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Two approaches for feedforward control and optimal design of underactuated multibody systems

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Abstract

An underactuated multibody system has less control inputs than degrees of freedom. For trajectory tracking, often a feedforward control is necessary. Two different approaches for feedforward control design are presented. The first approach is based on a coordinate transformation into the nonlinear input–output normal-form. The second approach uses servo-constraints and results in a set of differential algebraic equations. A comparison shows that both feedforward control designs have a similar structure. The analysis of the mechanical design of underactuated multibody systems might show that they are nonminimum phase, i.e., they have unstable internal dynamics. Then the feedforward control cannot be computed by time integration and output trajectory tracking becomes a very challenging task. Therefore, based on the two presented feedforward control design approaches, it is shown that through the use of an optimization procedure underactuated multibody systems can be designed in such a way that they are minimum phase. Thus, feedforward control design using the two approaches is significantly simplified.

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References

  1. Arnold, M., Brüls, O.: Convergence of the generalized-α scheme for constrained mechanical systems. Multibody Syst. Dyn. 18, 185–202 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  2. Betsch, P., Uhlar, S., Quasem, M.: On the incorporation of servo constraints into a rotationless formulation of flexible multibody dynamics. In: Proceedings of the ECCOMAS Thematic Conference Multibody Dynamics, Milano, Italy, 25–28 June 2007

    Google Scholar 

  3. Blajer, W.: Index of differential-algebraic equations governing the dynamics of constraint mechanical systems. Appl. Math. Model. 16, 70–77 (1992)

    Article  MATH  Google Scholar 

  4. Blajer, W., Kolodziejczyk, K.: A geometric approach to solving problems of control constraints: theory and a DAE framework. Multibody Syst. Dyn. 11, 343–364 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  5. Blajer, W., Kolodziejczyk, K.: Control of underactuated mechanical systems with servo-constraints. Nonlinear Dyn. 50, 781–791 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Blajer, W., Kolodziejczyk, K.: Improved DAE formulation for inverse dynamics simulation of cranes. Multibody Syst. Dyn. 25, 131–143 (2011)

    Article  Google Scholar 

  7. Blajer, W., Dziewiecki, K., Kolodziejczyk, K., Mazur, Z.: Inverse dynamics of underactuated mechanical systems: a simple case study and experimental verification. Commun. Nonlinear Sci. Numer. Simul. 16, 2265–2272 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  8. Brenan, K., Campbell, S., Petzold, L.: Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. SIAM, Philadelphia (1996)

    MATH  Google Scholar 

  9. Campbell, S.: High-index differential algebraic equations. Mech. Based Des. Struct. Mach. 23, 199–222 (1995)

    Article  Google Scholar 

  10. De Luca, A., Book, W.: Robots with flexible elements. In: Handbook of Robotics, pp. 287–319. Springer, Berlin (2008), Chap. 13

    Chapter  Google Scholar 

  11. Devasia, S., Chen, D., Paden, B.: Nonlinear inversion-based output tracking. IEEE Trans. Autom. Control 41, 930–942 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  12. Fliess, M., Lévine, J., Martin, P., Rouchon, P.: Flatness and defect of nonlinear systems: introductory theory and examples. Int. J. Control 61, 1327–1361 (1995)

    Article  MATH  Google Scholar 

  13. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II—Stiff and Differential Algebraic Problems. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  14. Heyden, T., Woernle, C.: Dynamics and flatness-based control of a kinematically undetermined cable suspension manipulator. Multibody Syst. Dyn. 16, 155–177 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hirschorn, R.: Invertibility of multivariable nonlinear control systems. IEEE Trans. Autom. Control 24, 855–865 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  16. Isidori, A.: Nonlinear Control Systems. Springer, London (1995)

    MATH  Google Scholar 

  17. Jankowski, K., Van Brussel, H.: An approach to discrete inverse dynamics control of flexible-joint robots. IEEE Trans. Robot. Autom. 8, 651–658 (1992)

    Article  Google Scholar 

  18. Kennedy, J., Eberhart, R.: Particle swarm optimization. In: Proceedings of the International Conference on Neural Networks, Perth, Australia, pp. 1942–1948 (1995)

    Google Scholar 

  19. Leyendecker, S., Marsen, J., Oritz, M.: Variational integrators for constraint dynamical systems. Z. Angew. Math. Mech. 88, 677–708 (2008)

    Article  MATH  Google Scholar 

  20. Moallem, M., Patel, R., Khorasani, K.: Flexible-Link Robot Manipulators: Control Techniques and Structural Design. Springer, London (2000)

    MATH  Google Scholar 

  21. Moberg, S., Hanssen, S.: A DAE approach to feedforward control of flexible manipulators. In: Proceedings of the 2007 IEEE International Conference on Robotics and Automation, pp. 3439–3444 (2007)

    Chapter  Google Scholar 

  22. Montambault, S., Gosselin, C.: Analysis of underactuated mechanical grippers. J. Mech. Des. 123, 367–374 (2001)

    Article  Google Scholar 

  23. Rouchon, P.: Flatness based control of oscillators. J. Appl. Math. Mech. 85, 411–421 (2005)

    MATH  MathSciNet  Google Scholar 

  24. Roy, B., Asada, H.: Nonlinear feedback control of a gravity-assisted underactuated manipulator with application to aircraft assembly. IEEE Trans. Robot. 25, 1125–1133 (2009)

    Article  Google Scholar 

  25. Sastry, S.: Nonlinear Systems: Analysis, Stability and Control. Springer, New York (1999)

    MATH  Google Scholar 

  26. Sedlaczek, K., Eberhard, P.: Using augmented Lagrangian particle swarm optimization for constrained problems in engineering. Struct. Multidiscip. Optim. 32, 277–286 (2006)

    Article  Google Scholar 

  27. Seifried, R.: Optimization-based design of feedback linearizable underactuated multibody systems. In: Proceedings of the ECCOMAS Thematic Conference Multibody Dynamics 2009, Warsaw, Poland, 29 June–2 July (2009), paper ID 121

    Google Scholar 

  28. Seifried, R.: Two approaches for designing minimum phase underactuated multibody systems. In: Proceedings of the First Joint International Conference on Multibody System Dynamics, Lappeenranta, Finland (2010)

    Google Scholar 

  29. Seifried, R.: Optimization-based design of minimum phase underactuated multibody systems. In: Arczewski, K., Blajer, W., Fraczek, J., Wojtyra, M. (eds.) Multibody Dynamics: Computational Methods and Applications, pp. 261–282. Springer, Berlin (2011)

    Google Scholar 

  30. Seifried, R.: Integrated mechanical and control design of underactuated multibody systems. Nonlinear Dyn. (accepted for publication). doi:10.1007/s11071-011-0087-2

  31. Seifried, R., Eberhard, P.: Design of feed-forward control for underactuated multibody systems with kinematic redundancy. In: Ulbrich, H., Ginzinger, L. (eds.) Motion and Vibration Control: Selected Papers from MOVIC 2008, pp. 275–284. Springer, Berlin (2009)

    Chapter  Google Scholar 

  32. Seifried, R., Held, A., Dietmann, F.: Analysis of feed-forward control design approaches for flexible multibody systems. J. Syst. Des. Dyn. 5, 429–440 (2011)

    Google Scholar 

  33. Shampine, L., Reichelt, M., Kierzenka, J.: Solving index-1 DAEs in MATLAB and Simulink. SIAM Rev. 18, 538–552 (1999)

    Article  MathSciNet  Google Scholar 

  34. Soltine, J.-J., Li, W.: Applied Nonlinear Control. Prentice-Hall, Englewood Cliffs (1991)

    Google Scholar 

  35. Spong, M., Hutchinson, S., Vidyasagar, M.: Robot Modeling and Control. Wiley, New York (2006)

    Google Scholar 

  36. Taylor, D., Li, S.: Stable inversion of continuous-time nonlinear systems by finite-difference methods. IEEE Trans. Autom. Control 47, 537–542 (2002)

    Article  MathSciNet  Google Scholar 

  37. Zhou, K., Doyle, J., Glover, K.: Robust and Optimal Control. Prentice-Hall, Upper Saddle River (1996)

    MATH  Google Scholar 

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Seifried, R. Two approaches for feedforward control and optimal design of underactuated multibody systems. Multibody Syst Dyn 27, 75–93 (2012). https://doi.org/10.1007/s11044-011-9261-z

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