We present methods for offline generation of sparse roadmap spanners that result in graphs 79 smaller than existing approaches while returning solutions of equivalent path quality. Our method uses a hybrid approach to sampling that combines traditional graph discretization with random sampling. We present techniques that optimize the graph for the L1-norm metric function commonly used in joint-based robotic planning, purposefully choosing a @math -stretch factor based on the geometry of the space, and removing redundant edges that do not contribute to the graph quality. A high-quality pre-processed sparse roadmap is then available for re-use across many different planning scenarios using standard repair and re-plan methods. Pre-computing the roadmap offline results in more deterministic solutions, reduces the memory requirements by affording complex rejection criteria, and increases the speed of planning in high-dimensional spaces allowing more complex problems to be solved such as multi-modal task planning. Our method is validated through simulated benchmarks against the SPARS2 algorithm. The source code is freely available online as an open source extension to OMPL.
Sparse roadmap spanners, instead, have recently been proven to provide asymptotically near-optimal guarantees within a @math -stretch factor. For example, if the @math -stretch factor is 1.1, then the maximum length a path can be from its optimal solution is 10 In order to have the same asymptotic optimality guarantees as PRM* within a @math -stretch factor, a number of checks are required to determine which potential vertices and edges should be saved to have coverage across a robot's free space. Only configurations that are useful for 1) coverage, 2) connectivity, or 3) improving the quality of paths on the sparse roadmap relative to the optimal paths in the c-space. Two parameters @math and the sparse delta factor @math control the sparsity of the graph. For more background on these criteria the reader is encouraged to reference @cite_2 .
@cite_2
2108232656 2072127296 1986004895 2122724557
Asymptotically optimal planners, such as PRM*, guarantee that solutions approach optimal as the number of iterations increases. Roadmaps with this property, however, may grow too large for storing on resource-constrained robots and for achieving efficient online query resolution. By relaxing optimality, asymptotically near-optimal planners produce sparser graphs by not including all edges. The idea stems from graph spanners, which produce sparse subgraphs that guarantee near-optimality. Existing asymptotically optimal and near-optimal planners, however, include all sampled configurations as roadmap nodes, meaning only infinite-size graphs have the desired properties. To address this limitation, this work describes SPARS, an algorithm that returns a sparse roadmap spanner. The method provides the following properties: (a) probabilistic completeness, (b) asymptotic near-optimality and (c) the probability of adding nodes to the spanner converges to zero as iterations increase. The last point suggests that finite-size data structures with asymptotic near-optimality in continuous spaces may indeed exist. The approach builds simultaneously a dense graph similar to PRM* and its roadmap spanner, meaning that upon construction an infinite-size graph is still needed asymptotically. An extension of SPARS is also presented, termed SPARS2, which removes the dependency on building a dense graph for constructing the sparse roadmap spanner and for which it is shown that the same desirable properties hold. Simulations for rigid-body motion planning show that algorithms for constructing sparse roadmap spanners indeed provide small data structures and result in faster query resolution. The rate of node addition is shown to decrease over time and practically the quality of solutions is considerably better than the theoretical bounds. Upon construction, the memory requirements of SPARS2 are significantly smaller but there is a small sacrifice in the size of the final spanner relative to SPARS. Roadmap spanners provide a way to acquire sparse data structures that efficiently answer motion planning queries with probabilistic completeness and asymptotic near-optimality. The current SPARS method provides these properties by building two graphs in parallel: a dense asymptotically-optimal roadmap based on PRM* and its spanner. This paper shows that it is possible to relax the conditions under which a sample is added to the spanner and provide guarantees, while not requiring the use of a dense graph. A key aspect of SPARS is that the probability of adding nodes to the roadmap goes to zero as iterations increase, which is maintained in the proposed extension. The paper describes the new algorithm, argues its theoretical properties and evaluates it against PRM* and the original SPARS algorithm. The experimental results show that the memory requirements of the method upon construction are dramatically reduced, while returning competitive quality paths with PRM*. There is a small sacrifice in the size of the final spanner relative to SPARS but the new method still returns graphs orders of magnitudes smaller than PRM*, leading to very efficient online query resolution. Sampling-based algorithms have proven practical in solving motion planning challenges in relatively high-dimensional instances in geometrically complex workspaces. Early work focused on quickly returning feasible solutions. Only recently was it shown under which conditions these algorithms asymptotically return optimal or near-optimal solutions. These methods yield desired properties only in an asymptotic fashion, i.e., the properties are attained after infinite computation time. This work studies the finite-time properties of sampling-based planners in terms of path quality. The focus is on roadmap-based methods, due to their simplicity. This work illustrates that existing sampling-based planners which construct roadmaps in an asymptotically (near-)optimal manner exhibit a “probably near-optimal” property in finite time. This means that it is possible to compute a confidence value, i.e. a probability, regarding the existence of upper bounds for the length of the path returned by the roadmap as a function of the number of configuration space samples. This property can result in useful tools for determining existence of solutions and a probabilistic stopping criterion for PRM-like methods. These properties are validated through experimental trials. 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.