Minimum Mixed Time–Energy Trajectory Planning of a Nonlinear Vehicle Subject to 2D Disturbances

The problem of a planar vehicle moving on a surface, such as aerial drones or small naval vessels, can be treated as a series of trajectory planning problems between way-points. While nominally the movement between each two fourth-dimensional points (positions and velocities) can be treated as a 1D projection of the movement on the vector connecting the two points, in the presence of arbitrary disturbance the full problem on a plane must be considered. The mixed minimum time–energy optimal solution is now dependent on the value and direction of the disturbance, which naturally affects the structure and completion of the movement task. In this work, we address the minimum time–energy problem of a movement on a 2D plane with quadratic drag, under norm state (velocity) and norm control (acceleration) constraints. The structure and properties of the optimal solution are found and analyzed. The Pontryagin’s maximum principle (PMP) with control and state constraints is utilized. Simulations supporting the results are provided and compared with those of the open-source academic optimal control solver Falcon.m.