Journal of Dynamical and Control Systems

, Volume 24, Issue 3, pp 497–510 | Cite as

Controlled Motion of a Spherical Robot with Feedback. I

  • Tatyana B. Ivanova
  • Alexander A. Kilin
  • Elena N. Pivovarova


In this paper, we develop a model of a controlled spherical robot with an axisymmetric pendulum-type actuator with a feedback system suppressing the pendulum’s oscillations at the final stage of motion. According to the proposed approach, the feedback depends on phase variables (the current position and velocities) and does not depend on the type of trajectory. We present integrals of motion and partial solutions, analyze their stability, and give examples of computer simulation of motion using feedback to illustrate compensation of the pendulum’s oscillations.


Spherical robot Nonholonomic constraint Control Feedback 

Mathematics Subject Classification (2010)

70F25 70Q05 93D15 



The authors extend their gratitude to A. V. Borisov, I. S. Mamaev and Yu. L. Karavaev for fruitful discussions of the results obtained.

The work of T. B. Ivanova (Sections 1 and 2) was carried out within the framework of the state assignment to the Ministry of Education and Science of Russia (1.2404.2017/4.6) and was supported by a grant of RFBR (15-08-09261-a). The work of A. A. Kilin (Section 3) was carried out at MIPT under project 5–100 for state support for leading universities of the Russian Federation. The work of E. N. Pivovarova (Section 4) was supported by the grant of RSF (15-12-20035).


  1. 1.
    Balandin DV, Komarov MA, Osipov GV. A motion control for a spherical robot with pendulum drive. J Comput Syst Sci Int 2013;52(4):650–663.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bizyaev IA, Borisov AV, Mamaev IS. The dynamics of nonholonomic systems consisting of a spherical shell with a moving rigid body inside. Regul Chaotic Dyn 2014;19(2):198–213.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bloch A. Nonholonomic mechanics and control. New York: Springer; 2015.CrossRefGoogle Scholar
  4. 4.
    Bloch AM, Krishnaprasad PS, Marsden JE, Sanchez de Alvarez G. Stabilization of rigid body dynamics by internal and external torques. Automatica 1992; 28:745–756.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bloch AM, Leonard NE, Marsden JE. Stabilization of mechanical systems using controlled lagrangians. Proceedings 36th conference on decision and control; 1997. p. 2356–2361.Google Scholar
  6. 6.
    Bloch AM, Reyhanoglu M, McClamroch NH. Control and stabilization of nonholonomic dynamic systems. IEEE Trans Autom Control 1992;37:1746–1757.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Borisov AV, Kilin AA, Mamaev IS. How to control Chaplygin’s sphere using rotors. Regul Chaotic Dyn 2012;17(3–4):258–272.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Borisov AV, Kilin AA, Mamaev IS. How to control the Chaplygin’s ball using rotors. II. Regul Chaotic Dyn 2013;18(1–2):144–158.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Borisov AV, Mamaev IS. Rigid body dynamics. Hamiltonian methods, integrability, chaos. Moscow–Izhevsk: Institute of Computer Science; 2005.zbMATHGoogle Scholar
  10. 10.
    Borisov AV, Mamaev IS. Two non-holonomic integrable problems tracing back to Chaplygin’s. Regul Chaotic Dyn 2012;17(2):191–198.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Borisov AV, Mamaev IS, Karavaev YL. Mobile robots: robot-wheel and robot-ball. Russian: Moscow–Izhevsk: Institute of Computer Science; 2013.Google Scholar
  12. 12.
    Chaplygin SA. On Some Generalization of the Area Theorem with Applications to the Problem of Rolling Balls. Regul Chaotic Dyn 2012;17(2):199–217.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Chase R, Pandya A. A review of active mechanical driving principles of spherical robots. Robotics 2012;1(1):3–23.CrossRefGoogle Scholar
  14. 14.
    Chowdhury AR, Vibhute A, Soh GS, Foong SH, Wood KL. Implementing caterpillar inspired roll control of a spherical robot. Proceedings 2017 IEEE international conference on robotics and automation; 2017. p. 4167–4174.Google Scholar
  15. 15.
    Gajbhiye S, Banavar RN. Geometric tracking control for a nonholonomic system: a spherical robot. IFAC-PapersOnLine 2016;49(18):820–825.CrossRefzbMATHGoogle Scholar
  16. 16.
    Halme A, Schonberg T, Wang Y. Motion control of a spherical mobile robot. Proceedings 4th international workshop on advanced motion control; 1996. p. 259–264.Google Scholar
  17. 17.
    Ivanov AP. On the control of a robot ball using two omniwheels. Regul Chaotic Dyn 2015;20(4):441–448.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ivanova TB, Pivovarova EN. Dynamics and control of a spherical robot with an axisymmetric pendulum actuator. Rus J Nonlin Dyn 2013;9(3):507–520. (Russian).Google Scholar
  19. 19.
    Ivanova TB, Pivovarova EN. 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;19(1):140–143.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Karavaev YL, Kilin AA. Nonholonomic Dynamics and Control of a spherical robot with an internal omniwheel platform: theory and experiments. Proc Steklov Inst Math 2016;295(1):158–167.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Karavaev YL, Kilin AA. The dynamics and control of a spherical robot with an internal omniwheel platform. Regul Chaotic Dyn 2015;20(2):134–152.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kayacan E, Bayraktaroglu ZY, Saeys W. Modeling and control of a spherical rolling robot a decoupled dynamics approach. Robotica 2012;30(4):671–680.CrossRefGoogle Scholar
  23. 23.
    Kayacan E, Kayacan E, Ramon H, Saeys W. Adaptive Neuro-Fuzzy control of a spherical rolling robot using sliding mode control Theory-Based online learning algorithm. IEEE Trans Cybern 2013;43(1):170–179.CrossRefGoogle Scholar
  24. 24.
    Kilin AA, Karavaev YL. Experimental research of dynamic of spherical robot of combined type. Rus J Nonlin Dyn 2015;11(4):721–734. (Russian).zbMATHGoogle Scholar
  25. 25.
    Kilin AA, Pivovarova EN, Ivanova TB. Spherical robot of combined type: dynamics and control. Regul Chaotic Dyn 2015;20(6):716–728.MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Lewis AD, Ostrowski JP, Burdickz JW, Murray RM. Nonholonomic mechanics and locomotion: the Snakeboard example. Proc 1994 IEEE Int Conf Robot Autom. 1994;3:2391–2397.Google Scholar
  27. 27.
    Madhushani TWU, Maithripala DHS, Berg JM. 2017. Feedback Regularization and Geometric PID Control for Trajectory Tracking of Coupled Mechanical Systems: Hoop Robots on an Inclined Plane. arXiv:1609.09557v2.
  28. 28.
    Madhushani TWU, Maithripala DHS, Wijayakulasooriya JV, Berg JM. 2017. Semi-globally Exponential Trajectory Tracking for a Class of Spherical Robots. arXiv:1608.01494v2.
  29. 29.
    Morinaga A, Svinin M, Yamamoto M. A motion planning strategy for a spherical rolling robot driven by two internal rotors. IEEE Trans Robot 2014;30(4): 993–1002.CrossRefGoogle Scholar
  30. 30.
    Muraleedharan N, Cohen DS, Isenberg DR. 2016. Omnidirectional locomotion control of a pendulum driven spherical robot. SoutheastCon.
  31. 31.
    Ostrowski JP, Desai JP, Kumar V. Optimal gait selection for nonholonomic locomotion systems. Int J Robot Res 2000;19(3):225–237.CrossRefGoogle Scholar
  32. 32.
    Pivovarova EN, Ivanova TB. Stability analysis of periodic solutions in the problem of the rolling of a ball with a pendulum. Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp’yuternye Nauki 2012;4:146–155. (Russian).CrossRefzbMATHGoogle Scholar
  33. 33.
    Puopolo MG, Jacob JD. Velocity control of a cylindrical rolling robot by shape changing. Adv Robotics 2016;30(23):1484–1494.CrossRefGoogle Scholar
  34. 34.
    Roozegar M, Mahjoob MJ. Modelling and control of a non-holonomic pendulum-driven spherical robot moving on an inclined plane: simulation and experimental results. IET Control Theory Appl 2017;11(4):541–549.MathSciNetCrossRefGoogle Scholar
  35. 35.
    Roozegar M, Mahjoob MJ, Ayati M. Adaptive estimation of nonlinear parameters of a nonholonomic spherical robot using a modified fuzzy-based speed gradient algorithm. Regul Chaotic Dyn 2017;22(3):226–238.MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Sugiyama Y, Shiotsu A, Yamanaka M, Hirai S. Circular/Spherical Robots for crawling and jumping. Proceedings 2005 IEEE international conference on robotics and automation; 2005. p. 3595–3600.Google Scholar
  37. 37.
    Svinin M, Bai Y, Yamamoto M. Dynamic model and motion planning for a pendulum-actuated spherical rolling robot. Proceedings 2015 IEEE international conference on robotics and automation; 2015. p. 656–661.Google Scholar
  38. 38.
    Svinin M, Morinaga A, Yamamoto M. On the dynamic model and motion planning for a spherical rolling robot actuated by orthogonal internal rotors. Regul Chaotic Dyn 2013;18(1–2):126–143.MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Terekhov G, Pavlovsky V. Controlling spherical mobile robot in a two-parametric friction model. MATEC Web Conf 2017;113:02007. (5 pages).CrossRefGoogle Scholar
  40. 40.
    Ylikorpi TJ, Halme AJ, Forsman PJ. Dynamic modeling and obstacle-crossing capability of flexible pendulum-driven ball-shaped robots. Robot Auton Syst 2017;87: 269–280.CrossRefGoogle Scholar
  41. 41.
    Ylikorpi T, Forsman P, Halme A. Dynamic obstacle overcoming capability of pendulum-driven ball-shaped robots. 17th IASTED international conference on robotics and applications; 2014. p. 329–338.Google Scholar
  42. 42.
    Yu T, Sun H, Jia Q, Zhang Y, Zhao W. Stabilization and control of a spherical robot on an inclined plane. Res J Appl Sci Eng Technol 2013;5(6):2289–2296.Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  • Tatyana B. Ivanova
    • 1
  • Alexander A. Kilin
    • 2
  • Elena N. Pivovarova
    • 1
  1. 1.Udmurt State UniversityIzhevskRussia
  2. 2.Moscow Institute of Physics and TechnologyDolgoprudnyRussia

Personalised recommendations