On the Power of Manifold Samples in Exploring Configuration Spaces and the Dimensionality of Narrow Passages

We extend our study of Motion Planning via Manifold Samples (MMS), a general algorithmic framework that combines geometric methods for the exact and complete analysis of low-dimensional configuration spaces with sampling-based approaches that are appropriate for higher dimensions. The framework explores the configuration space by taking samples that are entire low-dimensional manifolds of the configuration space capturing its connectivity much better than isolated point samples. The contributions of this paper are as follows: (i) We present a recursive application of MMS in a six-dimensional configuration space, enabling the coordination of two polygonal robots translating and rotating amidst polygonal obstacles. In the adduced experiments for the more demanding test cases MMS clearly outperforms PRM, with over 20-fold speedup in a coordination-tight setting. (ii) A probabilistic completeness proof for the most prevalent case, namely MMS with samples that are affine subspaces. (iii) A closer examination of the test cases reveals that MMS has, in comparison to standard sampling-based algorithms, a significant advantage in scenarios containing high-dimensional narrow passages. This provokes a novel characterization of narrow passages which attempts to capture their dimensionality, an attribute that had been (to a large extent) unattended in previous definitions.Comment: 20 page

Motion Planning via Manifold Samples

We present a general and modular algorithmic framework for path planning of robots. Our framework combines geometric methods for exact and complete analysis of low-dimensional configuration spaces, together with practical, considerably simpler sampling-based approaches that are appropriate for higher dimensions. In order to facilitate the transfer of advanced geometric algorithms into practical use, we suggest taking samples that are entire low-dimensional manifolds of the configuration space that capture the connectivity of the configuration space much better than isolated point samples. Geometric algorithms for analysis of low-dimensional manifolds then provide powerful primitive operations. The modular design of the framework enables independent optimization of each modular component. Indeed, we have developed, implemented and optimized a primitive operation for complete and exact combinatorial analysis of a certain set of manifolds, using arrangements of curves of rational functions and concepts of generic programming. This in turn enabled us to implement our framework for the concrete case of a polygonal robot translating and rotating amidst polygonal obstacles. We demonstrate that the integration of several carefully engineered components leads to significant speedup over the popular PRM sampling-based algorithm, which represents the more simplistic approach that is prevalent in practice. We foresee possible extensions of our framework to solving high-dimensional problems beyond motion planning.Comment: 18 page

Efficient Path Planning in Narrow Passages via Closed-Form Minkowski Operations

Path planning has long been one of the major research areas in robotics, with PRM and RRT being two of the most effective classes of path planners. Though generally very efficient, these sampling-based planners can become computationally expensive in the important case of "narrow passages". This paper develops a path planning paradigm specifically formulated for narrow passage problems. The core is based on planning for rigid-body robots encapsulated by unions of ellipsoids. The environmental features are enclosed geometrically using convex differentiable surfaces (e.g., superquadrics). The main benefit of doing this is that configuration-space obstacles can be parameterized explicitly in closed form, thereby allowing prior knowledge to be used to avoid sampling infeasible configurations. Then, by characterizing a tight volume bound for multiple ellipsoids, robot transitions involving rotations are guaranteed to be collision-free without traditional collision detection. Furthermore, combining the stochastic sampling strategy, the proposed planning framework can be extended to solving higher dimensional problems in which the robot has a moving base and articulated appendages. Benchmark results show that, remarkably, the proposed framework outperforms the popular sampling-based planners in terms of computational time and success rate in finding a path through narrow corridors and in higher dimensional configuration spaces

Multi-robot Caravanning

We study multi-robot caravanning, which is loosely defined as the problem of a heterogeneous team of robots visiting specific areas of an environment (waypoints) as a group. After formally defining this problem, we propose a novel solution that requires minimal communication and scales with the number of waypoints and robots. Our approach restricts explicit communication and coordination to occur only when robots reach waypoints, and relies on implicit coordination when moving between a given pair of waypoints. At the heart of our algorithm is the use of leader election to efficiently exploit the unique environmental knowledge available to each robot in order to plan paths for the group, which makes it general enough to work with robots that have heterogeneous representations of the environment. We implement our approach both in simulation and on a physical platform, and characterize the performance of the approach under various scenarios. We demonstrate that our approach can successfully be used to combine the planning capabilities of different agents

Sampling Based Motion Planning with Reachable Volumes

Motion planning for constrained systems is a version of the motion planning problem in which the motion of a robot is limited by constraints. For example, one can require that a humanoid robot such as a PR2 remain upright by constraining its torso to be above its base or require that an object such as a bucket of water remain upright by constraining the vertices of the object to be parallel to the robot’s base. Grasping can be modeled by requiring that the end effectors of the robot be located at specified handle positions. Constraints might require that the robot remain in contact with a surface, or that certain joints of the robot remain in contact with each other (e.g., closed chains). Such problems are particularly difficult because the constraints form a manifold in C-space, and planning must be restricted to this manifold. High degree of freedom motion planning and motion planning for constrained systems has applications in parallel robotics, grasping and manipulation, computational biology and molecular simulations, and animation. In this work, we introduce a new concept, reachable volumes, that are a geometric representation of the regions the joints and end effectors of a robot can reach, and use it to define a new planning space, called RV-space, where all points automatically satisfy a problem’s constraints. Visualizations of reachable volumes can enable operators to see the regions of workspace that different parts of the robot can reach. Samples and paths generated in RV-space naturally conform to constraints, making planning for constrained systems no more difficult than planning for unconstrained systems. Consequently, constrained motion planning problems that were previously difficult or unsolvable become manageable and in many cases trivial. We provide tools and techniques to extend the state of the art sampling based motion planning algorithms to RV-space. We define a reachable volume sampler, a reachable volume local planner and a reachable volume distance metric. We showcase the effectiveness of RV-space by applying these tools to motion planning problems for robots with constraints on the end effectors and/or internal joints of the robot. We show that RV-based planners are more efficient than existing methods, particularly for higher dimensional problems, solving problems with 1000+ degrees of freedom for multi-loop, and tree-like linkages