Receding Horizon Optimization Method for Solving the Cops and Robbers Problems in a Complex Environment with Obstacles

Abstract

Cops and Robbers problems are classical examples of pursuit and evasion problems which are parts of key researches in the field of robotics. This study shall specifically focus on the evasion strategies of robbers. This study presents the receding horizon optimization method to obtain such strategies of robbers and solves the Cops and Robbers problems in a complex environment with obstacles. In this method, the robbers estimate the control variables of the cops in real time to address the difficulties in obtaining the complete pursuit strategies of the latter for solving the evasion strategies. This method also guarantees the real-time solutions of receding horizon optimization problems. Orthogonal collocation is utilized to discretize the Cops and Robbers dynamic model, and then the resulting nonlinear programming problem is solved to obtain the optimal control. To improve the accuracy of the solution, we propose an iterative hp-adaptive mesh refinement strategy to satisfy the optimality conditions by adjusting the number of finite elements and the order of Lagrange polynomials. This mesh refinement strategy also iteratively uses finite elements and collocation points as well as applies the finite element merging strategy to improve the solution efficiency. The proposed method also provides a framework for solving other pursuit and evasion problems in a complex environment with obstacles.