Constrained Optimal Trajectory Tracking on the Group of Rigid Body Motions

In this paper we study optimality conditions for the finite time horizon, constrained optimal trajectory tracking problem on the group of rigid body motions SE(3). We treat SE(3) as a differentiable manifold and use a geometric approach to derive the necessary optimality conditions. To do so, we begin by studying simple optimal control problems on SE(3), including deriving the equations of motion of a rigid body (i.e., Euler's equations) by formulating the dynamics as a constrained variational optimal control problem. The main contribution of the paper is the derivation of the necessary optimality conditions for constrained optimal trajectory tracking on SE(3) and SO(3). A simple example on SO(2) is given.