arXiv subjects

Maksim Buzikov

Publications and source records attributed to Maksim Buzikov.

3 recordsLinked to original sources

arXiv2023

Computing the Minimum-Time Interception of a Moving Target

In this study, we propose an algorithmic framework for solving a class of optimal control problems. This class is associated with the minimum-time interception of moving target problems, where a plant with a given state equation must approach a moving target whose trajectory is known a priori. Our framework employs an analytical description of the distance from an arbitrary point to the reachable set of the plant. The proposed algorithm is always convergent and cannot be improved without losing the guarantee of its convergence to the correct solution for arbitrary Lipschitz continuous trajectories of the moving target. In practice, it is difficult to obtain an analytical description of the distance to the reachable set for the sophisticated state equation of the plant. Nevertheless, it was shown that the distance can be obtained for some widely used models, such as the Dubins car, in an explicit form. Finally, we illustrate the generality and effectiveness of the proposed framework for simple motions and the Dubins model.

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arXiv2022

The Game of Two Identical Cars: an Analytical Description of the Barrier

In this paper, a pursuit-evasion game of two players known as the game of two identical cars is examined. It is assumed that the game proceeds on a two-dimensional plane. Both players have constant speeds and limited turn radii. The goal of the first player (pursuer) is to ensure that the second player (evader) is guaranteed to be within a capture circle as quickly as possible. The goal of the evader is to avoid the capture or delay it as long as possible. The kinematics of both players are described by the same equations. Thus, the game has only one free parameter, the capture radius. This work aims at a fully analytical description of the barrier surface for all values of the capture radius. Previously, A.W.~Merz analytically investigated the barrier of the game of two identical cars. In this work, it is found that there is a certain critical value of the capture radius, above which the barrier occurs qualitatively differently from Merz's example. Also, we obtained an analytical description of the optimal feedback controls on the barrier.

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arXiv2021

Minimum-time lateral interception of a moving target by a Dubins car

This paper presents the problem of lateral interception by a Dubins car of a target that moves along an a priori known trajectory. This trajectory is given by two coordinates of a planar location and one angle of a heading orientation, every one of them is a continuous function of time. The optimal trajectory planning problem of constructing minimum-time trajectories for a Dubins car in the presence of a priory known time-dependent wind vector field is a special case of the presented problem. Using the properties of the three-dimensional reachable set of a Dubins car, it is proved that the optimal interception point belongs to a part of an analytically described surface in the three-dimensional space. The analytical description of the surface makes it possible to obtain 10 algebraic equations for calculating parameters of the optimal control that implements the minimum-time lateral interception. These equations are generally transcendental and can be simplified for particular cases of target motion (e.g. resting target, straight-line uniform target motion). Finally, some particular cases of the optimal lateral interception validate developments of the paper and highlight the necessity to consider each of 10 algebraic equations in general case.

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