Data-Driven Modeling with Hybrid Dynamical Systems

Abstract

Hybrid dynamical systems are used to describe systems that can instantaneously change state and dynamics. At small timescales, continuous electrodynamics govern the interaction of rigid bodies. Simulating the corresponding stiff differential equation introduces unnecessary complexity when the restitution of velocities post-impact is the phenomenon of interest. Although classical mathematics and physics deals primarily with smooth physical processes, the dynamics of real-world systems can and does abruptly change. We can learn from data to inform the structure and fit the parameters of hybrid dynamical models for such systems. These data-driven methods leverage developments in sensing and computation and are a natural progression in the study of modeling and controlling systems. Continuously collecting data can yield interactive systems that adapt towards a target behavior. An accurate computational model can also verify the safety and efficacy of engineered systems. This thesis seeks to further the practical application of data-driven hybrid dynamical systems - to control robotic systems and assistive devices. In the first aim, hybrid dynamical systems are commonly used to model mechanical systems subject to unilateral constraints, \textit{e.g.} legged locomotion. We demonstrated that nonsmoothness can cause standard optimization techniques to lose convergence guarantees and contribute to poor performance for the resulting control policy. The second aim seeks to predict rhythmic human locomotion with a motive to improve the clinical prescription of Ankle Foot Orthoses (AFO). We created subject-specific models that can predict how an individual will respond to an untested AFO torque profile. These aims tie together advancements in data science with the inherent ability of hybrid dynamical systems to represent phenomena of interest in the real world.