Fast Sample-Based Planning for Dynamic Systems by Zero-Control Linearization-Based Steering

Chapter
First Online:
Part of the Springer Proceedings in Advanced Robotics book series (SPAR, volume 3)

Abstract

We propose linearizing about zero-control trajectories of the dynamics and cost instead of linearizing about a single point for steering and distance computations in RRT-like motion planning for highly dynamic systems. Formulated as a time-varying Linear Quadratic Regulator problem, the proposed steering is designed to be efficient and numerically tractable. We describe the computational trade-offs that arise when compared to solving a conventional time-invariant LQR, and provide numerical results for a 3-link inverted pendulum on a cart for a wide range of look-aheads (from hundredths of a second to a second). We find that planning with longer time horizons for the cart-pendulum system requires fewer total vertices, leading to faster exploration than short look-aheads as are customary when linearizing around a single state depending on the density of the obstacles.

This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

This work has been supported by NASA Early Career Faculty fellowship NNX12AQ47GS02 and ARO grant W911NF1410203. This work utilized the Janus supercomputer, which is supported by the National Science Foundation (award number CNS-0821794) and the University of Colorado Boulder. The Janus supercomputer is a joint effort of the University of Colorado Boulder, the University of Colorado Denver and the National Center for Atmospheric Research. Janus is operated by the University of Colorado Boulder

References

  1. 1.
    Anderson, B.D.O., Moore, J.B.: Optimal Control: Linear Quadratic Methods. Dover Publications INC (1990)Google Scholar
  2. 2.
    Glassman, E., Tedrake, R.: A quadratic regulator-based heuristic for rapidly exploring state space. In: IEEE International Conference on Robotics and Automation, pp. 5021–5028 (2010)Google Scholar
  3. 3.
    Goretkin, G., Perez, A., Platt Jr. R., Konidaris, G.: Optimal sampling-based planning for linear-quadratic kinodynamic systems. In: IEEE International Conference on Robotics and Automation, pp. 2429–2436 (2013)Google Scholar
  4. 4.
    Hauser, J.: A projection operator approach to the optimization of trajectory functionals. In: IFAC World Congress (2002)Google Scholar
  5. 5.
    Hauser, J.: On the computation of optimal state transfers with application to the control of quantum spin systems. In: American Control Conference, pp. 2169 – 2174 (2003)Google Scholar
  6. 6.
    Hauser, J., Meyer, D.G.: The trajectory manifold of a nonlinear control system. In: IEEE Conference on Decision and Control, pp. 1034–1039 (1998)Google Scholar
  7. 7.
    Hespanha, J.P.: Linear Systems Theory. Princeton university press, Princeton (2009)Google Scholar
  8. 8.
    Janson, L., Pavone, M.: Fast marching trees: a fast marching sampling-based method for optimal motion planning in many dimensions. In: International Symposium on Robotics Research (2013)Google Scholar
  9. 9.
    Jeon, J.H., Karaman, H., Frazzoli, E.: Anytime computation of time-optimal off-road vehicle maneuvers using the rrt*. In: IEEE Conference on Decision and Control, pp. 3276–3282 (2011)Google Scholar
  10. 10.
    Karaman, S., Frazzoli, E.: Sampling-based algorithms for optimal motion planning. In: The International Journal of Robotics Research, pp. 846–894 (2011)Google Scholar
  11. 11.
    Kunz, T., Stilman, M.: Kinodynamic rrts with fixed time step and best-input extension are not probabilistically complete. In: International Workshop on the Algorithmic Foundations of Robotics (2014)Google Scholar
  12. 12.
    Lavalle, S.M., Kuffner, J.J.: Randomized kinodynamic planning. In: The International Journal of Robotics Research, pp. 378–400 (2001)Google Scholar
  13. 13.
    Li, Y., Littlefield, Z., Bekris, K.E.: Sparse methods for efficient asymptotically optimal kinodynamic planning. In: Workshop on the Algorithmic Foundations of Robotics (2014)Google Scholar
  14. 14.
    Littlefield, Z., Li, Y., Bekris, K.E.: Efficient sampling-based motion planning with asymptotic near-optimality guarantees for systems with dynamics. In: IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 1779–1785 (2013)Google Scholar
  15. 15.
    Perez, A., Platt, R., Konidaris, G., Kaelbling, L., Lozano-Perez, T.: LQR-RRT*: optimal sampling-based motion planning with automatically derived extension heuristics. In: IEEE International Conference on Robotics and Automation, pp. 2537–2542 (2012)Google Scholar
  16. 16.
    Schmerling, E., Janson, L., Pavone, M.: Optimal sampling-based motion planning under differential constraints: the drift case with linear affine dynamics. In: IEEE Conference on Decision and Control, pp. 2574–2581 (2015)Google Scholar
  17. 17.
    Webb, D.J., van den Berg, J.: Kinodynamic RRT*: asymptotically optimal motion planning for robots with linear dynamics. In: IEEE International Conference on Robotics and Automation, pp. 5054–5061 (2013)Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.University of ColoradoBoulderUSA

Personalised recommendations