Skip to main content
SpringerLink
Log in

The dynamics and control of a spherical robot with an internal omniwheel platform

  • Published:
Regular and Chaotic Dynamics Aims and scope Submit manuscript

Abstract

This paper deals with the problem of a spherical robot propelled by an internal omniwheel platform and rolling without slipping on a plane. The problem of control of spherical robot motion along an arbitrary trajectory is solved within the framework of a kinematic model and a dynamic model. A number of particular cases of motion are identified, and their stability is investigated. An algorithm for constructing elementary maneuvers (gaits) providing the transition from one steady-state motion to another is presented for the dynamic model. A number of experiments have been carried out confirming the adequacy of the proposed kinematic model.

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. Kilin, A.A., The Dynamics of Chaplygin Ball: The Qualitative and Computer Analysis, Regul. Chaotic Dyn., 2001, vol. 6, no. 3, pp. 291–306.

    Article  MATH  MathSciNet  Google Scholar 

  2. Borisov, A. V., Mamaev, I. S., and Kilin, A.A., The Rolling Motion of a Ball on a Surface: New Integrals and Hierarchy of Dynamics, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 201–219.

    Article  MATH  MathSciNet  Google Scholar 

  3. Chase, R. and Pandya, A., A Review of Active Mechanical Driving Principles of Spherical Robots, Robotics, 2012, vol. 1, no. 1, pp. 3–23.

    Article  Google Scholar 

  4. Crossley, V.A., A Literature Review on the Design of Spherical Rolling Robots, Pittsburgh, Pa., 2006.

    Google Scholar 

  5. Ylikorpi, T. and Suomela, J., Ball-Shaped Robots, in Climbing and Walking Robots: Towards New Applications, H. Zhang (Ed.), Vienna: InTech, 2007.

    Google Scholar 

  6. Mobile Robots: Ball-Shaped Robot and Wheel Robot, A. V. Borisov, I. S. Mamaev, Yu. L. Karavaev (Eds.), Izhevsk: R&C Dynamics, Institute of Computer Science, 2013 (Russian).

    Google Scholar 

  7. Borisov, A. V., Kilin, A.A., and Mamaev, I. S., An Omni-Wheel Vehicle on a Plane and a Sphere, Rus. J. Nonlin. Dyn., 2011, vol. 7, no. 4, pp. 785–801 (Russian).

    Google Scholar 

  8. Chen, W.-H., Chen, Ch.-P., Yu, W.-Sh., Lin, Ch.-H., and Lin, P.-Ch., Design and Implementation of an Omnidirectional Spherical Robot Omnicron, in IEEE/ASME Internat. Conf. on Advanced Intelligent Mechatronics (Kachsiung, Taiwan, 2012), pp. 719–724.

    Google Scholar 

  9. 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.

    Article  MATH  MathSciNet  Google Scholar 

  10. 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.

    Article  MATH  MathSciNet  Google Scholar 

  11. Svinin, M., Morinaga, A., and Yamamoto, M., On the Dynamic Model and Motion Planning for a Spherical Rolling Robot Actuated by Orthogonal Internal Rotors, Regul. Chaotic Dyn., 2013, vol. 18, nos. 1–2, pp. 126–143.

    Article  MATH  MathSciNet  Google Scholar 

  12. Morinaga, A., Svinin, M., and Yamamoto, M., A Motion Planning Strategy for a Spherical Rolling Robot Driven by Two Internal Rotors, IEEE Trans. on Robotics, 2014, vol. 30, no. 4, pp. 993–1002.

    Article  Google Scholar 

  13. Svinin, M., Morinaga, A., and Yamamoto, M., On the Dynamic Model and Motion Planning for a Class of Spherical Rolling Robots, in Proc. of the IEEE Internat. Conf. on Robotics and Automation (ICRA, 14–18 May, 2012), pp. 3226–3231.

  14. Kazakov, A. O., Strange Attractors and Mixed Dynamics in the Problem of an Unbalanced Rubber Ball Rolling on a Plane, Regul. Chaotic Dyn., 2013, vol. 18, no. 5, pp. 508–520.

    Article  MATH  MathSciNet  Google Scholar 

  15. Ivanova, T. B. and Pivovarova, E. N., Dynamics and Control of a Spherical Robot with an Axisymmetric Pendulum Actuator, Rus. J. Nonlin. Dyn., 2013, vol. 9, no. 3, pp. 507–520 (Russian).

    Google Scholar 

  16. Koshiyama, A. and Yamafuji, K., Design and Control of an All-Direction Steering Type Mobile Robot, Int. J. Robot. Res., 1993, vol. 12, no. 5, pp. 411–419.

    Article  Google Scholar 

  17. 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.

    Article  MATH  MathSciNet  Google Scholar 

  18. Kayacan, E., Bayraktaroglu, Z. Y., and Saeys, W., Modeling and Control of a Spherical Rolling Robot: A Decoupled Dynamics Approach, Robotica, 2012, vol. 30, no. 12, pp. 671–680.

    Article  Google Scholar 

  19. Yoon, J.-C., Ahn, S.-S., and Lee, Y.-J., Spherical Robot with New Type of Two-Pendulum Driving Mechanism, in Proc. 15th IEEE Internat. Conf. on Intelligent Engineering Systems (INES) (Poprad, High Tatras, Slovakia, 2011), pp. 275–279.

  20. Zhao, B., Li, M., Yu, H., Hu, H., and Sun, L., Dynamics and Motion Control of a Two Pendulums Driven Spherical Robot, in Proc. of the 2010 IEEE/RSJ Internat. Conf. on Intelligent Robots and Systems (IROS) (Taipei, Taiwan, October 2010), pp. 147–153.

  21. Bolsinov, A.V., Borisov, A. V., and Mamaev, I. S., Rolling of a Ball without Spinning on a Plane: The Absence of an Invariant Measure in a System with a Complete Set of Integrals, Regul. Chaotic Dyn., 2012, vol. 17, no. 6, pp. 571–579.

    Article  MATH  MathSciNet  Google Scholar 

  22. Borisov, A. V., Fedorov, Yu. N., and Mamaev, I. S., Chaplygin Ball over a Fixed Sphere: An Explicit Integration, Regul. Chaotic Dyn., 2008, vol. 13, no. 6, pp. 557–571.

    Article  MATH  MathSciNet  Google Scholar 

  23. Borisov, A. V. and Mamaev, I. S., Rolling of a Non-Homogeneous Ball over a Sphere without Slipping and Twisting, Regul. Chaotic Dyn., 2007, vol. 12, no. 2, pp. 153–159.

    Article  MATH  MathSciNet  Google Scholar 

  24. Borisov, A.V. and Mamaev, I. S., Isomorphism and Hamilton Representation of Some Nonholonomic Systems, Siberian Math. J., 2007, vol. 48, no. 1, pp. 26–36; see also: Sibirsk. Mat. Zh., 2007, vol. 48, no. 1, pp. 33–45.

    Article  MathSciNet  Google Scholar 

  25. Ahn, S.-S. and Lee, Y.-J., Novel Spherical Robot with Hybrid Pendulum Driving Mechanism, Adv. Mech. Eng., 2014, vol. 2014, 456727, 14 pp.

    Google Scholar 

  26. Forbes, J.R., Barfoot, T. D., and Damaren, Ch. J., Dynamic Modeling and Stability Analysis of a Power-Generating Tumbleweed Rover, Multibody Syst. Dyn., 2010, vol. 24, no. 4, pp. 413–439.

    Article  MATH  Google Scholar 

  27. Hartl, A.E. and Mazzoleni, A.P., Dynamic Modeling of a Wind-Driven Tumbleweed Rover Including Atmospheric Effects, J. of Spacecraft and Rockets, 2010, vol. 47, no. 3, pp. 493–502.

    Article  Google Scholar 

  28. Hartl, A. E. and Mazzoleni, A.P., Parametric Study of Spherical Rovers Crossing a Valley, J. Guid. Control Dynam., 2008, vol. 31, no. 3, pp. 775–779.

    Article  Google Scholar 

  29. Hogan, F.R. and Forbes, J.R., Modeling of Spherical Robots Rolling on Generic Surfaces, Multibody Syst. Dyn., 2014, 19 pp.

    Google Scholar 

  30. Hogan, F.R., Forbes, J.R., and Barfoot, T.D., Rolling Stability of a Power-Generating Tumbleweed Rover, J. of Spacecraft and Rockets, 2014, vol. 51, no. 6, pp. 1895–1906.

    Article  Google Scholar 

  31. Lee, J. and Park, W., Design and Path Planning for a Spherical Rolling Robot, in ASME Internat. Mechanical Engineering Congress and Exposition (San Diego, Calif., Nov. 15–21, 2013): Vol. 4A. Dynamics, Vibration and Control, IMECE2013-64994, 8 pp.

  32. Yu, T., Sun, H., Jia, Q., Zhang, Y., and Zhao, W., Stabilization and Control of a Spherical Robot on an Inclined Plane, Res. J. Appl. Sci. Eng. Technology, 2013, vol. 5, no. 6, pp. 2289–2296.

    Google Scholar 

  33. Kilin, A.A. and Karavaev, Yu. L., The Kinematic Control Model for a Spherical Robot with an Unbalanced Internal Omniwheel Platform, Rus. J. Nonlin. Dyn., 2014, vol. 10, no. 4, pp. 497–511 (Russian).

    Google Scholar 

  34. Borisov, A. V., Kilin, A.A., and Mamaev, I. S., Generalized Chaplygin’s Transformation and Explicit Integration of a System with a Spherical Support, Regul. Chaotic Dyn., 2012, vol. 17, no. 2, pp. 170–190.

    Article  MATH  MathSciNet  Google Scholar 

  35. Borisov, A. V., Kilin, A.A., and Mamaev, I. S., Rolling of a Homogeneous Ball over a Dynamically Asymmetric Sphere, Regul. Chaotic Dyn., 2011, vol. 16, no. 5, pp. 465–483.

    Article  MathSciNet  Google Scholar 

  36. Borisov, A.V. and Mamaev, I. S., Rigid Body Dynamics: Hamiltonian Methods, Integrability, Chaos, Izhevsk: R&C Dynamics, Institute of Computer Science, 2005 (Russian).

    Google Scholar 

  37. Kilin, A.A., Karavaev, Yu. L., and Klekovkin, A. V., Kinematic Control of a High Manoeuvrable Mobile Spherical Robot with Internal Omni-Wheeled Platform, Rus. J. Nonlin. Dyn., 2014, vol. 10, no. 1, pp. 113–126 (Russian).

    Google Scholar 

  38. Borisov, A. V. and Mamaev, I. S., Rolling of a Non-Homogeneous Ball over a Sphere without Slipping and Twisting, Regul. Chaotic Dyn., 2007, vol. 12, no. 2, pp. 153–159.

    Article  MATH  MathSciNet  Google Scholar 

  39. Borisov, A. V., Mamaev, I. S., and Bizyaev, I. A., The Hierarchy of Dynamics of a Rigid Body Rolling without Slipping and Spinning on a Plane and a Sphere, Regul. Chaotic Dyn., 2013, vol. 8, no. 3, pp. 277–328; see also: Rus. J. Nonlin. Dyn., 2013, vol. 9, no. 2, pp. 141–202.

    Article  MathSciNet  Google Scholar 

  40. Bolsinov, A.V., Borisov, A. V., and Mamaev, I. S., Rolling of a Ball without Spinning on a Plane: The Absence of an Invariant Measure in a System with a Complete Set of Integrals, Regul. Chaotic Dyn., 2012, vol. 17, no. 6, pp. 571–579; see also: Rus. J. Nonlin. Dyn., 2012, vol. 8, no. 3, pp. 605–616.

    Article  MATH  MathSciNet  Google Scholar 

  41. Koiller, J. and Ehlers, K. M., Rubber Rolling over a Sphere, Regul. Chaotic Dyn., 2007, vol. 12, no. 2, pp. 127–152.

    Article  MATH  MathSciNet  Google Scholar 

  42. Borisov, A.V., Kazakov, A.O., and Sataev, I.R., The Reversal and Chaotic Attractor in the Nonholonomic Model of Chaplygin’s Top, Regul. Chaotic Dyn., 2014, vol. 19, no. 6, pp. 718–733.

    Article  MathSciNet  Google Scholar 

  43. Borisov, A.V., Kazakov, A.O., and Kuznetsov, S.P., Nonlinear Dynamics of the Rattleback: A Nonholonomic Model, Physics-Uspekhi, 2014, vol. 57, no. 5, pp. 453–460; see also: Uspekhi Fiz. Nauk, 2014, vol. 184, no. 5, pp. 493–500.

    Article  Google Scholar 

  44. Borisov, A.V. and Mamaev, I. S., Strange Attractors in Rattleback Dynamics, Physics-Uspekhi, 2003, vol. 46, no. 4, pp. 393–403; see also: Uspekhi Fiz. Nauk, 2003, vol. 173, no. 4, pp. 407–418.

    Article  MathSciNet  Google Scholar 

  45. Borisov, A. V., Kilin, A.A., Mamaev, I. S., New Effects in Dynamics of Rattlebacks, Dokl. Phys., 2006, vol. 51, no. 5, pp. 272–275; see also: Dokl. Akad. Nauk, 2006, vol. 408, no. 2, pp. 192–195.

    Article  MATH  MathSciNet  Google Scholar 

  46. Vetchanin, E. V., Mamaev, I. S., and Tenenev, V. A., The Self-Propulsion of a Body with Moving Internal Masses in a Viscous Fluid, Regul. Chaotic Dyn., 2013, vol. 18, nos. 1–2, pp. 100–117.

    Article  MATH  MathSciNet  Google Scholar 

  47. Bolotin, S. V. and Popova, T. V., On the Motion of a Mechanical System inside a Rolling Ball, Regul. Chaotic Dyn., 2013, vol. 18, nos. 1–2, pp. 159–165.

    Article  MATH  MathSciNet  Google Scholar 

  48. Rutstam, N., High Frequency Behavior of a Rolling Ball and Simplification of the Separation Equation, Regul. Chaotic Dyn., 2013, vol. 18, no. 3, pp. 226–236.

    Article  MATH  MathSciNet  Google Scholar 

  49. Borisov, A. V. and Mamaev, I. S., Topological Analysis of an Integrable System Related to the Rolling of a Ball on a Sphere, Regul. Chaotic Dyn., 2013, vol. 18, no. 4, pp. 356–371.

    Article  MATH  MathSciNet  Google Scholar 

  50. Borisov, A. V. and Mamaev, I. S., The Dynamics of the Chaplygin Ball with a Fluid-Filled Cavity, Regul. Chaotic Dyn., 2013, vol. 18, no. 5, pp. 490–496.

    Article  MATH  MathSciNet  Google Scholar 

  51. Gonchenko, A. S., Gonchenko, S.V., and Kazakov, A.O., Richness of Chaotic Dynamics in Nonholonomic Models of a Celtic Stone, Regul. Chaotic Dyn., 2013, vol. 18, no. 5, pp. 521–538.

    Article  MATH  MathSciNet  Google Scholar 

  52. Borisov, A. V., Kilin, A.A., and Mamaev, I. S., The Problem of Drift and Recurrence for the Rolling Chaplygin Ball, Regul. Chaotic Dyn., 2013, vol. 18, no. 6, pp. 832–859.

    Article  MATH  MathSciNet  Google Scholar 

  53. Takano, H., Spin Reversal of a Rattleback with Viscous Friction, Regul. Chaotic Dyn., 2014, vol. 19, no. 1, pp. 81–99.

    Article  MathSciNet  Google Scholar 

  54. Mamaev, I. S. and Ivanova, T. B., The Dynamics of a Rigid Body with a Sharp Edge in Contact with an Inclined Surface in the Presence of Dry Friction, Regul. Chaotic Dyn., 2014, vol. 19, no. 1, pp. 116–139.

    Article  MathSciNet  Google Scholar 

  55. 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.

    Article  MathSciNet  Google Scholar 

  56. Bizyaev, I.A., Borisov, A.V., and Mamaev, I. S., The Dynamics of Nonholonomic Systems Consisting of a Spherical Shell with a Moving Rigid Body Inside, Regul. Chaotic Dyn., 2014, vol. 19, no. 2, pp. 198–213.

    Article  MATH  MathSciNet  Google Scholar 

  57. Burlakov, D. and Treschev, D., A Rigid Body on a Surface with Random Roughness, Regul. Chaotic Dyn., 2014, vol. 18, no. 3, pp. 296–309.

    Article  MathSciNet  Google Scholar 

  58. Borisov, A.V., Erdakova, N.N., Ivanova, T.B., and Mamaev, I. S., The Dynamics of a Body with an Axisymmetric Base Sliding on a Rough Plane, Regul. Chaotic Dyn., 2014, vol. 19, no. 6, pp. 607–634.

    Article  MathSciNet  Google Scholar 

  59. Borisov, A. V., Mamaev, I. S., and Kilin, A.A., Stability of Steady Rotations in the Nonholonomic Routh Problem, Regul. Chaotic Dyn., 2008, vol. 13, no. 4, pp. 239–249.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. M. T. Kalashnikov Izhevsk State Technical University, ul. Studencheskaya 7, Izhevsk, 426069, Russia

    Yury L. Karavaev

  2. Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034, Russia

    Alexander A. Kilin

Authors
  1. Yury L. Karavaev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexander A. Kilin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yury L. Karavaev.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Karavaev, Y.L., Kilin, A.A. The dynamics and control of a spherical robot with an internal omniwheel platform. Regul. Chaot. Dyn. 20, 134–152 (2015). https://doi.org/10.1134/S1560354715020033

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1560354715020033

MSC2010 numbers

Keywords

Search

Navigation