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

TL;DR

This paper develops an analytical method for planning minimum-time lateral interception trajectories for a Dubins car targeting a moving object along a known path, incorporating wind effects and providing algebraic solutions.

Contribution

It introduces a novel analytical framework for optimal interception by a Dubins car with a known moving target, including equations for control parameters and special case simplifications.

Findings

Optimal interception point lies on a specific analytical surface.

Derived 10 algebraic equations for control parameters.

Special cases allow simplification of equations.

Abstract

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.