During the last decade, sampling-based path planning algorithms, such as probabilistic roadmaps (PRM) and rapidly exploring random trees (RRT), have been shown to work well in practice and possess theoretical guarantees such as probabilistic completeness. However, little effort has been devoted to the formal analysis of the quality of the solution returned by such algorithms, e.g. as a function of the number of samples. The purpose of this paper is to fill this gap, by rigorously analyzing the asymptotic behavior of the cost of the solution returned by stochastic sampling-based algorithms as the number of samples increases. A number of negative results are provided, characterizing existing algorithms, e.g. showing that, under mild technical conditions, the cost of the solution returned by broadly used sampling-based algorithms converges almost surely to a non-optimal value. The main contribution of the paper is the introduction of new algorithms, namely, PRM* and RRT*, which are provably asymptotically optimal, i.e. such that the cost of the returned solution converges almost surely to the optimum. Moreover, it is shown that the computational complexity of the new algorithms is within a constant factor of that of their probabilistically complete (but not asymptotically optimal) counterparts. The analysis in this paper hinges on novel connections between stochastic sampling-based path planning algorithms and the theory of random geometric graphs.

Sampling-based algorithms for optimal motion planning

During the last decade, sampling-based path planning algorithms, such as Probabilistic RoadMaps (PRM) and Rapidly-exploring Random Trees (RRT), have been shown to work well in practice and possess theoretical guarantees such as probabilistic completeness. However, little effort has been devoted to the formal analysis of the quality of the solution returned by such algorithms, e.g., as a function of the number of samples. The purpose of this paper is to fill this gap, by rigorously analyzing the asymptotic behavior of the cost of the solution returned by stochastic sampling-based algorithms as the number of samples increases. A number of negative results are provided, characterizing existing algorithms, e.g., showing that, under mild technical conditions, the cost of the solution returned by broadly used sampling-based algorithms converges almost surely to a non-optimal value. The main contribution of the paper is the introduction of new algorithms, namely, PRM* and RRT*, which are provably asymptotically optimal, i.e., such that the cost of the returned solution converges almost surely to the optimum. Moreover, it is shown that the computational complexity of the new algorithms is within a constant factor of that of their probabilistically complete (but not asymptotically optimal) counterparts. The analysis in this paper hinges on novel connections between stochastic sampling-based path planning algorithms and the theory of random geometric graphs.

Sampling-based Algorithms for Optimal Motion Planning

The open motion planning library (OMPL) is a new library for sampling-based motion planning, which contains implementations of many state-of-the-art planning algorithms. The library is designed in a way that it allows the user to easily solve a variety of complex motion planning problems with minimal input. OMPL facilitates the addition of new motion planning algorithms, and it can be conveniently interfaced with other software components. A simple graphical user interface (GUI) built on top of the library, a number of tutorials, demos, and programming assignments are designed to teach students about sampling-based motion planning. The library is also available for use through Robot Operating System (ROS).

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The Open Motion Planning Library

A survey on coverage path planning for robotics

Statistical dialog systems (SDSs) are motivated by the need for a data-driven framework that reduces the cost of laboriously handcrafting complex dialog managers and that provides robustness against the errors created by speech recognizers operating in noisy environments. By including an explicit Bayesian model of uncertainty and by optimizing the policy via a reward-driven process, partially observable Markov decision processes (POMDPs) provide such a framework. However, exact model representation and optimization is computationally intractable. Hence, the practical application of POMDP-based systems requires efficient algorithms and carefully constructed approximations. This review article provides an overview of the current state of the art in the development of POMDP-based spoken dialog systems.

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POMDP-Based Statistical Spoken Dialog Systems: A Review

IN Proc. Robotics: Science & Systems, 2008 Abstract—Motion planning in uncertain and dynamic environ- ments is an essential capability for autonomous robots. Partially observable Markov decision processes (POMDPs) provide a principled mathematical framework for solving such problems, but they are often avoided in robotics due to high computational complexity. Our goal is to create practical POMDP algorithms and software for common robotic tasks. To this end, we have developed a new point-based POMDP algorithm that exploits the notion of optimally reachable belief spaces to improve com- putational efficiency. In simulation, we successfully applied the algorithm to a set of common robotic tasks, including instances of coastal navigation, grasping, mobile robot exploration, and target tracking, all modeled as POMDPs with a large number of states. In most of the instances studied, our algorithm substantially outperformed one of the fastest existing point-based algorithms. A software package implementing our algorithm will soon be released at http://motion.comp.nus.edu.sg/ projects/pomdp/pomdp.html.

http://www.roboticsproceedings.org/rss04/p9.pdf

SARSOP: Efficient Point-Based POMDP Planning by Approximating Optimally Reachable Belief Spaces

Why is probabilistic roadmap (PRM) planning probabilistic? How does the probability measure used for sampling a robot's configuration space affect the performance of a PRM planner? These questions have received little attention to date. This paper tries to fill this gap and identify promising directions to improve future planners. It introduces the probabilistic foundations of PRM planning and examines previous work in this context. It shows that the success of PRM planning depends mainly and critically on favorable "visibility" properties of a robot's configuration space. A promising direction for speeding up PRM planners is to infer partial knowledge of such properties from both workspace geometry and information gathered during roadmap construction, and to use this knowledge to adapt the probability measure for sampling. This paper also shows that the choice of the sampling source--pseudo-random or deterministic--has small impact on a PRM planner's performance, compared with that of the sampling measure. These conclusions are supported by both theoretical and empirical results.

On the Probabilistic Foundations of Probabilistic Roadmap Planning.

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On the Probabilistic Foundations of Probabilistic Roadmap Planning

Probabilistic roadmap (PRM) planners have been successful in path planning of robots with many degrees of freedom, but sampling narrow passages in a robot's configuration space remains a challenge for PRM planners. This paper presents a hybrid sampling strategy in the PRM framework for finding paths through narrow passages. A key ingredient of the new strategy is the bridge test, which reduces sample density in many unimportant parts of a configuration space, resulting in increased sample density in narrow passages. The bridge test can be implemented efficiently in high-dimensional configuration spaces using only simple tests of local geometry. The strengths of the bridge test and uniform sampling complement each other naturally. The two sampling strategies are combined to construct the hybrid sampling strategy for our planner. We implemented the planner and tested it on rigid and articulated robots in 2-D and 3-D environments. Experiments show that the hybrid sampling strategy enables relatively small roadmaps to reliably capture the connectivity of configuration spaces with difficult narrow passages.

/pdf/narrow-passage-sampling-for-probabilistic-roadmap-planning-rd0cyd2lz8.pdf

Narrow passage sampling for probabilistic roadmap planning

Motion planning under uncertainty is important for reliable robot operations in uncertain and dynamic environments. Partially Observable Markov Decision Process (POMDP) is a general and systematic framework for motion planning under uncertainty. To cope with dynamic environment well, we often need to modify the POMDP model during runtime. However, despite recent tremendous advances in POMDP planning, most solvers are not fast enough to generate a good solution when the POMDP model changes during runtime. Recent progress in online POMDP solvers have shown promising results. However, most online solvers are based on replanning, which recompute a solution from scratch at each step, discarding any solution that has been computed so far, and hence wasting valuable computational resources. In this paper, we propose a new online POMDP solver, called Adaptive Belief Tree (ABT), that can reuse and improve existing solution, and update the solution as needed whenever the POMDP model changes. Given enough time, ABT converges to the optimal solution of the current POMDP model in probability. Preliminary results on three distinct robotics tasks in dynamic environments are promising. In all test scenarios, ABT generates similar or better solutions faster than the fastest online POMDP solver today; using an average of less than 50 ms of computation time per step.

Hanna Kurniawati

Papers

SARSOP: Efficient Point-Based POMDP Planning by Approximating Optimally Reachable Belief Spaces

On the Probabilistic Foundations of Probabilistic Roadmap Planning.

On the Probabilistic Foundations of Probabilistic Roadmap Planning

Narrow passage sampling for probabilistic roadmap planning

An Online POMDP Solver for Uncertainty Planning in Dynamic Environment