Smooth trajectory generation on SE(3) for a free flying space robot

Abstract

We propose a new optimal trajectory generation technique on SE(3) which avoids known obstacles. We leverage techniques from differential geometry and Lie algebra to formulate a cost functional which is intrinsic to the geometric structure of this space and makes physical sense. We propose an approximation technique to generate trajectories on the subgroup SO(3) and use Semidefinite Programming (SDP) to approximate an NP-Hard problem with one which is tractable to compute. From this trajectory on the subgroup, the trajectory generation on the other dimensions of the group becomes a Quadratic Program (QP). For obstacle avoidance, we use a computational geometric technique to decompose the environment into overlapping convex regions to confine the trajectory. We show how this motion planning technique can be used to generate feasible trajectories for a space robot in SE(3) and describe controllers that enable the execution of the generated trajectory. We compare our method to other geometric techniques for calculating trajectories on SO(3) and SE(3), but in an obstacle-free environment.