Riemannian Metric Preconditioning for Trajectory Tracking
For researchers in geometric control and robotics, this work offers a method to improve trajectory tracking by preconditioning the Riemannian metric, though it is incremental and limited to specific conditions.
The paper introduces a rank-one Riemannian cometric update that modifies the metric to make specific directions cheaper, and proposes a preconditioned trajectory-tracking strategy using a connection-difference term. Simulation results on a maze navigation experiment show improved tracking when the nominal trajectory aligns with the reward metric.
We introduce a rank-one Riemannian cometric update inducing a modification of the Riemannian metric that makes specific directions of motion cheaper to travel along. We establish basic completeness properties of this reward metric, and give an explicit characterization of its Levi--Civita connection. We propose a preconditioned trajectory-tracking strategy by adding the connection-difference term to a standard intrinsic PD control, and illustrate the construction on a connection control-affine system on the Special Euclidean group with a maze navigation experiment. When the nominal trajectory is an integral curve of the vector field used to define the reward metric, our methodology improves the overall tracking, which is demonstrated through simulation results.