Use of relaxation methods in sampling-based algorithms for optimal motion planning

Abstract

Motion planning problems have been studied by both the robotics and the controls research communities for a long time, and many algorithms have been developed for their solution. Among them, incremental sampling-based motion planning algorithms, such as the Rapidlyexploring Random Trees (RRT), and the Probabilistic Road Maps (PRM) have become very popular recently, owing to their implementation simplicity and their advantages in handling high-dimensional problems. Although these algorithms work very well in practice, the quality of the computed solution is often not good, i.e., the solution can be far from the optimal one. A recent variation of RRT, namely the RRT * algorithm, bypasses this drawback of the traditional RRT algorithm, by ensuring asymptotic optimality as the number of samples tends to infinity. Nonetheless, the convergence rate to the optimal solution may still be slow. This paper presents a new incremental sampling-based motion planning algorithm based on Rapidly-exploring Random Graphs (RRG), denoted RRT # (RRT "sharp") which also guarantees asymptotic optimality but, in addition, it also ensures that the constructed spanning tree of the geometric graph is consistent after each iteration. In consistent trees, the vertices which have the potential to be part of the optimal solution have the minimum cost-come-value. This implies that the best possible solution is readily computed if there are some vertices in the current graph that are already in the goal region. Numerical results compare with the RRT * algorithm.require efficient low-level collision detection and trajectory planning algorithms to find collision-free trajectories between different samples [17].Incremental sampling-based algorithms were first proposed by Kavraki during the late 1990s. The so-called Probabilistic Road Map (PRM) was successfully implemented to solve multi-query motion planning problems and gained a lot of attention, both in industry and academia [12]. In PRM a graph of the environment is constructed by taking random samples from the configuration space of the robot and testing them to determine whether they belong to the free space. The PRM algorithm uses a local planner that attempts to find a feasible path between the sampled points. Once a reasonable graph is constructed, the initial and the goal states are added to the graph, and the optimal path is computed using a graph search algorithm.Another important class of incremental sampled-based motion planning algorithm is the Rapidlyexploring Random Tree (RRT) and its numerous variants [17]. RRTs have achieved great success in solving single-query motion planning problems in many real-time applications. However, the quality of RRT-based algorithms is often poor (i.e., highly suboptimal). As a result, a lot of effort has been devoted to the development of heuristic techniques in order to refine the quality of the solution obtained from RRTs. However, it has been recently shown that the best path returned by RRTs when the algorithm converges is almost always (i.e., wit...

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