A new mathematical model and algorithm for the optimal path to intercept a moving target

TL;DR

This work addresses the problem of intercepting a moving target with a pursuer traveling at constant speed. It introduces a Dubins-path–inspired optimal-control framework that concatenates a circular arc with a straight segment and analyzes it via Pontryagin’s Maximum Principle, yielding a bang-bang heading-rate control. An explicit optimization model with arc-length–based variables $\xi_i$ and interception constraints is developed, along with an efficient algorithm to compute $LS+S_T$ or $RS+S_T$ trajectories and turning-circle geometry. Numerical experiments show fast convergence and substantial gains over a prior method (Looker2008), validating the geometric interpretation and its applicability to practical path-planning tasks in robotics and UAV guidance.

Abstract

This paper is concerned with determining the shortest path for a pursuer aiming to intercept a moving target travelling at a constant speed. To address this challenge, we introduce an efficient mathematical model outlined as an optimal control problem. The proposed model is based on Dubin's path, where we concatenate two possible paths: a left-circular curve or a right-circular curve followed by a straight line. We develop and explore this model, providing a comprehensive geometric interpretation, and design an algorithm tailored to implement the proposed mathematical approach efficiently. Extensive numerical experiments involving diverse target positions highlight the strength of the model. The method exhibits a remarkably high convergence rate in finding solutions. We compare the proposed model and demonstrate its advantages through examples. For experiment purposes, we utilized the modelling software AMPL, employing a range of solvers to solve the problem. Subsequently, we simulated the obtained solutions using MATLAB, demonstrating the efficiency of the model in intercepting a moving target. The proposed model distinguishes itself by employing fewer parameters and making fewer assumptions, setting the model simplifies the complexities, and thus, makes it easier for experts to design optimal path plans.

Figures (8)

• Figure 1: Feasible optimal paths of interception of a moving target and a pursuer.
• Figure 2: Area where pursuer can not intercept target.
• Figure 3: Optimizing the path of the pursuer, starting at position $P(0,0)$ with a heading angle of $\theta_P = 2\pi/3$, to intercept target $T(-5,0)$, which is located to the left of the pursuer and heading in the direction of $\theta_T = \pi/2$.
• Figure 4: Optimizing the path of the pursuer, starting at position $P(0,0)$ with a heading angle of $\theta_P = \pi/3$, to intercept target $T(8,-2)$, which is located to the right of the pursuer and heading in the direction of $\theta_T = 2\pi/3$.
• Figure 5: Optimizing the path of the pursuer, starting at position $P(0,0)$ with a heading angle of $\theta_P = \pi/3$, to intercept target $T(8,3)$, which is located to the right of the pursuer and heading in the direction of $\theta_T = \pi$.

Theorems & Definitions (2)

• Proposition 5.1
• Remark 6.1