A Geometric Analysis of Trajectory Design for Underwater Vehicles∗†

Designing trajectories for a submerged rigid body motivates this paper. Two approaches are addressed: the time optimal approach and the motion planning approach using concatenation of kinematic motions. We focus on the structure of singular extremals and their relation to the existence of rank-one kinematic reductions; thereby linking the optimization problem to the inherent geometric framework. Using these kinematic reductions, we provide a solution to the motion planning problem in the under-actuated scenario, or equivalently, in the case of actuator failures. We finish the paper comparing a time optimal trajectory to one formed by concatenation of pure motions.