Abstract
In this work, the energy-optimal motion planning problem for planar robot manipulators with two revolute joints is studied, in which the end-effector of the robot manipulator is constrained to pass through a set of waypoints, whose sequence is not predefined. This multi-goal motion planning problem has been solved as a mixed-integer optimal control problem in which, given the dynamic model of the robot manipulator, the initial and final configurations of the robot, and a set of waypoints inside the workspace of the manipulator, one has to find the control inputs, the sequence of waypoints with the corresponding passage times, and the resulting trajectory of the robot that minimizes the energy consumption during the motion. The presence of the waypoint constraints makes this optimal control problem particularly difficult to solve. The mixed-integer optimal control problem has been converted into a mixed-integer nonlinear programming problem first making the unknown passage times through the waypoints part of the state, then introducing binary variables to enforce the constraint of passing once through each waypoint, and finally applying a fifth-degree Gauss–Lobatto direct collocation method to tackle the dynamic constraints. High-degree interpolation polynomials allow the number of variables of the problem to be reduced for a given numerical precision. The resulting mixed-integer nonlinear programming problem has been solved using a nonlinear programming-based branch-and-bound algorithm specifically tailored to the problem. The results of the numerical experiments have shown the effectiveness of the approach.
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Choset, H., Lynch, K.M., Hutchinson, S., Kantor, G., Burgard, W., Kavraki, L.E., Thrun, S.: Principles of Robot Motion: Theory, Algorithms, and Implementations. The MIT Press, Boston (2005)
La Valle, S.: Planning Algorithms. Cambridge University Press, Cambridge (2006)
Siciliano, B., Khatib, O. (eds.): Handbook of Robotics. Springer, Berlin (2008)
Sager, S.: Numerical methods for mixed-integer optimal control problems. Ph.D.Thesis, Universität Heidelberg (2006)
Siciliano, B., Sciavicco, L., Villani, L., Oriolo, G.: Robotics: Modelling, Planning and Control. Springer, Berlin (2009)
Sager, S.: Reformulations and algorithms for the optimization of switching decisions in nonlinear optimal control. J. Process Control 19(8), 1238–1247 (2009)
Gerdts, M.: Solving mixed-integer optimal control problems by branch & bound: a case study from automobile test-driving with gear shift. Optim. Control Appl. Methods 26(1), 1–18 (2005)
Gerdts, M.: A variable time transformation method for mixed-integer optimal control problems. Optim. Control Appl. Methods 27(3), 169–182 (2006)
Buss, M., Glocker, M., Hardt, M., von Stryk, O., Bulirsch, R., Schmidt, G.: Nonlinear hybrid dynamical systems: Modeling, optimal control, and applications. In: S. Engell, G. Frehse, E. Schnieder (eds.) Modelling, Analysis and Design of Hybrid Systems, Lecture Notes in Control and Information Science, vol. 279, pp. 331–335. Springer, Berlin (2002)
Loxton, R., Teo, K., Rehbock, V., Ling, W.: Optimal switching instants for a switched-capacitor DC/DC power converter. Automatica 45(4), 973–980 (2009)
Xu, X., Antsaklis, P.J.: Optimal control of switched systems via nonlinear optimization based on direct differentiations of value functions. Int. J. Control 75(16/17), 1406–1426 (2002)
Xu, X., Antsaklis, P.J.: Optimal control of switched systems based on parameterization of the switching instants. IEEE Trans. Autom. Control 49(1), 2–16 (2004)
Shaikh, M., Caines, P.: On the hybrid optimal control problem: theory and algorithms. IEEE Trans. Autom. Control 52(9), 1587–1603 (2007)
Sussmann, H.J.: Maximum principle for hybrid optimal control problems. In: Proceedings of the IEEE Conference on Decision and Control (1999)
Borrelli, F., Baotic, M., Bemporad, A., Morari, M.: Dynamic programming for constrained optimal control of discrete-time linear hybrid systems. Automatica 41(10), 1709–1721 (2005)
Hedlund, S., Rantzer, A.: Convex dynamic programming for hybrid systems. IEEE Trans. Autom. Control 47(9), 1536–1540 (2002)
Gapaillard, M.: Continuous representation and control of hybrid systems. Int. J. Control 81(1), 1–20 (2008)
Bengea, S., DeCarlo, R.: Optimal control of switching systems. Automatica 41(1), 11–27 (2005)
Wei, S., Uthaichana, K., Žefran, M., DeCarlo, R., Bengea, S.: Applications of numerical optimal control to nonlinear hybrid systems. Nonlinear Anal. 1(2), 264–279 (2007)
Axelsson, H., Wardi, Y., Egerstedt, M., Verriest, E.: Gradient descent approach to optimal mode scheduling in hybrid dynamical systems. J. Optim. Theory Appl. 136(2), 167–186 (2008)
Bemporad, A., Morari, M.: Control of systems integrating logic, dynamics, and constraints. Automatica 35(3), 407–427 (1999)
Herman, A.L., Conway, B.A.: Direct optimization using collocation based on high-order Gauss–Lobatto quadrature rules. J. Guid. Control Dyn. 19(3), 592–599 (1996)
Hargraves, C.R., Paris, S.W.: Direct trajectory optimization using nonlinear programming and collocation. J. Guid. Control Dyn. 10(4), 338–342 (1987)
Bonami, P., Biegler, L.T., Conn, A.R., Cornuéjols, G., Grossmann, I.E., Laird, C.D., Lee, J., Lodi, A., Margot, F., Sawaya, N., Wächter, A.: An algorithmic framework for convex mixed integer nonlinear programs. Discret. Optim. 5(2), 186–204 (2008)
Wächter, A., Biegler, L.T.: On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming. Math. Progr. 106(1), 25–57 (2006)
Soler, M., Olivares, A., Staffetti, E.: Hybrid optimal control approach to commercial aircraft trajectory optimization. J. Guid. Control Dyn. 33(3), 985–991 (2010)
Sahinidis, N.V.: BARON: A general purpose global optimization software package. J. Glob. Optim. 8, 201–205 (1996)
Tawarmalani, M., Sahinidis, N.V.: Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Algorithms, Software, and Applications. Kluwer Academic Publishers, Boston (2002)
Belotti, P., Lee, J., Liberti, L., Margot, F., Waechter, A.: Branching and bounds tightening techniques for non-convex MINLP. Optim. Methods Softw. 24, 597 (2009)
Dakin, R.J.: A tree search algorithm for mixed programming problems. Comput. J. 8, 250–255 (1965)
Land, A.H., Doig, A.G.: An automatic method for solving discrete programming problems. Econometrica 28, 497–520 (1960)
Leyffer, S.: User Manual for MINLP-BB. University of Dundee, Dundee (1998)
Bussieck, M.R., Drud, A.: SBB: A new solver for mixed integer nonlinear programming. Talk, OR 2001, Section “Continuous Optimization” (2001)
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Communicated by Felix L. Chernousko.
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Bonami, P., Olivares, A. & Staffetti, E. Energy-Optimal Multi-Goal Motion Planning for Planar Robot Manipulators. J Optim Theory Appl 163, 80–104 (2014). https://doi.org/10.1007/s10957-013-0516-0
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DOI: https://doi.org/10.1007/s10957-013-0516-0
Keywords
- Multi-goal motion planning
- Robot manipulators
- Mixed-integer optimal control problem
- Mixed-integer nonlinear programming