Optimal control of robotic systems and biased Riemannian splines

Abstract

In this paper, we study mechanical optimal control problems on a given Riemannian manifold ( Q , g ) in which the cost is defined by a general cometric g̃ . This investigation is motivated by our studies in robotics, in which we observed that the mathematically natural choice of cometric g̃ = g * – the dual of g – does not always capture the true cost of the motion. We then, first, discuss how to encode the system’s torque-based actuators configuration into a cometric g̃ . Second, we provide and prove our main theorem, which characterizes the optimal solutions of the problem associated to general triples ( Q , g , g̃ ) in terms of a 4th order differential equation. We also identify a tensor appearing in this equation as the geometric source of “biasing” of the solutions away from ordinary Riemannian splines and geodesics for ( Q , g ). Finally, we provide illustrative examples and practical demonstration of the biased splines as providing the true optimizers in a concrete robotics system.