Optimal Control of Differentially Flat Systems is Surprisingly Simple

As we move to increasingly complex cyber-physical systems (CPS), new approaches are needed to plan efficient state trajectories in real-time. In this paper, we propose an approach to significantly reduce the complexity of solving optimal control problems for a class of CPS. We exploit the property of differential flatness to simplify the Euler-Lagrange equations, and this simplification eliminates the numerical instabilities that arise in the general case. We also present an explicit differential equation that describes the evolution of the optimal state trajectory, and we extend our results to consider both the unconstrained and constrained cases. Furthermore, we demonstrate the performance of our approach by generating the optimal trajectory for a double-integrator agent in an environment with an obstacle. In simulation, our approach shows a 30% cost reduction and nearly a 3-fold increase in computational speed compared to existing collocation-based optimal control libraries.