Thesis Defense: Variational Methods for Control and Design of Bipedal Robot Models

David Pekarek

This thesis investigates the nonsmooth mechanics surrounding rigid body impacts using variational methods. The main goal is to model the nonsmooth dynamics of bipedal walking robots in a structured way. This in turn enables algorithms for the optimal control and optimal design of these robots.

First, the theory of Lagrangian mechanics is extended to capture a variety of nonsmooth collision behaviors in rigid body systems. Notably, a variational impact model is presented for the transition of constraints behavior that describes a biped switching stance feet at the conclusion of a step.

Next, discretizations of the impact mechanics are developed using the framework of variational discrete mechanics. The resulting variational collision integrators are consistent with the continuous time theory and have an underlying symplectic structure.

In addition to their role as integrators, the discrete equations of motion capturing nonsmooth dynamics enable a direct method for trajectory optimization. Upon specifically defining the optimal control problem for nonsmooth systems, examples demonstrate this optimization method in the task of determining periodic gaits for a variety of rigid body biped models.

An additional effort is made to optimize bipedal walking motions through modifications in system design. A method for determining optimal designs using a combination of trajectory optimization and surrogate function optimization is defined. This method is demonstrated in the task of determining knee joint placement in a given biped model.