Direct Collocation for Numerical Optimal Control of Second-Order ODE

Mechanical systems are usually modeled by second-order Ordinary Differential Equations (ODE) which take the form $\ddot{q}=f(t,\ q,\ \dot{q})$. While simulation methods tailored to these equations have been studied, using them in direct optimal control methods is rare. Indeed, the standard approach is to perform a state augmentation, adding the velocities to the state. The main drawback of this approach is that the number of decision variables is doubled, which could harm the performance of the resulting optimization problem. In this paper, we present an approach tailored to second-order ODE. We compare it with the standard one, both on theoretical aspects and in a numerical example. Notably, we show that the tailored formulation is likely to improve the performance of a direct collocation method, for solving optimal control problems with second-order ODE of the more restrictive form $\ddot{q}=f(t,\ q)$.