A Hybrid Dynamical Systems Theory for Legged Locomotion by Samuel A. Burden A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Engineering|Electrical Engineering and Computer Sciences in the Graduate Division of the University of California, Berkeley Committee in charge: Professor S. Shankar Sastry, Chair Professor Robert J. Full Professor Ruzena Bajcsy Professor Maciej Zworski Fall 2014 A Hybrid Dynamical Systems Theory for Legged Locomotion Copyright 2014 by Samuel A. Burden 1 Abstract A Hybrid Dynamical Systems Theory for Legged Locomotion by Samuel A. Burden Doctor of Philosophy in Engineering|Electrical Engineering and Computer Sciences University of California, Berkeley Professor S. Shankar Sastry, Chair Legged locomotion arises from intermittent contact between limbs and terrain. Since it emerges from a closed{loop interaction, reductionist study of body mechanics and terrestrial dynamics in isolation have failed to yield comprehensive strategies for forward{ or reverse{ engineering locomotion. Progress in locomotion science stands to benefit a diverse array of engineers, scientists, and clinicians working in robotics, neuromechanics, and rehabilitation. Eschewing reductionism in favor of a holistic study, we seek a systems{level theory tailored to the dynamics of legged locomotion. Parsimonious mathematical models for legged locomotion are hybrid, as the system state undergoes continuous flow through limb stance and swing phases punctuated by instanta- neous reset at discrete touchdown and liftoff events. In their full generality, hybrid systems can exhibit properties such as nondeterminism and orbital instability that are inconsistent with observations of organismal biomechanics. By specializing to a class of intrinsically self{ consistent dynamical models, we exclude such pathologies while retaining emergent phenom- ena that arise in closed{loop studies of locomotion. Beginning with a general class of hybrid control systems, we construct an intrinsic state{ space metric and derive a provably{convergent numerical simulation algorithm. This resolves two longstanding problems in hybrid systems theory: non{trivial comparison of states from distinct discrete modes, and accurate simulation up to and including Zeno events. Focusing on models for periodic gaits, we prove that isolated discrete transitions generically lead the hybrid dynamical system to reduce to an equivalent classical (smooth) dynamical system. This novel route to reduction in models of rhythmic phenomena demonstrates that the closed{loop interaction between limbs and terrain is generally simpler than either taken in isolation. Finally, we show that the non{smooth flow resulting from arbitrary footfall timing possesses a non{classical (Bouligand) derivative. This provides a foundation for design and control of multi{legged maneuvers. Taken together, these contributions yield a unified analytical and computational framework|a hybrid dynamical systems theory| applicable to legged locomotion. i to Mary Louise ii Contents Contents ii List of Figuresv List of Tables viii 1 Introduction1 1.1 Robotics, Neuromechanics, and Rehabilitation................. 2 1.1.1 Common Cause.............................. 3 1.1.2 Common Challenge............................ 4 1.2 Systems Theory for Sensorimotor Control.................... 4 1.2.1 Justification for a Systems{Level Approach............... 5 1.2.2 Elements of a Systems Theory...................... 6 1.2.3 Linear Systems Theory.......................... 7 1.2.4 Other Systems Sciences.......................... 7 1.3 Dynamical Systems Theory for Locomotion................... 8 1.3.1 Focus on Legged Locomotion ...................... 8 1.3.2 Hybrid Dynamical Systems Theory Foundations............ 8 1.3.3 Our Contributions ............................ 9 2 Metrization and Simulation 10 2.1 Preliminaries ................................... 11 2.1.1 Topology [Kel75] ............................. 11 2.1.2 Length Metrics [Bur+01]......................... 12 2.1.3 Hybrid Control Systems ......................... 14 2.2 Metrization and Relaxation ........................... 16 2.2.1 Hybrid Quotient Space.......................... 17 2.2.2 Relaxation of a Hybrid Control System................. 18 2.2.3 Relaxed Hybrid Quotient Space..................... 21 2.3 Numerical Simulation............................... 22 2.3.1 Execution of a Hybrid System...................... 23 2.3.2 Relaxed Execution of a Hybrid System................. 25 iii 2.3.3 Discrete Approximations......................... 29 2.4 Applications.................................... 32 2.4.1 Particle in a Box ............................. 32 2.4.2 Forced Linear Oscillator with Stop ................... 33 2.4.3 Navigation Benchmark for Hybrid System Verification......... 36 2.4.4 Simultaneous Transitions in Models of Legged Locomotion . 37 2.5 Discussion..................................... 40 3 Reduction and Smoothing 41 3.1 Preliminaries ................................... 42 3.1.1 Linear Algebra [CD91].......................... 42 3.1.2 Topology [Lee12] ............................. 43 3.1.3 Differential Topology [Hir76]....................... 43 3.1.4 Hybrid Differential Topology....................... 44 3.1.5 Hybrid Dynamical Systems ....................... 45 3.2 Periodic Orbits and Poincar´eMaps....................... 47 3.3 Exact Reduction ................................. 51 3.4 Approximate Reduction ............................. 55 3.5 Smoothing..................................... 58 3.6 Applications.................................... 60 3.6.1 Spontaneous Reduction in a Vertical Hopper.............. 60 3.6.2 Reducing a (3 + 2n) DOF Polyped to a 3 DOF LLS.......... 62 3.6.3 Deadbeat Control of Rhythmic Hybrid Systems............ 66 3.6.4 Hybrid Floquet Coordinates....................... 68 3.7 Discussion..................................... 69 3.A Continuous{Time Dynamical Systems...................... 71 3.A.1 Time{to{Impact ............................. 71 3.A.2 Smoothing Flows............................. 71 3.B Discrete{time Dynamical Systems........................ 73 3.B.1 Exact Reduction ............................. 73 3.B.2 Approximate Reduction ......................... 74 3.B.3 Superstability............................... 75 3.C C1 Linearization of a Noninvertible Map.................... 77 4 Piecewise–Differentiable Flow 79 4.1 Preliminaries ................................... 80 4.1.1 Topology [Fol99] ............................. 80 4.1.2 Differential Topology [Lee12]....................... 80 4.1.3 Non{Smooth Dynamical Systems [Fil88] ................ 80 4.1.4 Piecewise Differentiable Functions [Sch12] ............... 80 4.2 Local and Global Flow.............................. 81 4.2.1 Event{Selected Vector Fields Discontinuities.............. 81 iv 4.2.2 Construction of the Piecewise–Differentiable Flow........... 82 4.2.3 Piecewise–Differentiable Flow ...................... 86 4.3 Time{to{Impact and Poincar´eMaps ...................... 88 4.3.1 Piecewise–Differentiable Time{to{Impact Map............. 88 4.3.2 Piecewise–Differentiable Poincar´eMap................. 90 4.4 Perturbed Flow.................................. 90 4.4.1 Perturbation of Vector Fields ...................... 90 4.4.2 Perturbation of Event Functions..................... 91 4.5 Applications.................................... 92 4.5.1 Stability.................................. 92 4.5.2 Optimality................................. 93 4.5.3 Controllability............................... 94 4.6 Discussion..................................... 95 4.A Global Piecewise–Differentiable Flow...................... 96 4.B Perturbation of Differential Inclusions...................... 98 5 Conclusion and Future Directions 99 5.1 Towards Sensorimotor Control Theory ..................... 100 5.2 Frontiers in Robotics, Neuromechanics, and Rehabilitation . 101 5.2.1 Dynamic and Dexterous Robotics.................... 101 5.2.2 Principles of Neuromechanical Sensorimotor Control . 102 5.2.3 Automated Tools for Diagnosis and Rehabilitation . 102 5.3 Interdisciplinary Collaboration and Training.................. 103 Bibliography 104 v List of Figures 1.1 Sensorimotor control loop. Autonomous agents interact continually with their environments through a closed loop that transforms perception to action. The coupled system can exhibit emergent behavior not predicted by a reductionist analysis of the agent or its environment in isolation. ............... 2 1.2 False dichotomies in the sensorimotor control loop. (a) Biomechanics focuses on the dynamic interaction between an agent's body and the environment, whereas neurophysiology studies the internal transformation of sensory data to motor commands. (b) Cognitive neuroscience focuses on representations in the ner- vous system formed from impinging sensory data, whereas motor control seeks principles underlying behavior synthesis....................... 3 4 2.1 A g{connected curve γ with partition ti i=0, where Sα = [a; a + 1] [0; 1], Sα¯ = [b; b + 1] [0; 1], and g : a + 1 [0; 1]f g b [0; 1] is defined by ×g(a + 1; x) = g(b; x)..........................................× f g × ! f g × 13 2.2 Illustration of a hybrid control system with three modes.............. 14 2.3 (left) Disjoint union of D1 and D2. (right) Hybrid quotient space M obtained from the relation ΛRb.................................. 17 " 2.4 Disjoint union of D1 and the strips in its neighborhood, Se e2N1 (left), and the " f g relaxed domain