Skip to main content
SpringerLink
Log in

Stability analysis of a waveboard multibody model with toroidal wheels

  • Published:
Multibody System Dynamics Aims and scope Submit manuscript

Abstract

This paper analyses the stability of a waveboard, the skateboard consisting in two articulated platforms, coupled by a torsion bar and supported of two caster wheels. The waveboard presents an interesting propelling mechanism, since the rider can achieve a forward motion by means of an oscillatory lateral motion of the platforms. In this paper, the system is described using a multibody model with nonholonomic constraints.

To study the stability of the forward upright motion with constant speed, the equations of motion are linearized with respect to a reference motion. The employed numerical linearization procedure is valid for general multibody systems with holonomic and nonholonomic constraints. In practice, the employed approach makes use of the Jacobian matrix, which is expressed in terms of any of the main design parameters of the waveboard. This paper introduces a sensitivity analysis of the eigenvalues with respect to the forward speed, the casters’ inclination angle, the torsional stiffness of the torsion bar, the aspect ratio of the toroidal wheels and the mass of the human rider. Lastly, a summary with the influence of these design parameters on the external actuations exerted by the rider in the waveboard maneuvering is shown.

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
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27
Fig. 28
Fig. 29
Fig. 30
Fig. 31
Fig. 32
Fig. 33

Similar content being viewed by others

References

  1. Kinugasa, K., Ishikawa, M., Sugimoto, Y., Osuka, K.: Modeling and control of casterboard robot. IFAC Proc. Vol. 46(23), 785–790 (2013)

    Article  Google Scholar 

  2. Agrawal, A., Zaini, H.M., Dear, T., Choset, H.: Experimental gait analysis of waveboard locomotion. In: ASME 2016 Dynamic Systems and Control Conference, pp. V002T22A011–V002T22A011. Am. Soc. Mech. Eng., New York (2016)

    Google Scholar 

  3. Su, B., Wang, T., Wang, J., Kuang, S.: Kinematic mechanism and path planning of the essboard. Sci. China, Technol. Sci. 56(6), 1499–1516 (2013)

    Article  Google Scholar 

  4. Wang, T., Su, B., Kuang, S., Wang, J.: On kinematic mechanism of a two-wheel skateboard: the essboard. J. Mech. Robot. 5(3), 034503 (2013)

    Article  Google Scholar 

  5. Zeng, Z., Chen, D., Zhang, T., Guan, Y.: Kinematic modelling and analysis of an ess-board-like robot. In: 2018 IEEE International Conference on Robotics and Biomimetics (ROBIO), pp. 1371–1376. IEEE Press, New York (2018)

    Chapter  Google Scholar 

  6. Ito, S., Niwa, K., Sugiura, S., Morita, R.: An autonomous mobile robot with passive wheels propelled by a single motor. Robot. Auton. Syst. 122, 103310 (2019)

    Article  Google Scholar 

  7. Ito, S., Sugiura, S., Masuda, Y., Nohara, S., Morita, R.: Mechanism and control of a one-actuator mobile robot incorporating a torque limiter. J. Intell. Robot. Syst. 97(2), 431–448 (2020)

    Article  Google Scholar 

  8. Gadzhiev, M.M., Kuleshov, A.S., Bukanov, A.I.: Geometric constraints in the problem of motion of a caster board. J. Math. Sci. 248, 1–5 (2020)

    Article  MathSciNet  Google Scholar 

  9. García-Agúndez, A., García-Vallejo, D., Freire, E.: Study of the forward locomotion of a three-dimensional multibody model of a waveboard by inverse dynamics. Mech. Mach. Theory 149, 103826 (2020)

    Article  Google Scholar 

  10. Fukai, R., Yagi, K., Mori, Y.: Dynamic model for using casterboard by a humanoid robot. Adv. Robot. 34(10), 648–660 (2020)

    Article  Google Scholar 

  11. DasGupta, A.: Dynamics of a waveboard simplified. Proc. R. Soc. A 476(2244), 20200486 (2020)

    Article  MathSciNet  Google Scholar 

  12. Ruina, A.: Nonholonomic stability aspects of piecewise holonomic systems. Rep. Math. Phys. 42(1–2), 91–100 (1998)

    Article  MathSciNet  Google Scholar 

  13. Zenkov, D.V., Bloch, A.M., Marsden, J.E.: The energy–momentum method for the stability of non-holonomic systems. Dyn. Stab. Syst. 13(2), 123–165 (1998)

    Article  MathSciNet  Google Scholar 

  14. Coleman, M.J., Holmes, P.J.: Motions and stability of a piecewise holonomic system: the discrete Chaplygin sleigh. Regul. Chaotic Dyn. 4(2), 55–77 (1999)

    Article  MathSciNet  Google Scholar 

  15. Bloch, A.M.: Asymptotic Hamiltonian dynamics: the Toda lattice, the three-wave interaction and the non-holonomic Chaplygin sleigh. Physica D 141(3–4), 297–315 (2000)

    Article  MathSciNet  Google Scholar 

  16. Bizyaev, I.A., Borisov, A.V., Mamaev, I.S.: The Chaplygin sleigh with parametric excitation: chaotic dynamics and nonholonomic acceleration. Regul. Chaotic Dyn. 22(8), 955–975 (2017)

    Article  MathSciNet  Google Scholar 

  17. Hubbard, M.: Mechanics of skate boards. J. Appl. Mech. 46, 931 (1979)

    Article  Google Scholar 

  18. Kremnev, A.V., Kuleshov, A.S.: Nonlinear dynamics and stability of the skateboard. Discrete Contin. Dyn. Syst., Ser. S 3(1), 85 (2010)

    MathSciNet  MATH  Google Scholar 

  19. Ostrowski, J., Lewis, A., Murray, R., Burdick, J.: Nonholonomic mechanics and locomotion: the snakeboard example. In: Proceedings of the 1994 IEEE International Conference on Robotics and Automation, pp. 2391–2397. IEEE Press, New York (1994)

    Chapter  Google Scholar 

  20. Ostrowski, J.P., Desai, J.P., Kumar, V.: Optimal gait selection for nonholonomic locomotion systems. Int. J. Robot. Res. 19(3), 225–237 (2000)

    Article  Google Scholar 

  21. Kuleshov, A.S.: Further development of the mathematical model of a snakeboard. Regul. Chaotic Dyn. 12(3), 321–334 (2007)

    Article  MathSciNet  Google Scholar 

  22. Basu-Mandal, P., Chatterjee, A., Papadopoulos, J.M.: Hands-free circular motions of a benchmark bicycle. Proc. R. Soc. A, Math. Phys. Eng. Sci. 463(2084), 1983–2003 (2007)

    MathSciNet  MATH  Google Scholar 

  23. Meijaard, J.P., Papadopoulos, J.M., Ruina, A., Schwab, A.L.: Linearized dynamics equations for the balance and steer of a bicycle: a benchmark and review. Proc. R. Soc. A, Math. Phys. Eng. Sci. 463(2084), 1955–1982 (2007)

    MathSciNet  MATH  Google Scholar 

  24. Escalona, J.L., Recuero, A.M.: A bicycle model for education in multibody dynamics and real-time interactive simulation. Multibody Syst. Dyn. 27(3), 383–402 (2012)

    Article  MathSciNet  Google Scholar 

  25. Xiong, J., Wang, N., Liu, C.: Stability analysis for the Whipple bicycle dynamics. Multibody Syst. Dyn. 48(3), 311–335 (2020)

    Article  MathSciNet  Google Scholar 

  26. Xiong, J., Wang, N., Liu, C.: Bicycle dynamics and its circular solution on a revolution surface. Acta Mech. Sin. 36(1), 220–233 (2020)

    Article  MathSciNet  Google Scholar 

  27. Sharp, R.S.: The stability and control of motorcycles. J. Mech. Eng. Sci. 13(5), 316–329 (1971)

    Article  Google Scholar 

  28. Sharp, R.S.: Stability, control and steering responses of motorcycles. Veh. Syst. Dyn. 35(4–5), 291–318 (2001)

    Article  Google Scholar 

  29. Sharp, R.S., Limebeer, D.J.N.: A motorcycle model for stability and control analysis. Multibody Syst. Dyn. 6(2), 123–142 (2001)

    Article  Google Scholar 

  30. Schiehlen, W.: Multibody system dynamics: roots and perspectives. Multibody Syst. Dyn. 1(2), 149–188 (1997)

    Article  MathSciNet  Google Scholar 

  31. Nayfeh, A.: Nonlinear Interactions: Analytical, Computational, and Experimental Methods. Wiley, New York (2000)

    MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by Grant FPU18/05598 of the Spanish Ministry of Science, Innovation and Universities.

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering and Manufacturing, Universidad de Sevilla, Seville, Spain

    A. G. Agúndez & D. García-Vallejo

  2. Department of Applied Mathematics II, Universidad de Sevilla, Seville, Spain

    E. Freire

  3. Department of Mechanical Engineering, Lappeenranta University of Technology, Lappeenranta, Finland

    A. M. Mikkola

Authors
  1. A. G. Agúndez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. García-Vallejo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. E. Freire
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. A. M. Mikkola
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. G. Agúndez.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Agúndez, A.G., García-Vallejo, D., Freire, E. et al. Stability analysis of a waveboard multibody model with toroidal wheels. Multibody Syst Dyn 53, 173–203 (2021). https://doi.org/10.1007/s11044-021-09780-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11044-021-09780-2

Keywords

Search

Navigation