Application of methods of multidimensional descriptive geometry in the analysis of velocities in the problem of group pursuit of a set of targets

This article is devoted to the topic of how, in the problem of group pursuit, to reach simultaneous achievement of targets by pursuers. In the pursuit model considered in the article, the pursuer strives to achieve the goal by adhering to a network of predicted trajectories. The predicted trajectory of movement is formed at each moment of time. This trajectory is a compound curve, taking into account curvature constraints. The pursuers reach their targets over a period of time, depending on the pursuer's velocity modulus and the minimum radius of curvature of the pursuer's trajectory. The article analyzes the velocity of movement of pursuers and the curvature constraints of their trajectories for the simultaneous achievement of targets. In multivariate analysis in the problem of group pursuit of a set of targets, the methods of multidimensional descriptive geometry are used. On the radius of curvature - velocity projection plane in the Radishchev drawing, a family of parallel horizontal lines, corresponding to the pursuer's velocity range, is displayed. Although according to the data of the problem, the velocity of the pursuer is constant, the range of velocity is introduced to obtain the function of the dependence of the velocity on the radius of curvature. Next, on the projection plane (radius of curvature, time to reach the target), the corresponding images of the family of horizontal velocity lines are constructed. The appointed time for achieving the target by the pursuer is one of the optimizing factors. So, on the radius of curvature - time to reach the target projection plane, a set of intersection points with the velocity lines with the level line of the designated time of reaching the target by the pursuer is formed. Next, along the lines of communication, the images of these points on the projection plane (radius of curvature, velocity) with subsequent polynomial regression are formed. As a result, the function of the dependence of the velocity on the radius of curvature of the pursuer trajectory in order to reach the target in a specified fixed time is obtained. Next, on the projection plane (radius of curvature, velocity), a line of the velocity level as the second optimizing factor is built. The point obtained as a result of the intersection of the lines is the value of the radius of curvature and the velocity to reach the target at the appointed time. The proposed method of analyzing velocities in problems of group pursuit of a set of targets allows to obtain the result in an automated mode without an operator and may be of interest to developers of UAVs equipped with elements of artificial intelligence.