Optimal control of robotic systems and biased Riemannian splines

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 g. This investigation is motivated by our studies in robotics, in which we observed that the mathematically natural choice of cometric g = 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 g. Second, we provide and prove our main theorem, which characterizes the optimal solutions of the problem associated to general triples (Q, g, g) 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.