In this paper, an optimal coordinated motion planning problem is formulated where multiple agents have to reach given destination positions starting from given initial positions, subject to constraints on the admissible formation patterns. Solutions to the problem are reinterpreted as distance-minimizing geodesics on a certain manifold with boundary. A geodesic on this manifold that possesses conjugate points will not be distance-minimizing beyond its first conjugate point. We study a particular instance of the problem, where a number of initially aligned agents tries to switch, positions by rotating around their common centroid. We characterize analytically the complete set of conjugate points of a geodesic that naturally arises as a candidate solution. This allows us to prove that the geodesic does 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 characterized by a family of orthogonal polynomials.
