Structural Analysis of Combined Time and Energy Optimal Trajectory Planning with Quadratic Drag

The problem of mixed discrete-continuous task planning for mechanical systems, such as aerial drones or other autonomous units, can be often treated as a sequence of point-to-point trajectories. The combined time-energy optimal solution between points in the state space is critical not only for the calculation of the trajectory but also for the feasibility checking of the plan. In this work, we extend from previous results, a combined criterion for a second-order system with quadratic drag, under state (velocity) and control (acceleration) constraints. We show that the solution has a special structure that is maintained even in the marginal cases. We prove this result using the Pontryagin Maximum Principle. Simulations supporting the results are provided for different parameters.