Differential Flatness as a Sufficient Condition to Generate Optimal Trajectories in Real Time

As robotic systems increase in autonomy, there is a strong need to plan efficient trajectories in real-time. In this paper, we propose an approach to significantly reduce the complexity of solving optimal control problems both numerically and analytically. We exploit the property of differential flatness to show that it is always possible to decouple the forward dynamics of the system's state from the backward dynamics that emerge from the Euler-Lagrange equations. This coupling generally leads to instabilities in numerical approaches; thus, we expect our method to make traditional "shooting" methods a viable choice for optimal trajectory planning in differentially flat systems. To provide intuition for our approach, we also present an illustrative example of generating minimum-thrust trajectories for a quadrotor. Furthermore, we employ quaternions to track the quadrotor's orientation, which, unlike the Euler-angle representation, do not introduce additional singularities into the model.