Robotic Path Planning for High-Level Tasks in Discrete Environments

Abstract

This thesis proposes two techniques for solving high-level multi-robot motion planning problems with discrete environments. We focus on an important class of problems that require an allocation of spatially distributed tasks to robots, along with a set of efficient paths for the robots to visit their task locations. The first technique, SAT-TSP, models the problem with a framework that allows a natural coupling between the allocation problem and the path planning problem. The allocation problem is encoded as a Boolean Satisfiability problem (SAT) and the path planning problem is encoded as a Travelling Salesman Problem (TSP). In addition, this framework can handle complex constraints such as battery life limitations, robot carrying capacities, and robot-task incompatibilities. We propose an algorithm that leverages recent advances in Satisfiability Modulo Theory to combine state-of-the-art SAT and TSP solvers. We characterize the correctness of our algorithm and evaluate it in simulation on a series of patrolling, periodic routing, and multi-robot sample collection problems. The results show that our algorithm outperforms a state-of-the-art mathematical programming solver on a majority of the problems in our benchmark, especially the more difficult problems.

The second technique, Gamma-Clustering, is used to reduce the computational effort of finding good solutions for metric discrete path planning problems. This technique can be used on the set of allocation path planning problems that do not have ordering constraints (ordering only affects the cost of the solution, not its feasibility). To obtain the computational savings, we find Gamma-Clusters within the problem's environment and then restrict how feasible paths visit these clusters. We prove that solutions found using this approach are within a constant factor of the optimal. By increasing the parameter Gamma we can improve the quality of the bound but we do so with less computational savings. We provide a simple polynomial-time algorithm for finding the optimal Gamma-Clustering and show that for a given Gamma the clustering is unique. We provide two methods for using Gamma-Clusters on path planning problems, a coupled method and a hierarchical method. We demonstrate the effectiveness of these methods on travelling salesman instances, sample collection problems, and period routing problems. The results show that for many instances we obtain significant reductions in computation time with little to no reduction in solution quality. Comparing these methods to a standard integer programming approach reveals that as the problems become more difficult, the solution quality of the two methods degrade at a slower rate than the standard approach, thus for more difficult instances we can use Gamma-Clustering to find higher quality solutions.