Trajectory optimization on manifolds with applications to quadrotor systems
Michael Watterson, Sikang Liu, Ke Sun, Trey Smith, Vijay Kumar
Abstract
Manifolds are used in almost all robotics applications even if they are not modeled explicitly. 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 [Formula: see text] and a perception-aware planning example for visual–inertially guided robots navigating in three dimensions. Formulating field of view constraints naturally results in modeling with the manifold [Formula: see text], which cannot be modeled as a Lie group. We also demonstrate the example of planning trajectories on [Formula: see text] for a formation of quadrotors within an obstacle filled environment.