Abstract
The optimal trajectory with respect to some metric may require very many switches between controls, or even infinitely many, a phenomenon called chattering; this can be problematic for existing motion planning algorithms that plan using a finite set of motion primitives. One remedy is to add some penalty for switching between controls. This paper explores the implications of this switching cost for optimal trajectories, using rigid bodies in the plane (which have been studied extensively in the cost-free-switch model) as an example system. Blatt’s Indifference Principle (BIP) is used to derive necessary conditions on optimal trajectories; Lipschitzian optimization techniques together with an A* search yield an algorithm for finding trajectories that can arbitrarily approximate the optimal trajectories.
This work was supported in part by NSF grant IIS-0643476.
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References
Agarwal, P.K., Biedl, T., Lazard, S., Robbins, S., Suri, S., Whitesides, S.: Curvature-constrained shortest paths in a convex polygon. SIAM J. Comput. 31(6), 1814–1851 (2002)
Barraquand, J., Latombe, J.-C.: Robot motion planning: a distributed representation approach. Int. J. Robot. Res. 10(6), 628–649 (1991)
Blatt, J.M.: Optimal control with a cost of switching control. J. Aust. Math. Soc. 19, 316–332 (1976)
Chitsaz, H.R.: Geodesic problems for mobile robots. Ph.D. thesis, University of Illinois at Urbana-Champaign (2008)
Chitsaz, H.R., La Valle, S.M., Balkcom, D.J., Mason, M.T.: Minimum wheel-rotation paths for differential-drive mobile robots. Int. J. Robot. Res. 28(1), 66–80 (2009)
Chyba, M., Haberkorn, T.: Autonomous underwater vehicles: singular extremals and chattering. In: Ceragioli, F., Dontchev, A., Furuta, H., Marti, K., Pandolfi, L. (eds.) Systems, Control, Modeling and Optimization, vol. 202 of IFIP International Federation for Information Processing, pp. 103–113. Springer, Berlin (2006)
Cockayne, E.J., Hall, G.W.C.: Plane motion of a particle subject to curvature constraints. SIAM J. Control 13(1), 197–220 (1975)
Desaulniers, G.: On shortest paths for a car-like robot maneuvering around obstacles. Robot. Auton. Syst. 17(3), 139–148 (1996)
Dubins, L.E.: On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents. Am. J. Math. 79(3), 497–516 (1957)
Furtuna, A.: Minimum time kinematic trajectories for self-propelled rigid bodies in the unobstructed plane. Ph.D. thesis, Dartmouth College, June 2011
Kibalczyc, K., Walczak, S.: Necessary optimality conditions for a problem with costs of rapid variation of control. J. Aust. Math. Soc. 26, 45–55 (1984)
Loxton, R., Lin, Q., Lay Teo, K.: Minimizing control variation in nonlinear optimal control. Automatica 49(9), 2652–2664 (2013)
Lyu, Y.-H., Furtuna, A., Wang, W., Balkcom. D.: The bench mover’s problem: minimum-time trajectories, with cost for switching between controls. In: IEEE International Conference on Robotics and Automation (2014)
Mason, M.T.: Mechanics of Robotic Manipulation. MIT Press, Cambridge (2001)
Matula, J.: On an extremum problem. J. Aust. Math. Soc. 28, 376–392 (1987)
Noussair, E.S.: On the existence of piecewise continuous optimal controls. J. Aust. Math. Soc. 20, 31–37 (1977)
Pintér, J.D.: Global Optimization in Action: Continuous and Lipschitz Optimization: Algorithms, Implementations and Applications (Nonconvex Optimization and Its Applications, 2nd edn. Springer, Berlin (2010)
Piyavskii, S.A.: An algorithm for finding the absolute minimum of a function. USSR Comput. Math. Math. Phys. 12, 13–24 (1967)
Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: Mathematical Theory of Optimal Processes. Wiley, New York (1962)
Reeds, J.A., Shepp, L.A.: Optimal paths for a car that goes both forwards and backwards. Pac. J. Math. 145(2), 367–393 (1990)
Reister, D.B., Pin, F.G.: Time-optimal trajectories for mobile robots with two independently driven wheels. Int. J. Robot. Res. 13(1), 38–54 (1994)
Renaud, M., Fourquet, J.-Y.: Minimum time motion of a mobile robot with two independent, acceleration-driven wheels. In: IEEE International Conference on Robotics and Automation, vol. 3, pp. 2608–2613, April 1997
Slotine, J.J., Sastry, S.S.: Tracking control of non-linear systems using sliding surfaces with application to robot manipulators. In: American Control Conference 1983, pp. 132–135, June 1983
Souères, P., Boissonnat, J.-D.: Optimal trajectories for nonholonomic mobile robots. In: Laumond, J.-P. (ed.) Robot Motion Planning and Control, pp. 93–170. Springer, Berlin (1998)
Stewart, D.E.: A numerical algorithm for optimal control problems with switching costs. J. Aust. Math. Soc. 34, 212–228 (1992)
Sussmann, H.J., Tang, G.: Shortest paths for the reeds-shepp car: a worked out example of the use of geometric techniques in nonlinear optimal control. Department of Mathematics, Rutgers University, Technical report (1991)
Teo, K.L., Jennings, L.S.: Optimal control with a cost on changing control. J. Optim. Theory Appl. 68(2), 335–357 (1991)
Wang, W., Balkcom, D.: Sampling extremal trajectories for planar rigid bodies. In: Frazzoli, E., Lozano-Perez, T., Roy, N., Rus, D. (eds.) Algorithmic Foundations of Robotics X, vol. 86 of Springer Tracts in Advanced Robotics, pp. 331–347. Springer, Berlin (2013)
Wang, W., Balkcom, D.J.: Analytical time-optimal trajectories for an omni-directional vehicle. In: IEEE International Conference on Robotics and Automation, pp. 4519–4524, May 2012
Yu, C., Lay Teo, K., Tiow Tay, T.: Optimal control with a cost of changing control. In: Australian Control Conference, pp. 20–25, Nov 2013
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Lyu, YH., Balkcom, D. (2015). Optimal Trajectories for Planar Rigid Bodies with Switching Costs. In: Akin, H., Amato, N., Isler, V., van der Stappen, A. (eds) Algorithmic Foundations of Robotics XI. Springer Tracts in Advanced Robotics, vol 107. Springer, Cham. https://doi.org/10.1007/978-3-319-16595-0_22
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