoe,a (2.5) where Jheel/toe is the impulse vector at the heels and toes on each leg and vbefore/after are the linear velocities before and after the collision respectively. We also write the angular momentum balance equations in the form:

X X (Iωafter +rg ×mvafter) = (Iωbefore +rg ×mvbefore)+rheel ×Jheel +rtoe ×Jtoe,

(2.6) where ωbefore/after are the angular velocities before and after collision. Then to calculate the new velocities, we set the velocity of any point where where a new contact is made to zero as:

vheel/toe,after = 0. (2.7)

For any contact phase transition where no new contact is made, there is no collision and therefore no instantaneous change in velocities. For these ‘smooth-contact’ transitions we simply set vafter = vbefore and ωafter = ωbefore for all segments.

2.2 Optimization problem set-up

2.2.1 Periodic gait deﬁned through multiple shooting

It is possible to simulate all contact phases in sequence, starting from a single initial condition, and transition from one contact phase to the next using ‘event

19 detection’. However, such ‘single-shooting simulation’ is not ideal for complex trajectory optimization. For long contact phases, small shifts in the initial conditions and muscle forces can have a large eﬀect on the end state due to the intrinsic insta- bility of the dynamics. Therefore, each contact phase is divided into a number of equal duration segments with their own initial conditions, thereby using a multiple shooting-like method for the optimization. This means that the optimization must solve for initial conditions and control for a large number of independent short segments. A graphical representation of this break down is shown in Figure 2.3. For a typical optimization, we chose Nsegments = 6to8 time-segments for each of the eight contact phases, Nphases = 8, with Nunknowns = 32 unknowns for each segment (initial positions and velocities for all body segment, piecewise linear unknown muscle forces, and piecewise linear prosthesis motor torques), producing 1554 to 2056 unknowns to be determined by optimization through:

Ntotal = Nphases + Nphases Nunknowns Nsegments. (2.8)

2.2.2 Objective function

To predict the motion of an amputee with a prosthesis using optimization, we need to have a sense for the goals (conscious or unconscious) that the person is seeking in their gait. While people tend to walk in a manner that minimizes their metabolic cost, there are multiple competing models of metabolic cost. We also have to concede that people may have other implicit goals as well, which may include walking stability, reduced joint pain, walking speed, or bilateral symmetry. On top of that, since we wish to use this tool to design prostheses, we must pay attention to certain prosthesis

20 A) Walking cycle divisions HT00 0T00 0TH0 0THT 00HT 000T H00T HT0T

B) Multiple shooting optimization structure

qend q 0 q q q 0 0 0 qend q 0 q end qend q qend end qend q0 q0 initial guess optimized result

Figure 2.3: Structure of the optimization. A) We divide the walking gait into the contact phases defined by the points in contact with the ground. These contact phases are divided into six to eight segments depending on the test. B) Each segment is given a set of initial conditions, which are used for the given ODE over short time segment. Then the state variables at the end of each segment are constrained to be equal to the state variables at the beginning of the next. This way, each short time segment can be optimized with unknowns that are local to that segment, while still producing a continuous walking cycle once the optimization finds a minimum and satisfies the constraints.

21 objectives such as energy usage and required power. We account for these goals

through a variety of objective functions for the optimization and test how the results

from using these objectives compare with experimental data.

Activation based metabolic cost

Our human metabolic cost function is adapted from Alexander and Minetti [30,31]: T " # 1 Z X C˙ = a + a2 + a φ (¯v ) F v dt, (2.9) met T i i i i iso,i max,i 0 i where the metabolic cost for each muscle is summed and integrated over a single

periodic stride. This cost function depends on activation ai, shortening velocity vi, maximum shortening velocity vmax,i, and maximum isometric force Fiso,i for each muscle; = 0.05 is an activation-related cost coeﬃcient and φ(¯vi) is a function describing the metabolic dependence on normalized muscle shortening velocityv ¯i = vi/vmax,i [31]

. This cost is divided by the stride time to produce a metabolic rate, which is analogous to the cost of transport (normalized cost per unit distance) given that the walking gait is assumed to have constant average velocity. A graphical representation of this metabolic cost function can be found in Figure 2.4.

Because the simulated muscles are assumed to be piecewise linear force actuators in the body of our model, we calculate the activation of the muscle using the max isometric force, force-velocity, and force-length relationships based on the hill-type muscle model [58] :

Fi ai = , (2.10) FisoFv(vi)Fl(l) While we could calculate the length of the muscles using joint angles, muscle moment arms, optimal muscle lengths, and tendon stiﬀness, for the calculations presented here, we ignore the force length relationship. This is because when the muscle is far from

22 Individual muscle cost

(C i / Fiso vmax) 10

0 1 Shorten0.5 1 0.8 ing ve 0 0.6 lo cit -0.5 0.4 y (v 0.2 m /v -1 0 Activation (a) max)

Figure 2.4: Individual muscle metabolic cost details. The metabolic cost ex- 2 pression, Ci = [ (ai + ai ) + ai φ (¯vi)] Fiso,ivmax,i, adapted from [30,31], has two terms: an activation cost and a muscle-shortening related term φ. The muscle-shortening related term φ, is based on empirical heat and ATPase activity [30] and approximated by φ = 0.1 + 0.9(¯v)+ + 0.2(¯v)− + 9(¯v)+(¯v)+, wherev ¯ is the muscle contractile velocity over the maximum contractile velocity, (¯v)+ is the positive part of thev ¯, and (¯v)− is the negative part ofv ¯. This function is used to model the differing metabolic cost between a contracting muscle and an extending muscle. See [31] for more details. The 2 activation cost, (ai + ai ) where = 0.05, is a function that captures the empirical result that about 40% of isometric muscle exertion cost is the activation cost [56]. Because we optimize for a fixed walking speed, we do not include a resting cost to the total metabolic cost. Adding a fixed resting metabolic rate will simply add a constant term to all the metabolic costs and does not change any of the optimal strategies [57].

23 its optimal length (a parameter that can be determined using experimental data),

the activation required to produce a given force is much greater than the maximum

muscle activation of 1. Since the objective function is based on the square of the

˙ 20 activation, this causes the objective function to predict large costs (e.g. Cmet > 10 )

while exploring solutions far from the optimum and prevents convergence towards the

optimum.

Muscle force, force rate, and work costs

While we represent the metabolic cost using the aforementioned objective function

throughout this study, for select optimizations we replace or modify this function with

other terms that are correlated with increasing muscle eﬀort. A common cost function

to use is a scaled muscle force squared cost of the form:

T Z " 2 # ˙ 1 X Fi CF = 2 · Fisovmax dt, (2.11) T Fiso 0 i

th where the sum is over all muscles, Fi is the force produced by the i muscle, T is

the total time of one stride, and the product Fisovmax has units of power and provides appropriate scaling of the cost for various muscles as in [30].

Even though Cmet does have some dependence on muscle work, we also deﬁne a cost function that is explicitly based on muscle work alone:

1 X C˙ = (4W + + 0.83W − ), (2.12) W T m,i m,i i

+ − where the sum is over all muscles, Wm,i is the positive muscle work, and Wm,i is the negative muscle work. For this cost function, we use the coeﬃcients of 4 and 0.83

for the positive and negative work respectively based on experimentally determined

muscle eﬃciencies during concentric and eccentric muscle contraction [31,59].

24 Another cost function that is useful is one that takes into account the rate of force

production in each of the muscles:

T Z " ˙ 2 # ˙ 1 X Fi CFR = 2 · Fisovmax dt, (2.13) T Fiso 0 i ˙ where Fi is the derivative of the muscle force with respect to time (the muscle force rate). This cost function produces optima with smooth muscle forces given that rapid changes in force would increase cost. This force rate cost would likely not be used in isolation given that it predicts the same cost for a muscle that is fully activated the entire gait cycle and one that is never used.

Prosthesis cost

To observe the metabolic benefit of added prosthesis effort, we also define a prosthesis cost function. We use an objective function of the form:

T 1 Z C = (τ/r)2 + (τ/r ˙ )2 dt, (2.14) pros T 0 where τ is the motor torque,τ ˙ is the torque-rate, and r = 0.2 is a scaling constant

(equal to typical muscle moment arm); the torque-squared term is a model of motor

electrical losses [24] and the torque-rate-squared term with a small pre-multiplier ( =

0.01) is used to model torque production limitations by penalizing rapid activation

and deactivation [31,60].

2.2.3 Linear and nonlinear constraints

For our optimization to produce a physically meaningful walking cycle, we add

a series of linear and non-linear constraints. First, we constrain the biped to use a

set average velocity for the periodic stride. This allows us to ignore the resting cost

25 Joint Angle lower bound Angle upper bound Hip 1.571 6.2831 Knee 0.05 1.885 Ankle -0.9599 0.3491 Prosthesis -0.9599 0.3491

Table 2.3: Biped joint range of motion bounds. The joint angles are displayed graphically in Figure 2.1d. This table presents the bounds on these angles (in radians). Some bounds are assumed to be larger than anatomical, but such bounds are never active at the optimal solution, as is clear from the optimal kinematics depicted in Chapters 3 and 4.

in the optimization. Throughout our tests, we set the speed to v =1.3 m/s (non- p dimensional speedv ¯ = v/ g`leg = 0.42), except for calculations that varied speed systematically. We constrain positions, velocities, forces and torques to be continuous across diﬀerent time segments. For collisional transitions, the results of the collision equations (applied to the end of the contact phase) are equal to the initial conditions for the next contact phase. Further constraints include ground clearance for swing legs and ground contact for stance legs. Finally, we apply range of motion constraints for each joint to prevent hyperextension or intersecting segments. These range of motion constraints are given Table 2.3. We solve the constrained optimization problem using sparse nonlinear programming (SNOPT [31]), with equality constraint satisfaction of

10−7. An example of the optimization in progress is given in Figure 2.5.

26 0.25 Stance prosthesis torque Biped 10 Previous cost 3 10 0.2 2.5 0.15 2 10 5 0.1 0.05 1.5 0 1 10 0 -0.05 0.5 -0.1 0 10 -5 0 0.25 0.5 0.75 1 01 2 3 0 20 40 60 80 100 120

Torque vs angle -4 Constraint violations Previous constraint 0.25 10 10 1 0.2 1 0.15 0.5 10 0 0.1 0 10 -1 0.05 -0.5 0 -1 10 -2 -0.05 -1.5 -0.1 -3 -0.4-0.2 0 0.2 0.4 0.6 0 500 1000 1500 10 0 20 40 60 80 100 120

Figure 2.5: Illustration of optimization in progress. These ﬁgures give an estimate of current optimization progress by displaying the prosthesis toque proﬁle (both with respect to time and prosthesis angle), the biped position at the start of each contact phase, the current constraint violations, and a sample of the past objective function and constraint function values.

27 Chapter 3: Prosthesis control with time-dependent torques

3.1 Optimal human-prosthesis trade-oﬀs

One of the primary goals of active prostheses is to reduce the metabolic cost of

the user, thus making the walking gait more comfortable and less tiring. This goal

however, is likely at odds with the cost of the prosthesis in that prostheses that are

more powerful may be able to reduce metabolic cost but will also require more energy

to operate. So if we wish to design prostheses that are capable of reducing metabolic

cost and still use a reasonable about of energy in the prosthesis, it would be useful

to know the trade-oﬀs. To determine the trade-oﬀ, we set up the objective function

to include both a cost for the human and a cost for the prosthesis, namely:

˙ ˙ ˙ C = λCmet + (1 − λ)Cpros, (3.1)

˙ ˙ where Cmet is the metabolic rate, Cpros is the prosthesis cost rate, and 0 ≤ λ ≤ 1 is a

ﬁxed weighting factor: λ = 1 makes C˙ identical to human cost and λ = 0 gives the prosthesis cost. In practice we use 0.1 ≤ λ ≤ 0.95. Setting λ = 1 caused convergence issues in the numerical optimization, presumably because this limit λ = 1 implies a truly ‘zero cost prosthesis.’ The optimization tends to explore large and erratic prosthesis motor torques and torque rates, resulting in the numerical diﬃculties. A

28 similar situation occurs when λ = 0 as all muscles would be zero cost. Optimizing

with multiple λ values allows us vary the relative importance of the two costs in the

optimization and observe what happens to the gait when one cost is favored over the

other. In each case, the optimization produces feedforward prosthesis torques for a

periodic gait.

3.2 Results

3.2.1 Optimizing mostly just the human metabolic cost

We minimized the composite cost C˙ strongly weighted towards human cost (λ =

0.95), to obtain an optimum that would produce the lowest metabolic cost for human and mostly ignore the prosthesis cost. In doing so, we found a gait with a metabolic cost of 0.31 and a prosthesis cost of 0.24 (non-dimensional) Figure 3.1. As shown in Figures 3.2A-B, the corresponding optimal kinematics are bilaterally asymmetric

(the kinematics of the affected side differ from those on the unaffected side).

3.2.2 Comparison with non-amputee gait

We minimized the same human metabolic cost for a non-amputee human with symmetric legs and musculature, giving a symmetric optimal gait (Figure 3.2A-B).

The metabolic cost for the amputee with a prosthesis is lower than that for the non- amputee (Figure 3.1). In fact, this cost is even lower than the non-amputee cost if we made the muscles crossing a single ankle joint ‘free.’ These muscles contribute 41 % of the optimized non-amputee metabolic cost. Thus, if ideal motor torques replaced all muscles crossing one ankle while maintaining identical kinematics, the human cost could theoretically be reduced by 41% by this ‘muscle replacement strategy’.

Remarkably, for λ = 0.95, the human cost with a unilateral prosthesis is 73% lower

29 Pareto curves: human vs prosthesis cost trade-os Passive Optima 6 K1 K2

) Cost weighting -1 5 factor, λ K3 0.95 4 K4 Non-amputee (Able-bodied) 3

Muscle replacement strategy Symmetric 2 optima

Human metabolic rate (WHuman metabolic rate kg 1 Asymmetric 0.1 optima 0 0 0.1 0.2 0.3 Prosthesis cost rate

Figure 3.1: Optimal trade-offs (Pareto curves) between human and prosthesis costs. Results for optima without a symmetry constraint (blue line) and with a symmetry constraint (red line) are shown. Different markers (legend) denote results from different λ’s (0.1-0.95). Optima from passive prostheses with four non- dimensional stiffnesses (K1 = 0.1, K2 = 0.5, K3 = 1.0, K4 = 1.5), as are non-amputee optimum and the non-amputee optimum with muscle replacement strategy (cost-free ankle muscle costs).

30 Amputee with prosthesis Amputee with prosthesis A) (without symmetry contraint) (with symmetry contraint) Non-amputee

-0.50 0.5 1.0 1.5 -0.50 0.5 1.0 1.5 -0.50 0.5 1.0 1.5 B) 4 Joint Angles: Hip 1.5 Joint Angles: Knee 0.5 Joint Angles: Ankle

3.5 1 0

3 0.5 -0.5

2.5 0 -1 0 1000 1000 100 Percent gait cycle Percent gait cycle Percent gait cycle Amputee, Amputee, Legend: Non-amputee asymmetric symmetric

Figure 3.2: Comparison with non-amputee data. A) Optimal gait kinematics for amputee walking cycle (with and without a bilateral symmetry constraint) with cost function weighting λ = 0.95 and non-amputee walking cycle. The aﬀected side (or analogous side for non-amputee) is signiﬁed by the blue prosthesis. B) Optimal joint angles for all three conditions over one gait cycle with periods of 2.876 (asymmetric), 2.412 (symmetric), and 2.857 (non-amputee).

31 than the non-amputee cost. This greater cost reduction arises from allowing amputee

kinematics to be diﬀerent from non-amputee kinematics, thus allowing the prosthesis

ankle motor to perform much more work than the replaced ankle muscles.

3.2.3 Optimal trade-oﬀs between human and prosthesis cost.

By optimizing with different λ’s between 0.1 and 0.95, we obtain the optimal cost trade-off between human and prosthesis costs (Figure 3.1). This trade-off curve, often called a ‘Pareto curve’ [61], shows that increasing prosthesis cost decreases human cost. Here, even though it is obtained by minimizing a weighted sum of human and prosthesis cost, the Pareto curve (if ‘convex’) also has the following interpretation: this curve gives the lowest human cost for a given prosthesis cost and vice versa. Any other walking strategy will have either a higher human cost or a higher prosthesis cost compared to every point on this Pareto curve. Figure 3.3 shows that the optimal prosthesis actuation has most of prosthesis action at the end of stance phase, providing large push-off power; the human cost is reduced through an increase in prosthesis push-off torque and impulse. While the trade-off in Figure 3.1 is specific to the assumed human and prosthesis costs, we find that almost identical trade-off curves arise when substantially different cost functions are used (Figure 3.4)

3.2.4 Symmetry is expensive

Every Pareto-optimal gait we found was asymmetric irrespective of λ. However, walking with symmetry likely has physiological and psychological benefits for prosthesis users [62], so we repeated the optimizations while requiring approximate left-right symmetric kinematics. Symmetry was enforced as a constraint requiring the affected leg joint angles and angular rates during one step be nearly equal to the unaffected

32 Prosthesis torque over a complete gait period 0.3 Without symmetry constraint

0.2

0.1

Cost weighting

Prosthesis torques Prosthesis 0 factor, λ 0.95

-0.1 0 Fraction of gait period 1

0.3 With symmetry constraint

0.2 0.1

0.1

Prosthesis torques Prosthesis 0

-0.1 0 Fraction of gait period 1

Figure 3.3: Prosthesis torque over one stride with gait symmetry unconstrained and constrained. Each curve represents an optimization with different λ (0.1-0.95) as shown in the figure legend. The stride for each gait starts on the contact phase following the prosthesis toe-off (HT00).

33 leg joint angles and angular rates one step later (within 0.05 rad or 0.05 rad s−1).

The resulting symmetric optimal gaits had much higher human and prosthesis costs

(Figure 3.1) for each λ, a longer push-off phase, and different timing for dorsiflexion torques during swing phase (Figure 3.3). When we compared the cost of the optimal symmetric gait at λ = 0.95 to the non-amputee condition, we found only a 27% reduction in the metabolic cost. This reduction in cost is smaller than the 41% metabolic cost reduction achieved by the replacement strategy. The optimal symmetric gait with the prosthesis is worse than the simple replacement strategy, even though the replacement strategy would also result in a symmetric gait. While we do not know the reason for these relative costs, presumably the optimal symmetric gait with the prosthesis has a higher cost because the prosthesis is unable to produce knee torques similar to the missing gastrocnemius, requiring other muscles to compensate.

3.2.5 Lighter feet are less expensive.

Given that mass has a large eﬀect on metabolic cost, we perform optimizations with prosthetic foot masses from 50% to 150% that of the intact foot, scaling the prosthesis moments of inertia similarly. Figure 3.5 shows that by ﬁxing either the human or the prosthesis cost, we can reduce the other by reducing prosthesis mass.

Further, we see that cost reductions from reducing foot mass are much smaller than those obtained by increasing the prosthesis cost. Both human and prosthesis costs seem well-approximated by a linear dependence on the prosthesis mass for each λ

(Figure 3.5), but the coeﬃcients of the linear ﬁt depends on λ.

34 3.2.6 Greater human cost reduction at higher speeds.

We performed optimizations with ﬁve diﬀerent walking speeds between 0.7 and

1.5 m/s, computing the Pareto curves with λ = 0.1 to 0.9. As is true for non-amputee

walking, the human metabolic rate increases with increasing speed (Figure 3.6). In

particular, the whole Pareto curve for a lower speed is below and to the left of that for

a higher speed, implying that for a given Prosthesis cost rate, a lower speed implies

a lower human metabolic rate and vice versa. Further, for lower speeds, we predict

lower percentage and absolute reduction to the human metabolic rate while using an

optimal active prosthesis with most human beneﬁt (here λ = 0.9).

3.2.7 Passive prosthesis can be metabolically expensive

We performed optimizations constraining the prosthesis to be a linear torsional

spring and damper, simulating a passive prosthesis. We considered four stiﬀnesses:

0.10, 0.50, 1.00, and 1.50 non-dimensional stiﬀness normalized by bodyweight, chosen

to capture the torque-angle relationship of optima derived earlier. Damping was cho- p sen to make the foot over-damped during swing; we used Bpros = 1.73 · 2 IprosKpros, where Bpros is the damping, Ipros is the prosthesis moment of inertia, Kpros is the p torsional stiﬀness, and 2 IprosKpros is the critical damping value for a second order

linear system. While the metabolic cost decreases as the stiﬀness increased, all of

these passive devices produced asymmetric and metabolically expensive human gaits

(Figure 3.1). Human cost with the passive device is comparable to active prosthesis

with a symmetry constraint, except for high λ’s. Thus, the symmetry constraint is

so detrimental as to make active robotic prostheses have little to no energetic beneﬁt

over passive devices.

35 3.3 Discussion

We have obtained the optimal trade-oﬀs between human and prosthesis costs

for a robotic unilateral prosthesis, and have suggested that we can reduce amputee

metabolic cost by increasing prosthesis eﬀort, allowing asymmetry and decreasing

prosthesis mass. These relationships can inform prosthesis design, e.g., by selecting

a desired metabolic cost, we could predict the prosthesis cost at various prosthesis

masses; the prosthesis torques and costs along with information on number of steps

walked daily will allow us to pick motor and battery speciﬁcations for the prosthesis.

We have created a design tool that translates a speciﬁcation of the level of assistance to

be provided by the prosthesis (namely, λ) into appropriate prosthesis torque proﬁles.

Here, we have not compared predicted human kinematics and performance with experiment as such a comparison requires implementing our optimized feedforward control in a physical prosthesis. Highly accurate prediction of even non-amputee walking kinematics in a wide variety of novel situations remains an open problem; recent attempts have fallen short of quantitatively predicting the correct kinematics, kinetics, and/or metabolic costs [38,39,63], even though having qualitatively similar kinematics, analogous to our non-amputee optimization here.

Our optimization predicts asymmetric gaits for both robotic and passive prostheses but the mechanism that causes the asymmetry is likely diﬀerent in these two settings. When using a passive prosthesis, the gait becomes asymmetric since the prosthesis cannot add positive work to the system and the human is compensating with the intact limb. However, for robotic prostheses, our model suggests asymmetric gaits are energy optimal with symmetric gaits costing vastly more. In these asymmetric optimal gaits, the leg with the prosthesis spends more time on the ground,

36 compared to the biological leg, allowing the robotic prosthesis to provide greater assistance; we predict that optimal prosthesis actuation could reduce the human cost, possibly much below non-amputee levels (by over 70%). This energy reduction prediction is considerably higher than observed in experimental studies, which have at best reduced the users’ metabolic costs to about equal to non-amputee metabolic cost [18] (about 14% reduction compared to the subjects’ passive prostheses).

This discrepancy in energy costs is likely because our predictions are based on simultaneous optimization of human and prosthesis control and therefore a ‘best case scenario’, whereas the controllers in current robotic prostheses are not generally optimized to the person and the person may also not have had enough time to adapt to the prosthesis. Our optimal prosthesis actuation as a function of time has a large torque impulse near the end of the stance phase; it may be that current robotic prostheses, with their simpler ankle-state-based feedback controllers, are unable to allow the prosthesis to produce a large torque impulse right at the end. Most current prostheses also have a lower peak torque than what is allowed in our model, as our peak torque was based on a recent physical prosthesis emulator [6], reported as having the higher torque output among current prostheses. Further, whereas our prosthesis actuation is speciﬁc to walking at a particular speed on level ground, the feedback controllers in current robotic prostheses may have been the result of compromises made for use at diﬀerent walking speeds, slopes, etc. Producing large energy reductions in experiment may require a prosthesis capable of optimizing its torque output to the person and the environment (perhaps over a training period).

Other issues that compound the discrepancy in costs between current prostheses and our predictions include insuﬃcient torque motors, secondary goals sought by the

37 user (e.g. gait symmetry, pain reduction, lateral-stability, etc.), or other un-modeled eﬀects.

Alternatively, it may be that the large energy reduction we predict is due to our speciﬁc metabolic cost model. Experimental studies in non-amputees suggest that each ankle contributes about 13% of the metabolic cost of walking [64] while our metabolic cost model predicts a total of 41%. Repeating the non-amputee calculation with a work-based metabolic cost, we found that only about 18% of the total work cost is due to one ankle – this is the reduction predicted by a muscle replacement strategy for this work-based cost (Figure 3.4). For this work-based cost, the robotic prosthesis (with λ = 0.95) produced only a 23% reduction in cost, still greater than the 18% from the muscle replacement strategy, but much lower than 73%. On the other hand, for a scaled muscle-force-squared cost, the ankle contribution was about

43% for a non-amputee walk and the reduction from the robotic prosthesis was about

74% (Figure 3.4). Thus, it appears that some of these speciﬁc numerical predictions may rely on the metabolic cost model and may be improved with a much more accurate metabolic cost model, which remains an open problem [31,38].

One potential source of error in the model used for these optimizations is in the ground reaction forces and impulses. The ground reaction force was constrained to be positive at the beginning of every segment throughout the walking cycle, and given that each of the segments are constrained to be continuous, this was assumed to be a strict enough constraint. However, this does allow the biped to pull on the ground brieﬂy at the end of segments and during the impulse of collisions. Such unphysical forces would be small but possibly signiﬁcant to the prediction of metabolic cost. In future tests, we add GRF constraints and impulse constraints to additional segments

38 to prevent the biped from being able to pull on the ground, thereby avoiding this source of error.

Having performed over a hundred different optimization calculations under different parameter conditions (in contrast to other optimization-based studies [38,39,63]), we have demonstrated feasibility of using such large-scale optimizations in a formal design procedure with user-specific model parameters.

39 Optimal costs for three cost functions

0.05 0.08 0.16 Cost weighting 0.045 0.06 Non-amputee optimal factor, λ 0.12 0.95

0.04 0.04 0.08 Muscle replacement strategy

Human cost 0.035 0.02 0.04 Pareto-optimal amputee with robotic prosthesis 0.03 0 0 0.1 Alexander-Minetti 0.020.06 0.10 0.14 0.18 0.22

Force-squared 0.01 0.03 0.05 0.07 0.09 0.11 Work-based 0.005 0.01 0.015 0.02 0.025 0.03 Prosthesis cost

Figure 3.4: Pareto curves for two other cost models. Pareto curves for three different cost functions are shown (solid curves): (1) using Alexander-Minetti metabolic cost for human and smoothed torque-squared cost for prosthesis, as in the main manuscript (Figure 3A), (2) using a scaled muscle-force-squared cost for the human and a scaled torque-squared cost for the prosthesis, (3) using a muscle work-based cost for the human and a motor work cost for the prosthesis. All costs shown are normalized by body weight and leg length. Thus, these Pareto curves show that all these diﬀerent costs give qualitatively similar trade-oﬀs between human and prosthesis energy costs for the amputee. We also show the optimal costs for a non-amputee walker (long-dashed line) and the “muscle replacement strategy” (short-dashed line) for the three cost functions. The optimal robotic prosthesis actuation reduces amputee energy cost below that of a non-amputee and a non-amputee walker with cost-free muscles crossing an ankle.

40 Human vs prosthesis cost trade-o for dierent prosthesis masses

0.16 Cost weighting Relative prosthesis foot mass (m m " pros / foot factor, λ

0.12 0.5 0.75 1.0 1.25 1.5 0.9

0.08 Human metabolic rate 0.04 0.1

0 0 0.1 0.2 Prosthesis cost rate

Figure 3.5: Trade-offs between human and prosthesis cost for five different prosthetic foot inertial parameters. Masses / moment of inertias range from 50% to 150% unaffected foot mass. Lowering prosthetic foot mass decreases metabolic and prosthesis costs for a given cost weighting factor λ.

41 Human vs prosthesis cost trade-o! for di!erent speeds 0.20

Walking speed (m/s) Cost weighting 0.16 factor, λ 0.7 0.9 1.1 1.3 1.5 0.9

0.12

0.08 Human metabolic rate

0.04 0.1

0 0 0.05 0.10 0.15 0.20 0.25 Prosthesis cost rate

Figure 3.6: Trade-offs between human and prosthesis cost with five different walking speeds. Speeds range from 0.7 m/s to 1.5 m/s. As shown, high prosthesis effort is more important to reducing metabolic cost at higher speeds than at lower speeds.

42 Chapter 4: Human and prosthesis optimization with feedback control

4.1 State-based prosthesis controllers

In the previous chapter, we considered the idealized case of the prosthesis capable of an arbitrary torque as a function of time at the ankle. The high dimensionality of the prosthesis actuation as a piecewise linear function of time allows us to discover close to the greatest possible metabolic reduction in a mathematical model. However, such an idealized calculation may only be useful as a benchmark and may not be practical for a real world device since it produces purely time-based feedforward control for a perfectly periodic gait at a single walking speed.

Here, we attempt to predict the eﬀects of constraining the prosthesis actuation to four simple feedback controllers: three active controllers based on ankle ‘torque- angle relationships’ [6, 19, 51, 52, 53] and one passive prosthesis based on the SACH foot [2]. Each of the three active controllers is designed with a diﬀerent torque-angle loop and a single parameter that varies the amount of work being applied by the prosthesis. We examine the implications of the assumed torque-angle loops and test whether increasing net prosthesis work always reduces net metabolic rate. We also determine a relationship between bilateral asymmetry and net metabolic rate using

43 these controllers both with and without bilateral symmetry constraints. Finally, we

compare our optimization results with experimental data [8] which used a similar

prosthesis controller.

4.1.1 Variable work feedback control

The prosthesis torque is determined by state feedback using prosthesis ankle angle

α, angular rateα ˙ , and the contact phase. We consider three families of torque-angle relationships during stance (Figure 4.1). Controller 1 (Figure 4.1a) is based on a simple controller used on the BioM prosthesis [19]. Controllers 2 and 3 (Figures 4.1b and 4.1c) are based on those used in prosthesis emulators by Caputo and Collins [6].

The three controllers are parameterized by one variable each, allowing us to change the area within the torque-angle work-loop, thereby changing the net prosthesis work.

For a typical gait cycle, all three controllers are identical during the swing and initial stance phases, but diverge in how they apply power during late stance, as described below (see Figure 4.1e for more detail on the various stance phases).

Initial stance phase control is spring-like. The ‘initial stance phase’ for the

prosthesis starts at heel-strike, continues into the ﬂat-foot contact phase (with both

heel and toe on the ground), and then transitions into the push-oﬀ phase (deﬁned

in the next paragraph, see Figure 4.1e). During this initial stance phase, all three

controllers act like linear torsional springs with stiﬀness 290 N rad−1, no damping,

and rest angle αrest = 0.062 rad (3.5 deg) of plantar-ﬂexion.

Three strategies for active push-oﬀ during late stance. At the onset of push-

oﬀ (when ankle angleα ˙ changes from negative to positive after the heel has left the

44 ground), the three controller affect the stance work of the prosthesis through three distinct strategies. Controller 1 adds a constant torque offset ∆τ to the original stiffness during initial stance by changing the rest angle ∆αrest while keeping the stiffness constant (Figure 4.1a). Controller 2 uses a new linear torque-angle relation by changing the rest angle ∆αrest, with the new stiffness selected so as to produce a continuous torque throughout the stance phase (Figure 4.1b). Controller 3 maintains the max torque produced when dorsiflexion ends and holds it for a specified change in prosthesis angle ∆αrest before returning to the original stiffness. Thus, all three controllers are parameterized by the single variable ∆αrest.

PD control during swing phase. During the ‘prosthesis swing phase,’ when the prosthesis is not in contact with the ground, all three controllers use a proportional- derivative (PD) controller. This PD controller resets the prosthetic foot rest angle to zero (αrest = 0), thus re-positioning the prosthetic foot perpendicular to the shank.

We use a proportional gain (torsional stiﬀness) of 36.25 N rad−1 and a derivative gain

(damping) of 2.29 Ns rad−1. These gain values prevent underdamped foot oscillations during and reset the prosthesis angle in time for the next heel strike.

4.1.2 Comparing results of trajectory optimization to experiment

In order to verify if our optimization produces results consistent with real world experiments, we compare our results with human subject testing. Using data from a variable work prosthesis emulator [8], we look for both qualitative and quantitative similarities between the joint torque and angle data and the optimization results at similar prosthesis work rates. We quantify the error using the standard deviation of

45 Torque-angle ralationships for prosthesis controllers

A) Controller 1 B) Controller 2 C) Controller 3 D) Passive prosthesis (c1) (c2) (c3) (SACH) Torque Torque Torque Torque ∆α rest ∆τ 4 4 4 4 3 3 3 3

Ankle Ankle Ankle Ankle 1 5 1 5 1 5 5 1 angle angle angle angle ∆α ∆α rest rest 2 2 2 2

E) Phases of prosthesis stance

1. Heel strike 2. Toe strike 3. Braking 4. Push-o$ 5. Toe-o$

Swing Initial stance Late stance Swing

Figure 4.1: Torque-angle work-loops during stance for three prosthesis controllers. A) Controller 1. Push-off work is controlled through the addition of a constant torque ∆τ by increasing the rest angle by ∆αrest while maintaining stiffness. B) Controller 2. Push-off work is controlled through adjusting the rest angle (by ∆αrest) and stiffness to create a continuous torque throughout stance. C) Controller 3. Push-off work is controlled through maintaining the maximum torque produced during dorsiflexion over a range of ankle angles ∆αrest, then returns to the linear stiffness used in the initial stance phase. D) Passive controller. The work-loop shown is taken from experimental data while walking with a SACH foot [2]. E) Prosthesis stance phase sequence. Heel strike, toe strike, and braking all belong to the initial stance phase. Initial stance is followed by push-off, which lasts until the toe leaves the ground. The leg then continues into the swing phase. Numbers 1-5 are overlaid on panels A-D, corresponding to the contact phase sequence shown.

46 the diﬀerence between the value from data (either joint torque or joint angle) and the

value produced by the optimization throughout one stride. This error quantity cap-

tures the root-mean-square error, but subtracts out any systematic constant oﬀset.

Thus, if the diﬀerence is a constant oﬀset, this quantity will report zero. This mea-

surement does not account for diﬀerences in step time or contact phase information.

4.2 Changes to the optimization setup

Compared to previous calculations in Chapter 3, we made some changes to the

objective function and the ground constraints. For the new objective function, we ˙ minimize is a combination of the metabolic rate model Cmet suggested by Alexander ˙ and Minetti [30, 31, 65] and a small muscle force rate cost CFR, summed over all muscles: ˙ ˙ ˙ Ctotal = Cmet + αCFR. (4.1)

Here, α is a small weighting coeﬃcient equal to 5 × 10−4. This additional small force rate term penalizes rapid force changes, avoids convergence to local minima with erratic muscle forces, but is weighted to be so small that it does not increase ˙ the optimal metabolic rate Cmet. Some prior experiments also suggest that muscle metabolic cost has such a force-rate related term [66], although we have not attempted to incorporate this term quantitatively.

For the ground contact constraints, we have added extra ground force constraints such that the vertical force must be positive at the end of each segment as well as at the beginning. This is necessary since the ground forces are not constrained to be continuous throughout the gait. Without this additional constraint, the ground reaction forces at the end of a contact phase may drop below zero (thus brieﬂy pulling

47 on the ground). We also added a constraint in which we enforce any impulse applied

to the ground during collisions must be positive as well.

4.3 Results

4.3.1 Increase in prosthesis work rate reduces metabolic rate

We obtained gaits that minimized metabolic energy with the three prosthesis

˙ −1 controllers for net prosthesis work rates Wpros ranging between -0.3 and 1.0 W kg , by considering a range of rest angle changes (∆αrest) for the prosthesis. Specifying

∆αrest constrains the work approximately and not exactly. This is because, like humans, the model can vary stance time, push-oﬀ angle and push-oﬀ timing, even if the overall torque-angle-loop is constrained. A plot of each controller’s torque- angle-loop can be found in Figure 4.2. As prosthesis work rate increases, there is a decrease in metabolic rate down to a minimum. Any further increase in work led to increased metabolic rate. We highlight this non-monotonic behavior in metabolic

˙ ˙ ˙ 2 rate reduction by the quadratic ﬁt: C = 2.4 − 4.0Wpros + 3.2Wpros (shown in Figure 4.3). At the optimal net prosthesis work rate, the predicted human metabolic rate

due to all three controllers was about 1.0 W kg−1.

4.3.2 Simple feedback is worse than optimized control but both are better than SACH foot

As expected, the metabolic rates from using the simple feedback controllers were

higher than when the prosthesis torque was optimized as an arbitrary function of time

(that is, unconstrained by a simple torque-angle feedback relation). Minimizing the

composite human-plus-prosthesis cost function weighted mostly toward the human

metabolic rate (λ = 0.95), we ﬁnd an optimal metabolic rate around 0.86 W kg−1 for

48 Torque-angle-loops for all controllers

A) Controller 1 D) Optimal 200 200

100 100

0 0 torque (Nm) torque (Nm) -100 -100 -1 -0.50 0.5 1 1.5 -1 -0.50 0.5 1 1.5 angle (rad) angle (rad)

B) Controller 2 E) SACH 200 200

100 100

0 0 torque (Nm) torque (Nm) -100 -100 -1 -0.50 0.5 1 1.5 -1 -0.50 0.5 1 1.5 angle (rad) angle (rad)

C) Controller 3 200 Increasing work rate 100

0 torque (Nm) -100 -1 -0.50 0.5 1 1.5 angle (rad)

Figure 4.2: Controller torque-angle-loops for increasing work rate A-C) The torque angle relationship of Controller 1, 2, and 3 over a full gait cycle. The gradient from blue to red indicates the controller with increasing net prosthesis work rate. D-E) The optimized controller and SACH torque-angle-loops. Only a single color is shown as neither have variable work.

49 E!ects of prosthesis work on metabolic rate

4 SACH

exp ) -1 3

non-amp 2

c2 c3 c1 1 Net metabolic rate (WNet metabolic rate kg optimized

0 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Net prosthesis work rate (W kg -1 )

Figure 4.3: Eﬀects of prosthesis work on metabolic rate. For Controller 1 (c1, blue), Controller 2 (c2, red), and Controller 3 (c3, yellow), the relationship between net metabolic rate and net prosthesis work rate is non-monotonic and has a minimum. ˙ ˙ ˙ 2 A quadratic ﬁt to pooled data from all three controllers (C = 2.4−4.0Wpros+3.2Wpros) is shown. The simulated SACH foot and the optimized prosthesis controller (both shown in green) occupy two extremes of high metabolic rate / low prosthesis work for the SACH foot and low metabolic rate / high prosthesis work for the optimized controller. Metabolic rate versus work rate for human subject experiments shows no systematic trend and has high variability (mean ± standard deviation, shown in black and gray) [8].

50 such optimized control. Thus, this optimized control produces about an 14% lower metabolic rate than the torque-angle-based feedback controllers at their optimum

(Figure 4.3). In contrast, simulating a passive prosthesis, namely, the SACH foot, we found a cost of about 3.8 W kg−1, a much higher cost than all other controllers tested at similar work levels.

4.3.3 Zero work prostheses can give near-able-bodied costs

All three controllers have a zero net stance work condition, in which the controllers behave like an undamped spring and perform equal amounts of positive and negative work over a stance phase. These zero work conditions resulted in amputee metabolic rates that were comparable to our model predictions for able-bodied optimizations

(Figure 4.3). This result echoes the recent experiments in which the metabolic rates of physically active and well-trained amputees with passive energy-storage-and-return

(ESR) prostheses were negligibly diﬀerent from that of able-bodied controls [14, 15].

This result may also be related to the reduction of muscle work requirements in walking and running via energy storage and return in tendons, other elastic structures, or passive exoskeletons [67,68] and the ability of simple springy-legged models to walk without actuator work [69].

4.3.4 All energy-optimal gaits are asymmetric gaits

When there were no explicit constraints on symmetry, all energy optimal gaits were left-right asymmetric. Figure 4.4 shows the stance time percent difference between the unaffected and affected stance times. This stance time difference is a necessary condition for symmetry but not a sufficient condition. Non-zero stance time difference implies asymmetry, but zero stance time difference admits asymmetry in other

51 iue4.4: Figure AHfo eut nahgl smercgi ihmr ieseto h unaf- in the asymmetry on large spent in time results circles). more prosthesis (green with direction The robotic gait opposite optimized comparison. asymmetric the the for highly whereas shown a limb, bars) are in fected error circle) gray results to (pink and (black foot corresponds result [8] SACH generally non-amputee data pro- Experimental work simulated prosthesis negative side. the robotic affected contrast, and the the In as on side spent power. time affected less work. positive the prosthesis on net step affected time percent positive more more The the vides increased spends on rates. with work biped spent linearly, various the stride at roughly Thus, c3) of decreases, c2, percent (c1, difference the controller time each stride minus for the side shown of unaffected is percent side), (the the difference on time spent step percent time by quantified as symmetry, eral Bilateral symmetry (unaected - aectedBilateral limb) symmetry

% Step-time dierence -30 -20 -10 10 20 30 0 04-. . . . . 1.2 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 iaea ymtyi orltdwt rshsswork. prosthesis with correlated is symmetry Bilateral non-amp SACH Net prosthesis work rate (W kgNet prosthesis work rate (W exp 52 c2 c3 -1 ) c1 optimized h bilat- The Joint angles over one stride period for all controllers C1 C2 C3 Hip Knee Ankle Hip Knee Ankle Hip Knee Ankle 1

0 Zero work Decreasing work

Angle -1 1 0

Angle -1 1

0 Increasing work

Angle -1 1 0

Angle -1 % Gait % Gait % Gait % Gait % Gait % Gait % Gait % Gait % Gait

Figure 4.5: Joint angles over one stride have bilateral asymmetry for all controllers. The angles (in radians) for the hip, knee, and ankle are shown for each controller at various levels of work. Each group of plots belongs to the same controller and work level. The aﬀected and unaﬀected sides are shown in blue and gray respectively. All plots start at heel strike and display extension as positive.

53 aspects of the gait (See Figure 4.5). For all controllers considered, increase in net positive prosthesis work was associated with more stance time on the aﬀected side.

The optimizations predicted that walking with a SACH foot would have the opposite asymmetry, so that that person spends more time on the unaﬀected foot, as also seen in experiments [70, 71]. Experiments involving amputees wearing robotic prostheses [8] did not show strong correlations between symmetry and prosthesis power, whereas in other non-amputee prosthesis emulator experiments [6], the prosthesis stance fraction increased with prosthesis power, as predicted by our model.

4.3.5 Symmetry constraints increase cost but promote kinematics closer to experiment

We found that the metabolic rates increased with symmetry constraints (Figure

4.6). For step time symmetry, the metabolic rate increased slightly (by about 6% over the unconstrained condition at zero work). The metabolic rate increase was similar for constraining just the kinematics or the ground reaction forces (by about

60% over the unconstrained condition at zero work), but constraining symmetry in both kinematics and ground reaction forces increased the costs much more (by about

120% over the unconstrained condition at zero work). We ﬁnd that the addition of the symmetry constraint improves correspondence between optimization predictions and experimental observations [8], for metabolic rates (Figure 4.6), joint kinematics

(Figure 4.7), and joint torques (Figure 4.8). We do find significant differences in affected knee joint torques regardless of the symmetry constraints used, see Figure

4.9. Specifically, the experimental data shows knee flexion torques on the affected side during mid-stance, whereas the optimizations show no such torques. However, when

54 Eects of constrained symmetry (controller 1) 7

6 ) -1 5 GRF & kinematic

4 GRF kinematic exp 3

2 non-amp

Net metabolic rate (WNet metabolic rate kg step time unconstrained 1

0 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 Net prosthesis work rate (W kg -1 )

Figure 4.6: Controller 1 with various symmetry constraints. Applying step time, kinematic, or ground reaction force (GRF) symmetry constraints to the optimization increases the metabolic rate over the unconstrained condition. When both kinematic and GRF constraints are enforced, the metabolic rate is much higher than any constraint alone. Experimental metabolic data [8] (black and gray error bars) and simulated non-amputee metabolic rate (pink circle and line) are shown for comparison.

55 we compare the muscle forces with experimental EMG [15,72,73], we ﬁnd considerable overlap in the activation timings (Figure 4.10).

4.3.6 Reduced limb mass or limb muscle strength do not affect qualitative features

We performed two additional optimizations at various levels of prosthesis work, one with 30% reduced limb mass and one with 30% reduced muscle strength, as recently considered in [15]. These calculations resulted in negligible quantitative differences in torque profiles and joint angle profiles, and small differences in metabolic costs. We did not observe qualitative differences that suggested that such reduced mass or strength could improve model predictions.

4.4 Discussion

In this study, we have presented a computational framework for predicting human walking with a robotic or passive prosthesis and have considered multiple controllers.

We have shown that the relationship between prosthesis work and metabolic rate is non-monotonic: the metabolic rate of an amputee first decreases with increasing prosthesis work rate and then increases, roughly with a quadratic relationship. This relationship between prosthesis work and metabolic rate was quantitatively similar for all tested controllers. While it is possible that this relationship is common for a wide range of controllers (as suggested by our results), it is also possible that the controllers used here were not sufficiently different from one another in the space of all controllers.

The non-monotonic dependence of metabolic rate with net prosthesis work suggests that simple work-based heuristics such as the ‘augmentation factor’ [74] may

56 Joint angles over one stride Una!ected

0.5 Hip 0.5 Knee 1 Ankle 0 0 0.5 -0.5 -0.5 0

Angle (rad) Angle -1 -1 -1.5 -0.5 00.5 1 00.5 1 00.5 1 Stride fraction Stride fraction Stride fraction

A!ected

0.5 Hip 0.5 Knee 1 Prosthesis 0 0 0.5 -0.5 -0.5 0

Angle (rad) Angle -1 -1 -1.5 -0.5 00.5 1 00.5 1 00.5 1 Stride fraction Stride fraction Stride fraction

experimental data step time GRF unconstrained kinematic GRF & kinematic

Figure 4.7: Joint angle comparisons with experiment with and without symmetry constraints. This figure presents the joint angles for selected tests (with similar levels of net prosthesis work) using controller 1 with various step time, kinematic, and/or ground reaction force (GRF) symmetry constraints, overlaid with amputee data [8], depicted as a mean curve and 95% confidence interval band (black solid line and gray band). Positive angle values refer to extension and negative refer to flexion; plantar flexion and dorsiflexion are positive and negative respectively for the ankle and prosthesis. Each stride begins with the heel-strike of the corresponding leg being plotted.

57 Joint torques over one stride Una!ected 150 Hip 100 Knee 200 Ankle 100 50 150 50 100 0 0 50 -50 -50 0

Torque (Nm) Torque -100 -100 -50 -150 -150 -100 00.5 1 00.5 1 00.5 1 Stride fraction Stride fraction Stride fraction

A!ected 150 Hip 100 Knee 200 Prosthesis 100 50 150 50 100 0 0 50 -50 -50 0

Torque (Nm) Torque -100 -100 -50 -150 -150 -100 00.5 1 00.5 1 00.5 1 Stride fraction Stride fraction Stride fraction experimental data step time GRF unconstrained kinematic GRF & kinematic

Figure 4.8: Joint torque comparisons with experiment with and without symmetry constraints. This figure presents the joint torques for selected tests (with similar levels of net prosthesis work) using controller 1 with various step time, kinematic, and/or ground reaction force (GRF) symmetry constraints, overlaid with amputee data [8], depicted as a mean curve and 95% confidence interval band (black solid line and gray band). Positive torque values refer to extension and negative refer to flexion; plantar flexion and dorsiflexion are positive and negative respectively for the ankle and prosthesis. Each stride begins with the heel-strike of the corresponding leg being plotted.

58 Angle Error 0.3 unconstrained step time 0.2 kinematic GRF 0.1 GRF & kinematic

0 θhip,u θknee,u θankle,u θhip,a θknee,a θankle,a Torque Error 60

20 Torque (Nm)Torque Angle (rad) 0 τhip,u τknee,u τankle,u τhip,a τknee,a τankle,a

Figure 4.9: Error between joint torques/angles from optimization and experimental data. Error is measured using the standard deviation of the diﬀerence between experiment and optimization results. Joint torque and angle data is taken from selected tests (with similar levels of net prosthesis work) using controller 1 with various step time, kinematic, and/or ground reaction force (GRF) symmetry constraints. The legend refers to the symmetry constraints applied during each optimization.

59 Muscle forces with bilateral symmetry constraints A ected Una ected 1 1

Illio Gas 0 0 1 1

Glut Sol 0 0 1 1

Ham TA 0 0 1 0Stride fraction 1 EMG data RF Dierent symmetry 0 constraints 1 unconstrained step time Vas kinematic 0 GRF 0Stride fraction 1 0Stride fraction 1 GRF & kinematic

Figure 4.10: Normalized muscle forces with and without symmetry constraints. Normalized muscle forces are presented over one stride period, as a fraction of maximum isometric force. The curves shown are from optimizations that differ only by the symmetry constraints used (including no symmetry constraints). All optimizations had similar net prosthesis work rate and used prosthesis controller 1. Each stride shown starts from the heel-strike of the leg in which the corresponding muscle is present. Horizontal black bars atop each figure panel are an on/off representation of EMG data taken from experiment [15,71,72,73].

60 not be able to completely capture metabolic reductions at high work rates. Further, augmentation factor (as defined in [74]) may be more appropriate for exoskeletons and not as appropriate for prostheses (without modifications). For instance, we find that the augmentation factor predicts metabolic rate benefit for energy neutral or even energy negative passive prostheses. Nevertheless, the initial slope of the metabolic rate reduction versus work rate is about -4.0, corresponding to the standard 25% efficiency for muscle positive work (Figure 4.3, see linear term in quadratic fit). This suggests that mechanical-work-based heuristic reasoning [74] may be applicable for smaller work-rates.

Most metabolic measurement studies have shown that amputees experience about

10-30% higher metabolic rate using passive prostheses than their non-amputee control subjects [14, 75, 76, 77]. Our optimizations predict that a SACH foot, which has considerable damping, produces 35% higher metabolic costs than for non-amputees, qualitatively similar to the older experimental studies. However, when we tested a controller that produces zero net prosthesis work during stance (analogous to passive a prosthesis with no damping), we found that it is possible to produce non-amputee levels of metabolic rates. Such low metabolic costs seem inconsistent with some older experimental studies. However, the prostheses used in the older experimental studies are not ‘zero work’ prosthesis. There is always some energy loss in any passive prosthesis, whether it is a SACH foot or an ESR prosthesis like the Flex Foot. The higher costs measured in experiment may also be due to the subjects’ physical ﬁtness or training on their prostheses. Indeed, a recent study showed that when physical

ﬁtness is not an issue, amputees use about the same metabolic rate as non-amputee control subjects [14], similar to our results from the zero-work-zero-damping condition

61 of the prosthesis controllers. Qualitatively similar conclusions were reached by another recent simulation study [15], albeit by including tracking of normal kinematics in the optimized cost function.

At net positive prosthesis work rates, we ﬁnd metabolic rates lower than that of an analogous able-bodied (non-amputee) walking optimization (computed in [65]). Such large metabolic rate reductions do not seem to match those found in experiments thus far. Reductions in metabolic rates in experiment are often small [7,78] or insigniﬁcant

[8] and have never produced a metabolic rate below that of a non-amputee.

One source of these large metabolic reductions could be the bilateral asymmetries we see at high work rates in our optimization. In the absence of explicit symmetry constraints, the simulated biped tends to spend more time in stance on the affected side than on the unaffected side (Figure 4.4), making use of the ‘free’ power of the prosthesis. When we constrain the biped to walk with equal stance time on each leg, relative metabolic benefit of the high prosthesis work is reduced. If we further constrain the joint angles and/or ground reaction forces to be left-right symmetric, the metabolic rate increases further and better matches experiment at similar work rates. This suggests that gait symmetry could be a conscious or subconscious goal for the user. Therefore, it may be necessary to design prostheses such that the energy optimal gait naturally has high symmetry. This is an open problem. Further, we see that these bilateral symmetry constraints lead to more human like joint torques.

For example without any constraints, the push oﬀ torque in the unaﬀected ankle occur more suddenly at the end of stance (Figure 4.8). However, with constrained kinematic symmetry, the ankle torque ramps up gradually over the stance phase, similar to experiment. Our results here and in a previous study [41, 65] suggest that

62 such improvements in gait symmetry will have a metabolic penalty, but could equalize joint stresses [41,79]. Based on these results, we speculate that simply implementing the controller inferred from normal human foot function [64,80,81,82] will still result in substantial gait asymmetry, as the prosthetic ankle will still be an asymmetric source of free propulsive power to the user. We hypothesize that promoting gait symmetry without metabolic penalty may require augmenting the unilateral prosthesis with an exoskeleton on the unaﬀected side [74,83]. We further hypothesize that a human-in- the-loop optimization that systematically moves the gait away from symmetry may produce metabolic reductions greater than that observed so far [83].

Discrepancies between model-predicted and experimental metabolic rates (Figure

4.6) could be due to model simplicity, including the simplified metabolic model, the simplified socket interaction that ignore effects of interfacial forces between the prosthesis and the residual limb [41], the simplified ground contact, and the perfectly periodic gait that ignores gait stability. We also ignored the muscle force-length relationship and ignored any potential dependence of metabolic rate on muscle length. Fu- ture work may consider other metabolic cost models, such as [84], but to do so would require more complex muscle models which include activation dynamics and force length relationships. One simple modification could be incorporation of a musculo- tendon-length-dependence of metabolic rate, without implementing the force-length relationship. Incorporating such a cost may potentially limit the motion to smaller joint angle excursions, closer to where force and work production is optimal for the muscle. Similar to discrepancies in metabolic rates, we found that the optimizations and data also differed somewhat for kinematics and much more for joint torques (Fig- ure 4.8, Figure 4.7, and Figure 4.9). Specifically, our optimizations lack knee flexion

63 torques on the affected limb during late stance, but such knee flexion torques are present in experimental data. One possibility is that in our model, the unaffected side does not have residual gastrocnemius, which is known to have some activity in transtibial amputees [85, 86]. Alternatively, the lack of knee flexion torques may represent an alternate walking strategy in which the knees are more flexed during mid-stance.

Given that detailed quantitative agreement between model predictions and walking data (without any ﬁtting) is an open problem, our goal here was to mainly es- tablish qualitative predictions for humans walking with robotic prostheses similar to past non-amputee simulations [38, 39]. Past work on even non-amputee models also contained large errors in predicted forces and torques and smaller errors in predicted kinematics [39]. However, with explicit ﬁtting to experimental data, it has been possible to obtain better agreement between model and data [15,38], but even then, the kinematic match has been better than the kinetic (force) match.

One possible interpretation of our results is as a negative result: that is, amputees do not walk in a manner that just minimizes metabolic rate, subject to the above caveats about model simplicity. For instance, amputees may artiﬁcially co-contract to improve stability [87, 88, 89], increasing metabolic expenditure while keeping mechanical work similar. Amputee walking may also be adapted to reduce aﬀected limb joint loading, socket loading, or more generally, increase comfort and reduce pain. So, in future work, we will consider adding such loading terms as part of the objective to be minimized (e.g., [90]).

64 Amputees are known to have reduced muscle strength on the aﬀected side [91,

92,93], a feature we have mostly ignored in our modeling here except for a few optimization calculations with reduced maximum isometric force. The eﬀects of retaining muscle strength was considered in more detail in a recent simulation study [15]. The gait optimizations in that simulation study [15] had an objective to track normal kinematics in addition to a metabolic rate term. They found that reduction of muscle strength resulted in increased metabolic rate, whereas retaining normal muscle strength resulted in a normal metabolic rate.

Our model of the human sensorimotor system has been particularly simple: we assumed that the human motor system will eventually converge to the metabolic minimum, as humans are known to do in the presence of prostheses [33] or exoskeletons

[42, 83]. By focusing exclusively on metabolic rate, we have ignored active feedback control of human walking by the human sensorimotor system. This active feedback control can be in the form of fast reﬂexes or longer latency feedback control. Inclusion of such human motor control would be desirable in our simulation, as that would further constrain the space of possible muscle activations and walking strategies.

Unfortunately, we do not yet have a good enough characterization of the human walking sensorimotor system at the level of each muscle. For instance, Geyer and colleagues [78, 94] have proposed a reﬂex-based controller for walking that has been used both for prosthesis control [78] and as a hypothesis of human motor control

[94]. However, the detailed structure and parameters of such feedback controller models have not been validated using joint-level or muscle-level human experiments.

Nevertheless, constraining our optimizations by such simple models of human motor control may provide valuable insight into the qualitative eﬀect of such control. One

65 subtlety in modeling the amputee motor control system is that the amputated limb is not an explicit part of the human sensorimotor loop. For instance, the human nervous system does not have direct sensory information of the prosthesis state and torque. However, it is known that humans can adapt to forceful interactions with exoskeletons whose internal state are not directly accessible to the user [42,83].

We have focused here on unilateral prosthesis with a speciﬁc class of feedback controllers. We could apply the same techniques to obtain predictions about a broad class of prosthesis and exoskeleton controllers, including those that rely on other human body and myoelectric state variables [95], qualitatively inspired by neuromuscular control [78], or to explore and optimize the eﬀect of geometry of the prosthesis [96]. The long-term goal is a general framework for model-based rational design of biomechatronic devices. Our approach can test high dimensional prosthesis controllers with thousands of parameters, and therefore could be a complement to the human-in-the-loop optimization approaches, which are usually constrained to low- dimensional prosthesis controllers.

66 Chapter 5: Improving model predictions through controller and cost function modiﬁcation

Two main predictions of our model are: 1) a prosthesis capable of producing positive work will produce a gait with bilateral asymmetry favoring the aﬀected side and 2) adding prosthesis work can decrease the metabolic cost below that of non- amputee gaits. These two predictions are not always seen in experimental data.

When we add a symmetry constraint, as shown previously, the metabolic cost is higher than non-amputee cost. However, symmetry may not necessarily be the goal of the prosthesis users, rather the result of various other goals. In this section, we will explore diﬀerent controller morphology and various objective functions to attempt to reduce diﬀerences between model predictions and experimental data.

5.1 Methods

In this section, we describe the various modifications to our model. First, we define a controller with a non-linear spring that is meant to better approximate the function of the unaffected ankle joint. Next, we describe the various mechanical costs that we add to the metabolic cost. Finally, we describe the socket interaction costs, which penalize large forces and moments at the interface between the residual limb and the prosthesis socket.

67 5.1.1 Controller with nonlinear initial stiﬀness

Even though all the active controllers presented in Chapter 4 used a diﬀerent

methods for adding work during the push-oﬀ phase of the gait (see Figure 4.1), they

all produced the same cost-vs-work relationship with only minor diﬀerences in gait

kinematics. This could be due to the controllers being too similar in function: all used

the same linear stiﬀness during dorisﬂexion and had the same rest angle. Here we

perform optimizations with a controller that has a non-linear dorsiﬂexion and plantar

flexion stiffness, meant to mimic the stiffening properties of an intact ankle [80]. To

produce a controller similar to that used in [6,7,8] we use a cubic feedback controller

of the form:

3 τ = klinear(θ − θrest) + kcubic(θ − θrest) , (5.1)

where klinear = 0.1 and kcubic = 7 are the normalized linear and cubic stiﬀness respectively, θ is the prosthesis joint angle, and θrest is the prosthesis joint rest angle.

Similar to controller 1 (Figure 4.1A), we add a constant torque to the push-oﬀ in order to perform net positive or negative prosthesis work during stance phase. The torque angle relationship for this cubic spring controller (csc) is shown in Figure 5.1.

5.1.2 Cost function sweep

The results of the optimizations in previous chapters differ from experiment in some important ways beyond the controller used. The optimizations predict that if people only optimized metabolic cost, they should spend a majority of their stride on their affected side. Using this strategy, amputees should be able to produce metabolic cost below that of a non-amputee when the prosthesis work rate is sufficiently high.

68 A) Cubic spring controller (csc)

∆τ 3 Torque

Cubic spring Cubic spring 4 with torque o!set

Ankle 1 5 angle ∆α rest

B) Phases of prosthesis stance

1. Heel strike 2. Toe strike 3. Braking 4. Push-o! 5. Toe-o!

Swing Initial stance Late stance Swing

Figure 5.1: Simple active controller with cubic stiffness during dorsiflexion and plantar flexion A) The ideal torque loop for the cubic spring controller. Work is added using a constant torque input at the onset of push-off. The numbers 1-5 refer to the stage of the gait cycle. B) Stages of the gait cycle, same as in Chapter 4.

69 However, the experiments show much higher metabolic cost than the optimization at similar levels of prosthesis work, and none of the amputees in the experimental study clearly favored their affected side over their unaffected side. This may be due to additional cost functions that the amputees are seeking to minimize. In Chapter 2, we introduced a variety of cost functions, which could be used in our optimization to simulate a human cost. Given that human metabolic rate is linked to muscle activation, muscle work, muscle force, and muscle force rate [57], we test the effects of the different mechanical costs when used in a weighted cost function. This leads to a cost function of the form:

˙ ˙ ˙ ˙ ˙ Ctotal = Cmet + λWCw + λFCF + (λFR + α)CFR, (5.2)

where λW, λF, and λFR, are weighting coefficients for each of the mechanical cost terms. The force rate cost is always given a small non-zero coefficient, 5 × 10−4, for the same reasons discussed in Chapter 4. Using this equation, we run an optimization for each λ in equation 5.2 ranging from zero to one in order to observe the effect of each cost function in conjunction with the metabolic function while keeping all other

λs equal to zero. These optimizations are performed using controller 1 with a torque input of 42 N m and thus produce around a 0.3 W kg−1 work rate. From these optimizations we choose one of the λ from each cost function and set it to a constant and varied the prosthesis work rate.

70 5.1.3 Socket interaction cost

Amputees may be concerned with more than just energy optimality while using a prosthesis. Unlike non-amputees, an amputee cannot interact with the ground directly, but instead they must interface with the prosthesis socket through their residual limb. High stresses at this interface may result in discomfort or injury. Therefore, amputees may prefer to walk in a manner that reduces the interfacial stresses. As shown in Figure 5.2, the shank of the aﬀected side of the body is represented by two separate components: the residual limb, which resides within the socket, and the pylon that attaches to the prosthetic foot. We assume that the two are attached rigidly at a single point and determine the forces and moments at that location by sectioning the rigid body of the shank (Figure 5.2) and writing the equations of motion for the section. For these tests, we assume the residual limb makes up 1/3 of the shank

(and thus the pylon would make up the remainder) and scale the mass assuming it is proportional to the length and the moment of inertia proportional to the length cubed. Namely:

Lresidual mresidual = mshank , (5.3) Lshank 3 Lresidual Iresidual = Ishank . (5.4) Lshank Once we have access to these internal socket forces, we deﬁne a cost function related to each force: T T T 1 Z 1 Z 1 Z C˙ = N 2 dt, C˙ = V 2 dt, and C˙ = M 2 dt, (5.5) N T V T M T 0 0 0 where N is the normal force along the axis of the shank, V is the shear force perpendicular to the shank, and M is the bending moment about the axis out of the page. Implementation of any of these cost functions has the eﬀect of minimizing the

71 Computing socket interactions

A) Anatomy of the B) Free body diagrams socket interface Fknee τ Residual limb knee N in socket M V

Prosthesis Prosthesis shank pylon Gshank Gpylon

Gfoot Gfoot Ftoe Ftoe Fheel Fheel

Figure 5.2: Aﬀected shank structure and internal loads. A) The shank of the aﬀected side consists of the residual limb, which resides in the socket, and the pylon, which attaches to the base. B) By making a section cut at the end of the socket, we expose the interfacial loading. This includes the force normal to the cross- sectional area of the cut (and axial to the shank) N, the shear force planar to the surface (perpendicular to the axis of the shank) V , and the bending moment M. We calculate these forces and moment using the ground forces on the prosthesis, the gravitational forces on the prosthesis and pylon, and the accelerations of those bodies.

72 magnitude of the corresponding internal load. We combine these cost functions with ˙ the original metabolic cost function Cmet using the equation:

˙ ˙ ˙ ˙ ˙ ˙ Ctotal = (Cmet + αCFR) + λNCN + λVCN + λMCN. (5.6)

By changing the various λs, we weigh the importance of each cost function. Since this has the same issue of dimensionality as the previous section, we chose to test each cost in isolation rather than in combination with one another.

5.2 Results

5.2.1 A stiﬀ cubic controller produces higher metabolic cost regardless of kinematic symmetry constraint

Similar to Chapter 4, we performed trajectory optimizations with the cubic spring controller over a range of prosthesis work between -0.2 and 0.8 W kg−1 based on the control input. These optimizations produced the torque-angle work loops shown in

Figure 5.3. As the prosthesis work rate increases, the maximum plantar ﬂexion angle increases during swing and the max torque stays around 200 Nm for all but the highest prosthesis work rate. For all of these prosthesis work rate conditions, we ﬁnd optimal metabolic costs which were higher for the cubic spring controller than for controller 1

(Figure 5.4). This higher cost is observed both with unconstrained bilateral symmetry and with bilateral symmetry constrained. Even with the increase in metabolic rate, there still exists a quadratic relationship between the prosthesis work rate and the metabolic rate (Figure 5.4).

73 Cubic spring controller torque-angle work loop

200

100

Work Rate Work

Torque (Nm) 0

-100 -0.50 0.5 1 Angle (rad)

Figure 5.3: Cubic spring controller torque-angle relationships. Net prosthesis work rate ranging from -0.2 and 0.8 W kg−1 is displayed as a gradient from blue to red.

74 Cubic spring controller costs at various work levels

5 sym csc ) -1 4 SACH sym c1

exp 3 csc

2 non-amp

Net metabolic rate (WNet metabolic rate kg c1 1 optimized

0 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 Net prosthesis work rate (W kg -1 )

Figure 5.4: Eﬀects of net prosthesis work rate on metabolic rate. The relationship between net metabolic rate (W kg−1) and net prosthesis work rate (W kg−1) are shown for controller 1 (c1) and the cubic spring controller (csc) with symmetry unconstrained and constrained (sym c1 and sym csc respectively). The costs for the optimized controller, SACH foot, non-amputee, and experimental data [8] are also displayed for comparison.

75 5.2.2 Cubic controller changes aﬀected limb kinematics and dynamics

As shown in Figure 5.5 and Figure 5.6, the change to a stiﬀer nonlinear controller

produced altered gait dynamics for all joints, not just those of the aﬀected side. For

example, the prosthesis joint angle range is smaller than controller 1 and the aﬀected

knee during mid-stance is fully extended, both of which seem to match experimental

data well. However, the aﬀected hip is now has a positive ﬂexion angle throughout

stance and the prosthesis peak torque is pushed later into stance. Comparing the

error between simulated data and the experimental mean (Figure 5.11), most of the

joint angles had a smaller error than controller 1, excluding the aﬀected hip. In

contrast, the torque error was reduced for only a few joints.

5.2.3 Added force rate squared costs has a greater eﬀect on cost than added force squared or work costs

We ran a series of optimizations using a combination of Cmet and CF, CW, or CFR, setting each λ to 0.25, 0.5, 0.75, and 1 while keeping the others at zero. At each of these conditions, we track the change in the total objective function (Figure 5.8A) and the metabolic function by itself (Figure 5.8B). We ﬁnd that all the functions cause an increase in the total cost and in the metabolic cost. However, the addition of the force rate costs has a much larger eﬀect than the force squared and work costs at all values of λ. Since this could be due to the relative magnitude of the force rate when compared to force and work, we ran another test with a λFR = 0.1, also shown in Figure 5.8A-B.

76 Joint angles over one stride Unaected

0.5 Hip 0.5 Knee 1 Ankle 0 0 0.5 -0.5 -0.5 0

Angle (rad) Angle -1 -1 -1.5 -0.5 00.5 1 00.5 1 00.5 1 Stride fraction Stride fraction Stride fraction Aected 0.5 Hip 0.5 Knee 1 Prosthesis 0 0 0.5 -0.5 -0.5 0

Angle (rad) Angle -1 -1 -1.5 -0.5 00.5 1 00.5 1 00.5 1 Stride fraction Stride fraction Stride fraction

experimental data c1 sym c1 csc sym csc

Figure 5.5: Joint angle comparison between cubic spring controller optimizations and experiment. Joint angles for controller 1 (c1) and the cubic spring controller (csc) as well as the angles for both controllers with symmetry constrained (c1 sym and csc sym respectively). Experimental data [8] is shown in black and gray.

77 Joint torques over one stride Unaected 150 Hip 100 Knee 200 Ankle 100 50 150 50 100 0 0 50 -50 -50 0 -100 -100 -50 -150 -150 -100 00.5 1 00.5 1 00.5 1 Stride fraction Stride fraction Stride fraction

Aected 150 Hip 100 Knee 200 Prosthesis 100 50 150 50 100 0 0 50 -50 -50 0

Torque (Nm) Torque -100 (Nm) Torque -100 -50 -150 -150 -100 00.5 1 00.5 1 00.5 1 Stride fraction Stride fraction Stride fraction

experimental data c1 sym c1 csc sym csc

Figure 5.6: Joint torque comparison between cubic spring controller optimizations and experiment. Joint torques for controller 1 (c1) and the cubic spring controller (csc) as well as the torques for both controllers with symmetry constrained (c1 sym and csc sym respectively). Experimental data [8] is shown in black and gray.

78 Angle Error 0.3 c1 sym c1 0.2 csc 0.1 sym csc

0 θhip,u θknee,u θankle,u θhip,a θknee,a θankle,a Torque Error 60

20 Torque (Nm)Torque Angle (rad)

0 τhip,u τknee,u τankle,u τhip,a τknee,a τankle,a

Figure 5.7: Error between optimization and experimental joint torques and angles. The joint torque and angle data is from selected tests (with similar levels of net prosthesis work) using controller 1 with various step time, kinematic, and/or ground reaction force (GRF) symmetry constraints.

79 Augmented cost function sweeps A) Total Cost function B) Metabolic Cost function 20 7 λ 18 6 1 C +λC 0.75 16 ) met FR -1 5 Cmet +λC FR 0.50 14 0.25 4 0.10 0 12 3

Total Cost Total 10 2 C +λC met F Cmet 8 (WMetabolic Cost kg Cmet +λC F C +λC met W 1 C +λC 4 Cmet met W

0 0 0.26 0.27 0.28 0.29 0.3 0.26 0.27 0.28 0.29 0.3 Prosthesis Work (W kg -1 ) Prosthesis Work (W kg -1 )

Figure 5.8: Cost comparison results of augmented cost function optimizations. The resultant total cost (A) and the net metabolic cost (B) of various optimizations. Each test used equation 5.2 with one lambda set to a value ranging from 0 to 1 and the other two set to zero. All optimizations used a periodic stride with controller 1 set with a ∆τ = 42 N m. A) The total cost of all components summed together (e.g. Ctotal = Cmet + λFCF. B) The contribution of the metabolic cost alone for each test.

80 5.2.4 Force rate cost causes larger changes in stride kinematics and kinetics than work and force costs

As shown in Figures 5.9 and Figure 5.10 as λ increases from 0 to 1 for the force squared and work costs, there are small changes present in the hip and knee joint angles and joint torques on the unaffected side, most of which occur during the swing phase of the gait. However, the affected side is mostly unchanged. These results are also reflected in the error values when comparing the joint torques and angles from optimization to experimental data, see Figure 5.11. For optimizations with added force rate cost, the change in kinematics and kinetics is more pronounced. As shown in Figure 5.9, on the unaffected side, the knee angle during swing has a gradual, monotonic change in angular velocity and higher maximum knee flexion. In addition, the ankle angle during swing has a higher plantar flexion around 0.5 radians, similar to the amputee experimental results. As shown in Figure 5.10, the added force rate squared cost smooths out the joint torques on both the affected and unaffected sides.

The hip and knee torques on the aﬀected side now show ﬂexion torques throughout the stance phase, a feature that was not present in other cost function used.

5.2.5 Muscle forces change with additional mechanical costs

The muscle forces resulting from the optimizations with an additional work cost were very similar to the muscle forces to the metabolic cost alone with minor differences mostly concentrated in the illiosoas of the unaffected side and the rectus femoris on both limbs. The additional force cost reduced the max forces in the unaffected illiosoas and the rectus femoris of both legs, while increasing the activation time of both. The force rate cost changes most muscle force profiles: activation times are

81 Joint angles with added cost functions Una!ected 0.5 Hip0.5 Knee1 Ankle

0 0 0.5 -0.5 Angle -0.5 0 -1

-1 -1.5 -0.5 0 0.5 1 0 0.5 1 0 0.5 1 Stride fraction Stride fraction Stride fraction

A!ected 0.5 Hip0.5 Knee1 Prosthesis

0 0 0.5 -0.5 Angle -0.5 0 -1

-1 -1.5 -0.5 0 0.5 1 0 0.5 1 0 0.5 1 Stride fraction Stride fraction Stride fraction

Cost function Cmet Cmet +0.5CF Cmet +0.5CW Cmet +0.1CFR

Figure 5.9: Joint angles from optimizations with added mechanical costs. The results shown use controller 1 with a torque input of 42 N m and are calculated ˙ ˙ ˙ ˙ using Cmet (blue) plus 0.5CF (orange), 0.5CW (yellow), or 0.1CFR (purple). Joint angles from experiment [8] are shown in grey and black.

82 Joint torques with added cost functions Una!ected 150 Hip 100 Knee 250 Ankle 100 50 200 50 150 0 0 100 -50 -50 50 Torque (Nm) -100 -100 0 -150 -150 -50 0 0.5 1 0 0.5 1 0 0.5 1 Stride fraction Stride fraction Stride fraction

A!ected 150 Hip 100 Knee 250 Prosthesis 100 50 200 50 150 0 0 100 -50 -50 50 Torque (Nm) -100 -100 0 -150 -150 -50 0 0.5 1 0 0.5 1 0 0.5 1 Stride fraction Stride fraction Stride fraction

Cost function Cmet Cmet +0.5CF Cmet +0.5CW Cmet +0.1CFR

Figure 5.10: Joint torques from optimizations with added mechanical costs. The results shown use controller 1 with a torque input of 42 N m and are calculated ˙ ˙ ˙ ˙ using Cmet (blue) plus 0.5CF (orange), 0.5CW (yellow), or 0.1CFR (purple). Joint torques from experiment [8] are shown in grey and black.

83 Angle Error 0.3 Cost function C 0.2 met

Cmet +0.5CF 0.1 Cmet +0.5CW C +0.1C 0 met FR θhip,u θknee,u θankle,u θhip,a θknee,a θankle,a

Torque Error 60

20 Torque (Nm)Torque Angle (rad)

0 τhip,u τknee,u τankle,u τhip,a τknee,a τankle,a

Figure 5.11: Error between optimization and experimental results. The errors in both joint torques and angles are consistently reduced with the addition of a force rate cost. The addition of improves some predictions (e.g. unaffected knee angle) but has no significant effect on other joint angles or joint torques.

84 Muscle forces with added cost functions Aected Unaected 1 1

Illio Gas

0 0 1 1

Glut Sol

0 0 1 1

Ham TA

0 0 1 0Stride fraction 1

RF EMG data

0 Cost functions C 1 met C +0.5C Vas met F C +0.5C 0 met W C +0.1C 0Stride fraction 1 0Stride fraction 1 met FR

Figure 5.12: Muscle forces from various cost functions. The muscle forces (scaled by max isometric force) for the metabolic cost are shown in blue. All other lines depict the muscle forces produced by the metabolic cost plus the force squared cost (orange), the work cost (yellow), or the force rate squared cost (purple). The weighting coefficients for each cost function are shown in the figure legend. All muscle force profiles are compared to experimental EMG data [15,71,72,73], shown as black bars.

greatly increased for all muscles and the glutei and hamstrings are activated during the aﬀected stance phase. The muscle force proﬁles of selected tests can be found in

Figure 5.12.

85 5.2.6 Cost and symmetry relationships to prosthesis work rate for added costs

We characterize the cost vs. prosthesis work relationship for controller 1 with each of the three added costs. We used λF = 0.5, λW = 0.5, and λFR = 0.1 for the corresponding added cost and the prosthesis work was varied in the same manner as

Chapter 4. As shown in Figure 5.13, the total cost per work rate increases with all added cost functions but only the addition of a force rate cost results in a significant change in the metabolic rate (Figure 5.14). Figure 5.15 shows the effect of each added cost on the bilateral symmetry. The only cost to increase the percent step time difference was the force rate cost (by 10%).

5.2.7 Socket loading costs

For socket loading costs with similar λ, the normal force cost has a larger effect on the optimization than shear force or bending moment. We set λ to 0.5 for each socket load cost function (while setting the other costs to 0) and varied the net prosthesis work rate. Running the optimization at various levels of work produces the results shown in Figure 5.16. Under these conditions, the normal force cost increases the net metabolic rate while the shear force and bending moment costs have almost no effect on the net metabolic rate when the net prosthesis work rate is below 0.5 W kg−1.A similar relationship occurs with the percent step time difference shown in Figure 5.17.

5.2.8 Socket costs eﬀect on kinematics and dynamics.

As shown in Figure 5.18 and Figure 5.19, we see some changes to the joint angles and torques when socket costs are added, such as larger unaﬀected knee ﬂexion angles

86 E!ects of prosthesis work on total cost plus various costs 10

9 C +0.1C 8 met FR

7 ) -1 6

4 expexp C +0.5C met W C +0.5C Total cost (W cost kg Total 3 met F

2 C 1 met

0 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Net prosthesis work rate (W kg -1 )

Figure 5.13: Total cost of walking gait at various levels of work. The curves depict the total cost produced by the metabolic cost by itself (blue) and with the additional force squared cost (orange), the work cost (yellow), or the force rate squared cost (purple). The weighting coeﬃcients for each cost function are shown in the ﬁgure. All results are compared to the cost-vs-work relationship found in experiment (black and gray).

87 E!ects of prosthesis work on metabolic rate plus various costs 7

6 )

-1 5

4 Cmet +0.1CFR

3 expexp

2 Cmet +0.5CW Cmet +0.5CF Net metabolic rate (WNet metabolic rate kg 1 Cmet

0 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Net prosthesis work rate (W kg -1 )

Figure 5.14: Metabolic cost at various levels of work. The curves depict the metabolic cost produced when the objective function is defined as the metabolic cost by itself (blue) or with the additional force squared cost (orange), work cost (yellow), or force rate squared cost (purple). The weighting coefficients for each cost function are shown in the figure. All results are compared to the cost-vs-work relationship found in experiment (black and gray).

88 otfnto r hw ntefiue l eut r oprdt h cost-vs-work the to compared are results gray). and All each (black for figure. experiment coefficients in the weighting found in The relationship shown (purple). work are cost (orange), function squared cost squared rate cost the force force as additional defined or the (yellow), is with or function cost (blue) objective itself the by when cost metabolic produced symmetry bilateral the depict 5.15: Figure Bilateral symmetry (unaected -aectedBilateral limb) symmetry

-30 % Step-time dierence 30 0 04-. . . . . 1.2 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 tptm ieec ihec betv function. objective each with diﬀerence Step-time Net prosthesis work rate (W kgNet prosthesis work rate (W expexp 89 C met +0.5C F -1 C ) met +0.1C C FR met C met +0.5C h curves The W E!ects of prosthesis work on metabolic rate plus socket costs 7

6 )

-1 5

4 expexp 3 Cmet+0.5CN

2 Cmet+0.5CV Cmet+0.5CM Net metabolic rate (WNet metabolic rate kg 1 Cmet

0 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Net prosthesis work rate (W kg -1 )

Figure 5.16: Net metabolic rate with socket loading costs. Each line represents the net metabolic rate of the optimization using diﬀerent weighted cost functions, which include the net metabolic rate and a cost associated with one of the socket loads (normal force, shear force, or bending moment). An added normal force cost is shown to increase the metabolic cost more than and added shear or moment cost. Experimental data is shown in black and gray for comparison [8].

90 oc,adbnigmmn otfntos de omlfrecs rdcsstrides produces cost force times. normal stance Added unaffected shear longer functions. force, normal with cost the moment for bending shown is and side force, affected and unaffected the between difference 5.17: Figure Bilateral symmetry (unaected -aectedBilateral limb) symmetry

-30 % Step-time dierence 30 0 04-. . . . . 1.2 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 ecn tptm ihsce odn costs. loading socket with time step Percent Net prosthesis work rate (W kgNet prosthesis work rate (W expexp 91 C met C +0.5C met +0.5C N V -1 ) C h ecn tptime step percent The met +0.5C C met M and ankle dorsiflexion angles during swing. However, most of these differences do not improve model predictions of reality. This fact is further reinforced by the error between each test and experimental data (Figure 5.20). There are similar changes in the muscle forces, but again none that could be classified as ‘closer to experimental data’ (Figure 5.21).

5.3 Discussion

The different optimizations described in this chapter were a preliminary step in improving the predictive qualities of our optimization. We altered the controller to better approximate the non-linear stiffnesses used in the experiment to which we are comparing. We also attempted to improve the objective function of the optimization through adding scaled versions of three separate mechanical costs and three socket loading costs. Each of these modifications to the original optimization had some effect on the optimal gaits, but only the addition of a force rate cost seemed to improve model predictions consistently.

The cubic spring controller increased the predicted metabolic cost per prosthesis work rate for both symmetry constrained and symmetry unconstrained. This is likely less due to the non-linear stiffness and more due to the large stiffness chosen. The optimized prosthesis found in Chapter 4 (Figure 4.2D) also had a non-linear relationship with the prosthesis angle during dorsiflexion, but the maximum dorsiflexion angle was close to 0.45 rad, producing a max torque of 190 N m. Since this cubic spring controller has a relatively large cubic term, the prosthesis produces the same torque at only 0.26 rad. This effectively creates a limit on how far the prosthesis ankle can flex while on the ground (about 0.3 rad). To be able to say that such a non-linear

92 Joint angles with added socket costs Una!ected 0.5 Hip0.5 Knee1 Ankle

0 0 0.5 -0.5 Angle -0.5 0 -1

-1 -1.5 -0.5 0 0.5 1 0 0.5 1 0 0.5 1 Stride fraction Stride fraction Stride fraction

A!ected 0.5 Hip0.5 Knee1 Prosthesis

0 0 0.5 -0.5 Angle -0.5 0 -1

-1 -1.5 -0.5 0 0.5 1 0 0.5 1 0 0.5 1 Stride fraction Stride fraction Stride fraction

Cost function Cmet Cmet+0.5CN Cmet+0.5CV Cmet+0.5CM

Figure 5.18: Joint angles with socket loading costs. The curves represent the joint angles found using the standard metabolic cost (blue) or with the addition of the socket normal cost (orange), shear cost (yellow), or moment cost (purple). Each cost function produces similar joint angles for all affected joints and the unaffected hip and ankle. Added socket loading cost did produce larger unaffected knee flexions during swing.

93 Joint torques with added socket costs Una!ected 150 Hip 100 Knee 250 Ankle 100 50 200 50 150 0 0 100 -50 -50 50 Torque (Nm) -100 -100 0 -150 -150 -50 0 0.5 1 0 0.5 1 0 0.5 1 Stride fraction Stride fraction Stride fraction

A!ected 150 Hip 100 Knee 250 Prosthesis 100 50 200 50 150 0 0 100 -50 -50 50 Torque (Nm) -100 -100 0 -150 -150 -50 0 0.5 1 0 0.5 1 0 0.5 1 Stride fraction Stride fraction Stride fraction

Cost function Cmet Cmet+0.5CN Cmet+0.5CV Cmet+0.5CM

Figure 5.19: Joint torques with socket loading costs. The curves represent the joint torques found using the standard metabolic cost (blue) or with the addition of the socket normal cost (orange), shear cost (yellow), or moment cost (purple). Joint torques produced by each cost function are not significantly different from the results produced using metabolic cost alone except in minor differences in timing.

94 Angle Error 0.3 Cost function C 0.2 met

Cmet+0.5CN 0.1 Cmet+0.5CV C +0.5C 0 met M θhip,u θknee,u θankle,u θhip,a θknee,a θankle,a Torque Error 60

20 Torque (Nm)Torque Angle (rad)

0 τhip,u τknee,u τankle,u τhip,a τknee,a τankle,a

Figure 5.20: Error between simulated and experimental data with socket loading costs. There is no consistent decrease in error for any one socket loading function for either joint angles or joint torques.

95 Muscle forces with various socket costs

A ected Una ected 1 1

Illio Gas

0 0 1 1

Glut Sol

0 0 1 1

Ham TA

0 0 1 0Stride fraction 1

RF EMG data Cost functions 0 C 1 met C +0.5C Vas met N C +0.5C met V 0 C +0.5C 0Stride fraction 1 0Stride fraction 1 met M

Figure 5.21: Muscle forces produced by tests with socket loading costs. Each cost function changes the muscle activations throughout gait.

96 controller is ‘worse’ or ‘better’ for metabolic cost would require testing a variety of

linear and non-linear coeﬃcients.

The cubic spring controller results tell us is that prosthesis work is not the only

factor for determining metabolic cost. It is likely that the similarities in the metabolic

rates found in Chapter 4 were due to overly similar controllers. When we make

large changes, e.g. by changing stiﬀness, and torque-angle-loop shape, we produce a

diﬀerent metabolic rate curve. We therefore speculate that it is possible to produce a

ﬁxed feedback controller, which could produce a lower metabolic rate than controllers

1, 2, and 3. While these three controllers have produced the lowest metabolic costs

of any feedback controller we have tested thus far, these controllers still produce

kinematics which are known to cause higher metabolic costs, namely a bent knee

during stance.

Shown in Figure 5.5, we see that by changing to a non-linear spring for dorsiﬂexion,

there are large changes to the intact joints of the aﬀected side of the biped. For

example, the aﬀected knee fully extends during midstance instead of ﬂexing as is seen

in all other feedback controllers tested on this model. This change to the aﬀected knee

during stance only occurred in tests with the cubic spring controller while little to no

change was observed at the aﬀected knee when we made changes to the cost function.

Therefore, the cost function chosen to represent the users’ objectives may sometimes

have less impact on the joint kinematics of the aﬀected side than the structure of the

controller chosen.

When testing the additions of mechanical costs to the metabolic rate, for the λs used, force rate had the largest eﬀect on the optimization. The force squared and work costs increased the optimal metabolic rate by a small amount when compared

97 to the metabolic rate optimization on its own. These cost functions also led to small changes in activation timing, joint angles, and joint torques. In comparison, the force rate cost increased both the total objective cost and the metabolic rate for a given amount of prosthesis work. This additional force rate cost also smoothed all of the muscle forces during stride and encouraged activation of hip and knee on the affected side, similar to what is seen in amputees. This led to a decrease in the error for almost all of the joint angles and joint torques when compared to experiment. However, if we examine the individual plots, not all have better agreement with experiment as our definition of error suggests. For example, the hip torque is producing a flexion torque for the entire stride cycle on the unaffected side and produces no hip extension torques during early stance as is expected from experiment.

The addition of socket loading costs was used to simulate an avoidance of internal socket forces, which may lead to pain in residual limb of an amputee. We expected that an additional cost associated with the diﬀerent loads would decrease the amount that the user would use the prosthesis. This appears to be true with an additional normal force cost in that the biped spent much less time on its aﬀected side and experienced higher metabolic costs, both of which are consistent with experiment.

However, the other socket costs (shear and bending moment costs) had very little eﬀect on the metabolic rate or the step-time diﬀerence. When the gait dynamics are compared with experiment, the addition of any one of these costs did not seem to decrease the error measured consistently.

Our explorations of augmented cost functions were not exhaustive. To get a better grasp on how each of these cost functions aﬀects the system, we will have to perform

98 more than one-dimensional sweeps of various augmented costs. We could study combinations of the socket costs at vastly diﬀerent magnitudes. Each of the augmented cost functions used was only examined at a single weight in isolation. These weights were chosen based on the magnitude of the individual cost when compared to the metabolic cost, but that does not mean that the weights used would be better than any other. For an exhaustive search of all possible combinations of these cost functions, a λmet could be added to the function. We could then perform a grid search of all possible combinations of the cost functions over some range of λs. This has a dimensionality problem in that, if we chose n values of λ for each cost term and used a single controller at m levels of work, the number of tests required would total

(n + m)7. Since this would be a prohibitively large search area, such a search may be infeasible. Alternatively, we could perform an ‘inverse optimization,’ which would en- tail running an optimization which took the various λ as inputs and minimized some tracking objective function which would compare the results of each gait optimization to known gait kinematic or ground reaction force data.

99 Chapter 6: Point mass biped walking with a unilateral prosthesis or exoskeleton

Up to this point, we have used a muscle driven planar model of a human walking with a prosthesis. While such a model is a simpliﬁcation over the physical human and prosthesis system, it is still complex for a trajectory optimization. There are many unknowns to be optimized, many parameters that can be tuned, and there is the potential for local minima. In this chapter, we consider a much simpler model for comparison.

6.1 Simple point mass model

6.1.1 Past models

The seven-link planar model described in Chapter 2 is a simpliﬁcation of the real biological system: no three dimensional motion, fewer muscles, rigid bodies, etc. Further simpliﬁcations can be made such as removing the foot and ankle as well as removing actuation around the hip joint [31]. The simplest models which are commonly used to study human locomotion are inverted pendulum models [35,97,98].

These models assume that the body is a single point mass which is actuated via collision impulses, a spring, or more generally, a telescoping leg [31, 59]. Here, we represent the person as a point mass model with a telescoping leg.

100 6.1.2 Our model

The point mass model with a telescoping leg is shown in Figure 6.1A. This point mass model is actuated via a piecewise linear force that can be applied to the ground along the axis of the leg. By reducing the biped in this manner, we limit our state variables to the horizontal and vertical positions and velocities of the center of mass, q = (x, y, x,˙ y˙), and we simulate the movement of the center of mass through a single step using the following equation of motion:

ma = F − G, (6.1)

x − x mx¨ = F foot , (6.2) L y my¨ = F − mg, (6.3) L where m is the mass of the upper body, a is the acceleration of the upper body, F is the force along the leg as a piecewise function of time, G is the force of gravity,

L is the leg length, xfoot is the position of the foot, and g is the acceleration due to gravity. As shown in Figure 6.1A, we deﬁne the foot position xfoot for each step to calculate the direction of the leg force and to determine the current length of the leg.

These equations of motion are rewritten in non-dimensional form using quantities normalized by upper body mass m, maximum leg length Lmax and gravity g, then integrated forwards in time for two steps to simulate a full periodic stride (Figure

6.1B). Given that such a model does not make any distinction between the diﬀerent joints on each leg, it would not be possible to distinguish between a transtibial or transfemoral prosthesis. Therefore, instead of representing the prosthesis as a physical part of the system, we represent its inﬂuence as a free source of mechanical work inside of the optimization.

101 A) Point mass model with telescoping leg B) Periodic Stride

(x,y) Unaected Aected limb stance limb stance m

Stance leg FBD

F G (x foot ! mfoot

Figure 6.1: Point mass model. A) A point mass representation of a biped capable of pushing off the ground with one leg at a time. The free body diagram is shown with the gravitational force G and the leg force F. The direction of F is determined based on the mass position (x, y) and the foot position xfoot on the ground. The foot mass mfoot is used in calculating the cost of swing. B) A periodic stride consisting of two steps, one on the unaffected limb (without amputation) and one on the affected limb (with amputation and a prosthesis).

102 6.2 Point mass optimization

6.2.1 Work-based objective function

The total metabolic cost of the periodic gait is a sum of three parts: the stance cost Cstance, the swing cost Cswing, and the rest cost Crest [31].

C + C + C C = stance swing rest , (6.4) Lstride where Lstride is the total distance of the point mass model over the stride.

Stance Cost. For our initial optimizations, we define the stance cost Cstance as the cost of the positive and negative mechanical work for both the unaffected and affected limbs of the biped during their respective stance phase: Cstance = Cstance,u + Cstance,a.

For the unaﬀected side, the stance cost function becomes:

+ + − − Cstance,u = η Wu + η Wu , (6.5) where η+ = 4 and η− = 1.2 are the positive and negative work eﬃciencies respectively

+ − and Wu and Wu are the positive and negative work for the unaffected limb. For the affected side we use the same stance cost but add λ as the prosthesis work coefficient:

+ + − − Cstance,a = (1 − λ) η Wa + η Wa . (6.6)

Here, λ, which ranges from zero to one, represents the ‘level of assistance’. The value of λ deﬁnes what percentage of the work done during the second step of the stride cycle is performed by the prosthesis: when λ = 0 the prosthesis is providing 0% of the work and when λ = 1 the prosthesis is doing 100% of the work for that step, eﬀectively making that step-cost free to the human.

103 Swing Cost. Cswing is an estimate of the work required to move the non-stance leg.

For both the unaﬀected side and eﬀected side we used:

Lswing 2 Cswing = ηswingmfoot( ) , (6.7) tswing

+ − where ηswing = 0.5 (η + η ) is the swing eﬃciency, mfoot is the foot mass, Lswing is

the foot travel distance during swing, and tswing is the swing time.

Resting Cost. The resting cost Crest is the basal metabolic rate of a person, the

energy the person uses without performing an explicit task, and is deﬁned as:

Crest = ηresttstride, (6.8)

where ηrest = 0.05 is the resting rate coeﬃcient and tstride is the stride time. This

term creates a penalty for walking with zero velocity.

6.2.2 Constraints

To ensure that the optimization produces a physically meaningful walking cycle,

we apply continuity and periodicity constraints between the two steps and at the end

of the stride such that the state variables at the beginning of the current step are

equal to the state variables at the end of the previous. We also apply an inequality

constraint such that 0 ≤ L ≤ Lmax. Since we have included a resting cost in our

cost function, we do not require that the velocity be constant for all our tests but to

match our previous results with the complex model, we perform some tests with a

constant velocity constraint as well.

104 6.3 Results

We performed a series of optimizations with 0 ≤ λ ≤ 0.99 corresponding to a

prosthesis that is responsible for 0 to 99% of the work for the second stance phase

stance phase. Similar to the complex model, λ = 1 is not used as it would lead to

a ‘cost free’ step. This parameter sweep is performed with both velocity constrained

and with velocity unconstrained. All results are presented in normalized quantities.

6.3.1 Optimization discovers pendular walking for zero or low assistance

As shown in Figure 6.2, during the unaﬀected stance phase, the point mass model

discovers pendular walking, deﬁned by walking with a near constant leg length (Figure

6.3), despite the ability of the leg to telescope. This optimal pendular gait is also

seen during the aﬀected stance phase when the level of assistance is low (λ ≤ 0.6).

However, when the level of assistance is high enough, the leg compresses during the aﬀected stance phase and has a more compliant behavior.

6.3.2 Increasing assistance increases bilateral asymmetry.

When assistance λ is set to zero, the point mass model produces a perfectly symmetric gait. This is true for both the kinematics (Figure 6.4) and the leg forces

(Figure 6.5). As the assistance increases, the affected step time, step length and work become a larger percentage of the total stride. This trend continues for the step time and force, but not for the step length. As shown in Figure 6.4, when the assistance reaches 70% this trend reverses. At 99% assistance the point mass model has a shorter affected step length than the unaffected.

105 Velocity unconstrained Unaected Aected 1 1 1

y y

0 0 0 -0.5 0 0.5 -0.5 0 0.5 x x Velocity constrained Unaected Aected 1 1 1

y y

0 0 0 -0.5 0 0.5 -0.5 0 0.5 x x

Figure 6.2: Periodic stride. The plots show the mass position through each step of the periodic stride at various levels of prosthesis work assistance 0 ≤ λ ≤ 0.99. The positions are separated by the unaﬀected and aﬀect limb stance phases. The gaits with velocity unconstrained are shown in blue and red while those with velocity constrained are shown in blue and green.

106 Velocity unconstrained 1 1 Una!ected A!ected

0.5 Leg Length 0 0 0 25 50 75 100 % Stride

Velocity constrained 1 1 Una!ected A!ected

0.5 Leg Length

0 0 0 25 50 75 100 % Stride

Figure 6.3: Leg length throughout stride. The leg length is limited to the maximum length Lmax = 1. When the assistance level is low or zero, both legs have a mostly constant leg length. When the assistance level is high (λ ≥ 0.6) there is a noticeable decrease in length during the aﬀected limb stance phase. The gaits with velocity unconstrained are shown in blue and red while those with velocity constrained are shown in blue and green.

107 20

-20

-40

-60 unc.v con.v Stance time -80 Stance work Step length -100 % Di!erence, (una!ected - a!ected)/total % Di!erence, (una!ected 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 6.4: Symmetry changes in the presence of of greater levels of assistance. This plot displays the percent difference of the stance time, stance work, and step length, defined as the difference between the unaffected and affected sides over the stride total, at increasing levels of prosthesis work assistance, λ. As shown by the red and green lines, representing the unconstrained velocity (unc.v) and constrained velocity (con.v) conditions respectively, the stance time percent and the stance work percent increase with increasing assistance. The step length, shown by the dashed red and green lines, has a similar trend except at the highest values of λ.

108 Velocity unconstrained 2 Una!ected A!ected 1

1 Force

0 0 0 25 50 75 100 % Stride

Velocity constrained

2 Una!ected A!ected 1

1 Force

0 0 0 25 50 75 100 % Stride

Figure 6.5: Forces over the percent gait cycle. The leg forces for the optimization with velocity unconstrained (blue-red) and velocity constrained (blue-green) are shown. As the level of assistance increases, the aﬀected side forces increase while those on the unaﬀected side decrease.

109 6.3.3 Asymmetric gaits have a lower cost than symmetric

Figure 6.6 shows the total work cost of the biped at the various levels of assistance.

As expected, when the prosthesis accounts for a higher percentage of the work done, the work required of the person decreases and the total cost is reduced. This decrease is greater than it would be by forcing the point mass model to take two symmetric steps (Shown in Figure 6.6 by the dotted lines). This trend seems to mostly hold true for the movement cost (the total cost minus the resting cost) as well.

6.3.4 Trends when velocity is constrained are similar to when it is not

Due to the addition of the resting cost, the walking gait can be optimized without specifying speed. Through the unconstrained tests, we ﬁnd that the optimal velocity increases as the assistance increases as shown in Figure 6.7. However, because our more complex model is constrained by speed, we performed a similar optimization with the simpler model. Here, we constrained the speed to 1 m/s normalized. As can be seen in Figures 6.2-6.6, the trends are similar between the two velocity conditions.

6.3.5 Changing stance cost

Due to the assumed cost function, when we set to the level of assistance to λ =

0.99, the prosthesis will account for 99% of the work performed in the second step, regardless of what the total work of that step is. As shown in Figure 6.4, this has the effect of the second step performing close to 99% of the work for the entire stride cycle. To test a more realistic level of assistance for the prosthesis, we redefine the stance cost of the affected side as

+ + − − Cstance,a = η Wλ + η Wλ , (6.9)

110 Work cost with increasing prosthesis assistance

Total cost

0.3 Symmetric

0.2 Movement cost (no rest)

Symmetric 0.1

Work cost per distance

Velocity unconstrained Velocity constrained 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 6.6: Asymmetry leads to larger decreases in cost. Both the total work cost per unit distance and the movement cost (total cost minus the resting cost) are shown. In both measurements, the cost decreases as the prosthesis provides more assistance. The dotted lines display the cost of the model with a symmetric gait (found using the kinematic and dynamic results from the zero assistance condition) and applying the prosthesis assistance at the various levels. For the velocity-constrained condition (shown in green), the asymmetric gaits have a lower total and movement cost. For the velocity-unconstrained conditions (shown in red) the total cost of the asymmetric gates is always lower while the movement cost is only lower at high levels of assistance.

111 Velocity with increasing level of assistance 0.5

0.4

Velocity 0.3 Velocity unconstrained Velocity constrained 0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 6.7: Velocity increases with increased prosthesis assistance when unconstrained The red line displays the speed at all levels of assistance when velocity is unconstrained while the green line displays the ﬁxed velocity of 1 m/s normalized by gravity and leg length.

112 where

+/− +/− Wλ = Wa − λPavgtstance,a. (6.10)

−3 Here, Pavg = 4×10 is the positive or negative average power that can be supplied by the prosthesis, determined using the average power of one step during an unassisted walking gait and tstance,a is the stance time of the affected limb. For reference, the average positive power used by the prosthesis at λ = 0.99 with velocity unconstrained was 0.234 previously. This means the new average positive power is 1.7% of the previous maximum. By redefining the cost function in this way, the level of assistance accounts for a constant amount of average power during the affected step of the walking gait.

We test this new stance cost using λ ranging from 0 to 1 with the velocity unconstrained. The leg forces and leg lengths are similar between steps for all levels of assistance but there are some systematic trends of the unaffected leg shortening as the level of assistance increases (Figure 6.8). However, there is an increase in bilateral asymmetry for the stance time, stance work, and step length. All of these quantities increase on the affected side while they decrease on the unaffected (Figure

6.9). Finally, as was the case previously, we see a decrease in the metabolic cost over the symmetric cost replacement for each level of assistance (Figure 6.10). The overall velocity of the gait remains relatively constant but there is a small linear decrease from 0.292 to 0.278.

6.4 Discussion

Similar to the complex model presented in Chapter 2, the simple model adjusts its walking strategy to optimize its gait in the presence of a free source of work. The step

113 Force 2

Una!ected A!ected 1

1 0 0 25 50 75 100 % Stride Leg Length

1.000 0 Una!ected A!ected 0.998

0.996

0.994 0 25 50 75 100 % Stride

Figure 6.8: Changes in leg length and force with increasing average power assistance. Gradient from blue to red displays the increase in assistance from λ = 0 to λ = 1. As provided power increases, the magnitude of the leg force throughout gait does not change while the unaﬀected leg length gets systematically shorter.

114 10

0 Step Length

-5 Stance time Stance Work

-10

% Di!erence, (una!ected - a!ected)/total % Di!erence, (una!ected 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 6.9: Percent difference in stance work, stance time, and step length with added power. As the prosthesis adds more power to the affected side, the stance work, stance time, and step length asymmetry increases and favors the affected side.

115 Prosthesis with power assistance decreases cost as well Total cost

0.3 Symmetric

0.2 Movement cost (no rest)

Symmetric 0.1 Work cost per distance

Power Assistance % Work Assistance 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 6.10: Metabolic cost with added power assistance. Decrease in metabolic cost is associated with increase assistance similar to previous tests.

116 time of the aﬀected side increases with increasing prosthesis work-rate and decreasing

cost. These trends coincide with higher shortening in the leg at these high levels of

assistance. The leg likely shortens to allow the model to produce the majority of the

positive and negative work on the aﬀected side, which is capable of providing free

work through the prosthesis. Rather than producing the symmetric pendular gait

with strong impulsive forces at collisions that we see in the low assistance conditions,

the point mass model tends to use the aﬀected stance to ‘catch and launch’ the point

mass. Using this strategy, the unaﬀected stance does not need to brake or push oﬀ,

only maintain the leg length at Lmax. We speculate that the only reason why the biological leg was used at all was due to the modeled constraints: the biped was forced to take a step with both legs, limit the height oﬀ the ground by the maximum leg length, and could not have a ﬂight phase. Given the option, at high levels of assistance, the biped would have likely adopted a hopping strategy.

−3 When the assistance provided by the prosthesis was limited to Pavg = 4 × 10

rather than allowed to make up a percentage of the stance work (regardless of how

much work was used), the eﬀects on the forces and leg length were greatly reduced.

While there was a systematic trend in the shortening of the leg during the unaﬀected

stance phase, the magnitude of the change was only by less than 0.5% of the maximum

leg length. The force proﬁles for both legs also remained relatively similar. The more

pronounced change to the model was in bilateral symmetry, where more time and

work were used on the aﬀected side when assistance was given.

A weakness of the model used here is that it assumes that the prosthesis will

replace the work done by the aﬀected limb without any detriment. Further, we

ignore any cost for producing forces with the biological part of the aﬀected leg: this

117 biological part is in series with the prosthesis and needs to produce the same force as the prosthesis. Therefore, there is no reduction of force requirements, even if there is oﬀ-loading of work requirements. In reality, prostheses are currently not capable of perfectly matching the function of an intact human limb and most amputees cannot produce a bilaterally symmetric gait with a prosthesis that contributes zero work.

Because the prosthesis is not physically represented in the simulation and is only treated as a source of work, this model may be thought of as a representation of a unilateral exoskeleton rather than a prosthesis.

118 Chapter 7: Contributions and Future Work

7.1 Contributions

We provided a framework for predicting human gait dynamics in the presence of a prosthetic device through energy optimization. This optimization can be set up to study unilateral or bilateral amputation at any joint along either leg. We have used this optimization to suggest that human metabolic cost and prosthesis cost are related in optimized prostheses. We have also shown that this relationship changes with prosthesis mass and walking speed.

We have shown that, independent of the cost function minimized (unless it is a tracking objective), the optimal gait of the amputee diﬀers from that of a symmetric non-amputee gait, usually resulting in an asymmetric gait. By assuming an amputee will use the same gait as an able-bodied user, previous researchers may have missed bilateral asymmetries and other issues that may be caused by the controller being tested.

Using our model with speciﬁed active and passive prosthesis controllers, we have provided evidence that human metabolic rate during walking tasks are dependent on prosthesis work rate and bilateral symmetry. We have shown that if the ultimate goal of an amputee is energy minimization, they will tend to favor their aﬀected side

119 when it is capable of high work rates and avoid using it if there is too much energy dissipation.

We have attempted to predict the walking kinematics and joint torques seen in experiment by using a controller with a non-linear cubic spring and changing the cost function in our optimization. While our explorations with additional cost functions were not exhaustive, we have found evidence that a cost associated with gait symmetry, muscle force rate, and normal forces in the prosthesis socket could improve our qualitative predictions.

Finally, we have built a simple point mass model of a biped that predicts similar trends as our more complex seven-segment model. As the prosthesis is able to produce more net work, the metabolic cost to the person is reduced and the bilateral asymmetry of the gait increases; favoring the aﬀected side.

7.2 Future work

In future work, we will model the prosthesis morphology and controller used in speciﬁc experiments [6, 8] with greater ﬁdelity, so that a better comparison may be made. For instance, the prosthesis emulator in [6,8] is designed with a separated toe

(actuated) and heel (passive), whereas the simulated prosthesis is a single rigid body.

We can also model the socket with greater ﬁdelity as our current model assumes that the prosthesis is rigidly attached to the leg. Amputees, however, interface with their prosthesis through soft tissues at the socket. This interaction may be better modeled as a three degree of freedom socket joint.

We will consider a wider range of behaviors. Such behaviors could include walking or running at diﬀerent speeds, walking on diﬀerent slopes or uneven terrain, or moving

120 with accelerations and decelerations including starting and stopping. We could also consider expanding our model to include out of plane dynamics. Such a 3D model could be used to study change in direction. It is possible that a 3D model will improve the predictions found in the sagittal plane model as sagittal plane dynamics are not entirely decoupled from out-of-plane dynamics.

Finally, we will consider other types of prosthetic and orthotic devices including multi-joint uni- and bilateral prostheses and orthoses. Such systems would only require a few changes to our existing model (e.g. replacing the knee muscles with a motor to simulate a transfemoral prosthesis or keeping all muscles and adding parallel actuators for orthoses).

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