Skip to main content
SpringerLink
Log in

On variational integrators for optimal control of mechanical control systems

  • Original Paper
  • Published:
Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas Aims and scope Submit manuscript

Abstract

In this paper we study discrete second-order vakonomic mechanics, that is, constrained variational problems for second-order lagrangian systems. One of the main applications of the presented theory will be optimal control of underactuated mechanical control systems. We derive geometric integrators which are symplectic and preserve the momentum map. Additional, we show the applicability of the proposed theory in an example, the planar rigid body.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abraham R., Marsden J.: Foundations of Mechanics, 2nd edn. Benjamin/Cummings, New York (1978)

    MATH  Google Scholar 

  2. Benito R., Martín de Diego D.: Discrete Vakonomic Mechanics. J. Math. Phys. 46, 083521 (2005)

    Article  MathSciNet  Google Scholar 

  3. Benito R., de León M., Martín de Diego D.: Higher order discrete Lagrangian mechanics. Int. J. Geom. Methods Mod. Phys. 3, 421–436 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bloch A.M.: Nonholonomic Mechanics and Control. Interdisciplinary Applied Mathematics Series, 24. Springer-Verlag, New York (2003)

    Google Scholar 

  5. Bullo F, Lewis A.: Geometric Control of Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical Control Systems. Texts in Applied Mathematics. Springer Verlag, New York (2005)

    Google Scholar 

  6. Colombo L., Martin de Diego D., Zuccalli M.: Optimal control of underactuated mechanical systems: a geometrical approach. J. Math. Phys. 51, 083519 (2010)

    Article  MathSciNet  Google Scholar 

  7. Colombo, L., Martin de Diego, D., Zuccalli, M.: Higher-order Discrete vakonomic mechanics. Preprint (2011)

  8. Cortés J.: Geometric, Control and Numerical Aspects of Nonholonomic Systems. Lec. Notes in Math., 1793. Springer-Verlag, Berlin (2002)

    Google Scholar 

  9. Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations. Springer Series in Computational Mathematics, 31, Springer-Verlag, Berlin (2002)

  10. de León, M., Rodrigues, P.R.: Generalized Classical Mechanics and Field Theory. North-Holland Mathematical Studies 112. North-Holland, Amsterdam (1985)

Download references

Author information

Authors and Affiliations

  1. Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), C/Nicolas Cabrera, 15-Campus de Cantoblanco, UAM, 28049, Madrid, Spain

    Leonardo Colombo & David Martín de Diego

  2. Departamento de Matemáticas, Universidad Nacional de La Plata, Calle 50 y 115, 1900, La Plata, Buenos Aires, Argentina

    Marcela Zuccalli

Authors
  1. Leonardo Colombo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David Martín de Diego
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Marcela Zuccalli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Leonardo Colombo.

Additional information

This work has been supported by MICINN (Spain) Grant MTM2010-21186-C02-01, MTM2009-08166-E, project “Ingenio Mathematica” (i-MATH) No. CSD 2006-00032 (Consolider-Ingenio 2010) and IRSES-project “Geomech-246981”. L. Colombo also wants to thank CSIC and JAE program for a JAE-Pre grant.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Colombo, L., de Diego, D.M. & Zuccalli, M. On variational integrators for optimal control of mechanical control systems. RACSAM 106, 161–171 (2012). https://doi.org/10.1007/s13398-011-0032-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13398-011-0032-8

Keywords

Mathematics Subject Classification (2000)

Search

Navigation