Incremental Sampling-based Algorithms for Optimal Motion Planning

During the last decade, incremental sampling-based motion planning algorithms, such as the Rapidly-exploring Random Trees (RRTs) have been shown to work well in practice and to possess theoretical guarantees such as probabilistic completeness. However, no theoretical bounds on the quality of the solution obtained by these algorithms have been established so far. The first contribution of this paper is a negative result: it is proven that, under mild technical conditions, the cost of the best path in the RRT converges almost surely to a non-optimal value. Second, a new algorithm is considered, called the Rapidly-exploring Random Graph (RRG), and it is shown that the cost of the best path in the RRG converges to the optimum almost surely. Third, a tree version of RRG is introduced, called the RRT$^*$ algorithm, which preserves the asymptotic optimality of RRG while maintaining a tree structure like RRT. The analysis of the new algorithms hinges on novel connections between sampling-based motion planning algorithms and the theory of random geometric graphs. In terms of computational complexity, it is shown that the number of simple operations required by both the RRG and RRT$^*$ algorithms is asymptotically within a constant factor of that required by RRT.Comment: 20 pages, 10 figures, this manuscript is submitted to the International Journal of Robotics Research, a short version is to appear at the 2010 Robotics: Science and Systems Conference

Massively parallelizing the RRT and the RRT*

In recent years, the growth of the computational power available in the Central Processing Units (CPUs) of consumer computers has tapered significantly. At the same time, growth in the computational power available in the Graphics Processing Units (GPUs) has remained strong. Algorithms that can be implemented on GPUs today are not only limited to graphics processing, but include scientific computation and beyond. This paper is concerned with massively parallel implementations of incremental sampling-based robot motion planning algorithms, namely the widely-used Rapidly-exploring Random Tree (RRT) algorithm and its asymptotically-optimal counterpart called RRT*. We demonstrate an example implementation of RRT and RRT* motion-planning algorithm for a high-dimensional robotic manipulator that takes advantage of an NVidia CUDA-enabled GPU. We focus on parallelizing the collision-checking procedure, which is generally recognized as the computationally expensive component of sampling-based motion planning algorithms. Our experimental results indicate significant speedup when compared to CPU implementations, leading to practical algorithms for optimal motion planning in high-dimensional configuration spaces

Sampling-based algorithm for filtering using Markov chain approximations

In this paper, the filtering problem for a large class of continuous-time, continuous-state stochastic dynamical systems is considered. Inspired by recent advances in asymptotically-optimal sampling-based motion planning algorithms, such as the PRM* and the RRT*, an incremental sampling-based algorithm is proposed. Using incremental sampling, this approach constructs a sequence of Markov chain approximations, and solves the filtering problem, in an incremental manner, on these discrete approximations. It is shown that the trajectories of the Markov chain approximations converge in distribution to the trajectories of the original stochastic system; moreover, the optimal filter calculated on these Markov chains converges to the optimal continuous-time nonlinear filter. The convergence results are verified in a number of simulation examples.United States. Army Research Office. Multidisciplinary University Research Initiative (Grant W911NF-11-1-0046

Toward Asymptotically-Optimal Inspection Planning via Efficient Near-Optimal Graph Search

Inspection planning, the task of planning motions that allow a robot to inspect a set of points of interest, has applications in domains such as industrial, field, and medical robotics. Inspection planning can be computationally challenging, as the search space over motion plans that inspect the points of interest grows exponentially with the number of inspected points. We propose a novel method, Incremental Random Inspection-roadmap Search (IRIS), that computes inspection plans whose length and set of inspected points asymptotically converge to those of an optimal inspection plan. IRIS incrementally densifies a motion planning roadmap using sampling-based algorithms, and performs efficient near-optimal graph search over the resulting roadmap as it is generated. We demonstrate IRIS's efficacy on a simulated planar 5DOF manipulator inspection task and on a medical endoscopic inspection task for a continuum parallel surgical robot in anatomy segmented from patient CT data. We show that IRIS computes higher-quality inspection paths orders of magnitudes faster than a prior state-of-the-art method.Comment: RSS 201

Sampling-based algorithms for optimal path planning problems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2012.Cataloged from PDF version of thesis.Includes bibliographical references (p. 141-152).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 pt-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 p-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.by Sertac Karaman.Ph.D