Abstract
For a mechanical system it often arises that its planned motion will need to be corrected either to refine an approximate plan or to deal with disturbances. This paper develops an algorithmic framework allowing for fast and elegant path correction for nonholonomic underactuated systems with Lie group symmetries, which operates without the explicit need for control strategies. These systems occur frequently in robotics, particularly in locomotion, be it ground, underwater, airborne, or surgical domains. Instead of reintegrating an entire trajectory, the method alters small segments of an initial trajectory in a consistent way so as to transform it via symmetry operations. This approach is demonstrated for the cases of a kinematic car and for flexible bevel tip needle steering, showing a prudent and simple, yet computationally tractable, trajectory correction.
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References
Bloch, A.M.: Nonholonomic Mechanics and Control. In: Interdisciplinary Applied Mathematics, vol. 24. Springer, New York (2003)
Byrd, R.H., Nocedal, J., Waltz, R.A.: Knitro: An integrated package for nonlinear optimization. In: Large Scale Nonlinear Optimization, pp. 35–59. Springer, Heidelberg (2006)
Canny, J.F.: The complexity of robot motion planning. MIT Press, Cambridge (1988)
Cheng, P., Frazzoli, E., LaValle, S.M.: Improving the performance of sampling-based motion planning with symmetry-based gap reduction. IEEE Transactions on Robotics 24(2), 488–494 (2008)
Kanayama, Y., Kimura, Y., Miyazaki, F., Noguchi, T.: A stable tracking control method for an autonomous robot. In: Proceedings of the IEEE International Conference on Robotics and Automation, vol. 1, pp. 384–389 (1990)
Kelly, A., Nagy, B.: Reactive nonholonomic trajectory generation via parametric optimal control. The International Journal of Robotics Research 22(7-8), 583–601 (2003)
Kobilarov, M., Desbrun, M., Marsden, J.E., Sukhatme, G.S.: A discrete geometric optimal control framework for systems with symmetries. In: Proceedings of Robotics: Science and Systems, Atlanta, GA, USA (2007)
Koon, W.S., Marsden, J.E.: Optimal control for holonomic and nonholonomic mechanical systems with symmetry and Lagrangian reduction. SIAM Journal on Control and Optimization 35(3), 901–929 (1995)
Lamiraux, F., Bonnafous, D., Lefebvre, O.: Reactive path deformation for nonholonomic mobile robots. IEEE Transactions on Robotics 20(6), 967–977 (2004)
Latombe, J.C.: Robot Motion Planning. Kluwer, Boston (1991)
LaValle, S.M.: Planning Algorithms. Cambridge University Press, Cambridge (2006)
Murray, R.M., Li, Z., Sastry, S.S.: A Mathematical Introduction to Robotic Manipulation. CRC Press, Boca Raton (1994)
Ollero, A., Heredia, G.: Stability analysis of mobile robot path tracking. In: Proceedings of IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 461–466 (1995)
Ostrowski, J.P.: Computing reduced equations for robotic systems with constraintsand symmetries. IEEE Transactions on Robotics and Automation 15(1), 111–123 (1999)
Park, W., Reed, K.B., Chirikjian, G.S.: Estimation of model parameters for steerable needles. In: IEEE International Conference on Robotics and Automation, pp. 3703–3708 (2010)
Shewchuk, J.R.: An introduction to the conjugate gradient method without the agonizing pain. Tech. rep., Carnegie Mellon University Pittsburgh (1994)
Webster III, R.J., Cowan, N.J., Chirikjian, G.S., Okamura, A.M.: Nonholonomic modeling of needle steering. In: Proc. 9th International Symposium on Experimental Robotics (2004)
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Seiler, K., Singh, S.P.N., Durrant-Whyte, H. (2010). Using Lie Group Symmetries for Fast Corrective Motion Planning. In: Hsu, D., Isler, V., Latombe, JC., Lin, M.C. (eds) Algorithmic Foundations of Robotics IX. Springer Tracts in Advanced Robotics, vol 68. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17452-0_3
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DOI: https://doi.org/10.1007/978-3-642-17452-0_3
Publisher Name: Springer, Berlin, Heidelberg
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