Variational Methods for Control and Design of Bipedal Robot Models

Abstract

This thesis investigates nonsmooth mechanics using variational methods for the modeling, control, and design of bipedal robots. 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 two 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 methods and surrogate function optimization methods is defined. This method is demonstrated in the task of determining knee joint placement in a given biped model.