Minimum Time Optimal Control of Second Order System With Quadratic Drag and State Constraints

The problem of mixed discrete-continuous task planning for mechanical systems, such as aerial drones or other autonomous units, can often be treated as a sequence of point-to-point trajectories. The minimum time optimal solution between points in the plan is critical not only for the calculation of the trajectory in cases where the goal has to be achieved quickly but also for the feasibility checking of the plan and the planning process itself, especially in the presence of deadlines and temporal constraint. In this work, we address the minimum time problem for a second-order system with quadratic drag, under state (velocity) and control (acceleration) constraints. Closed-form expressions for the trajectory are derived and the optimality is proven using the Pontryagin Maximum Principle. Simulations supporting the results are provided and compared with those of an open source academic optimal control solver.