Interception and Rendezvous Between Autonomous Vehicles

Abstract

The problem of pursuit and evasion is a classic problem that has intrigued mathematicians for many generations. Suppose a target or evader is moving along a given curve in the plane. A pursuer or chaser is moving such that its line of sight is always pointing towards the target. The classic problem is to determine the trajectory of the pursuer such that it eventually captures the target. There is extensive literature on this problem, see for example the recent book by Nahin (Nahin, 2007). A special case of the pursuit problem is the case where the curve of the target is a circle. This is a classic problem which was first treated by Hathaway (Hathaway, 1921), see also the book by Davis (Davis, 1962). In the classic problems, the line of sight is always pointing towards the target. This method of pursuit is also known as 'pure pursuit' or 'dog pursuit' or 'courbe de chien' in French. Although there have been some modern works on extensions of the classic problem of pursuit, see for example (Marshall, 2005), here we completely abandon the 'pure pursuit' restriction and we treat the problem where the pursuer is required to capture or intercept the target in a given prescribed time or in minimum time. Also, in the classic problem, the velocities of the target and pursuer are assumed constant and there is no consideration of the physics of the motion, such as thrust forces, hydrodynamic or aerodynamic drag and other forces that might be acting on the target and pursuer. In an active-passive rendezvous problem between two vehicles, the passive or target vehicle moves passively along its trajectory. The active or chaser vehicle is controlled or guided such as to meet the passive vehicle at a later time, matching both the location and the velocity of the target vehicle. An interception problem is similar to the rendezvous problem, except that there is no need to match the final velocities of the two vehicles. On the other hand, in a cooperative rendezvous problem, the two vehicles are active and maneuver such as to meet at a later time, at the same location with the same velocity. The two vehicles start the motion from different initial locations and might have different initial velocities. An optimal control problem consists of finding the control histories, i.e., the controls as a function of time, and the state variables of the dynamical system such as to minimize a performance index. The differential equations of motion of the vehicles are treated as dynamical constraints. One possible approach to the solution of the rendezvous problem is to formulate it as an optimal control problem in which one is seeking the controls such as to