Research Papers

A Stable Inversion Method for Feedforward Control of Constrained Flexible Multibody Systems

[+] Author and Article Information
Olivier Brüls

e-mail: o.bruls@ulg.ac.be

Guaraci Jr. Bastos

e-mail: g.bastos@ulg.ac.be

Department of Aerospace
and Mechanical Engineering (LTAS),
University of Liège,
Chemin des chevreuils 1 (B52),
Liège 4000, Belgium

Robert Seifried

Institute of Engineering
and Computational Mechanics,
University of Stuttgart,
Pfaffenwaldring 9,
Stuttgart 70569, Germany
e-mail: robert.seifried@itm.uni-stuttgart.de

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received February 11, 2013; final manuscript received May 23, 2013; published online October 15, 2013. Assoc. Editor: Hiroyuki Sugiyama.

J. Comput. Nonlinear Dynam 9(1), 011014 (Oct 15, 2013) (9 pages) Paper No: CND-13-1034; doi: 10.1115/1.4025476 History: Received February 11, 2013; Revised May 23, 2013

The inverse dynamics of flexible multibody systems is formulated as a two-point boundary value problem for an index-3 differential-algebraic equation (DAE). This DAE represents the equation of motion with kinematic and trajectory constraints. For so-called nonminimum phase systems, the remaining dynamics of the inverse model is unstable. Therefore, boundary conditions are imposed not only at the initial time but also at the final time in order to obtain a bounded solution of the inverse model. The numerical solution strategy is based on a reformulation of the DAE in index-2 form and a multiple shooting algorithm, which is known for its robustness and its ability to solve unstable problems. The paper also describes the time integration and sensitivity analysis methods that are used in each shooting phase. The proposed approach does not require a reformulation of the problem in input-output normal form, which is known from nonlinear control theory. It can deal with serial and parallel kinematic topology, minimum phase and nonminimum phase systems, and rigid and flexible mechanisms.

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Cannon, H., and Schmitz, E., 1984, “Initial Experiments on the End-Point Control of a Flexible One-Link Robot,” Int. J. Robot. Res., 3(3), pp. 62–75. [CrossRef]
Book, W., 1993, “Controlled Motion in an Elastic World,” ASME J. Dyn. Syst., Measu., Control, 115(2B), pp. 252–261. [CrossRef]
Da Silva, M., Brüls, O., Swevers, J., Desmet, W., and Van Brussel, H., 2009, “Computer-Aided Integrated Design for Machines With Varying Dynamics,” Mech. Mach. Theory, 44, pp. 1733–1745. [CrossRef]
Fliess, M., Lévine, J., Martin, P., and Rouchon, P., 1995, “Flatness and Defect of Nonlinear Systems: Introductory Theory and Examples,” Int. J. Control, 61, pp. 1327–1361. [CrossRef]
Isidori, A., 1995, Nonlinear Control Systems, 3rd ed., Springer, London.
Sastry, S., 1999, Nonlinear Systems: Analysis, Stability and Control, Springer, New York.
Van Nieuwstadt, M. and Murray, R., 1998, “Real-Time Trajectory Generation for Differentially Flat Systems,” Int. J. Robust Nonlinear Control, 18(11), pp. 995–1020. [CrossRef]
Asada, H., and Slotine, J.-J., 1986, Robot Analysis and Control, Wiley-Interscience, New York.
Spong, M., 1987, “Modeling and Control of Elastic Joint Robots,” ASME J. Dyn. Syst., Meas., Control, 109(4), pp. 310–318. [CrossRef]
Kwon, D., and Book, W., 1994, “A Time-Domain Inverse Dynamic Tracking Control of a Single-Link Flexible Manipulator,” ASME J. Dyn. Syst., Meas., Control, 116(2), pp. 193–200. [CrossRef]
Devasia, S., and Bayo, E., 1994, “Inverse Dynamics of Articulated Flexible Structures: Simultaneous Trajectory Tracking and Vibration Reduction,” J. Dyn. Control, 4(3), pp. 299–309. [CrossRef]
Seifried, R., Held, A., and Dietmann, F., 2011, “Analysis of Feed-Forward Control Design Approaches for Flexible Multibody Systems,” J. Syst. Des. Dyn., 5(3), pp. 429–440.
Seifried, R., Burkhardt, M., and Held, A., 2013, “Trajectory Control of Serial and Parallel Flexible Manipulators Using Model Inversion,” Multibody Dynamics: Computational Methods and Applications, Computational Methods in Applied Sciences, Vol. 28, J.Samin and P.Fisette, eds., Springer, New York.
Devasia, S., Chen, D., and Paden, B., 1996, “Nonlinear Inversion-Based Output Tracking,” IEEE Trans. Autom. Control, 41(7), pp. 930–942. [CrossRef]
Taylor, D., and Li, S., 2002, “Stable Inversion of Continuous-Time Nonlinear Systems by Finite-Difference Methods,” IEEE Trans. Autom. Control, 47(3), pp. 537–542. [CrossRef]
Seifried, R., 2012, “Two Approaches for Feedforward Control and Optimal Design of Underactuated Multibody Systems,” Multibody Syst. Dyn., 27(1), pp. 75-93. [CrossRef]
Seifried, R., 2012, “Integrated Mechanical and Control Design of Underactuated Multibody Systems,” Nonlinear Dyn., 67, pp. 1539–1557. [CrossRef]
Seifried, R., and Eberhard, P., 2009, “Design of Feed-Forward Control for Underactuated Multibody Systems With Kinematic Redundancy,” Motion and Vibration Control: Selected Papers from MOVIC 2008, H.Ulbrich and L.Ginzinger, eds., Springer, New York.
Blajer, W., and Kolodziejczyk, K., 2004, “A Geometric Approach to Solving Problems of Control Constraints: Theory and a DAE Framework Theory and a DAE Framework,” Multibody Syst. Dyn., 11, pp. 343–364. [CrossRef]
Blajer, W., and Kolodziejczyk, K., 2007, “Control of Underactuated Mechanical Systems With Servo-Constraints,” Nonlinear Dyn., 50, pp. 781–791. [CrossRef]
Seifried, R., and Blajer, W., 2013, “Analysis of Servo-Constraint Problems for Underactuated Multibody Systems,” Mech. Sci., 4, pp. 113–129. [CrossRef]
Bastos, G., Seifried, R., and Brüls, O., 2013, “Inverse Dynamics of Serial and Parallel Underactuated Multibody Systems Using a DAE Optimal Control Approach,” Multibody Syst. Dyn., 30, pp. 359–376. [CrossRef]
Morrison, D., Riley, J., and Zancarano, J., 1962, “Multiple Shooting Methods for Two-Point Boundary Value Problems,” Commun. ACM, 5, pp. 613–614. [CrossRef]
Keller, H., 1968, Numerical Methods for Two-Point Boundary-Value Problems, Blaisdell, Waltham, MA.
Roberts, S., and Shipman, J., 1972, Two-Point Boundary Value Problems: Shooting Methods, Elsevier, New York.
Chung, J., and Hulbert, G., 1993, “A Time Integration Algorithm for Structural Dynamics With Improved Numerical Dissipation: The Generalized-α Method,” ASME J. Appl. Mech., 60(2), pp. 371–375. [CrossRef]
Newmark, N., 1959, “A Method of Computation for Structural Dynamics,” J. Eng. Mech. Div., Am. Soc. Civ. Eng.,, 85, pp. 67–94.
Arnold, M., and Brüls, O., 2007, “Convergence of the Generalized-α Scheme for Constrained Mechanical Systems,” Multibody Syst. Dyn., 18(2), pp. 185–202. [CrossRef]
Arnold, M., Brüls, O., and Cardona, A., 2011, “Convergence Analysis of Generalized-α Lie Group Integrators for Constrained Systems,” Proceedings of the Multibody Dynamics ECCOMAS Thematic Conference.
Gear, C., Leimkuhler, B., and Gupta, G., 1985, “Automatic Integration of Euler–Lagrange Equations With Constraints,” J. Comput. Appl. Math., 12–13, pp. 77–90. [CrossRef]
Brüls, O. E. L., Duysinx, P., and Eberhard, P., 2011, “Optimization of Multibody Systems and Their Structural Components,” Multibody Dynamics: Computational Methods and Applications, Computational Methods in Applied Sciences, Vol. 23, W.Blajer, J.Arczewski, K.Fraczek, and M.Wojtyra, eds., Springer, New York, pp. 49–68.
Wasfy, T., and Noor, A., 2003, “Computational Strategies for Flexible Multibody Systems,” Appl. Mech. Rev., 56(6), pp. 553–613. [CrossRef]
Bayo, E., and Ledesma, R., 1996, “Augmented Lagrangian and Mass-Orthogonal Projection Methods for Constrained Multibody Dynamics,” Nonlinear Dyn., 9, pp. 113–130. [CrossRef]
Jay, L., and Negrut, D., 2007, “Extensions of the HHT-Method to Differential-Algebraic Equations in Mechanics,” Electron. Trans. Numer. Anal., 26, pp. 190–208.
Géradin, M., and Cardona, A., 2001, Flexible Multibody Dynamics: A Finite Element Approach, John Wiley and Sons, Chichester, UK.
Bastos, G., Seifried, R., and Brüls, O., 2011, “Inverse Dynamics of Underactuated Multibody Systems Using a DAE Optimal Control Approach,” Proceedings of the Multibody Dynamics ECCOMAS Conference.
Wenger, P., and Chablat, D., 2009, “Kinematic Analysis of a Class of Analytic Planar 3-RPR Parallel Manipulators,” Computational Kinematics: Proceedings of the 5th International Workshop on Computational Kinematics, pp. 43–50.


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Fig. 1

Dynamic system with feedforward and feedback controller

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Fig. 2

Boundary conditions for the zero-dynamics

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Fig. 3

Multiple shooting method

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Fig. 4

Planar serial manipulator with one passive joint

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Fig. 5

Underactuated serial manipulator: discretization in absolute coordinates

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Fig. 6

Serial manipulator: desired trajectory

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Fig. 7

Serial manipulator: results for c = 50 Nm/rad and d = 0.25 Nms/rad

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Fig. 8

Serial manipulator: results for c = 5 Nm/rad and d = 0 Nm s/rad

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Fig. 9

Flexible parallel manipulator

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Fig. 10

Results for the parallel manipulator (solid line: 1 finite element, and dashed line: 2 finite elements)




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