This paper presents an equivalence between feasible kinodynamic planning and optimal kinodynamic planning, in that any optimal planning problem can be transformed into a series of feasible planning problems in a state-cost space whose solutions approach the optimum. This transformation gives rise to a meta-algorithm that produces an asymptotically optimal planner, given any feasible kinodynamic planner as a subroutine. The meta-algorithm is proven to be asymptotically optimal, and a formula is derived relating expected running time and solution suboptimality. It is directly applicable to a wide range of optimal planning problems because it does not resort to the use of steering functions or numerical boundary-value problem solvers. On a set of benchmark problems, it is demonstrated to perform, using the EST and RRT algorithms as subroutines, at a superior or comparable level to related planners.
The prior work with closest relation to ours in terms of generality of applicability is the Sparse-Stable-RRT planner @cite_23 . Like our work, it avoids the use of a steering function entirely and samples directly in control space. Approximate rewiring is performed by allowing connections to points that are near enough'' according to a state-space distance metric. This scheme was proven to satisfy asymptotic near-optimality, which is the property of converging toward a path with bounded suboptimality @cite_10 . In a more recent paper, the same authors have extended it to an asymptotically-optimal planner, SST*, by progressively shrinking nearness threshold parameters @cite_12 . However, Sparse-Stable-RRT and SST* and have many parameters to tune, and our experiments suggest that AO- @math planners in general outperform both planners.
@cite_10 @cite_12 @cite_23
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Recent motion planners, such as RRT*, that achieve asymptotic optimality require a local planner, which connects two states with a trajectory. For systems with dynamics, the local planner corresponds to a two-point boundary value problem (BVP) solver, which is not always available. Furthermore, asymptotically optimal solutions tend to increase computational costs relative to alternatives, such as RRT, that focus on feasibility. This paper describes a sampling-based solution with the following desirable properties: a) it does not require a BVP solver but only uses a forward propagation model, b) it employs a single propagation per iteration similar to RRT, making it very efficient, c) it is asymptotically near-optimal, and d) provides a sparse data structure for answering path queries, which further improves computational performance. Simulations on prototypical dynamical systems show the method is able to improve the quality of feasible solutions over time and that it is computationally efficient. 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. 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. Asymptotically-optimal sampling-based motion planners, like RRT*, perform vast amounts of collision checking, and are hence rather slow to converge in complex problems where collision checking is relatively expensive. This paper presents two novel motion planners, Lazy-PRM* and Lazy-RRG*, that eliminate the majority of collision checks using a lazy strategy. They are sampling-based, any-time, and asymptotically complete algorithms that grow a network of feasible vertices connected by edges. Edges are not immediately checked for collision, but rather are checked only when a better path to the goal is found. This strategy avoids checking the vast majority of edges that have no chance of being on an optimal path. Experiments show that the new methods converge toward the optimum substantially faster than existing planners on rigid body path planning and robot manipulation problems.