A well-known weakness of the probabilistic path planners is the so-called narrow passage problem, where a region with a relatively low probability of being sampled must be explored to find a solution path. Many strategies have been proposed to alleviate this problem, most of them based on biasing the sampling distribution. When kinematic constraints appear in the problem, the configuration space typically becomes a non-parametrizable, implicit manifold. Unfortunately, this invalidates most of the existing sampling bias approaches, which rely on an explicit parametrization of the space to explore. In this paper, we propose and evaluate three novel strategies to bias the sampling under the presence of narrow passages in kinematically-constrained systems.
To detect narrow passages, the bridge-test @cite_9 focuses on the generation of pairs of samples in collision where the central point between these samples is collision-free. Even if the configurations could be generated easily, this test implicitly assumes that the linear interpolation between two samples is also a valid configuration, which is not the case when the configuration space is an arbitrary manifold. The assumption is also taken by the approaches that use linear interpolation to displace the random samples toward the medial axis of the free configuration space @cite_5 . Moreover, approaches that use linear dimensionality reduction of the nodes already in the RRT to bias the sampling @cite_24 are also affected by this issue.
@cite_24 @cite_5 @cite_9
2029652111 2163670193 2122724557
Several motion planning methods using networks of randomly generated nodes in the free space have been shown to perform well in a number of cases, however their performance degrades when paths are required to pass through narrow passages in the free space. In previous work we proposed MAPRM, a method of sampling the configuration space in which randomly generated configurations, free or not, are retracted onto the medial axis of the free space without having to first compute the medial axis; this was shown to increase sampling in narrow passages. In this paper we give details of the MAPRM algorithm for the case of a rigid body moving in three dimensions, and show that the retraction may be carried out without explicitly computing the C-obstacles or the medial axis. We give theoretical and experimental results to show this improves performance on problems involving narrow corridors and compare the performance to uniform random sampling from the free space. 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. We propose a general framework for sampling the configuration space in which randomly generated configurations, free or not, are retracted onto the medial axis of the free space. Generalizing our previous work, this framework provides a template encompassing all possible retraction approaches. It also removes the requirement of exactly computing distance metrics thereby enabling application to more realistic high dimensional problems. In particular, our framework supports methods that retract a given configuration exactly or approximately onto the medial axis. As in our previous work, exact methods provide fast and accurate retraction in low (2 or 3) dimensional space. We also propose new approximate methods that can be applied to high dimensional problems, such as many DOF articulated robots. Theoretical and experimental results show improved performance on problems requiring traversal of narrow passages. We also study tradeoffs between accuracy and efficiency for different levels of approximation, and how the level of approximation effects the quality of the resulting roadmap. Recent progress in motion planning has made it possible to determine homotopy inequivalent trajectories between an initial and terminal configuration in a robot configuration space. Current approaches have however either assumed the knowledge of differential one-forms related to a skeletonization of the collision space, or have relied on a simplicial representation of the free space. Both of these approaches are currently however not yet practical for higher dimensional configuration spaces. We propose 2D topological task projections (TTPs): mappings from the configuration space to 2-dimensional spaces where simplicial complex filtrations and persistent homology can identify topological properties of the high-dimensional free configuration space. Our approach only requires the availability of collision free samples to identify winding centers that can be used to determine homotopy inequivalent trajectories. We propose the Winding Augmented RRT and RRT* (WA-RRT RRT*) algorithms using which homotopy inequivalent trajectories can be found. We evaluate our approach in experiments with configuration spaces of planar linkages with 2–10 degrees of freedom. Results indicate that our approach can reliably identify suitable topological task projections and our proposed WA-RRT and WA-RRT* algorithms were able to identify a collection of homotopy inequivalent trajectories in each considered configuration space dimension. The situation arising in path planning under kinematic constraints, where the valid configurations define a manifold embedded in the joint ambient space, can be seen as a limit case of the well-known narrow corridor problem. With kinematic constraints, the probability of obtaining a valid configuration by sampling in the joint ambient space is not low but null, which complicates the direct application of sampling-based path planners. This paper presents the AtlasRRT algorithm, which is a planner especially tailored for such constrained systems that builds on recently developed tools for higher-dimensional continuation. These tools provide procedures to define charts that locally parametrize a manifold and to coordinate the charts, forming an atlas that fully covers it. AtlasRRT simultaneously builds an atlas and a bidirectional rapidly exploring random tree (RRT), using the atlas to sample configurations and to grow the branches of the RRTs, and the RRTs to devise directions of expansion for the atlas. The efficiency of AtlasRRT is evaluated in several benchmarks involving high-dimensional manifolds embedded in large ambient spaces. The results show that the combined use of the atlas and the RRTs produces a more rapid exploration of the configuration space manifolds than existing approaches. In this paper, we address the problem determining the connectivity of a robot's free configuration space. Our method iteratively builds a constructive proof that two configurations lie in disjoint components of the free configuration space. Our algorithm first generates samples that correspond to configurations for which the robot is in collision with an obstacle. These samples are then weighted by their generalized penetration distance, and used to construct alpha shapes. The alpha shape defines a collection of simplices that are fully contained within the configuration space obstacle region. These simplices can be used to quickly solve connectivity queries, which in turn can be used to define termination conditions for sampling-based planners. Such planners, while typically either resolution complete or probabilistically complete, are not able to determine when a path does not exist, and therefore would otherwise rely on heuristics to determine when the search for a free path should be abandoned. An implementation of the algorithm is provided for the case of a 3D Euclidean configuration space, and a proof of correctness is provided. Despite the significant advances in path planning methods, highly constrained problems are still challenging. In some situations, the presence of constraints defines a configuration space that is a non-parametrizable manifold embedded in a high-dimensional ambient space. In these cases, the use of sampling-based path planners is cumbersome since samples in the ambient space have low probability to lay on the configuration space manifold. In this paper, we present a new path planning algorithm specially tailored for highly constrained systems. The proposed planner builds on recently developed tools for higher-dimensional continuation, which provide numerical procedures to describe an implicitly defined manifold using a set of local charts. We propose to extend these methods focusing the generation of charts on the path between the two configurations to connect and randomizing the process to find alternative paths in the presence of obstacles. The advantage of this planner comes from the fact that it directly operates into the configuration space and not into the higher-dimensional ambient space, as most of the existing methods do. In this paper we present a novel probabilistic sampling-based motion planning algorithm called the Fast Marching Tree algorithm FMT*. The algorithm is specifically aimed at solving complex motion planning problems in high-dimensional configuration spaces. This algorithm is proven to be asymptotically optimal and is shown to converge to an optimal solution faster than its state-of-the-art counterparts, chiefly PRM* and RRT*. The FMT* algorithm performs a 'lazy' dynamic programming recursion on a predetermined number of probabilistically drawn samples to grow a tree of paths, which moves steadily outward in cost-to-arrive space. As such, this algorithm combines features of both single-query algorithms chiefly RRT and multiple-query algorithms chiefly PRM, and is reminiscent of the Fast Marching Method for the solution of Eikonal equations. As a departure from previous analysis approaches that are based on the notion of almost sure convergence, the FMT* algorithm is analyzed under the notion of convergence in probability: the extra mathematical flexibility of this approach allows for convergence rate bounds-the first in the field of optimal sampling-based motion planning. Specifically, for a certain selection of tuning parameters and configuration spaces, we obtain a convergence rate bound of order On −1 d+? , where n is the number of sampled points, d is the dimension of the configuration space, and ? is an arbitrarily small constant. We go on to demonstrate asymptotic optimality for a number of variations on FMT*, namely when the configuration space is sampled non-uniformly, when the cost is not arc length, and when connections are made based on the number of nearest neighbors instead of a fixed connection radius. Numerical experiments over a range of dimensions and obstacle configurations confirm our theoretical and heuristic arguments by showing that FMT*, for a given execution time, returns substantially better solutions than either PRM* or RRT*, especially in high-dimensional configuration spaces and in scenarios where collision-checking is expensive. We present a novel trajectory computation algorithm to smooth piecewise linear collision-free trajectories computed by sample-based motion planners. Our approach uses cubic B-splines to generate trajectories that are C2 almost everywhere, except on a few isolated points. The algorithm performs local spline refinement to compute smooth, collision-free trajectories and it works well even in environments with narrow passages. We also present a fast and reliable algorithm for collision checking between a robot and the environment along the B-spline trajectories. We highlight the performance of our algorithm on complex benchmarks, including path computation for rigid and articulated models in cluttered environments. ABSTRACT The configuration space is a fundamental concept that is widely used in algorithmic robotics. Many applications in robotics, computer-aided design, and related areas can be reduced to computational problems in terms of configuration spaces. In this paper, we survey some of our recent work on solving two important challenges related to configuration spaces: how to efficiently compute an approximate representation of high-dimensional configuration spaces; and how to efficiently perform geometric proximity and motion planning queries in high-dimensional configuration spaces. We present new configuration space construction algorithms based on machine learning and geometric approximation techniques. These algorithms perform collision queries on many configuration samples. The collision query results are used to compute an approximate representation for the configuration space, which quickly converges to the exact configuration space. We also present parallel GPU-based algorithms to accelerate the performance of optimization and search computations in configuration spaces. In particular, we design efficient GPU-based parallel k -nearest neighbor and parallel collision detection algorithms and use these algorithms to accelerate motion planning. A simple and efficient randomized algorithm is presented for solving single-query path planning problems in high-dimensional configuration spaces. The method works by incrementally building two rapidly-exploring random trees (RRTs) rooted at the start and the goal configurations. The trees each explore space around them and also advance towards each other through, the use of a simple greedy heuristic. Although originally designed to plan motions for a human arm (modeled as a 7-DOF kinematic chain) for the automatic graphic animation of collision-free grasping and manipulation tasks, the algorithm has been successfully applied to a variety of path planning problems. Computed examples include generating collision-free motions for rigid objects in 2D and 3D, and collision-free manipulation motions for a 6-DOF PUMA arm in a 3D workspace. Some basic theoretical analysis is also presented. In this paper we consider the motion planning problem of humanoid with articulated-spine for object manipulation in double support phase. Complexity of motion planning increases due to higher degrees-of-freedom (DOF), redundancy arising due to articulated spine, inherent underactuation and stability constraint. Additionally, loop-closure constraints arise during double-support phase, and self-collision constraints exist independent of the task to be performed. These problems make motion planning of humanoids a challenging task. In this work, we address the above issues by proposing a sampling based approach for planning the motion of the humanoid. Our approach to the problem is based on RRT* which can generate asymptotically optimal paths. The proposed approach deals with the stability constraints by rejective sampling approach which divides random configurations generated into valid and invalid configurations. Additionally, the loop-closure constraints are tackled by separating the humanoid into two open-kinematic sub-chains, and then generating random configurations in one sub-chain whereas the remaining sub-chain uses inverse kinematics for closure. Efficacy of the proposed approach is demonstrated for whole-body motion planning of a 25-DOF humanoid in generating asymptotically optimal end-effector paths. The widespread success of sampling-based planning algorithms stems from their ability to rapidly discover the connectivity of a configuration space. Past research has found that non-uniform sampling in the configuration space can significantly outperform uniform sampling; one important strategy is to bias the sampling distribution based on features present in the underlying workspace. In this paper, we unite several previous approaches to workspace biasing into a general framework for automatically discovering useful sampling distributions. We present a novel algorithm, based on the REINFORCE family of stochastic policy gradient algorithms, which automatically discovers a locally-optimal weighting of workspace features to produce a distribution which performs well for a given class of sampling-based motion planning queries. We present as well a novel set of workspace features that our adaptive algorithm can leverage for improved configuration space sampling. Experimental results show our algorithm to be effective across a variety of robotic platforms and high- dimensional configuration spaces. In this paper we develop a systematic topological approach to motion planning for a planar 2-R manipulator with point obstacles. By considering components in the free space for the second joint as the first joint varies, we build a two-dimensional array representing the cells of the free space and an associated graph representing the boundaries of those cells. Using this graph, we derive a closed formula for the number of components of the free space. At the same time we solve the motion existence problem, namely, when are two arbitrary configurations in the same component? If so, we develop two explicit algorithms for constructing the path - a middle path method and a linear interpolation method. These algorithms give complete solutions to the path planning problem. Extensive examples are worked out which verify the correctness and efficiency of the resulting program. Then we briefly discuss how these methods generalize to a 3-R planar manipulator. Many approaches for testing configurable software systems start from the same assumption: it is impossible to test all configurations. This motivated the definition of variability-aware abstractions and sampling techniques to cope with large configuration spaces. Yet, there is no theoretical barrier that prevents the exhaustive testing of all configurations by simply enumerating them if the effort required to do so remains acceptable. Not only this: we believe there is a lot to be learned by systematically and exhaustively testing a configurable system. In this case study, we report on the first ever endeavour to test all possible configurations of the industry-strength, open source configurable software system JHipster, a popular code generator for web applications. We built a testing scaffold for the 26,000+ configurations of JHipster using a cluster of 80 machines during 4 nights for a total of 4,376 hours (182 days) CPU time. We find that 35.70 configurations fail and we identify the feature interactions that cause the errors. We show that sampling strategies (like dissimilarity and 2-wise): (1) are more effective to find faults than the 12 default configurations used in the JHipster continuous integration; (2) can be too costly and exceed the available testing budget. We cross this quantitative analysis with the qualitative assessment of JHipster’s lead developers. Interpolation between pairs of values, typically vectors, is a fundamental operation in many computer graphics applications. In some cases simple linear interpolation yields meaningful results without requiring domain knowledge. However, interpolation between pairs of distributions or pairs of functions often demands more care because features may exhibit translational motion between exemplars. This property is not captured by linear interpolation. This paper develops the use of displacement interpolation for this class of problem, which provides a generic method for interpolating between distributions or functions based on advection instead of blending. The functions can be non-uniformly sampled, high-dimensional, and defined on non-Euclidean manifolds, e.g., spheres and tori. Our method decomposes distributions or functions into sums of radial basis functions (RBFs). We solve a mass transport problem to pair the RBFs and apply partial transport to obtain the interpolated function. We describe practical methods for computing the RBF decomposition and solving the transport problem. We demonstrate the interpolation approach on synthetic examples, BRDFs, color distributions, environment maps, stipple patterns, and value functions. Non-Euclidean plates are a subset of the class of elastic bodies having no stress-free configuration. Such bodies exhibit residual stress when relaxed from all external constraints, and may assume complicated equilibrium shapes even in the absence of external forces. In this work we present a mathematical framework for such bodies in terms of a covariant theory of linear elasticity, valid for large displacements. We propose the concept of non-Euclidean plates to approximate many naturally formed thin elastic structures. We derive a thin plate theory, which is a generalization of existing linear plate theories, valid for large displacements but small strains, and arbitrary intrinsic geometry. We study a particular example of a hemispherical plate. We show the occurrence of a spontaneous buckling transition from a stretching dominated configuration to bending dominated configurations, under variation of the plate thickness. We introduce several techniques for sampling and visualizing the latent spaces of generative models. Replacing linear interpolation with spherical linear interpolation prevents diverging from a model's prior distribution and produces sharper samples. J-Diagrams and MINE grids are introduced as visualizations of manifolds created by analogies and nearest neighbors. We demonstrate two new techniques for deriving attribute vectors: bias-corrected vectors with data replication and synthetic vectors with data augmentation. Binary classification using attribute vectors is presented as a technique supporting quantitative analysis of the latent space. Most techniques are intended to be independent of model type and examples are shown on both Variational Autoencoders and Generative Adversarial Networks. We present a method for estimating eye gaze direction, which represents a departure from conventional eye gaze estimation methods, the majority of which are based on tracking specific optical phenomena like corneal reflection and the Purkinje images. We employ an appearance manifold model, but instead of using a densely sampled spline to perform the nearest manifold point query, we retain the original set of sparse appearance samples and use linear interpolation among a small subset of samples to approximate the nearest manifold point. The advantage of this approach is that since we are only storing a sparse set of samples, each sample can be a high dimensional vector that retains more representational accuracy than short vectors produced with dimensionality reduction methods. The algorithm was tested with a set of eye images labelled with ground truth point-of-regard coordinates. We have found that the algorithm is capable of estimating eye gaze with a mean angular error of 0.38 degrees, which is comparable to that obtained by commercially available eye trackers. Abstract Irregular sampling and “stable sampling” of band-limited functions have been studied by H.J. Landau [H.J. Landau, Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math. 117 (1967) 37–52]. We prove that quasicrystals are sets of stable sampling. To cite this article: B. Matei, Y. Meyer, C. R. Acad. Sci. Paris, Ser. I 346 (2008).