Abstract
This paper presents an approach to finding the time-optimal trajectories for a simple rigid-body model of a mobile robot in an obstacle-free plane. Previous work has used Pontryagin’s Principle to find strong necessary conditions on time-optimal trajectories of the rigid body; trajectories satisfying these conditions are called extremal trajectories. The main contribution of this paper is a method for sampling the extremal trajectories sufficiently densely to guarantee that for any pair of start and goal configurations, a trajectory can be found that (provably) approximately reaches the goal, approximately optimally; the quality of the approximation is only limited by the availability of computational resources.
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References
Balkcom, D.J., Kavathekar, P.A., Mason, M.T.: Time-optimal trajectories for an omni-directional vehicle. International Journal of Robotics Research 25(10), 985–999 (2006)
Balkcom, D.J., Mason, M.T.: Time optimal trajectories for differential drive vehicles. International Journal of Robotics Research 21(3), 199–217 (2002)
Barraquand, J., Latombe, J.-C.: Nonholonomic multibody mobile robots: Controllability and motion planning in the presence of obstacles. In: IEEE International Conference on Robotics and Automation, Sacramento, CA, pp. 2328–2335 (1991)
Dubins, L.E.: On curves of minimal length with a constraint on average curvature and with prescribed initial and terminal positions and tangents. American Journal of Mathematics 79, 497–516 (1957)
Fernandes, C., Gurvits, L., Li, Z.X.: A variational approach to optimal nonholonomic motion planning. In: IEEE International Conference on Robotics and Automation, pp. 680–685 (April 1991)
Furtuna, A.A.: Minimum time kinematic trajectories for self-propelled rigid bodies in the unobstructed plane. Dartmouth PhD thesis, Dartmouth College (2004)
Furtuna, A.A., Balkcom, D.J.: Generalizing Dubins curves: Minimum-time sequences of body-fixed rotations and translations in the plane. International Journal of Robotics Research 29(6), 703–726 (2010)
Furtuna, A.A., Lu, W., Wang, W., Balkcom, D.J.: Minimum-time trajectories for kinematic mobile robots and other planar rigid bodies with finite control sets. In: IROS, pp. 4321–4328 (2011)
Hayet, J.-B., Esteves, C., Murrieta-Cid, R.: A Motion Planner for Maintaining Landmark Visibility with a Differential Drive Robot. In: Chirikjian, G.S., Choset, H., Morales, M., Murphey, T. (eds.) Algorithmic Foundation of Robotics VIII. STAR, vol. 57, pp. 333–347. Springer, Heidelberg (2009)
Kavraki, L.E., Svestka, P., Latombe, J.-C., Overmars, M.H.: Probabilistic roadmaps for path planning in high-dimensional configuration spaces. IEEE Transactions on Robotics and Automation 12(4), 566–580 (1996)
Laumond, J.-P.: Feasible trajectories for mobile robots with kinematic and environment constraints. In: International Conference on Intelligent Autonomous Systems, pp. 346–354 (1986)
LaValle, S.M.: Planning Algorithms. Cambridge Press (2006), also freely available online at http://planning.cs.uiuc.edu
Lynch, K.M.: The mechanics of fine manipulation by pushing. In: IEEE International Conference on Robotics and Automation, Nice, France, pp. 2269–2276 (1992)
Oberle, H.J., Grimm, W.: BNDSCO: a program for the numerical solution of optimal control problems (1989)
Ratliff, N., Zucker, M., Andrew (Drew) Bagnell, J., Srinivasa, S.: CHOMP: Gradient optimization techniques for efficient motion planning. In: IEEE International Conference on Robotics and Automation, ICRA (May 2009)
Reeds, J.A., Shepp, L.A.: Optimal paths for a car that goes both forwards and backwards. Pacific Journal of Mathematics 145(2), 367–393 (1990)
Souères, P., Boissonnat, J.-D.: Optimal Trajectories for Nonholonomic Mobile Robots. In: Laumond, J.-P. (ed.) Robot Motion Planning and Control. LNCIS, vol. 229, pp. 93–170. Springer, Heidelberg (1998)
Sussmann, H., Tang, G.: Shortest paths for the Reeds-Shepp car: a worked out example of the use of geometric techniques in nonlinear optimal control. SYCON 91-10, Department of Mathematics, Rutgers University, New Brunswick, NJ 08903 (1991)
Vendittelli, M., Laumond, J.P., Nissoux, C.: Obstacle distance for car-like robots. IEEE Transactions on Robotics and Automation 15(4), 678–691 (1999)
Wang, W., Balkcom, D.J.: Analytical time-optimal trajectories for an omni-directional vehicle. In: IEEE International Conference on Robotics and Automation (2012)
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Wang, W., Balkcom, D. (2013). Sampling Extremal Trajectories for Planar Rigid Bodies. 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_20
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DOI: https://doi.org/10.1007/978-3-642-36279-8_20
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-36278-1
Online ISBN: 978-3-642-36279-8
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