Advertisement

Regular and Chaotic Dynamics

, Volume 24, Issue 2, pp 145–170 | Cite as

On the Normal Force and Static Friction Acting on a Rolling Ball Actuated by Internal Point Masses

  • Vakhtang PutkaradzeEmail author
  • Stuart M. Rogers
Article
  • 9 Downloads

Abstract

The goal of this paper is to investigate the normal and tangential forces acting at the point of contact between a horizontal surface and a rolling ball actuated by internal point masses moving in the ball’s frame of reference. The normal force and static friction are derived from the equations of motion for a rolling ball actuated by internal point masses that move inside the ball’s frame of reference, and, as a special case, a rolling disk actuated by internal point masses. The masses may move along one-dimensional trajectories fixed in the ball’s and disk’s frame. The dynamics of a ball and disk actuated by masses moving along one-dimensional trajectories are simulated numerically and the minimum coefficients of static friction required to prevent slippage are computed.

Keywords

nonholonomic mechanics holonomic mechanics rolling balls rolling disks 

MSC2010 numbers

37J60 70E18 70E60 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., How to Control Chaplygin’s Sphere Using Rotors, Regul. Chaotic Dyn., 2012, vol. 17, nos. 3–4, pp. 258–272.MathSciNetzbMATHGoogle Scholar
  2. 2.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., How to Control Chaplygin’s Sphere Using Rotors: 2, Regul. Chaotic Dyn., 2013, vol. 18, nos. 1–2, pp. 144–158.MathSciNetzbMATHGoogle Scholar
  3. 3.
    Burkhardt, M. R. and Burdick, J.W., Reduced Dynamical Equations for Barycentric Spherical Robots, in Proc. of the IEEE Internat. Conf. on Robotics and Automation (Stockholm, Sweden, 2016), pp. 2725–2732.Google Scholar
  4. 4.
    Kilin, A.A., Pivovarova, E.N., and Ivanova, T.B., Spherical Robot of Combined Type: Dynamics and Control, Regul. Chaotic Dyn., 2015, vol. 20, no. 6, pp. 716–728.MathSciNetzbMATHGoogle Scholar
  5. 5.
    Gajbhiye, S. and Banavar, R. N., Geometric Modeling and Local Controllability of a Spherical Mobile Robot Actuated by an Internal Pendulum, Internat. J. Robust Nonlinear Control, 2016, vol. 26, no. 11, pp. 2436–2454.MathSciNetzbMATHGoogle Scholar
  6. 6.
    Das, T., Mukherjee, R., and Yuksel, H., Design Considerations in the Development of a Spherical Mobile Robot, in Proc. of the 15th SPIE Annual International Symposium on Aerospace/Defense Sensing, Simulation, and Controls (Orlando, Fla., Apr 2001): Vol. 4364, pp. 61–71.Google Scholar
  7. 7.
    Javadi, A.H.A. and Mojabi, P., Introducing August: A Novel Strategy for an Omnidirectional Spherical Rolling Robot, in Proc. of the IEEE Internat. Conf. on Robotics and Automation (Washington,D.C., 2002), pp. 3527–3533.Google Scholar
  8. 8.
    Shen, J., Schneider, D.A., and Bloch, A.M., Controllability and Motion Planning of a Multibody Chaplygin’s Sphere and Chaplygin’s Top, Internat. J. Robust Nonlinear Control, 2008, vol. 18, no. 9, pp. 905–945.MathSciNetzbMATHGoogle Scholar
  9. 9.
    Ilin, K. I., Moffatt, H.K., and Vladimirov, V.A., Dynamics of a Rolling Robot, Proc. Natl. Acad. Sci. USA, 2017, vol. 114, no. 49, pp. 12858–12863.MathSciNetzbMATHGoogle Scholar
  10. 10.
    Putkaradze, V. and Rogers, S.M., On the Dynamics of a Rolling Ball Actuated by Internal Point Masses, Meccanica, 2018, vol. 53, no. 15, pp. 3839–3868.MathSciNetGoogle Scholar
  11. 11.
    Editorial Discussion on Some Papers by G.M.Rosenblat, Nelin. Dinam., 2009, vol. 5, no. 4, pp. 621–624 (Russian).Google Scholar
  12. 12.
    Ivanova, T. B. and Pivovarova, E. N., Comments on the Paper by A.V.Borisov, A.A.Kilin, I. S.Mamaev “How To Control the Chaplygin Ball Using Rotors: 2”, Regul. Chaotic Dyn., 2014, vol. 19, no. 1, pp. 140–143.zbMATHGoogle Scholar
  13. 13.
    Vas’kin, V. V. and Naimushina, O. S., A study of the Motion of Axisymmetric Sphere with a Shifted Center of mass on a Rough Plane, Vestn. Udmurtsk. Univ. Fiz. Khim., 2012, vol. 2, pp. 10–17 (Russian).Google Scholar
  14. 14.
    Wagner, A., Heffel, E., Arrieta, A. F., Spelsberg-Korspeter, G., and Hagedorn, P., Analysis of an Oscillatory Painlevé–Klein Apparatus with a Nonholonomic Constraint, Differ. Equ. Dyn. Syst., 2013, vol. 21, nos. 1–2, pp. 149–157.MathSciNetzbMATHGoogle Scholar
  15. 15.
    Ivanova, T. B. and Mamaev, I. S., Dynamics of a Painlevé–Appel System, J. Appl. Math. Mech., 2016, vol. 80, no. 1, pp. 7–15; see also: Prikl. Mat. Mekh., 2016, vol. 80, no. 1, pp. 11–23.MathSciNetGoogle Scholar
  16. 16.
    Ivanov, A.P., On Detachment Conditions in the Problem on the Motion of a Rigid Body on a Rough Plane, Regul. Chaotic Dyn., 2008, vol. 13, no. 4, pp. 355–368.MathSciNetzbMATHGoogle Scholar
  17. 17.
    Ivanov, A.P., Geometric Representation of Detachment Conditions in Systems with Unilateral Constraints, Regul. Chaotic Dyn., 2008, vol. 13, no. 5, pp. 435–442.MathSciNetzbMATHGoogle Scholar
  18. 18.
    Blau, P. J., Friction Science and Technology: From Concepts to Applications, 2nd ed., Boca Raton, Fla.: CRC, 2008.Google Scholar
  19. 19.
    Kozlov, V.V., On the Dry-Friction Mechanism, Dokl. Phys., 2011, vol. 56, no. 4, pp. 256–257; see also: Dokl. Ross. Akad. Nauk, 2011, vol. 437, no. 6, pp. 766–767.Google Scholar
  20. 20.
    Kozlov, V.V., Friction by Painlevé and Lagrangian Mechanics, Dokl. Phys., 2011, vol. 56, no. 6, pp. 355–358; see also: Dokl. Ross. Akad. Nauk, 2011, vol. 438, no. 6, pp. 758–761.Google Scholar
  21. 21.
    Balandin, D. V., Komarov, M.A., and Osipov, G. V., A Motion Control for a Spherical Robot with Pendulum Drive, J. Comput. Sys. Sc. Int., 2013, vol. 52, no. 4, pp. 650–663; see also: Izv. Ross. Akad. Nauk. Teor. Sist. Upr., 2013, no. 4, pp. 150–163.zbMATHGoogle Scholar
  22. 22.
    Holm, D.D., Geometric Mechanics: P.2. Rotating, Translating and Rolling, 2nd ed., London: Imperial College Press, 2011.zbMATHGoogle Scholar
  23. 23.
    Putkaradze, V. and Rogers, S.M., On the Optimal Control of a Rolling Ball Robot Actuated by Internal Point Masses, arXiv:1708.03829v5 (2018).Google Scholar
  24. 24.
    Bai, Y., Svinin, M., and Yamamoto, M., Dynamics-Based Motion Planning for a Pendulum-Actuated Spherical Rolling Robot, Regul. Chaotic Dyn., 2018, vol. 23, no. 4, pp. 372–388.MathSciNetGoogle Scholar
  25. 25.
    Chaplygin, S.A., On a Motion of a Heavy Body of Revolution on a Horizontal Plane, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 119–130.MathSciNetzbMATHGoogle Scholar
  26. 26.
    Rozenblat, G.M., On the Separation-Free Motions of a Rigid Body on a Plane, Dokl. Phys., 2007, vol. 52, no. 8, pp. 447–449; see also: Dokl. Ross. Akad. Nauk, 2007, vol. 415, no. 5, pp. 622–624.zbMATHGoogle Scholar
  27. 27.
    Ascher, U.M., Mattheij, R. M. M., and Russell, R. D., Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Philadelphia,Pa.: SIAM, 1995.zbMATHGoogle Scholar
  28. 28.
    Hairer, E. and Wanner, G., Solving Ordinary Differential Equations:2. Stiff and Differential-Algebraic Problems, 2nd ed., Springer Series in Computational Mathematics, vol. 14, Berlin: Springer, 1996.zbMATHGoogle Scholar
  29. 29.
    Squire, W. and Trapp, G., Using Complex Variables to Estimate Derivatives of Real Functions, SIAM Rev., 1998, vol. 40, no. 1, pp. 110–112.MathSciNetzbMATHGoogle Scholar
  30. 30.
    Martins, J.R.R.A., Sturdza, P., and Alonso, J. J., The Connection between the Complex-Step Derivative Approximation and Algorithmic Differentiation, in Proc. of the 39th AIAA Aerospace Sciences Meeting (Reno,Nev., 2001): AIAA Paper 2001–0921, 11 pp.Google Scholar
  31. 31.
    Martins, J. R. R. A., Sturdza, P., and Alonso, J. J., The Complex-Step Derivative Approximation, ACM Trans. Math. Software, 2003, vol. 29, no. 3, pp. 245–262.MathSciNetzbMATHGoogle Scholar
  32. 32.
    ASM Handbook: Vol.18. Friction, Lubrication, and Wear Technology, G. E.Totten (Ed.), ASM, 2017.Google Scholar
  33. 33.
    Schröder, D., Transferring the Bearing Using a Strapdown Inertial Measurement Unit, in Applications of Geodesy to Engineering, K. Linkwitz, V.Eisele, H. J.Mönicke (Eds.), Berlin: Springer, 1993, pp. 25–38.Google Scholar
  34. 34.
    Stuelpnagel, J., On the Parametrization of the Three-Dimensional Rotation Group, SIAM Rev., 1964, vol. 6, no. 4, pp. 422–430.MathSciNetzbMATHGoogle Scholar
  35. 35.
    Frisvad, J.R., Building an Orthonormal Basis from a 3D Unit Vector without Normalization, J. Graph. Tools, 2012, vol. 16, no. 3, pp. 151–159.Google Scholar
  36. 36.
    Ivanova, T. B. and Pivovarova, E. N., Dynamics and Control of a Spherical Robot with an Axisymmetric Pendulum Actuator, Nonlinear Dynamics & Mobile Robotics, 2013, vol. 1, no. 1, pp. 71–85.Google Scholar
  37. 37.
    Ivanova, T. B., Kilin, A.A., and Pivovarova, E. N., Controlled Motion of a Spherical Robot with Feedback: 2, J. Dyn. Control Syst., 2019, vol. 25, no. 1, pp. 1–16.MathSciNetzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Department of Mathematical and Statistical Sciences, Faculty of ScienceUniversity of AlbertaEdmontonCanada
  2. 2.Institute for Mathematics and its Applications, College of Science and EngineeringUniversity of MinnesotaMinneapolisUSA

Personalised recommendations