Bifurcations and Symmetries of Optimal Solutions for Distributed Robotic Systems

In this thesis, we consider the motion planning problem for a symmetric distributed system which consists of a group of autonomous mobile robots operating in a two-dimensional obstacle-free environment. Each robot has a predefined initial state and final state and the problem is to find the optimal path between two states for every robot. The path is optimized with respect to the control effort and the deviation from a desired formation. Due to scaling issues, it becomes more and more difficult and sometimes infeasible to numerically find solutions to the problem as the number of robots increases. One goal of this thesis is to exploit symmetries in distributed control systems to reduce the computational effort to determine solutions for optimal control of such systems. One way to characterize a distributed system is that it is a control system in which the state space is naturally decomposed into multiple subsystems, each of which typically only interacts with a limited subset of the other subsystems. A symmetric distributed system can be defined when the subsystems are diffeomorphically related. The optimal control problem for distributed systems may not scale well with the size of the overall system; hence, our efforts are directed toward exactly solving the optimization problem for large scale systems by working with a reduced order model that is determined by considering invariance properties with respect to certain group actions of the governing equations of the overall system. This thesis also studies bifurcations and multiple solutions of the optimal control problem for mobile robotic systems. While the existence of multiple local solutions to a nonlinear optimization problem is not unexpected, the nature of the solutions are such that a relatively rich and interesting structure is present, which potentially could be exploited for controls purposes. The bifurcation parameter is the relative weight given to penalizing the deviation from the desired formation versus control effort. Numerically it is shown that as this number varies, bifurcations of solutions are obtained. Theoretic results of this paper relate to the symmetric properties of these bifurcations and the number and existence of multiple solutions for large and small values of the bifurcation parameter. Understanding the existence and nature of multiple solutions for optimization problems of this type is also of practical importance due to the ubiquity of gradient-based optimization methods where the search method will typically converge to the nearest local optimum.