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

Anytime Hierarchical Clustering

We propose a new anytime hierarchical clustering method that iteratively transforms an arbitrary initial hierarchy on the configuration of measurements along a sequence of trees we prove for a fixed data set must terminate in a chain of nested partitions that satisfies a natural homogeneity requirement. Each recursive step re-edits the tree so as to improve a local measure of cluster homogeneity that is compatible with a number of commonly used (e.g., single, average, complete) linkage functions. As an alternative to the standard batch algorithms, we present numerical evidence to suggest that appropriate adaptations of this method can yield decentralized, scalable algorithms suitable for distributed/parallel computation of clustering hierarchies and online tracking of clustering trees applicable to large, dynamically changing databases and anomaly detection.Comment: 13 pages, 6 figures, 5 tables, in preparation for submission to a conferenc

Scaling Robot Motion Planning to Multi-core Processors and the Cloud

Imagine a world in which robots safely interoperate with humans, gracefully and efficiently accomplishing everyday tasks. The robot's motions for these tasks, constrained by the design of the robot and task at hand, must avoid collisions with obstacles. Unfortunately, planning a constrained obstacle-free motion for a robot is computationally complex---often resulting in slow computation of inefficient motions. The methods in this dissertation speed up this motion plan computation with new algorithms and data structures that leverage readily available parallel processing, whether that processing power is on the robot or in the cloud, enabling robots to operate safer, more gracefully, and with improved efficiency. The contributions of this dissertation that enable faster motion planning are novel parallel lock-free algorithms, fast and concurrent nearest neighbor searching data structures, cache-aware operation, and split robot-cloud computation. Parallel lock-free algorithms avoid contention over shared data structures, resulting in empirical speedup proportional to the number of CPU cores working on the problem. Fast nearest neighbor data structures speed up searching in SO(3) and SE(3) metric spaces, which are needed for rigid body motion planning. Concurrent nearest neighbor data structures improve searching performance on metric spaces common to robot motion planning problems, while providing asymptotic wait-free concurrent operation. Cache-aware operation avoids long memory access times, allowing the algorithm to exhibit superlinear speedup. Split robot-cloud computation enables robots with low-power CPUs to react to changing environments by having the robot compute reactive paths in real-time from a set of motion plan options generated in a computationally intensive cloud-based algorithm. We demonstrate the scalability and effectiveness of our contributions in solving motion planning problems both in simulation and on physical robots of varying design and complexity. Problems include finding a solution to a complex motion planning problem, pre-computing motion plans that converge towards the optimal, and reactive interaction with dynamic environments. Robots include 2D holonomic robots, 3D rigid-body robots, a self-driving 1/10 scale car, articulated robot arms with and without mobile bases, and a small humanoid robot.Doctor of Philosoph

IBIA: An Incremental Build-Infer-Approximate Framework for Approximate Inference of Partition Function

Exact computation of the partition function is known to be intractable, necessitating approximate inference techniques. Existing methods for approximate inference are slow to converge for many benchmarks. The control of accuracy-complexity trade-off is also non-trivial in many of these methods. We propose a novel incremental build-infer-approximate (IBIA) framework for approximate inference that addresses these issues. In this framework, the probabilistic graphical model is converted into a sequence of clique tree forests (SCTF) with bounded clique sizes. We show that the SCTF can be used to efficiently compute the partition function. We propose two new algorithms which are used to construct the SCTF and prove the correctness of both. The first is an algorithm for incremental construction of CTFs that is guaranteed to give a valid CTF with bounded clique sizes and the second is an approximation algorithm that takes a calibrated CTF as input and yields a valid and calibrated CTF with reduced clique sizes as the output. We have evaluated our method using several benchmark sets from recent UAI competitions and our results show good accuracies with competitive runtimes

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

Graph Summarization

The continuous and rapid growth of highly interconnected datasets, which are both voluminous and complex, calls for the development of adequate processing and analytical techniques. One method for condensing and simplifying such datasets is graph summarization. It denotes a series of application-specific algorithms designed to transform graphs into more compact representations while preserving structural patterns, query answers, or specific property distributions. As this problem is common to several areas studying graph topologies, different approaches, such as clustering, compression, sampling, or influence detection, have been proposed, primarily based on statistical and optimization methods. The focus of our chapter is to pinpoint the main graph summarization methods, but especially to focus on the most recent approaches and novel research trends on this topic, not yet covered by previous surveys.Comment: To appear in the Encyclopedia of Big Data Technologie