Skip to main content
SpringerLink
Log in

The equivalence of Boltzmann–Hamel and Gibbs–Appell equations in modeling constrained systems

  • Published:
International Journal of Dynamics and Control Aims and scope Submit manuscript

Abstract

The model-based control design process for mechanical systems such as robotic vehicles and manipulators requires a clear mathematical model that describes the dynamics of the system to be available. Formulation of such mathematical models can be challenging especially if constraints are involved. There exist several methods of developing dynamic models for mechanical systems. The popular methods can be carried out in the natural inertial space, configuration space or in the quasi-coordinate space. Inertial space models such as those based on Newton-Euler formulation are only suitable for small systems because of the need to determine all unknown constraint reaction forces. Configuration space models such as Euler–Lagrange models are preferred to inertial space models because they reduce the number of unknown reaction forces into Lagrange multipliers where each constraint is associated with one Lagrange multiplier. However, these Lagrange multipliers also increase the dimension of the system and can be prohibitive if the system has many constraints. The most suitable alternative approach for multi-constrained systems is the use of quasi-coordinate space models such as the Maggi, Boltzmann–Hamel and the Gibbs–Appell formulations that eliminate the constraints altogether from the process. By doing so, they reduce the size of the working space compared to both the configuration space and the inertial space. Despite these advantages, quasi-coordinate space methods are not yet sufficiently popular in control design applications. The objective of this paper is to not only try to popularize the Boltzmann–Hamel and the Gibbs–Appell formulations but also to show that the two approaches are equivalent. Two examples are provided at the end to show this equivalence by yielding the same equations for the tested systems.

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

Fig. 1
Fig. 2

Similar content being viewed by others

Code Availability

The MATLAB code used in all examples is available by request to the authors.

References

  1. Euler L, Mechanica Vol. 2 (1736). English translation by Ian Bruce at http://www.17centurymaths.com/contents/mechanica2.html

  2. Lagrange JL, Mécanique analytique Vol. 2 (1789) English translation by Auguste Boissonnade and Victor N. Vagliente is available as Volume 191 of Boston Studies in Philosophy of Science Series released in 1997

  3. Maggi GA (1896) Principii della teoria matematica del movimento dei corpi: corso di meccanica razionale (Ulrico Hoepli)

  4. Maggi GA (1901) Di alcune nuove forme delle equazioni della dinamica applicabili ai sistemi anolonomi. Atti Acad Naz Lincei Rend Cl Fis Mat Nat 10(5):287–291

    MATH  Google Scholar 

  5. Kane TR, Levinson DA (1980) Formulation of equations of motion for complex spacecraft. J Guid Control 3(2):99–112. https://doi.org/10.2514/3.55956

    Article  MathSciNet  MATH  Google Scholar 

  6. Boltzmann L (1902) Über die Form der Lagrangeschen Gleichungen für nichtholonome, generalisierte Koordinaten, Vol. 3 of Cambridge Library Collection—Physical Sciences (Edited by F. Hasenöhrl), 682–692 (Cambridge University Press, 2012). Initially published in Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften in Wien, No. 111, pp 1603–1614

  7. Hamel G (1904) Die Lagrange-Eulerschen Gleichungen der Mechanik. Z Math Phys 50

  8. Gibbs JW (1879) On the fundamental formulae of dynamics. Am J Math 2(1):49–64

    Article  MathSciNet  MATH  Google Scholar 

  9. Appell P (1925) Sur une forme generale des equations de la dynamique. Mémorial des Dciences Mathématiques, Fascicule 1 (Gauthier-Villars). https://doi.org/10.1515/crll.1900.121.310

  10. Müller A (2021) On the Hamel coefficients and the Boltzmann–Hamel equations for the rigid body. J Nonlinear Sci 31(2):1–39. https://doi.org/10.1007/s00332-021-09692-7

    Article  MathSciNet  MATH  Google Scholar 

  11. Desloge EA (1988) The Gibbs–Appell equations of motion. Am J Phys 56(9):841–846. https://doi.org/10.1119/1.15463

    Article  MathSciNet  Google Scholar 

  12. Lewis AD (1996) The geometry of the Gibbs–Appell equations and Gauss’ principle of least constraint. Rep Math Phys 38(1):11–28. https://doi.org/10.1016/0034-4877(96)87675-0

    Article  MathSciNet  MATH  Google Scholar 

  13. Udwadia FE, Kalaba RE (1998) The explicit Gibbs–Appell equation and generalized inverse forms. Q Appl Math 56(2):277–288. https://doi.org/10.1090/qam/1622570

    Article  MathSciNet  MATH  Google Scholar 

  14. Greenwood DT (2006) Advanced dynamics. Cambridge University Press, ISBN-13:978-0521029933, https://isbndb.com/book/9780521029933

  15. Baruh H (1999) Analytical dynamics. WCB/McGraw-Hill Boston, ISBN-13:978-0073659770, https://isbndb.com/book/9780073659770

  16. Ginsberg JH (1998) Advanced engineering dynamics. Cambridge University Press, ISBN-13:978-0521646048, https://isbndb.com/book/9780521646048

  17. Arczewski K, Blajer W (1996) A unified approach to the modelling of holonomic and nonholonomic mechanical systems. Math Model Syst 2(3):157–174. https://doi.org/10.1080/13873959608837036

    Article  MathSciNet  MATH  Google Scholar 

  18. Cameron JM, Book WJ (1997) Modeling mechanisms with nonholonomic joints using the Boltzmann–Hamel equations. Int J Robot Res 16(1):47–59. https://doi.org/10.1177/027836499701600104

    Article  Google Scholar 

  19. Tarn TJ, Shoults GA, Yang SP (1996) A dynamic model of an underwater vehicle with a robotic manipulator using Kane’s method. Auton Robot 3(2):269–283. https://doi.org/10.1007/BF00141159

    Article  Google Scholar 

  20. Altuzarra O, Campa Gomes F, Roldan-Paraponiaris C, Pinto C (2015) Dynamic simulation of a tripod based in Boltzmann–Hamel equations. In: 39th Mechanisms and Robotics Conference of International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, vol 5C. https://doi.org/10.1115/DETC2015-47427

  21. Maruskin JM, Bloch AM (2011) The Boltzmann–Hamel equations for the optimal control of mechanical systems with nonholonomic constraints. Int J Robust Nonlinear Control 21(4):373–386. https://doi.org/10.1002/rnc.1598

    Article  MathSciNet  MATH  Google Scholar 

  22. Jarzębowska E (2009) Quasi-coordinates based dynamics modeling and control design for nonholonomic systems. Nonlinear Anal Theory Methods Appl 71(12):118–131. https://doi.org/10.1016/j.na.2008.10.049

  23. Jarzębowska E, Lewandowski R (2006) Modeling and control design using the Boltzmann–Hamel equations: a roller-racer example. IFAC Proc Vol 39(15):236–241. https://doi.org/10.3182/20060906-3-IT-2910.00041

    Article  Google Scholar 

  24. Papastavridis JG (1994) On the Boltzmann–Hamel equations of motion: a vectorial treatment. J Appl Mech 61(2):453–459. https://doi.org/10.1115/1.2901466

  25. Featherstone R, Orin D (2000) Robot dynamics: equations and algorithms. Proceedings 2000 ICRA. Millennium Conference. In: IEEE international conference on robotics and automation. Symposia proceedings (Cat. No. 00CH37065) vol 1, 826–834. https://doi.org/10.1109/ROBOT.2000.844153

  26. Korayem MH, Shafei AM (2013) Application of recursive Gibbs–Appell formulation in deriving the equations of motion of N-viscoelastic robotic manipulators in 3D space using Timoshenko beam theory. Acta Astronaut 83:273–294. https://doi.org/10.1016/j.actaastro.2012.10.026

    Article  Google Scholar 

  27. Korayem MH, Yousefzadeh M, Manteghi S (2017) Dynamics and input–output feedback linearization control of a wheeled mobile cable-driven parallel robot. Multibody SysDyn 40(1):55–73. https://doi.org/10.1007/s11044-016-9543-6

    Article  MathSciNet  MATH  Google Scholar 

  28. Mirzaeinejad H, Shafei AM (2018) Modeling and trajectory tracking control of a two-wheeled mobile robot: Gibbs–Appell and prediction-based approaches. Robotica 36(10):1551–1570. https://doi.org/10.1017/S0263574718000565

    Article  Google Scholar 

  29. Vossoughi G, Pendar H, Heidari Z, Mohammadi S (2008) Assisted passive snake-like robots: conception and dynamic modeling using Gibbs–Appell method. Robotica 26(3):267–276. https://doi.org/10.1017/S0263574707003864

    Article  Google Scholar 

  30. Náprstek J, Fischer C 2018) Appell–Gibbs approach in dynamics of non-holonomic systems. In: Nonlinear systems, Ch. 1. IntechOpen, Rijeka. https://doi.org/10.5772/intechopen.76258

  31. Xiong J et al (2021) Reduced dynamics and control for an autonomous bicycle. 2021 IEEE International Conference on Robotics and Automation (ICRA) 6775–6781. https://doi.org/10.1109/ICRA48506.2021.9560905

  32. Gauss CF (1877) Über Ein Neues Allgemeines Grundgesetz der Mechanik. In: Werke: Fünfter band. Springer, Berlin, pp 23–28. https://doi.org/10.1007/978-3-642-49319-5_2

  33. Tufillaro NB, Abbott TA, Griffiths DJ (1984) Swinging Atwood’s machine. Am J Phys 52(10):895–903. https://doi.org/10.1119/1.13791

    Article  Google Scholar 

Download references

Funding

The work reported in this paper was not funded by any governmental or private agency.

Author information

Author notes

  1. Pius Pius and Majura Selekwa contributed equally to this work.

Authors and Affiliations

  1. Mechatronics and Robotics Laboratory, Department of Mechanical Engineering, North Dakota State University, Dept 2490, PO Box 6050, Fargo, ND, 58108-6050, USA

    Pius Pius & Majura Selekwa

Authors
  1. Pius Pius
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Majura Selekwa
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

The two authors contributed equally in this work. Selekwa started investigating the equivalence of the two methods, and he assigned Pius to establish that equivalence by applying the two methods on different systems as part of a term project. Finally the two authors compiled the findings and decided to share them with the rest of the world through this manuscript.

Corresponding author

Correspondence to Majura Selekwa.

Ethics declarations

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pius, P., Selekwa, M. The equivalence of Boltzmann–Hamel and Gibbs–Appell equations in modeling constrained systems. Int. J. Dynam. Control 11, 2101–2111 (2023). https://doi.org/10.1007/s40435-023-01119-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40435-023-01119-3

Keywords

Search

Navigation