Sampling-based algorithms for optimal path planning problems

Sampling-based motion planning received increasing attention during the last decade. In particular, some of the leading paradigms, such the Probabilistic RoadMap (PRM) and the Rapidly-exploring Random Tree (RRT) algorithms, have been demonstrated on several robotic platforms, and found applications well outside the robotics domain. However, a large portion of this research effort has been limited to the classical feasible path planning problem, which asks for finding a path that starts from an initial configuration and reaches a goal configuration while avoiding collision with obstacles. The main contribution of this dissertation is a novel class of algorithms that extend the application domain of sampling-based methods to two new directions: optimal path planning and path planning with complex task specifications. Regarding the optimal path planning problem, we first show that the existing algorithms either lack asymptotic optimality, i.e., almost-sure convergence to optimal solutions, or they lack computational efficiency: on one hand, neither the RRT nor the k-nearest PRM (for any fixed k) is asymptotically optimal; on the other hand, the simple PRM algorithm, where the connections are sought within fixed radius balls, is not computationally as efficient as the RRT or the efficient PRM variants. Subsequently, we propose two novel algorithms, called PRM∗ and RRT∗, both of which guarantee asymptotic optimality without sacrificing computational efficiency. In fact, the proposed algorithms and the most efficient existing algorithms, such as the RRT, have the same asymptotic computational complexity. Regarding the path planning problem with complex task specifications, we propose an incremental sampling-based algorithm that is provably correct and probabilistically complete, i.e., it generates a correct-by-design path that satisfies a given deterministic μ-calculus specification, when such a path exists, with probability approaching to one as the number of samples approaches infinity. For this purpose, we develop two key ingredients. First, we propose an incremental sampling-based algorithm, called the RRG, that generates a representative set of paths in the form of a graph, with guaranteed almost-sure convergence towards feasible paths. Second, we propose an incremental local model-checking algorithm for the deterministic μ-calculus. Moreover, with the help of these tools and the ideas behind the RRT∗, we construct algorithms that also guarantee almost sure convergence to optimal solutions.