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Provably good approximation algorithms for optimal kinodynamic planning: Robots with decoupled dynamics bounds

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Abstract

We consider the following problem: given a robot system, find a minimal-time trajectory that goes from a start state to a goal state while avoiding obstacles by a speed-dependent safety margin and respecting dynamics bounds. In [1] we developed a provably good approximation algorithm for the minimum-time trajectory problem for a robot system with decoupled dynamics bounds (e.g., a point robot in ℝ3). This algorithm differs from previous work in three ways. It is possible (1) to bound the goodness of the approximation by an error termɛ; (2) to bound the computational complexity of our algorithm polynomially; and (3) to express the complexity as a polynomial function of the error term. Hence, given the geometric obstacles, dynamics bounds, and the error termɛ, the algorithm returns a solution that isɛ-close to optimal and requires only a polynomial (in (1/ɛ)) amount of time.

We extend the results of [1] in two ways. First, we modify it to halve the exponent in the polynomial bounds from 6d to 3d, so that the new algorithm isO(c d N 1/ɛ)3d), whereN is the geometric complexity of the obstacles andc is a robot-dependent constant. Second, the new algorithm finds a trajectory that matches the optimal in time with anɛ factor sacrificed in the obstacle-avoidance safety margin. Similar results hold for polyhedral Cartesian manipulators in polyhedral environments.

The new results indicate that an implementation of the algorithm could be reasonable, and a preliminary implementation has been done for the planar case.

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Authors and Affiliations

  1. Department of Computer Science, Cornell University, 14853-7501, Ithaca, NY, USA

    B. R. Donald

  2. Sandia National Laboratories, 87185-0951, Albuquerque, NM, USA

    P. Xavier

Authors
  1. B. R. Donald
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  2. P. Xavier
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Additional information

Communicated by J.-D. Boissonnat.

This paper describes research done at the Computer Science Robotics Laboratory at Cornell University. Support for our robotics research there is provided in part by the National Science Foundation under Grant Nos. IRI-8802390 and IRI-9000532, by a Presidential Young Investigator award, and in part by the Mathematical Sciences Institute, Intel Corporation, and AT&T Bell Laboratories.

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Donald, B.R., Xavier, P. Provably good approximation algorithms for optimal kinodynamic planning: Robots with decoupled dynamics bounds. Algorithmica 14, 443–479 (1995). https://doi.org/10.1007/BF01586636

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