Conjugate Points in Formation Constrained Optimal Multi-Agent Coordination: A Case Study

In this paper, an optimal coordinated motion planning problem for multiple agents subject to constraints on the admissible formation patterns is formulated. Solutions to the problem are reinterpreted as distance minimizing geodesics on a certain manifold with boundary. A geodesic on this manifold may fail to be a solution for different reasons. In particular, if a geodesic possesses conjugate points, then it will no longer be distance minimizing beyond its first conjugate point. We study a particular instance of the formation constrained optimal coordinated motion problem, where a number of initially aligned agents tries to switch positions by rotating around their common centroid. The complete set of conjugate points of a geodesic naturally associated to this problem is characterized analytically. This allows us to prove that the geodesic will not correspond to an optimal coordinated motion when the angle of rotation exceeds a threshold that decreases to zero as the number of agents increases. Moreover, infinitesimal perturbations that improve the performance of the geodesic after it passes the conjugate points are also determined, which interestingly are characterized by a certain family of orthogonal polynomials.