Archive of Applied Mechanics

, Volume 89, Issue 10, pp 2193–2208 | Cite as

Dynamics and motion control of a capsule robot with an opposing spring

  • Armen Nunuparov
  • Felix Becker
  • Nikolay Bolotnik
  • Igor ZeidisEmail author
  • Klaus Zimmermann


Non-classical locomotion systems have the perspective for a wide application in the vast fields of bio-medical and maintenance technology. Capsule bots are small, simple, and reliable realizations with a great potential for practical application. In this paper, the motion of a capsule-type mobile robot along a straight line on a rough horizontal plane is studied applying analytical and experimental methods. The robot consists of a housing and an internal body attached to the housing by a spring. The motion of the system is generated by a force that acts between the housing and the internal body and changes periodically in a pulse-width mode. The average velocity of the motion of the robot is studied as a function of the excitation parameters. The results from the model-based and experimental investigations agree with each other. It can be concluded that the presented robot design can be a basis for the creation of mobile robotic systems with locomotion properties that can be controlled by the parameters of a periodic actuation force.


Mobile robot Motion control Capsubot Locomotion Self-propulsion Vibration-driven system 



The research work reported here was partly supported by the Deutsche Forschungsgemeinschaft (Grant ZIM 540 / 19-2) and the Russian Foundation for Basic Research (Grant 17-51-12025).


  1. 1.
    Abbott, J.J., Nagy, Z., Beyeler, F., Nelson, B.: Robotics in the small. IEEE Robot. Autom. Mag. 14(2), 92–103 (2007)CrossRefGoogle Scholar
  2. 2.
    Becker, F., Zimmermann, K., Volkova, T., Minchenya, V.T.: An amphibious vibration-driven microrobot with a piezoelectric actuator. Regul. Chaot. Dyn. 18(1–2), 63–74 (2013)zbMATHCrossRefGoogle Scholar
  3. 3.
    Becker, F., Lysenko, V., Minchenya, V.T., Kunze, O., Zimmermann, K.: Locomotion principles for microrobots based on vibrations. Microactuators and Micromechanisms, pp. 91–102. Springer, Cham (2017)CrossRefGoogle Scholar
  4. 4.
    Bogue, R.: Miniature and microrobots: a review of recent developments. Ind. Robot. 42(2), 98–102 (2015)CrossRefGoogle Scholar
  5. 5.
    Bolotnik, N.N., Figurina, T.Y.: Optimal control of the rectilinear motion of a rigid body on a rough plane my means of the motion of two internal masses. J. Appl. Math. Mech. 72(2), 126–135 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bolotnik, N.N., Figurina, T.Y., Chernousko, F.L.: Optimal control of the rectilinear motion of a two-body system in a resistive medium. J. Appl. Math. Mech. 76(1), 1–4 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Bolotnik, N.N., Nunuparov, A.M., Chashchukhin, V.G.: Capsule-type vibration-driven robot with an electromagnetic actuator and an opposing spring: dynamics and control of motion. J. Comput. Syst. Sci. Int. 55(6), 986–1000 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Chashchukhin, V.G.: Simulation of dynamics and determination of control parameters of inpipe minirobot. J. Comput. Syst. Sci. Int. 47(5), 806–811 (2008)zbMATHCrossRefGoogle Scholar
  9. 9.
    Chernousko, F.L.: On the motion of a body containing a movable internal mass. Dokl. Phys. 50(11), 593–597 (2005)CrossRefGoogle Scholar
  10. 10.
    Chernousko, F.L.: Analysis and optimization of the motion of a body controlled by a movable internal mass. J. Appl. Math. Mech. 70(6), 915–941 (2006)MathSciNetGoogle Scholar
  11. 11.
    Chernousko, F.L.: The optimal periodic motions of a two-mass system in a resistant medium. J. Appl. Math. Mech. 72(2), 116–125 (2008)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Chernousko, F.L.: Motion of a body along a plane under the influence of movable internal masses. Dokl. Phys. 61(10), 494–498 (2016)CrossRefGoogle Scholar
  13. 13.
    Chernousko, F.L.: Two-dimensional motions of a robot under the influence of movable internal masses. In: Matveenko, V.P., Krommer, M., Belyaev, A.K., Irschik, H. (eds.) Dynamics and Control of Advanced Structures and Machines: Contributions from the 3rd International Workshop, Perm, Russia, pp. 49–56. Springer International Publishing, Cham (2019)CrossRefGoogle Scholar
  14. 14.
    Diller, E., Sitti, M.: Micro-scale mobile robotics. Found Trends Robot. 2(3), 143–259 (2013)CrossRefGoogle Scholar
  15. 15.
    Egorov, A.G., Zakharova, O.S.: The energy-optimal motion of a vibration-driven robot in a resistive medium. J. Appl. Math. Mech. 74(4), 443–451 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Egorov, A.G., Zakharova, O.S.: The energy-optimal motion of a vibration-driven robot in a medium with a inherited law of resistance. J. Comput. Syst. Sci. Int. 54(3), 495–503 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Fang, H.B., Xu, J.: Dynamic analysis and optimization of a three-phase control mode of a mobile system with an internal mass. J. Vib. Control 74(4), 443–451 (2011)MathSciNetGoogle Scholar
  18. 18.
    Farahani, A.A., Suratgar, A.A., Talebi, H.A.: Optimal controller design of legless piezo capsubot movement. Int. J. Adv. Robot. Syst. 10(2), 126 (2013)CrossRefGoogle Scholar
  19. 19.
    Figurina, T.Y.: Optimal control of the motion of a two-body system along a straight line. J. Comput. Syst. Sci. Int. 46(2), 227–233 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Gradetsky, V.G., Knyazkov, M.M., Fomin, M.M., Chashchukhin, V.G.: Mechanics of miniature robots. Nauka (2010). (in Russian)Google Scholar
  21. 21.
    Hariri, H.H., Soh, G.S., Foong, S., Wood, K.: Locomotion study of a standing wave driven piezoelectric miniature robot for bi-directional motion. IEEE T Robot 33(3), 742–747 (2017)CrossRefGoogle Scholar
  22. 22.
    Huda, M.N., Yu, H.: Modelling and motion control of a novel double parallel mass capsubot. In: 18th IFAC World Congress, IFAC Proceedings, vol. 44(1), pp. 8120–8125 (2011)Google Scholar
  23. 23.
    Huda, M.N., Yu, H.: Trajectory tracking control of an underactuated capsubot. Auton. Robots 39(2), 183–198 (2015)CrossRefGoogle Scholar
  24. 24.
    Huda, M.N., Yu, H., Wane, S.O.: Self-contained capsubot propulsion mechanism. Int. J. Autom. Comput. 8(3), 348–356 (2011)CrossRefGoogle Scholar
  25. 25.
    Huda, M.N., Yu, H., Goodwin, M.J.: Experimental study of a capsubot for two dimensional movements. In: Proceedings of 2012 UKACC International Conference on Control, pp 108–113 (2012)Google Scholar
  26. 26.
    Huda, M.N., Yu, H., Cang, S.: Behaviour-based control approach for the trajectory tracking of an underactuated planar capsule robot. IET Control Theory Appl. 9(2), 163–175 (2015)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Ivanov, A.P., Sakharov, A.V.: Dynamics of a rigid body carrying moving masses and a rotor on a rough plane. Nelineinaya Dinamika (Russ. J. Nonlinear Dyn.) 8(4), 763–772 (2012). (in Russian)CrossRefGoogle Scholar
  28. 28.
    Levenberg, K.: A method for the solution of certain non-linear problems in least squares. Q. Appl. Math. 2(2), 164–168 (1944)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Li, H., Furuta, K., Chernousko, F.L.: Motion generation of the capsubot using internal force and static friction. In: Proceedings of the 45th IEEE Conference on Decision and Control, pp 6575–6580 (2006)Google Scholar
  30. 30.
    Liu, P., Huda, M.N., Tang, Z., Sun, L.: A self-propelled robotic system with a visco-elastic joint: dynamics and motion analysis. Eng. Comput. (2019). CrossRefGoogle Scholar
  31. 31.
    Liu, Y., Yu, H., Yang, T.: Analysis and control of a capsubot. In: 17th IFAC World Congress, IFAC Proceedings, vol. 41(2), pp. 756 – 761 (2008)Google Scholar
  32. 32.
    Liu, Y., Pavlovskaya, E., Hendry, D., Wiercigroch, M.: Vibro-impact responses of a capsule systems with various friction models. Int. J. Mech. Sci. 72, 39–54 (2013a)CrossRefGoogle Scholar
  33. 33.
    Liu, Y., Wiercigroch, M., Pavlovskaya, E., Peng, Z.K.: Forward and backward motion control of a vibro-impact capsule system. Int. J. Mech. Sci. 74, 2–11 (2013b)CrossRefGoogle Scholar
  34. 34.
    Liu, Y., Wiercigroch, M., Pavlovskaya, E., Yu, H.: Modelling of a vibro-impact capsule system. Int. J. Non-Linear Mech. 70, 30–46 (2015)CrossRefGoogle Scholar
  35. 35.
    Liu, Y., Islam, S., Pavlovskaya, E., Wiercigroch, M.: Optimization of the vibro-impact capsule system. J. Mech. Eng. 62, 430–439 (2016a)CrossRefGoogle Scholar
  36. 36.
    Liu, Y., Pavlovskaya, E., Wiercigroch, M.: Experimental verification of the vibro-impact capsule model. Nonlinear Dyn. 83, 1029–1041 (2016b)CrossRefGoogle Scholar
  37. 37.
    Marquardt, D.W.: An algorithm for least-squares estimation of nonlinear parameters. J. Soc. Ind. Appl. Math. 11(2), 431–441 (1963)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Rao, C.R.: Linear Statistical Inference and Its Applications. Wiley, New York (1965)zbMATHGoogle Scholar
  39. 39.
    Rios, S.A., Fleming, A.J., Yong, Y.K.: Miniature resonant ambulatory robot. IEEE Robot. Autom. Lett. 2(1), 337–343 (2017)CrossRefGoogle Scholar
  40. 40.
    Sahu, B., Taylor, C., Leang, K.: Emerging challenges of microactuators for nanoscale positioning assembly and manipulation. J. Manuf. Sci. Eng. 132(3), 030917-1–030917-16 (2010)CrossRefGoogle Scholar
  41. 41.
    Sakharov, A.V.: Rotation of a body with two movable internal masses on a rough plane. J. Appl. Math. Mech. 79(2), 132–141 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Sitti, M., Ceylan, H., Hu, W., Giltinan, J., Turan, M., Yim, S., Diller, E.: Biomedical applications of untethered mobile milli/microrobots. Proc. IEEE 103(2), 205–224 (2015)CrossRefGoogle Scholar
  43. 43.
    Sun, L., Sun, P., Qin, X.: Study on micro robot in small pipe. In: Proc. of International Conference on Control’ 98, Swansea, pp 1212–1217 (1998)Google Scholar
  44. 44.
    Yan, Y., Liu, Y., Liao, M.: A comparative study of the vibro-impact capsule systems with one-sided and two-sided constraints. Nonlinear Dyn. 89, 1063–1087 (2015)CrossRefGoogle Scholar
  45. 45.
    Yu, H., Huda, M.N., Wane, S.O.: A novel acceleration profile for the motion control of capsubots. In: 2011 IEEE International Conference on Robotics and Automation, pp 2437–2442 (2011)Google Scholar
  46. 46.
    Zhan, X., Xu, J., Fang, H.: Planar locomotion of a vibration-driven system with two internal masses. Appl. Math. Model. 40(2), 871–885 (2016)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Zhan, X., Xu, J., Fang, H.: A vibration-driven planar locomotion robot–shell. Robotica 36(9), 1402–1420 (2018)CrossRefGoogle Scholar
  48. 48.
    Zimmermann, K., Zeidis, I., Behn, C.: Mechanics of Terrestrial Locomotion with a Focus on Nonpedal Motion Systems. Springer, Berlin (2009a)zbMATHGoogle Scholar
  49. 49.
    Zimmermann, K., Zeidis, I., Bolotnik, N., Pivovarov, M.: Dynamics of a two-module vibration-driven system moving along a rough horizontal plane. Multibody Syst. Dyn. 22, 199–219 (2009b)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Armen Nunuparov
    • 1
  • Felix Becker
    • 3
  • Nikolay Bolotnik
    • 2
  • Igor Zeidis
    • 3
    Email author
  • Klaus Zimmermann
    • 3
  1. 1.Moscow Institute of Physics and TechnologyDolgoprudnyRussia
  2. 2.Laboratory of Robotics and MechatronicsInstitute for Problems in Mechanics of the Russian Academy of SciencesMoscowRussia
  3. 3.Technical Mechanics GroupTechnische Universität IlmenauIlmenauGermany

Personalised recommendations