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