Abstract
A general approach for generating optimal movements of actuatedmulti-jointed systems is presented. The method is based on theimplementation of the Pontryagin Maximum Principle (PMP) used as amathematical optimization tool. It applies to mechanical systems withkinematic tree-like topology such as serial robots, walking machines,and articulated biosystems. Emphasis is put on the choice of anappropriate dynamic model of the multibody system, together with thechoice of relevant performance criteria to be minimized for generatingthe optimal motion. It is shown that the Hamiltonian formalism isperfectly suitable to deal with the optimization problem using the PMP.On the other hand, prominence is given to performance criteria ensuringsoft and efficient functioning of the articulated systems. Two computingtechniques for solving the optimization problem are presented. Threenumerical simulations demonstrate the applicability of the method.
Similar content being viewed by others
References
Bobrow, J.E., Dubowsky, S. and Gibson, J.S., 'On the optimal control of robotic manipulators with actuator constraints', in Proceedings of the American Control Conference, San Francisco, CA, 1983, 782-787.
Bobrow, J.E., Dubowsky S. and Gibson, J.S., 'Time optimal control of robotic manipulators along specified paths', International Journal of Robotic Research 4, 1985, 3-17.
Shiller, Z. and Dubowsky, S., 'On the optimal control of robotic manipulators with actuator and end-effector constraints', in Proceedings IEEE International Conference on Robotics and Automation, IEEE, New York, 1985, 614-620.
Shiller, Z. and Dubowsky, S., 'Robot path planning with obstacles, actuator, gripper and payload constraints', International Journal of Robotics Research 8, 1989, 3-18.
Shin, K.G. and McKay, N.D., 'Minimum-time control of robotic manipulators with geometric path constraints', IEEE Transactions on Automatic Control 30(6), 1985, 531-541.
Shin, K.G. and McKay, N.D., 'Minimum cost trajectory planning for industrial robots', Control and Dynamic Systems 9, 1991, 345-403.
Pfeiffer, F. and Johanni, R., 'A concept for manipulator trajectory planning', in Proceedings IEEE International Conference on Robotics and Automation, San Francisco, CA, April 7-10, IEEE, New York, 1986, 1139-1405.
Slotine, J.J.E. and Yang, H.S., 'Improving the efficiency of time optimal path following algorithms', IEEE Transactions on Robotics and Automation 5, 1989, 118-124.
Shiller, Z. and Dubowsky, S., 'On computing the global time-optimal motions of robotic manipulators in the presence of obstacles', IEEE Transactions on Robotics and Automation 7, 1991, 785-797.
Shiller, Z., 'Optimal robot motion planning and work-cell layout design', Robotica 15, 1997, 31-40.
Beletskii, V.V. and Chudinov, P.S., 'Parametric optimization in the problem of biped locomotion', Izvestiya Akademii Nauk SSSR Mekhanika Tverdogo Tela 12, 1977, 22-31.
Channon, P.H., Hopkins, S.H. and Pham, D.T., 'Derivation of optimal walking motions for a bipedal walking robot', Robotica 10, 1992, 165-172.
Chevallereau, C. and Aoustin, Y., 'Optimal reference trajectories for walking and running of a biped robot', Robotica 19, 2001, 557-569.
Schmitt, D., Soni, A.H., Srinivasan, V. and Naganathan, G., 'Optimal motion planning of robot manipulators', Journal of Mechanisms, Transmissions, and Automation in Design 107, 1985, 239-244.
Yamamoto, M., Ozaki, H. and Mohri, A. 'Planning of manipulator joint trajectories by an iterative method', Robotica 6, 1988, 101-105.
Martin, B.J. and Bobrow, J.E., 'Minimum effort motions for open chains manipulators with task-dependent end-effector constraints', International Journal of Robotics Research 18, 1999, 213-324.
Wang, C.-Y.E., Bobrow, J.E. and Reinkensmeyer, D. J., 'Swinging fromthe hip: Use of dynamic motion optimization in the design of robotic gait rehabilitation', in Proceedings of the IEEE International Conference on Robotics and Automation, Seoul, Korea, May 21-26, IEEE, New York, 2001, 1433-1438.
Kahn, M.E. and Roth, B., 'The near-minimum-time control of open-loop articulated kinematic chains', ASME Journal of Dynamic Systems, Measurement, and Control, September 1971, 164-172.
Weinreb, A. and Bryson, A.E., 'Optimal control of systems with hard control bounds', IEEE Transactions on Automatic Control 30(11), 1985, 1135-1138.
Geering, H.P., Guzella, L., Hepner, S.A.R. and Onder, C.H., 'Time-optimal motions of robots in assembly tasks', IEEE Transactions on Automatic Control 31(6), 1986, 512-518.
Chen, Y. and Desrochers, A.A., 'A proof of the structure of the minimum-time control law of robotic manipulators using a Hamiltonian formulation', IEEE Transactions on Robotics and Automation 6(3), 1990, 388-393.
Bessonnet, G. and Lallemand, J.P., 'Optimal trajectories of robot arms minimizing constrained actuators and traveling time', in IEEE International Conference on Robotics and Automation, Cincinnati, OH, IEEE, New York, 1990, 112-117.
Galicki, M., 'The planning of robotic optimal motions in the presence of obstacles', International Journal of Robotic Research 17, 1998, 248-259.
Galicki, M. and Ucinski, D., 'Time-optimal motions of robotic manipulators', Robotica 18, 2000, 659-667.
Lewis, F.L. and Syrmos, V.L., Optimal Control, John Wiley & Sons, New York, 1995.
Pontryagin, L., Boltiansky, V., Gamkrelitze, A. and Mishchenko, E., The Mathematical Theory of Optimal Processes, Wiley Intersciences, New York, 1962.
Bryson, A.E. and Ho, Y.C., Applied Optimal Control, Hemisphere, New York, 1975.
Ioffe, A.D. and Tihomirov, V.M., Theory of Extremal Problems, North-Holland, Amsterdam, 1979.
Skowronski, J.M., Control Dynamics of Robotic Manipulators, Academic Press, Orlando, FL, 1986.
Flashner, H. and Skowronski, J.M., 'Model tracking control of Hamiltonian systems', ASME Journal of Dynamic Systems, Measurement, and Control 111, 1989, 656-660.
Bessonnet, G., Sardain, P. and Danes, F., 'Reducing high order derivations to algebraic operations in dynamic modelling for motion optimization purpose', in Proceedings of 15th IMACS World Congress on Scientific Computer Modelling and Applied Mathematics, A. Sydow (ed.), Wissenschaft & Technik Verlag, Berlin, 1997, 337-342.
NAG Fortran Library, Routine D02RAF, 1992.
Pereyra V., 'Pasva 3: An adaptative finite difference fortran program for the first order nonlinear, ordinary boundary problem', in Codes for Boundary Value Problems in Ordinary Differential Equations, Lecture Notes in Computer Science, Vol. 76 B. Childs, M. Scott, J.W. Daniel, E. Denman and P. Nelson (eds), Springer-Verlag, Berlin, 1979, 67-88.
Danes, F., 'Critères et contraintes pour la synthèse optimale des mouvements de robots manipulateurs. Application à l'évitement d'obstacles', Ph.D. Thesis, University of Poitiers, France, 1998.
Blajer, W. and Schiehlen, W., 'Walking without impacts as a motion/force control problem', ASME Journal of Dynamic Systems, Measurement, and Control 114, 1992, 660-665.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bessonnet, G., Sardain, P. & Chessé, S. Optimal Motion Synthesis – Dynamic Modelling and Numerical Solving Aspects. Multibody System Dynamics 8, 257–278 (2002). https://doi.org/10.1023/A:1020928112173
Issue Date:
DOI: https://doi.org/10.1023/A:1020928112173