Trajectory optimization on manifolds with applications to quadrotor systems

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.

Global rates of convergence for nonconvex optimization on manifolds

We consider the minimization of a cost function f on a manifold $\mathcal{M}$ using Riemannian gradient descent and Riemannian trust regions (RTR). We focus on satisfying necessary optimality conditions within a tolerance ε. Specifically, we show that, under Lipschitz-type assumptions on the pullbacks of f to the tangent spaces of $\mathcal{M}$, both of these algorithms produce points with Riemannian gradient smaller than ε in $\mathcal{O}\big(1/\varepsilon ^{2}\big)$ iterations. Furthermore, RTR returns a point where also the Riemannian Hessian’s least eigenvalue is larger than −ε in $\mathcal{O} \big(1/\varepsilon ^{3}\big)$ iterations. There are no assumptions on initialization. The rates match their (sharp) unconstrained counterparts as a function of the accuracy ε (up to constants) and hence are sharp in that sense. These are the first deterministic results for global rates of convergence to approximate first- and second-order Karush-Kuhn-Tucker points on manifolds. They apply in particular for optimization constrained to compact submanifolds of ${\mathbb{R}^{n}}$, under simpler assumptions.