Regular and Chaotic Dynamics

, Volume 20, Issue 2, pp 153–172 | Cite as

Dynamics and control of an omniwheel vehicle

  • Alexey V. Borisov
  • Alexander A. Kilin
  • Ivan S. Mamaev
Article

Abstract

A nonholonomic model of the dynamics of an omniwheel vehicle on a plane and a sphere is considered. A derivation of equations is presented and the dynamics of a free system are investigated. An explicit motion control algorithm for the omniwheel vehicle moving along an arbitrary trajectory is obtained.

Keywords

omniwheel roller-bearing wheel nonholonomic constraint dynamical system invariant measure integrability controllability 

MSC2010 numbers

70F25 70E18 70E55 70E60 

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References

  1. 1.
    Alves, J. and Dias, J., Design and Control of a Spherical Mobile Robot, J. Syst. Control Eng., 2003, vol. 217, pp. 457–467.Google Scholar
  2. 2.
    Asama, H., Sato, M., Bogoni, L., Kaetsu, H., Mitsumoto, A., and Endo, I., Development of an Omni-Directional Mobile Robot with 3 DOF Decoupling Drive Mechanism, in Proc. of the 1995 IEEE Internat. Conf. on Robotics and Automation (Nagoya, Japan, 21–27 May 1995): Vol. 2, pp. 1925–1930.Google Scholar
  3. 3.
    Ashmore, M. and Barnes, N., Omni-Drive Robot Motion on Curved Paths: The Fastest Path between Two Points Is Not a Straight-Line, in AI 2002: Advances in Artificial Intelligence Lecture Notes in Comput. Sci., vol. 2557, Berlin: Springer, 2002, pp. 225–236.CrossRefGoogle Scholar
  4. 4.
    Bizyaev, I. A., Nonintegrability and Obstructions to the Hamiltonianization of a Nonholonomic Chaplygin Top, Dokl. Math., 2014, vol. 90, no. 2, pp. 631–634; see also: Dokl. Akad. Nauk, 2014, vol. 458, no. 4, pp. 398–401.CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    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.CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    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.CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    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: Nelin. Dinam., 2012, vol. 8, no. 3, pp. 605–616.CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    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.CrossRefMathSciNetGoogle Scholar
  9. 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; see also: Nelin. Dinam., 2012, vol. 8, no. 2, pp. 289–307.CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., How To Control the Chaplygin Ball Using Rotors: 2, Regul. Chaotic Dyn., 2013, vol. 18, nos. 1–2, pp. 144–158; see also: Nelin. Dinam., 2013, vol. 9, no. 1, pp. 59–76.CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., An Omni-Wheel Vehicle on a Plane and a Sphere, Nelin. Dinam., 2011, vol. 7, no. 4, pp. 785–801 (Russian).Google Scholar
  12. 12.
    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.CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Borisov, A.V. and Mamaev, I. S., Dynamics of a Rigid Body: Hamiltonian Methods, Integrability, Chaos, 2nd ed., Izhevsk: R&C Dynamics, Institute of Computer Science, 2005 (Russian).Google Scholar
  14. 14.
    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; see also: Nelin. Dinam., 2012, vol. 8, no. 1, pp. 103–111.CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    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; see also: Nelin. Dinam., 2012, vol. 8, no. 5, pp. 957–975.CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    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: Nelin. Dinam., 2013, vol. 9, no. 2, pp. 141–202.CrossRefMathSciNetGoogle Scholar
  17. 17.
    Borisov, A.V. and Tsygvintsev, A.V., Kowalewski Exponents and Integrable Systems of Classic Dynamics: 1, 2, Regul. Chaotic Dyn., 1996, vol. 1, no. 1, pp. 15–37 (Russian).MathSciNetGoogle Scholar
  18. 18.
    Borisov, A. V. and Tsygvintsev, A.V., Kovalevskaya’s Method in Rigid Body Dynamics, J. Appl. Math. Mech., 1997, vol. 61, no. 1, pp. 27–32; see also: Prikl. Mat. Mekh., 1997, vol. 61, no. 1, pp. 30–36.CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Campion, G., Bastin, G., and d’Andréa-Novel, B., Structural Properties and Classification of Kinematic and Dynamic Models of Wheeled Mobile Robots, IEEE Trans. Robot. Autom., 1996, vol. 12, no. 1, pp. 47–62.CrossRefGoogle Scholar
  20. 20.
    Ferrière, L., Campion, G., and Raucent, B., ROLLMOBS, a New Drive System for Omnimobile Robots, Robotica, 2001, vol. 19, pp. 1–9.CrossRefGoogle Scholar
  21. 21.
    Fedorov, Yu.N. and Kozlov, V.V., Various Aspects of n-Dimensional Rigid Body Dynamics, Amer. Math. Soc. Transl. Ser. 2, 1995, vol. 168, pp. 141–171.MathSciNetGoogle Scholar
  22. 22.
    Fedorov, Yu.N., Maciejewski, A. J., and Przybylska, M., The Generalized Euler — Poinsot Rigid Body Equations: Explicit Elliptic Solutions, J. Phys. A, 2013, vol. 46, no. 41, 415201, 26 pp.CrossRefMathSciNetGoogle Scholar
  23. 23.
    García-Naranjo, L.C., Maciejewski, A. J., Marrero, J.C., and Przybylska, M., The Inhomogeneous Suslov Problem, Phys. Lett. A, 2014, vol. 378, nos. 32–33, pp. 2389–2394.CrossRefMATHGoogle Scholar
  24. 24.
    García-Naranjo, L.C., Marrero, J.C., Non-Existence of an Invariant Measure for a Homogeneous Ellipsoid Rolling on the Plane, Regul. Chaotic Dyn., 2013, vol. 18, no. 4, pp. 372–379.CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Gonchenko, A. S., Gonchenko, S.V., and Kazakov, A.O., Richness of Chaotic Dynamics in the Nonholonomic Model of Celtic Stone, Regul. Chaotic Dyn., 2013, vol. 18, no. 5, pp. 521–538.CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Karavaev, Yu. L. and Kilin, A.A., The Dynamics and Control of a Spherical Robot with an Internal Omniwheel Platform, Regul. Chaotic Dyn., 2015, vol. 20, no. 2, pp. 134–152.CrossRefMathSciNetGoogle Scholar
  27. 27.
    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.CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Kilin, A.A. and Karavaev, Yu. L., The Kinematic Control Model for a Spherical Robot with an Unbalanced Internal Omniwheel Platform, Nelin. Dinam., 2014, vol. 10, no. 4, pp. 497–511 (Russian).Google Scholar
  29. 29.
    Kilin, A.A., Karavaev, Yu. L., and Klekovkin, A. V., Kinematic Control of a High Manoeuvrable Mobile Spherical Robot with Internal Omni-Wheeled Platform, Nelin. Dinam., 2014, vol. 10, no. 1, pp. 113–126 (Russian).Google Scholar
  30. 30.
    Lobas, L. G., Nonholonomic Models of Vehicles, Kiev: Naukova Dumka, 1986 (Russian).MATHGoogle Scholar
  31. 31.
    Martynenko, Yu. G., Stability of Steady Motions of a Mobile Robot with Roller-Carrying Wheels and a Displaced Centre of Mass, J. Appl. Math. Mech., 2010, vol. 74, no. 4, pp. 436–442; see also: Prikl. Mat. Mekh., 2010, vol. 74, no. 4, pp. 610–619.CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Martynenko, Yu. G. and Formal’skii, A. M., On the Motion of a Mobile Robot with Roller-Carrying Wheels, J. Comput. Sys. Sc. Int., 2007, vol. 46, no. 6, pp. 976–983; see also: Izv. Ross. Akad. Nauk. Teor. Sist. Upr., 2007, no. 6, pp. 142–149.CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Muir, P.F. and Neuman, C.P., Kinematic Modeling for Feedback Control of an Omnidirectional Wheeled Mobile Robot, in Autonomous Robot Vehicles, I. J. Cox, G.T. Wilfong (Eds.), New York: Springer, 1990, pp. 25–31.CrossRefGoogle Scholar
  34. 34.
    Nagarajan, U., Mampetta, A., Kantor, G.A., and Hollis, R. L., State Transition, Balancing, Station Keeping, and Yaw Control for a Dynamically Stable Single Spherical Wheel Mobile Robot, IEEE Internat. Conf. on Robotics and Automation (ICRA) (Kobe, Japan, 2009), pp. 998–1003.Google Scholar
  35. 35.
    Nagarajan, U., Kantor, G., and Hollis, R. L., Trajectory Planning and Control of an Underactuated Dynamically Stable Single Spherical Wheeled Mobile Robot, IEEE Internat. Conf. on Robotics and Automation (ICRA) (Kobe, Japan, 2009), pp. 3743–3748.Google Scholar
  36. 36.
    Ostrowski, J.P., The Mechanics and Control of Undulatory Robotic Locomotion: Ph.D.Thesis, Pasadena, CA, California Institute of Technology, 1995. 149 p.Google Scholar
  37. 37.
    Ostrowski, J.P., Desai, J.P., and Kumar, V., Optimal Gait Selection for Nonholonomic Locomotion Systems, Internat. J. Robotics Res., 2000, vol. 19, no. 3, pp. 225–237.CrossRefGoogle Scholar
  38. 38.
    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.CrossRefMATHMathSciNetGoogle Scholar
  39. 39.
    Tatarinov, Ya.V., Equations of Classical Mechanics in New Form, Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., 2003, no. 3, pp. 67–76 (Russian).Google Scholar
  40. 40.
    Tsiganov, A.V., On the Lie Integrability Theorem for the Chaplygin Ball, Regul. Chaotic Dyn., 2014, vol. 19, no. 2, pp. 185–197.CrossRefMATHMathSciNetGoogle Scholar
  41. 41.
    Tsiganov, A.V., One Invariant Measure and Different Poisson Brackets for Two Non-Holonomic Systems, Regul. Chaotic Dyn., 2012, vol. 17, no. 1, pp. 72–96.CrossRefMATHMathSciNetGoogle Scholar
  42. 42.
    Watanabe, K., Shiraishi, Y., Tzafestas, S. G., Tang, J., Fukuda, T., Feedback Control of an Omnidirectional Autonomous Platform for Mobile Service Robots, J. Intell. Robot. Syst., 1998, vol. 22, nos. 3–4, pp. 315–330.CrossRefGoogle Scholar
  43. 43.
    Yoon, J.-C., Ahn, S.-S., and Lee, Y.-J., Spherical Robot with New Type of Two-Pendulum Driving Mechanism, Proc. 15th IEEE Internat. Conf. on Intelligent Engineering Systems (INES) (Poprad, High Tatras, Slovakia, 2011), pp. 275–279.Google Scholar
  44. 44.
    Zhan, Q., Cai, Y., and Yan, C., Design, Analysis and Experiments of an Omni-Directional Spherical Robot, IEEE Internat. Conf. on Robotics and Automation (ICRA) (Shanghai, China, 2011), pp. 4921–4926.Google Scholar
  45. 45.
    Zobova, A.A., Application of Laconic Forms of the Equations of Motion in the Dynamics of Nonholonomic Mobile Robots, Nelin. Dinam., 2011, vol. 7, no. 4, pp. 771–783 (Russian).Google Scholar
  46. 46.
    Zobova, A.A. and Tatarinov, Ya.V., The Dynamics of an Omni-Mobile Vehicle, J. Appl. Math. Mech., 2009, vol. 73, no. 1, pp. 8–15; see also: Prikl. Mat. Mekh., 2009, vol. 73, no. 1, pp. 13–22.CrossRefMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  • Alexey V. Borisov
    • 1
    • 2
  • Alexander A. Kilin
    • 1
  • Ivan S. Mamaev
    • 1
    • 3
  1. 1.Udmurt State UniversityIzhevskRussia
  2. 2.Moscow Institute of Physics and TechnologyDolgoprudnyiRussia
  3. 3.Kalashnikov Izhevsk State Technical UniversityIzhevskRussia

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