Trajectory Optimization On Manifolds with Applications to SO(3) and R3XS2

Manifolds are used in almost all robotics applications even if they are not explicitly modeled. We propose a differential geometric approach for optimizing trajectories on a Riemannian manifold with obstacles. The optimization problem depends on a metric and collision function specific to a manifold. We then propose our Safe Corridor on Manifolds (SCM) method of computationally optimizing trajectories for robotics applications via a constrained optimization problem. Our method does not need equality constraints, which eliminates the need to project back to a feasible manifold during optimization. We then demonstrate how this algorithm works on an example problem on SO(3) and a perception-aware planning example for visual-inertially guided robots navigating in 3 dimensions. Formulating field of view constraints naturally results in modeling with the manifold R3XS2 which cannot be modeled as a Lie group.