Abstract
Underactuated mechanical systems are featured with less control inputs than degrees of freedom. Their performance goal may then be realization of specified in time outputs whose number coincides the number of inputs. A solution to the inverse simulation problem (servo-constraint problem), that is, determination of an input control strategy that forces the underactuated system to complete the partly specified motion, is a challenging task. Since systems may be “underactuated” in several ways and the servo-constraint realization may range from orthogonal to tangential, diverse formulations and analysis methods of the servo-constraint problem arise. The diversity is discussed with reference to some simple case studies. The governing equations are handled in two ways. A direct formulation in configuration coordinates is first motivated and is then compared to a setting in which the actuated coordinates are replaced with the outputs. The governing equations arise either as ODEs (ordinary differential equations) or DAEs (differential-algebraic equations). Some computational issues related to the ODE and DAE formulations are discussed, and simulation results for the sample case studies are reported.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Abdel-Rahman, E.M., Nayfeh, A.H., Masoud, Z.N.: Dynamics and control of cranes: a review. J. Vib. Contr. 9, 863–908 (2003)
Ascher, U.M., Petzold, L.R.: Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM, Philadelphia (1998)
Benosman, M., Le Vey, G.: Control of flexible manipulators: a survey. Robotica 22, 533–545 (2004)
Blajer, W.: A geometrical interpretation and uniform matrix formulation of multibody system dynamics. Z. Angew. Math. Mech. 81, 247–259 (2001)
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)
Blajer, W., Kołodziejczyk, K.: Control of underactuated mechanical systems with servo-constraints. Nonlinear Dyn. 50, 781–791 (2007)
Blajer, W., Kołodziejczyk, K.: Improved DAE formulation for inverse dynamics simulation of cranes. Multibody Syst. Dyn. 25, 131–143 (2011)
Blajer, W., Graffstein, J., Krawczyk, M.: Modeling of aircraft prescribed trajectory flight as an inverse simulation problem. In: Awrejcewicz, J. (ed.) Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems, pp. 153–162. Springer, Netherlands (2009)
Blajer, W., Seifried, R., Kołodziejczyk, K.: Diversity of servo-constraint problems for underactuated mechanical systems: a case study illustration. Solid State Phenom. 198, 473–482 (2013)
Campbell, S.L., Gear, C.W.: The index of general nonlinear DAEs. Numer. Math. 72, 173–196 (1995)
Chen, Y.-H.: Equations of motion of mechanical systems under servoconstraints: the Maggi approach. Mechatronics 18, 208–217 (2008)
De Luca, A.: Trajectory control of flexible manipulators. In: Siciliano, B., Valavanis, K.P. (eds.) Control Problems in Robotics and Automation, pp. 83–104. Springer, London (1998)
Fantoni, I., Lozano, R.: Non-linear Control for Underactuated Mechanical Systems. Springer, London (2002)
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)
Kirgetov, V.I.: The motion of controlled mechanical systems with prescribed constraints (servoconstraints). J. Appl. Math. Mech. USS 31, 433–466 (1967)
Lee, H.-H.: Modeling and control of a three-dimensional overhead crane. J Dyn. Syst. Meas. Control T ASME 120, 471–476 (1998)
Paul, R.P.: Robot Manipulators: Mathematics, Programming, and Control. MIT, Cambridge, MA (1981)
Rouchon, P.: Flatness based control of oscillators. Z. Angew. Math. Mech. 85, 411–421 (2005)
Sahinkaya, M.N.: Inverse dynamic analysis of multiphysics systems. Proc. Inst. Mech. Eng. I J. Syst. Control Eng. 218, 13–26 (2004)
Sara-Ramirez, H., Agrawal, S.K.: Differentially Flat Systems. Marcel Dekker, New York (2004)
Seifried, R.: Integrated mechanical and control design of underactuated multibody systems. Nonlinear Dyn. 67, 1539–1557 (2012)
Seifried, R.: Two approaches for feedforward control and optimal design of underactuated multibody systems. Multibody Syst. Dyn. 27, 75–93 (2012)
Seifried, R., Blajer, W.: Analysis of servo-constraint problems for underactuated multibody systems. Mech. Sci. 4, 113–129 (2013)
Spong, M.W.: Underactuated mechanical systems. In: Siciliano, B., Valavanis, K.P. (eds.) Control Problems in Robotics and Automation, pp. 135–150. Springer, London (1998)
Thompson, D., Bradley, R.: Inverse simulation as a tool for flight dynamics research: principles and applications. Progr. Aero. Sci. 42, 174–210 (2006)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Blajer, W. (2014). Diversities in the Inverse Dynamics Problem for Underactuated Mechanical Systems Subject to Servo-constraints. In: Awrejcewicz, J. (eds) Applied Non-Linear Dynamical Systems. Springer Proceedings in Mathematics & Statistics, vol 93. Springer, Cham. https://doi.org/10.1007/978-3-319-08266-0_14
Download citation
DOI: https://doi.org/10.1007/978-3-319-08266-0_14
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-08265-3
Online ISBN: 978-3-319-08266-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)