Numerical Integration of Underactuated Mechanical Systems Subjected to Mixed Holonomic and Servo Constraints

Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 42)


A new index reduction approach is developed for the inverse dynamics simulation of underactuated mechanical systems. The underlying equations of motion contain both holonomic and servo constraints. The proposed method is applied to a very general and versatile formulation of cranes. The numerical results demonstrate the functional efficiency of the method.


Underactuated mechanical systems Feedforward control Inverse dynamics Differentially flat systems 



The second author was supported by the ERC Advanced Grant ‘Modeling, Simulation and Control of Multi-Physics Systems’ MODSIMCONMP. The third author was supported by the Deutsche Forschungsgemeinschaft (DFG) under Grant BE 2285/12-1. This support is gratefully acknowledged.


  1. 1.
    Kirgetov VI (1967) The motion of controlled mechanical systems with prescribed constraints (servoconstraints). J Appl Math Mech 31(3):465–477MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Blajer W (1997) Dynamics and control of mechanical systems in partly specified motion. J Franklin Inst 334B(3):407–426MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Lam SH (1998) On Lagrangian dynamics and its control formulations. Appl Math Comput 91:259–284MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Uhlar S, Betsch P (2009) A rotationless formulation of multibody dynamics: Modeling of screw joints and incorporation of control constraints. Multibody Syst Dyn 22(1):69–95MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Blajer W, Seifried R, Kołodziejczyk K (2013) Diversity of servo-constraint problems for underactuated mechanical systems: a case study illustration. Solid State Phenom 198:473–482CrossRefGoogle Scholar
  6. 6.
    Seifried R, Blajer W (2013) Analysis of servo-constraint problems for underactuated multibody systems. Mech Sci 4(1):113–129CrossRefGoogle Scholar
  7. 7.
    Blajer W (2014) The use of servo-constraints in the inverse dynamics analysis of underactuated multibody systems. J Comput Nonlinear Dynam 9(4):041008/1-11Google Scholar
  8. 8.
    Kunkel P, Mehrmann V (2004) Index reduction for differential-algebraic equations by minimal extension. Z Angew Math Mech (ZAMM) 84(9):579–597MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Altmann R, Betsch P, Yang Y (2015) Index reduction by minimal extension for the inverse dynamics simulation of cranes. Accepted for Publication in Multibody System Dynamics. doi: 10.1007/s11044-015-9471-x
  10. 10.
    Blajer W, Kołodziejczyk K (2004) A geometric approach to solving problems of control constraints: theory and a DAE framework. Multibody Syst Dynam 11(4):343–364Google Scholar
  11. 11.
    Kiss B, Lévine J, Müllhaupt P (1999) Modelling, flatness and simulation of a class of cranes. Electr Eng 43(3):215–225Google Scholar
  12. 12.
    Ascher UM, Petzold LR (1998) Computer methods for ordinary differential equations and differential-algebraic equations. SIAM, PhiladelphiaGoogle Scholar
  13. 13.
    Blajer W, Kołodziejczyk K (2011) Improved DAE formulation for inverse dynamics simulation of cranes. Multibody Syst Dynam 25(2):131–143CrossRefGoogle Scholar
  14. 14.
    Blajer W, Kołodziejczyk K (2005) A computational framework for control design of rotary cranes. In: Goicolea JM, Cuadrado J, Garcia Orden JC (ed) Proceedings of ECCOMAS thematic conference on advances in computational multibody dynamics (CD-ROM), Madrid, June 21–24Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Institut für MechanikKarlsruher Institut für TechnologieKarlsruheGermany
  2. 2.Institut für MathematikTechnische Universität BerlinBerlinGermany

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