Abstract
In this paper we consider the problem of generating motion plans for a nonlinear dynamical system that are guaranteed to succeed despite uncertainty in the environment, parametric model uncertainty, disturbances, and/or errors in state estimation. Furthermore, we consider the case where these plans must be generated online, because constraints such as obstacles in the environment may not be known until they are perceived (with a noisy sensor) at runtime. Previous work on feedback motion planning for nonlinear systems was limited to offline planning due to the computational cost of safety verification. Here we take a trajectory library approach by designing controllers that stabilize the nominal trajectories in the library and precomputing regions of finite time invariance (”funnels”) for the resulting closed loop system. We leverage sums-of-squares programming in order to efficiently compute funnels which take into account bounded disturbances and uncertainty. The resulting funnel library is then used to sequentially compose motion plans at runtime while ensuring the safety of the robot. A major advantage of the work presented here is that by explicitly taking into account the effect of uncertainty, the robot can evaluate motion plans based on how vulnerable they are to disturbances.We demonstrate our method on a simulation of a plane flying through a two dimensional forest of polygonal trees with parametric uncertainty and disturbances in the form of a bounded ”cross-wind”.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Betts, J.T.: Practical Methods for Optimal Control Using Nonlinear Programming. SIAM Advances in Design and Control. Society for Industrial and Applied Mathematics (2001)
Burridge, R.R., Rizzi, A.A., Koditschek, D.E.: Sequential composition of dynamically dexterous robot behaviors. International Journal of Robotics Research 18(6), 534–555 (1999)
Camacho, E.F., Bordons, C.: Model Predictive Control, 2nd edn. Springer (2004)
Dey, D., Liu, T.Y., Sofman, B., Bagnell, D.: Efficient optimization of control libraries. Technical report, Technical Report, CMU-RI-TR-11-20 (2011)
Frazzoli, E., Dahleh, M., Feron, E.: Maneuver-based motion planning for nonlinear systems with symmetries. IEEE Transactions on Robotics 21(6), 1077–1091 (2005)
Green, C., Kelly, A.: Toward optimal sampling in the space of paths. In: 13th International Symposium of Robotics Research (November 2007)
Green, M., Limebeer, D.: Linear Robust Control. Prentice Hall (1995)
Hu, T.C., Kahng, A.B., Robins, G.: Optimal robust path planning in general environments. IEEE Transactions on Robotics and Automation 9(6), 775–784 (1993)
Jacobs, P., Canny, J.: Robust motion planning for mobile robots. In: Proceedings of the 1990 IEEE International Conference on Robotics and Automation, pp. 2–7. IEEE (1990)
Karaman, Frazzoli: High-speed flight through an ergodic forest. In: IEEE Conference on Robotics and Automation (submitted 2012)
Karaman, S., Frazzoli, E.: Sampling-based algorithms for optimal motion planning. Int. Journal of Robotics Research 30, 846–894 (2011)
Kuffner, J., Lavalle, S.: RRT-connect: An efficient approach to single-query path planning. In: Proceedings of the IEEE International Conference on Robotics and Automation (ICRA), pp. 995–1001 (2000)
Liu, C., Atkeson, C.G.: Standing balance control using a trajectory library. In: IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2009), pp. 3031–3036. IEEE (2009)
Majumdar, A., Tobenkin, M., Tedrake, R.: Algebraic verification for parameterized motion planning libraries. In: Proceedings of the 2012 American Control Conference (ACC) (2012)
Mayne, D.Q., Seron, M.M., Rakovic, S.V.: Robust model predictive control of constrained linear systems with bounded disturbances. Automatica 41(2), 219–224 (2005)
Mellinger, D., Kumar, V.: Minimum snap trajectory generation and control for quadrotors. In: 2011 IEEE International Conference on Robotics and Automation (ICRA), pp. 2520–2525. IEEE (2011)
Nesterov, Y., Nemirovskii, A.: Interior-Point Polynomial Algorithms in Convex Programming. Studies in Applied Mathematics. Society for Industrial Mathematics (1995)
Parrilo, P.A.: Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization. PhD thesis, California Institute of Technology (May 18, 2000)
Petersen, I.R., Ugrinovskii, V.A., Savkin, A.V.: Robust control design using H-8 methods. Communications and control engineering series. Springer (2000)
Schouwenaars, T., Mettler, B., Feron, E., How, J.P.: Robust motion planning using a maneuver automation with built-in uncertainties. In: Proceedings of the 2003 American Control Conference, vol. 3, pp. 2211–2216. IEEE (2003)
Sermanet, P., Scoffier, M., Crudele, C., Muller, U., LeCun, Y.: Learning maneuver dictionaries for ground robot planning. In: Proc. 39th International Symposium on Robotics (ISR 2008) (2008)
Shkolnik, A.: Sample-Based Motion Planning in High-Dimensional and Differentially-Constrained Systems. PhD thesis, MIT (February 2010)
Steinhardt, J., Tedrake, R.: Finite-time regional verification of stochastic nonlinear systems. In: Proceedings of Robotics: Science and Systems (RSS 2011), January 17 (2011)
Tedrake, R., Manchester, I.R., Tobenkin, M.M., Roberts, J.W.: LQR-Trees: Feedback motion planning via sums of squares verification. International Journal of Robotics Research 29, 1038–1052 (2010)
Tobenkin, M.M., Manchester, I.R., Tedrake, R.: Invariant funnels around trajectories using sum-of-squares programming. In: Proceedings of the 18th IFAC World Congress (2011), Extended Version Available Online: arXiv:1010.3013 [math.DS]
Topcu, U., Packard, A.K., Seiler, P., Balas, G.J.: Robust region-of-attraction estimation. IEEE Transactions on Automatic Control 55(1), 137–142 (2010)
Toussaint, G.J.: Robust control and motion planning for nonlinear underactuated systems using H-infinity Techniques. PhD thesis, University of Illinois at Urbana-Champaign, Urbana, IL (2000)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Majumdar, A., Tedrake, R. (2013). Robust Online Motion Planning with Regions of Finite Time Invariance. In: Frazzoli, E., Lozano-Perez, T., Roy, N., Rus, D. (eds) Algorithmic Foundations of Robotics X. Springer Tracts in Advanced Robotics, vol 86. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36279-8_33
Download citation
DOI: https://doi.org/10.1007/978-3-642-36279-8_33
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-36278-1
Online ISBN: 978-3-642-36279-8
eBook Packages: EngineeringEngineering (R0)