A different look at the optimal control of the Brockett integrator

Abstract

We propose a method to analyze the three-dimensional nonholonomic system known as the Brockett integrator and to derive the (energy) optimal trajectories between two given points. For systems with nonholonomic constraint, it is well-known that the energy optimal trajectories corresponds to sub-Riemannian geodesics under a proper sub-Riemannian metric. Our method uses symmetry reduction and an analysis of the quotient space associated with the action of a (symmetry) group on . By lifting the Riemannian geodesics with respect to an appropriate metric from the quotient space back to the original space , we derive the optimal trajectories of the original problem.