# Identification of Hybrid Dynamical Models for Human Movement via Switched System Optimal Control

### Ram Vasudevan

### EECS Department, University of California, Berkeley

### Technical Report No. UCB/EECS-2014-38

### May 1, 2014

### http://www2.eecs.berkeley.edu/Pubs/TechRpts/2014/EECS-2014-38.pdf

The empirical observation of human locomotion has found considerable utility in the diagnosis of numerous neuromuscular pathologies. Unfortunately without the construction of a dynamical system model of the measured gait, the effectualness of these observations is restricted to just the existing diagnostic variety rather than the prediction of potential instabilities in gait or guiding the construction of user specific prosthetics. In order to construct a dynamical system model of an observed gait in an automated fashion, one requires a family of representations rich enough to describe the dynamics of gait and an automated procedure to select a particular representation capable of describing a given observation from this family. The goal of this thesis is to address these two problems. First, a hybrid dynamical system representation is introduced that is shown to be capable of describing the discontinuities in dynamics that occur during locomotion. In particular, such a representation is constructible from observation given an unconstrained Lagrangian which is intrinsic to the biped after the identification of the sequence of contact points that are enforced during the observed motion. Second, a specific hybrid dynamical system representation is shown to be constructible from observed data by optimally switching between the set of vector fields corresponding to all possible combinations of contact point enforcements. At this point an algorithm for the computation of an optimal control of constrained nonlinear switched dynamical systems is devised. The control parameter for such systems include a continuous-valued input and discrete-valued input, where the latter corresponds to the mode of the switched system that is active at a particular instance in time. The presented approach, which this thesis proves converges to local minimizers of the constrained optimal control problem, first relaxes the discrete-valued input, performs traditional optimal control, and then projects the constructed relaxed discrete-valued input back to a pure discrete-valued input by employing an extension of the classical Chattering Lemma. This algorithm is extended by formulating a computationally implementable algorithm that works by discretizing the time interval over which the switched dynamical system is defined. Importantly, this thesis proves that the implementable algorithm constructs a sequence of points by recursive application that converge to the local minimizers of the original constrained optimal control problem. Four simulation experiments are included to validate the theoretical developments and illustrate its superiority when compared to standard mixed integer optimization techniques. The thesis concludes by applying the presented algorithm to perform the identification of a hybrid dynamical system representation of two classes of gaits. The first is a synthetic gait generated by the application of feedback linearization to a classical robotic bipedal model. For this synthetic observation, the presented identification scheme is able to correctly identify the correct model. The second set of gaits is one constructed from motion capture observations of $9$ subjects during a flat ground walking experiment. For each subject, the presented identification scheme determines a distinct hybrid dynamical system representation. Surprisingly, the identified models for each participant share an identical discrete structure, or an identical sequence of contact point enforcements.

**Advisors:** Ruzena Bajcsy

BibTeX citation:

@phdthesis{Vasudevan:EECS-2014-38, Author= {Vasudevan, Ram}, Title= {Identification of Hybrid Dynamical Models for Human Movement via Switched System Optimal Control}, School= {EECS Department, University of California, Berkeley}, Year= {2014}, Month= {May}, Url= {http://www2.eecs.berkeley.edu/Pubs/TechRpts/2014/EECS-2014-38.html}, Number= {UCB/EECS-2014-38}, Abstract= {The empirical observation of human locomotion has found considerable utility in the diagnosis of numerous neuromuscular pathologies. Unfortunately without the construction of a dynamical system model of the measured gait, the effectualness of these observations is restricted to just the existing diagnostic variety rather than the prediction of potential instabilities in gait or guiding the construction of user specific prosthetics. In order to construct a dynamical system model of an observed gait in an automated fashion, one requires a family of representations rich enough to describe the dynamics of gait and an automated procedure to select a particular representation capable of describing a given observation from this family. The goal of this thesis is to address these two problems. First, a hybrid dynamical system representation is introduced that is shown to be capable of describing the discontinuities in dynamics that occur during locomotion. In particular, such a representation is constructible from observation given an unconstrained Lagrangian which is intrinsic to the biped after the identification of the sequence of contact points that are enforced during the observed motion. Second, a specific hybrid dynamical system representation is shown to be constructible from observed data by optimally switching between the set of vector fields corresponding to all possible combinations of contact point enforcements. At this point an algorithm for the computation of an optimal control of constrained nonlinear switched dynamical systems is devised. The control parameter for such systems include a continuous-valued input and discrete-valued input, where the latter corresponds to the mode of the switched system that is active at a particular instance in time. The presented approach, which this thesis proves converges to local minimizers of the constrained optimal control problem, first relaxes the discrete-valued input, performs traditional optimal control, and then projects the constructed relaxed discrete-valued input back to a pure discrete-valued input by employing an extension of the classical Chattering Lemma. This algorithm is extended by formulating a computationally implementable algorithm that works by discretizing the time interval over which the switched dynamical system is defined. Importantly, this thesis proves that the implementable algorithm constructs a sequence of points by recursive application that converge to the local minimizers of the original constrained optimal control problem. Four simulation experiments are included to validate the theoretical developments and illustrate its superiority when compared to standard mixed integer optimization techniques. The thesis concludes by applying the presented algorithm to perform the identification of a hybrid dynamical system representation of two classes of gaits. The first is a synthetic gait generated by the application of feedback linearization to a classical robotic bipedal model. For this synthetic observation, the presented identification scheme is able to correctly identify the correct model. The second set of gaits is one constructed from motion capture observations of $9$ subjects during a flat ground walking experiment. For each subject, the presented identification scheme determines a distinct hybrid dynamical system representation. Surprisingly, the identified models for each participant share an identical discrete structure, or an identical sequence of contact point enforcements.}, }

EndNote citation:

%0 Thesis %A Vasudevan, Ram %T Identification of Hybrid Dynamical Models for Human Movement via Switched System Optimal Control %I EECS Department, University of California, Berkeley %D 2014 %8 May 1 %@ UCB/EECS-2014-38 %U http://www2.eecs.berkeley.edu/Pubs/TechRpts/2014/EECS-2014-38.html %F Vasudevan:EECS-2014-38