Second order collocation

Abstract

Collocation methods for optimal control commonly assume that the system dynamics is expressed as a first order ODE of the form dx/dt = f(x, u, t), where x is the state and u the control vector. However, in many cases, the dynamics involve the second order derivatives of the coordinates: d^2q/t^2 = g(q, dq/dt, u, t), so that, to preserve the first order form, the usual procedure is to introduce one velocity variable for each coordinate and define the state as x = [q,v]T, where q and v are treated as independent variables. As a consequence, the resulting trajectories do not fulfill the mandatory relationship v = dq/dt except at the collocation points, where it is explicitly imposed. We propose a formulation for Trapezoidal and Hermite-Simpson collocation methods adapted to deal directly with second order dynamics without the need to introduce v as independent from q, and granting the consistency of the trajectories for q and v.