Batch Informed Trees (BIT*): Informed Asymptotically Optimal Anytime Search

Path planning in robotics often requires finding high-quality solutions to continuously valued and/or high-dimensional problems. These problems are challenging and most planning algorithms instead solve simplified approximations. Popular approximations include graphs and random samples, as respectively used by informed graph-based searches and anytime sampling-based planners. Informed graph-based searches, such as A*, traditionally use heuristics to search a priori graphs in order of potential solution quality. This makes their search efficient but leaves their performance dependent on the chosen approximation. If its resolution is too low then they may not find a (suitable) solution but if it is too high then they may take a prohibitively long time to do so. Anytime sampling-based planners, such as RRT*, traditionally use random sampling to approximate the problem domain incrementally. This allows them to increase resolution until a suitable solution is found but makes their search dependent on the order of approximation. Arbitrary sequences of random samples approximate the problem domain in every direction simultaneously and but may be prohibitively inefficient at containing a solution. This paper unifies and extends these two approaches to develop Batch Informed Trees (BIT*), an informed, anytime sampling-based planner. BIT* solves continuous path planning problems efficiently by using sampling and heuristics to alternately approximate and search the problem domain. Its search is ordered by potential solution quality, as in A*, and its approximation improves indefinitely with additional computational time, as in RRT*. It is shown analytically to be almost-surely asymptotically optimal and experimentally to outperform existing sampling-based planners, especially on high-dimensional planning problems.Comment: International Journal of Robotics Research (IJRR). 32 Pages. 16 Figure

Batch Informed Trees (BIT*): Sampling-based Optimal Planning via the Heuristically Guided Search of Implicit Random Geometric Graphs

In this paper, we present Batch Informed Trees (BIT*), a planning algorithm based on unifying graph- and sampling-based planning techniques. By recognizing that a set of samples describes an implicit random geometric graph (RGG), we are able to combine the efficient ordered nature of graph-based techniques, such as A*, with the anytime scalability of sampling-based algorithms, such as Rapidly-exploring Random Trees (RRT). BIT* uses a heuristic to efficiently search a series of increasingly dense implicit RGGs while reusing previous information. It can be viewed as an extension of incremental graph-search techniques, such as Lifelong Planning A* (LPA*), to continuous problem domains as well as a generalization of existing sampling-based optimal planners. It is shown that it is probabilistically complete and asymptotically optimal. We demonstrate the utility of BIT* on simulated random worlds in $\mathbb{R}^2$ and $\mathbb{R}^8$ and manipulation problems on CMU's HERB, a 14-DOF two-armed robot. On these problems, BIT* finds better solutions faster than RRT, RRT*, Informed RRT*, and Fast Marching Trees (FMT*) with faster anytime convergence towards the optimum, especially in high dimensions.Comment: 8 Pages. 6 Figures. Video available at http://www.youtube.com/watch?v=TQIoCC48gp

Anytime Point-Based Approximations for Large POMDPs

The Partially Observable Markov Decision Process has long been recognized as a rich framework for real-world planning and control problems, especially in robotics. However exact solutions in this framework are typically computationally intractable for all but the smallest problems. A well-known technique for speeding up POMDP solving involves performing value backups at specific belief points, rather than over the entire belief simplex. The efficiency of this approach, however, depends greatly on the selection of points. This paper presents a set of novel techniques for selecting informative belief points which work well in practice. The point selection procedure is combined with point-based value backups to form an effective anytime POMDP algorithm called Point-Based Value Iteration (PBVI). The first aim of this paper is to introduce this algorithm and present a theoretical analysis justifying the choice of belief selection technique. The second aim of this paper is to provide a thorough empirical comparison between PBVI and other state-of-the-art POMDP methods, in particular the Perseus algorithm, in an effort to highlight their similarities and differences. Evaluation is performed using both standard POMDP domains and realistic robotic tasks

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

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.Comment: 76 pages, 26 figures, to appear in International Journal of Robotics Researc