Bifurcations and symmetries of optimal solutions for distributed robotic systems

This paper studies bifurcations and multiple solutions of the optimal control problem for mobile robotic systems. While the existence of multiple local solutions to an 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. In particular, this paper studies a group of unicycle-like autonomous mobile robots operating in a 2-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. 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.