Geodesic Problems for Mobile Robots

As mobile robots operate with limited resources which they carry onboard in large obstructed environments, their success is dependent on how efficiently they move while they avoid collision with obstacles and other robots. Moving optimally is the ultimate efficiency a mobile robot can achieve. Therefore, planning optimal motions and devising optimal coordination strategies are two important and challenging fundamental problems in mobile robotics, which have received significant attention in the last couple of decades. Both of those problems can be reduced to shortest path, or equivalently geodesic, problems in appropriate geometric settings. Geodesic problems have been studied in two disciplines: 1) optimal control theory, and 2) computational geometry. Optimal control theory has focused on the differential constraints of robotic systems, while computational geometry has focused on shortest path problems in an environment with obstacles. Optimal control theory has historically disregarded obstacles in the environment, and computational geometry does not consider dynamics of the robotic system, various optimality criteria, or multi-objective optimality. While each discipline has its own powerful tools to address some geodesic problems, there is a large class of problems that cannot be solved using existing algorithms and methods. We introduce a unified approach that is inspired by main results in both disciplines. In this dissertation, we demonstrate our technique, which combines the celebrated Pontryagin Maximum Principle from optimal control theory with visibility graph methods from computational geometry, by solving three geodesic problems for mobile robots: 1) geodesics for the differential drive among obstacles, 2) geodesics for a kinematic airplane, and 3) optimal coordination of two polygonal robots moving on a predetermined network of paths. We consider the differential drive because it is ubiquitous in mobile robotics. To ob-