On Vehicle Placement to Intercept Moving Targets

TL;DR

This paper studies optimal vehicle placement strategies to intercept moving targets on a line segment, using convex optimization and gradient-based algorithms for both single and multiple vehicle scenarios.

Contribution

It introduces convexity and smoothness properties of the cost function, enabling gradient-based optimization for vehicle placement in interception tasks.

Findings

Cost function is strictly convex for various target movement strategies.

Gradient-based methods effectively find optimal vehicle positions.

Algorithms converge to critical configurations under specified conditions.

Abstract

We address optimal placement of vehicles with simple motion to intercept a mobile target that arrives stochastically on a line segment. The optimality of vehicle placement is measured through a cost function associated with intercepting the target. With a single vehicle, we assume that the target moves (i) with fixed speed and in a fixed direction perpendicular to the line segment, or (ii) to maximize the distance from the line segment, or (iii) to maximize intercept time. In each case, we show that the cost function is strictly convex, its gradient is smooth, and the optimal vehicle placement is obtained by a standard gradient-based optimization technique. With multiple vehicles, we assume that the target moves with fixed speed and in a fixed direction perpendicular to the line segment. We present a discrete time partitioning and gradient-based algorithm, and characterize conditions under which the algorithm asymptotically leads the vehicles to a set of critical configurations of the cost function.