Towards some general results in bifurcations in optimal solutions for symmetric distributed robotic formation control

The modern trend of increased integration of formerly disparate systems is necessitating the development of advanced tools to study large-scale complex systems. This paper studies bifurcations in solution to an optimal control problem for robotic formation control. Robotic formation control is an excellent system-integration prototype problem because the scale of the problem can grow rapidly with increased numbers of robots, but the system retains some degree of homogeneity which makes its study manageable. Our prior efforts have numerically studied the bifurcations for particular systems and performed an asymptotic analysis on those systems which provided insight into the rich and complicated structure of the solution space for such systems. The main contribution of this paper is an extension of the asymptotic analysis beyond the specific systems studied previously to a general class of systems. In both the specific and general cases, we show that as a system parameter is varied, the number of solutions increases from an unique solution to an infinite number of expected solutions, which bears resemblance to the cascade of period-doubling bifurcations typical of a dynamical system that exhibits chaos.