  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.
