Consistent Approximations for the Optimal Control of Constrained Switched Systems

Though switched dynamical systems have shown great utility in modeling a variety of physical phenomena, the construction of an optimal control of such systems has proven difficult since it demands some type of optimal mode scheduling. In this paper, we devise an algorithm for the computation of an optimal control of constrained nonlinear switched dynamical systems. 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. Our approach, which we prove converges to local minimizers of the constrained optimal control problem, first relaxes the discrete-valued input, then performs traditional optimal control, and then projects the constructed relaxed discrete-valued input back to a pure discrete-valued input by employing an extension to the classical Chattering Lemma that we prove. We extend this algorithm by formulating a computationally implementable algorithm which works by discretizing the time interval over which the switched dynamical system is defined. Importantly, we prove that this 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

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

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

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

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

A Framework for Collaborative Real-Time 3D Teleimmersion in a Geographically Distributed Environment

In this paper, we present a framework for immersive 3D video conferencing and geographically distributed collab-oration. Our multi-camera system performs a full-body 3D reconstruction of users in real time and renders their image in a virtual space allowing remote interaction be-tween users and the virtual environment. The paper fea-tures an overview of the technology and algorithms used for calibration, capturing, and reconstruction. We introduce stereo mapping using adaptive triangulation which allows for fast (under 25 ms) and robust real-time 3D reconstruc-tion. The chosen representation of the data provides high compression ratios for transfer to a remote site. The algo-rithm produces partial 3D meshes, instead of dense point clouds, which are combined on the renderer to create a uni-fied model of the user. We have successfully demonstrated the use of our system in various applications such as remote dancing and immersive Tai Chi learning. 1