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On Energy-Optimal and Time-Optimal Precise Displacement of Rigid Body with Friction

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Abstract

Minimal time optimality and minimal energy optimality are usually considered as competing approaches for the trajectory planning for the precise movement of a rigid body. Theoretical and simulation results show that, with appropriate choice of constraints, these approaches are equivalent in the sense that they produce the same trajectory. This trajectory is globally optimal for both objectives. Constraints for velocity, driving force and jerk are taken into account. The model includes Coulomb and viscous friction. The optimal control solver was used as a numerical tool.

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References

  1. Lambrechts, P., Boerlage, M., Steinbuch, M.: Trajectory planning and feedforward design for electromechanical motion systems. Control Eng. Pract. 13(2), 145–157 (2005)

    Article  Google Scholar 

  2. Nguyen, K.D., Chen, I.M., Ng, T.C.: Planning algorithms for s-curve trajectories. In: Proceedings of IEEE/ASME International Conference on Advanced Intelligent Mechatronics 2007, pp 1–6 (2007)

  3. Heertjes, M., Hennekens, D., Steinbuch, M.: Mimo feed-forward design in wafer scanners using a gradient approximation-based algorithm. Control Eng. Pract. 18(5), 495–506 (2010)

    Article  Google Scholar 

  4. Atef, A.: Optimal trajectory planning of manipulators: a review. J. Eng. Sci. Technol. 2(1), 32–54 (2007)

    Google Scholar 

  5. Bobrow, J., Dubowsky, S., Gibson, J.: Time-optimal control of robotic manipulators along specified paths. Int. J. Robot. Res. 4(3), 3–17 (1985)

    Article  Google Scholar 

  6. Kyriakopoulos, K., Saridis, G.: Minimum jerk path generation. In: Proceedings of IEEE International Conference on Robotics and Automation, vol. 1, pp. 364–369 (1988)

  7. Shin, K., McKay, N.D.: Automatic generation of trajectory planners for industrial robots. In: Proceedings of IEEE International Conference on Robotics and Automation, vol. 3, pp. 260–266 (1986)

  8. Macfarlane, S., Croft, E.: Jerk-bounded manipulator trajectory planning: design for real-time applications. IEEE Trans. Robot. Autom. 19(1), 42–52 (2003)

    Article  Google Scholar 

  9. Gasparetto, A., Zanotto, V.: A new method for smooth trajectory planning of robot manipulators. Mech. Mach. Theory 42(4), 455–471 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Wang, Y., Ueda, K., Bortoff, S.A.: A Hamiltonian approach to compute an energy efficient trajectory for a servomotor system. Automatica 49, 3550–3561 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Zhao, Y., Tsiotras, P.: Speed profile optimization for optimal path tracking. Am. Control Conf. (ACC) 2013, 1171–1176 (2013)

    Google Scholar 

  12. Berger, A., Ioslovich, I., Gutman, P.O.: Time optimal trajectory planning with feedforward and friction compensation. In: American Control Conference (ACC 2015) Proceedings, Chicago, IL, USA, pp. 4143–4148 (2015)

  13. Khrustalev, M.M.: Necessary and sufficient dynamic programming conditions for optimal control problem with state constraints. In: Sebastian, H.J., Tammer, K. (eds.), Lecture Notes in Control and Information Sciences, vol. 143, pp. 311–320. Springer, Berlin (1990)

  14. Hartl, R., Sethi, S., Vickson, R.: A survey of the maximum principles for optimal control problems with state constraints. SIAM Rev. 37(2), 181–218 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ioslovich, I., Gutman, P.O.: Time-optimal control of wafer stage positioning using simplified models. Contemp. Math. 619, 99–107 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ioslovich, I., Moshenberg, S., Gutman, P.O.: On the energy-optimal precise wafer stage positioning: equivalence with minimal time optimality. In: Proceedings of 24th International Symposium on Industrial Electronics (IEEE ISIE 2015), June 3–5, Búzios, Brazil, pp. 95–99 (2015)

  17. Ross, I.M.: A beginners guide to DIDO (ver 7.3): a matlab application package for solving optimal control problems. Technical report, Elissar LLC, Monterey, Calif, USA (2007)

  18. Leitmann, G.: The Calculus of Variations and Optimal Control. Plenum Press, New York (1981)

    Book  MATH  Google Scholar 

  19. Armstrong-Helouvry, B., Dupont, P.: Friction modeling for control. Am. Control Conf. 1993, 1905–1909 (1993)

    Google Scholar 

  20. Krotov, V.F.: Global methods in optimal control theory. M. Dekker, New York (1996)

    MATH  Google Scholar 

  21. Al-Bender, F., Swevers, J.: Characterization of friction force dynamics. IEEE Control Syst. Mag. 28(6), 64–81 (2008)

    Article  MathSciNet  Google Scholar 

  22. Pontryagin, L., Boltyansky, V.G., Gamkrelidze, R., Mischenko, E.F.: Mathematical theory of optimal processes. Wiley, New York (1962)

    Google Scholar 

  23. Krotov, V.F., Bukreev, V.Z., Gurman, V.I.: New variational methods in flight dynamics (translated from Russian). NASA, TT F-657 (1971)

  24. Gurman, V.I.: Vyrojdennye zadachi optima’lnogo upravleniya (in Russian). Nauka, Moscow, USSR (1977)

  25. Gurman V., I.: Printsip rasshireniya v zadachakh upravleniya (Extension Principle in Control Problems), (in Russian). Nauka, Moscow, USSR (1985)

  26. Khrustalev, M.: Sufficient conditions for optimality in problems with constraints on phase coordinates. Autom. Remote Control 4, 18–29 (1967)

    MathSciNet  MATH  Google Scholar 

  27. Kelley, J., Kopp, R.E., Moyer, H.G.: Singular extremals. In: Leitmann, G. (ed.) Topics in optimization. Academic press, New York (1967)

    Google Scholar 

  28. Zelikin, M.I., Borisov, M.F.: Theory of chattering control with applications to astronautics, robotics, economics, and engineering. Birkhauser, Boston (1994)

    Book  MATH  Google Scholar 

  29. Milyutin, A.A., Illyutovich, A.E., Osmolovskii, N.P., Chukanov, S.B.: Optima’lnoe upravlenie v linejnyh systemah (in Russian). Nauka, Moscow (1993)

    Google Scholar 

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Acknowledgments

This research was partially supported by METRO 450 project from the Office of the Chief Scientist (OCS) in the Ministry of Economy, Israel. The authors are thankful to anonymous reviewer for his valuable notes.

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Authors and Affiliations

  1. Technion–Israel Institute of Technology, Haifa, Israel

    Ilya Ioslovich, Per-Olof Gutman, Ari Berger & Shai Moshenberg

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  1. Ilya Ioslovich
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  2. Per-Olof Gutman
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  4. Shai Moshenberg
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Correspondence to Ilya Ioslovich.

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Ioslovich, I., Gutman, PO., Berger, A. et al. On Energy-Optimal and Time-Optimal Precise Displacement of Rigid Body with Friction. J Optim Theory Appl 172, 466–480 (2017). https://doi.org/10.1007/s10957-016-0913-2

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  • DOI: https://doi.org/10.1007/s10957-016-0913-2

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