Construction of movement trajectories for objects based on the Dubins car problem, taking into account constant external influences

Abstract

The paper considers an approach in terms of optimal speed problem for Dubins cars to the formation of control trajectories of moving objects (airplanes, ships), that have control restrictions, under external influences that are constant in magnitude and direction and constant control values at each part of the trajectory. Instead of solving the Pontryagin maximum principle, it is proposed to use a simple comparison of possible control strategies in order to determine the best among them in terms of speed. For each strategy, the calculation of control switching points on the trajectory is based on minimizing the difference between the specified coordinates of the endpoint and the coordinates of the point at which the trajectory comes, depending on the choice of the parameters of two intermediate control switching points. The problem of finding the best speed trajectory for an object from one point to another is solved using the Dubins approach, and their coordinates and heading angles are given for both points. All calculations were carried out taking into account wind and water disturbances, which are constant in magnitude and direction and distort the trajectory. The problem of finding the Dubins paths is reduced to finding the parameters of two intermediate points at which the control changes. Different possibilities for changing controls are considered, taking into account the existing restrictions. The lengths of the trajectories are calculated and the best travel time is selected. The proposed method considers several trajectories acceptable in terms of constraints, taking into account the external influences, from which the optimal path is selected by comparison. Having multiple feasible strategies is beneficial when choosing a trajectory depending on the environment. Instead of solving the problem of nonlinear optimization of the Pontryagin maximum principle, a simple comparison of possible control strategies is used in order to determine the best among them in terms of speed, each of the possible strategies is sought from the condition of minimizing the residual of the analytical solution and the boundary condition at the end of the trajectory. When searching for possible trajectories, control constraints, the influence of external influences, that are constant in magnitude and direction, and the constancy of the control value at each part of the trajectory are taken into account. And all these factors make it possible to sufficiently and adequately simulate the movement of the ship. Physically, restrictions on control (turning radius) are associated with a limited steering angle. Restrictions can be associated not only with restrictions on the turning radius, but also with geographical features of a specific area: for unmanned aerial vehicles this may be due to the buildings and terrain, and for ships this may be due to the coastline, shoals, islands, etc. In this regard, it may turn out that the solution found optimal in terms of speed cannot be realizable in practice. Then the method proposed in the work has the ability to choose another trajectory among the less optimal in terms of speed.