Abstract
Cranes are underactuated systems with less control inputs than degrees of freedom. Dynamics and control of such systems is a challenging task, and the existence of solution to the inverse dynamics simulation problem in which an r-degree-of-freedom system with m actuators, m<r, is subject to m specified motion task (servo-constraints) is conditioned upon the system is differentially flat (all the system states and control inputs can be algebraically expressed in terms of the outputs and their time derivatives up to a certain order). The outputs are often designed as specified in time load coordinates to model a rest-to-rest maneuver along a trajectory in the working space, from the initial load position to its desired destination. The flatness-based methodology results then in the required control inputs determined in terms of the fourth time derivatives of the imposed outputs, and the derivations are featured by substantial complexity. The DAE formulation motivated in this contribution offers a more convenient approach to the prediction of dynamics and control of cranes executing prescribed load motions, and only the second time derivatives of the specified outputs are involved. While most of the inverse simulation formulations, both flatness-based and DAE ones, are performed using independent state variables, the use of dependent coordinates and velocities may lead to substantial modeling simplifications and gains in computational efficiency. An improved DAE formulation of this type is presented in this paper.
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Spong, M.W.: Underactuated mechanical systems. In: Siciliano, B., Valavanis, K.P. (eds.) Control Problems in Robotics and Automation. Lecture Notes in Control and Information Sciences, vol. 230, pp. 135–150. Springer, London (1998)
Fantoni, I., Lozano, R.: Non-linear Control for Underactuated Mechanical Systems. Springer, London (2002)
Sahinkaya, M.N.: Inverse dynamic analysis of multiphysics systems. Proc. IME, J. Syst. Control Eng. 218, 13–26 (2004)
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)
Blajer, W., Kołodziejczyk, K.: A geometric approach to solving problems of control constraints: theory and a DAE framework. Multibody Syst. Dyn. 11, 343–364 (2004)
Rouchon, P.: Flatness based control of oscillators. Z. Angew. Math. Mech. 85, 411–421 (2005)
Abdel-Rahman, E.M., Nayfeh, A.H., Masoud, Z.N.: Dynamics and control of cranes: a review. J. Vib. Control 9, 863–908 (2003)
Omar, H.M.: Control of gantry and tower cranes. PhD Thesis, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, USA (2003)
Lee, H.H.: Motion planning for three-dimensional overhead cranes with high-speed load hoisting. Int. J. Control 78, 875–886 (2005)
Blajer, W., Kołodziejczyk, K.: Dynamics and control of rotary cranes executing a load prescribed motion. J. Theor. Appl. Mech. 44, 929–948 (2006)
Blajer, W., Kołodziejczyk, K.: A DAE formulation for the dynamic analysis and control design of cranes executing prescribed motions of payloads. In: García Orden, J.C., Goicolea, J.M., Cuadrado, J. (eds.) Multibody Dynamics. Computational Methods and Applications. Computational Methods in Applied Sciences, vol. 4, pp. 91–112. Springer, Dordrecht (2007)
Blajer, W., Kołodziejczyk, K.: Control of underactuated mechanical systems with servo-constraints. Nonlinear Dyn. 50, 781–791 (2007)
Kirgetov, V.I.: The motion of controlled mechanical systems with prescribed constraints (servo-constraints). J. Appl. Math. Mech. 21, 433–466 (1967)
Bajodah, A.H., Hodges, D.H., Chen, Y.-H.: Inverse dynamics of servo-constraints based on the generalized inverse. Nonlinear Dyn. 39, 179–196 (2005)
Rosen, A.: Applying the Lagrange method to solve problems of control constraints. ASME Trans. J. Appl. Mech. 66, 1013–1015 (1999)
Woernle, Ch.: Control of robotic systems by exact linearization methods. In: Maisser, P., Tenberge, P. (eds.) Proc. of the First International Symposium on Mechatronics (ISoM 2002), pp. 207–218. Advanced Driving Systems, Chemnitz (2002)
Heyden, T., Woernle, Ch.: Dynamics and flatness-based control of a kinematically underdetermined cable suspension manipulator. Multibody Syst. Dyn. 16, 155–177 (2006)
Aschemann, H.: Optimale Trajektrienplanung sowie modelgestützte Steuerung für einen Brückenkran. Fortschritt-Berichte VDI, Reihe 8: Meß-, Steuerungs- und Regelungstechnik, Nr. 929. Düsseldorf, Germany (2002)
Blajer, W., Kołodziejczyk, K.: Motion planning and control of gantry cranes in cluttered work environment. IET Control Theory Appl. 1, 1370–1379 (2007)
Ascher, U.M., Petzold, L.R.: Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM, Philadelphia (1998)
Campbell, S.L., Gear, C.W.: The index of general nonlinear DAEs. Numer. Math. 72, 173–196 (1995)
Gear, C.W.: An introduction to numerical methods for ODEs and DAEs. In: Haug, E.J., Deyo, R.C. (eds.) Real-Time Integration Methods for Mechanical System Simulations. NATO ASI Series, vol. F69, pp. 115–126. Springer, Berlin (1990)
Petzold, L.R.: Methods and software for differential-algebraic systems. In: Haug, E.J., Deyo, R.C. (eds.) Real-Time Integration Methods for Mechanical System Simulations. NATO ASI Series, vol. F69, pp. 127–140. Springer, Berlin (1990)
Blajer, W.: A geometric unification of constrained system dynamics. Multibody Syst. Dyn. 1, 3–21 (1997)
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Blajer, W., Kołodziejczyk, K. Improved DAE formulation for inverse dynamics simulation of cranes. Multibody Syst Dyn 25, 131–143 (2011). https://doi.org/10.1007/s11044-010-9227-6
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DOI: https://doi.org/10.1007/s11044-010-9227-6