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 nearoptimality 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.
Many exciting robotic applications require multiple robots with many degrees of freedom, such as manipulators, to coordinate their motion in a shared workspace. Discovering high-quality paths in such scenarios can be achieved, in principle, by exploring the composite space of all robots. Sampling-based planners do so by building a roadmap or a tree data structure in the corresponding configuration space and can achieve asymptotic optimality. The hardness of motion planning, however, renders the explicit construction of such structures in the composite space of multiple robots impractical. This work proposes a scalable solution for such coupled multi-robot problems, which provides desirable path-quality guarantees and is also computationally efficient. In particular, the proposed dRRT * is an informed, asymptotically-optimal extension of a prior sampling-based multi-robot motion planner, dRRT. The prior approach introduced the idea of building roadmaps for each robot and implicitly searching the tensor product of these structures in the composite space. This work identifies the conditions for convergence to optimal paths in multi-robot problems, which the prior method was not achieving.A. Dobson, R. Shome and K.
Discovering high-quality paths for multirobot problems can be achieved, in principle, through asymptotically-optimal data structures in the composite space of all robots, such as a sampling-based roadmap or a tree. The hardness of motion planning, however, which depends exponentially on the number of robots, renders the explicit construction of such structures impractical. This work proposes a scalable, sampling-based planner for coupled multi-robot problems that provides desirable pathquality guarantees. The proposed dRRT * is an informed, asymptotically-optimal extension of a prior method dRRT, which introduced the idea of building roadmaps for each robot and implicitly searching the tensor product of these structures in the composite space. The paper describes the conditions for convergence to optimal paths in multirobot problems. Moreover, simulated experiments indicate dRRT * converges to high-quality paths and scales to higher numbers of robots where various alternatives fail. It can also be used on high-dimensional challenges, such as planning for robot manipulators.
We present a framework for deformable object manipulation that interleaves planning and control, enabling complex manipulation tasks without relying on high-fidelity modeling or simulation. The key question we address is when should we use planning and when should we use control to achieve the task? Planners are designed to find paths through complex configuration spaces, but for highly underactuated systems, such as deformable objects, achieving a specific configuration is very difficult even with high-fidelity models. Conversely, controllers can be designed to achieve specific configurations, but they can be trapped in undesirable local minima owing to obstacles. Our approach consists of three components: (1) a global motion planner to generate gross motion of the deformable object; (2) a local controller for refinement of the configuration of the deformable object; and (3) a novel deadlock prediction algorithm to determine when to use planning versus control. By separating planning from control we are able to use different representations of the deformable object, reducing overall complexity and enabling efficient computation of motion. We provide a detailed proof of probabilistic completeness for our planner, which is valid despite the fact that our system is underactuated and we do not have a steering function. We then demonstrate that our framework is able to successfully perform several manipulation tasks with rope and cloth in simulation, which cannot be performed using either our controller or planner alone. These experiments suggest that our planner can generate paths efficiently, taking under a second on average to find a feasible path in three out of four scenarios. We also show that our framework is effective on a 16-degree-of-freedom physical robot, where reachability and dual-arm constraints make the planning more difficult.
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