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 By 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 ex- plicitly 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 opti- mization 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 kine- matics 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 asymmet- ric 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 mus- cle 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 op- timal, 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 re- gardless 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 as- sistance . 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 seg- ment 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 an- kle (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.