Differential Game to Evaluate the Worst Case Influence of a Varying Disturbance on the Minimal Time Solution for a Small Vehicle Movement

The problem of an autonomous agent moving on a planar surface, such as an aerial drone or a small naval vessel can be treated as navigation between a series of points. In addition the robots operating in the real world often have to come up with a sequence of actions, a plan, which will take them from an initial state to the desired goal state. During planning, the robots have to take into account both discrete and continuous changes, as well as temporal constraints. While nominally the movement between each pair of points can be treated as a 1D projection of the movement on the vector connecting the two points, in the presence of disturbances, the full problem on the plane must be considered. The minimum-time optimal solution depends on the value and direction of the disturbance which in this paper is assumed to be a sum of a constant velocity of the medium (wind or current, respectively) and a bounded time-varying part with uncertain but bounded time derivative. We address the minimum time problem of a movement on a 2D plane with quadratic drag, under norm state (inertial vessel velocity), and norm control (acceleration) constraints. The worst case influence of an uncertain disturbance is evaluated by consideration of a differential game with state and control constraints. The structure and properties of the optimal solution were found and analyzed.