  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.
