Abstract
The paper deals with underactuated mechanical systems, featured by less control inputs m than degrees of freedom f, m < f, subject to m servo-constraints (specified in time outputs) on the system. The arising servo-constraint problem (inverse dynamics analysis) is discussed with an emphasis on the way the servo-constraints are realized, varying from orthogonal to tangential, and a geometrical illustration of the different realization types is provided. Depending on the way the servo-constraints are realized, the governing equations are formulated either as ordinary differential equations (ODEs) or differential-algebraic equations (DAEs), and some computational issues for the ODEs and DAEs are discussed. The existence or non-existence of an explicit solution to the governing equations is further discussed, related to so-called differentially flat problems (without internal dynamics) and non-flat problems (with internal dynamics). It is shown that in case of non-flat problems with orthogonal realization of servo-constraints, stability of the internal dynamics must be assured. Simple case studies are reported to illustrate the proposed formulations and methodologies.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
García de Jalón J., Bayo E.: Kinematic and Dynamic Simulation of Multibody Systems: The Real Time Challenge. Springer, New York (1994)
Sahinkaya M.N.: Inverse dynamic analysis of multiphysics systems. Proc. Inst. Mech. Eng. I J. Syst. Control Eng. 218, 13–26 (2004)
Kirgetov V.I.: The motion of controlled mechanical systems with prescribed constraints (servoconstraints). J. Appl. Math. Mech. USS. 31, 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)
Blajer W., Kołodziejczyk K.: Control of underactuated mechanical systems with servo-constraints. Nonlinear Dyn. 50, 781–791 (2007)
Blajer W.: Dynamics and control of mechanical systems in partly specified motion. J. Franklin Inst. 334, 407–426 (1997)
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)
Jankowski K.P.: Inverse Dynamics Control in Robotics Applications. Trafford Publishing, Victoria (2004)
Fumagalli A., Masarati P., Morandini M., Mantegazza P.: Control constraint realization for multibody systems. J. Comput. Nonlinear Dyn. 6, 011002 (2011)
Sastry S.: Nonlinear Systems: Analysis, Stability, and Control. Springer, New York (1999)
Paul R.P.: Robot Manipulators: Mathematics, Programming, and Control. MIT Press, Cambridge (1981)
Yamaguchi G.T.: Dynamic Modeling of Musculoskeletal Motion: A Vectorized Approach for Biomechanical Analysis in Three Dimensions. Kluwer, Dordrecht (2001)
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)
Olfati-Saber, R.: Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles. PhD Thesis, Massachusetts Institute of Technology, Cambridge, MA (2001)
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)
Sara-Ramirez H., Agrawal S.K.: Differentially Flat Systems. Marcel Dekker, New York (2004)
Rouchon P.: Flatness based control of oscillators. Z. Angew. Math. Mech. 85, 411–421 (2005)
Lévine J.: Analysis and Control of Nonlinear Systems A Flatness-Based Approach. Springer, Berlin (2009)
Blajer W.: A geometrical interpretation and uniform matrix formulation of multibody system dynamics. Z. Angew. Math. Mech. 81, 247–259 (2001)
Campbell S.L., Gear C.W.: The index of general nonlinear DAEs. Numer. Math. 72, 173–196 (1995)
Ascher U.M., Petzold L.R.: Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM, Philadelphia (1998)
Blajer W., Kołodziejczyk K.: Motion planning and control of gantry cranes in cluttered work environment. IET Control Theory Appl. 1, 1370–1379 (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, The Netherlands (2009)
De Luca, A.: Trajectory control of flexible manipulators. In: Siciliano, B., Valavanis, K.P. (eds.) Control Problems in Robotics and Automation. Springer, London, pp. 83–104 (1998)
Hagenmeyer V., Delaleau E.: Exact feedforward linearization based on differential flatness. Int. J. Control 76, 537–556 (2003)
Masarati, P., Morandini, M., Fumagalli, A.: Control constraint realization applied to underactuated aerospace systems. In: Proceedings of the IDETC/CIE 2011 (ASME 2011 International Design Engineering Technical Conference and Computers and Information in Engineering Conference), August 28–31, 2011, Washington, DC, paper ID DETC2011-47276, Vol. 4, 2011, pp. 305–316 (2011)
Seifried, R., Burkhardt, M.: Servo-constraints for control of flexible multibody systems with contact. In: Proceedings of the IDETC/CIE 2013 (ASME 2013 International Design Engineering Technical Conference and Computers and Information in Engineering Conference), August 4–7, 2013, Portland, OR, USA, paper ID DETC2012-12334 (2013)
Lee H.-H.: Modeling and control of a three-dimensional overhead crane. ASME. Trans. J. Dyn. Syst. Meas. Control 120, 471–476 (1998)
Abdel-Rahman E.M., Nayfeh A.H., Masoud Z.N.: Dynamics and control of cranes: a review. J. Vib. Control 9, 863–908 (2003)
Heyden T., Woernle C.: Dynamics and flatness-based control of a kinematically undetermined cable suspension manipulator. Multibody Syst. Dyn. 16, 155–177 (2006)
Forrest-Barlach M.G., Babcock S.M.: Inverse dynamics position control of a compliant manipulator. IEEE J. Robot. Autom. 3, 75–83 (1987)
Jankowski K.P.: Inverse Dynamics Control in Robotics Applications. Trafford Publishing, Victoria (2004)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
About this article
Cite this article
Blajer, W., Seifried, R. & Kołodziejczyk, K. Servo-constraint realization for underactuated mechanical systems. Arch Appl Mech 85, 1191–1207 (2015). https://doi.org/10.1007/s00419-014-0959-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00419-014-0959-2