phone +7 (3412) 91 60 92
mail imi@udsu.ru
ru-RU Russian
Home » Issues » Archive of Issues » 2019 - 3 issue » pp. 408-421

Archive of Issues

Russia Yekaterinburg
Year
2019
Volume
29
Issue
3
Pages
408-421
<<
>>
Section Mechanics
Title Quaternion model of programmed control over motion of a Chaplygin ball
Author(-s) Mityushov E.A.a, Misyura N.E.a, Berestova S.A.a
Affiliations Ural Federal Universitya
Abstract This paper deals with the problem of program control of the motion of a dynamically asymmetric balanced ball on the plane using three flywheel motors, provided that the ball rolls without slipping. The center of mass of the mechanical system coincides with the geometric center of the ball. Control laws are found to ensure the motion of the ball along the basic trajectories (line and circle), as well as along an arbitrarily given piecewise smooth trajectory on the plane. In this paper, we propose a quaternion model of ball motion. The model does not require using the traditional trigonometric functions. Kinematic equations are written in the form of linear differential equations eliminating the disadvantages associated with the use of Euler angles. The solution of the problem is carried out using the quaternion function of time, which is determined by the type of trajectory and the law of motion of the point of contact of the ball with the plane. An example of ball motion control is given and a visualization of the ball-flywheel system motion in a computer algebra package is presented.
Keywords quaternions, control, nonholonomic connection, geometric dynamics, smooth movement, spherical robot
UDC 531.133.1, 531.36, 514.758.3
MSC 70Q05, 34H05, 93C15
DOI 10.20537/vm190310
Received 27 June 2019
Language Russian
Citation Mityushov E.A., Misyura N.E., Berestova S.A. Quaternion model of programmed control over motion of a Chaplygin ball, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2019, vol. 29, issue 3, pp. 408-421.
References
  1. Bolotin S.V. The problem of optimal control of a Chaplygin ball by internal rotors, Nelin. Dinam., 2012, vol. 8, no. 4, pp. 837-852 (in Russian). https://doi.org/10.20537/nd1204011
  2. Borisov A.V., Kilin A.A., Mamaev I.S. How to control the Chaplygin sphere using rotors, Mobil'nye roboty: robot-koleso i robot-shar (Mobile robots: robot wheel and robot ball), Eds.: A.V. Borisov, I.S. Mamaev, Yu.L. Karavaev. Moscow-Izhevsk: Institute of Computer Science, 2013, pp. 131-168 (in Russian).
  3. Borisov A.V., Mamaev I.S. Dinamika tverdogo tela (Rigid body dynamics), Izhevsk: Regular and Chaotic Dynamics, 2001, 384 p.
  4. Borisov A.V., Kilin A.A., Mamaev I.S. How to control the Chaplygin sphere using rotors, Nelin. Dinam., 2012, vol. 8, no. 2, pp. 289-307 (in Russian). https://doi.org/10.20537/nd1202006
  5. Borisov A.V., Kilin A.A., Mamaev I.S. How to control the Chaplygin ball using rotors. II, Nelin. Dinam., 2013, vol. 9, no. 1, pp. 59-76 (in Russian). https://doi.org/10.20537/nd1301006
  6. Golubev Yu.F. Quaternion algebra in rigid body kinematics, Keldysh Institute Preprints, 2013, no. 39, 23 p. (In Russian). http://mi.mathnet.ru/eng/ipmp1789
  7. Mashtakov A.P., Sachkov Yu.L. Extremal trajectories and the asymptotics of the Maxwell time in the problem of the optimal rolling of a sphere on a plane, Sbornik: Mathematics, 2011, vol. 202, no. 9, pp. 1347-1371. https://doi.org/10.1070/SM2011v202n09ABEH004190
  8. Pavlovskii V.E., Terekhov G.P. Control of the mobile spherical information robot with three orthogonal flywheels, Spetstekhnika i Svyaz', 2012, no. 3, pp. 19-24 (in Russian). https://elibrary.ru/item.asp?id=17878482
  9. Terekhov G.P. Control of the ball with three wheels on orthogonal axes, Sovremennaya mekhatronika. Sbornik nauchnykh trudov Vserossiiskoi nauchnoi shkoly (Modern mechatronics. Collection of scientific papers of the All-Russian Scientific School), Orekhovo-Zuyevo, 2011, pp. 126-130 (in Russian).
  10. Terekhov G.P., Pavlovsky V.E. Control of the spherical robot by means of fly-wheels, Keldysh Institute Preprints, 2017, no. 16, 31 p. (In Russian). http://mi.mathnet.ru/eng/ipmp2232
  11. Chaplygin S.A. On a ball's rolling on a horizontal plane, Mat. Sb., 1903, vol. 24, no. 1, pp. 139-168 (in Russian). http://mi.mathnet.ru/eng/msb6665
  12. Furuse Y., Hirano T., Ishikawa M. Dynamical analysis of spherical mobile robot utilizing off-centered internal mass distribution, IFAC-PapersOnLine, 2015, vol. 48, issue 13, pp. 176-181. https://doi.org/10.1016/j.ifacol.2015.10.235
  13. Gajbhiye S., Banavar R.N. Geometric approach to tracking and stabilization for a spherical robot actuated by internal rotors, 2015, arXiv: 1511.00428v2 [cs.SY]. https://arxiv.org/abs/1511.00428v2
  14. Gajbhiye S., Banavar R.N. Geometric tracking control for a nonholonomic system: a spherical robot, IFAC-PapersOnLine, 2016, vol. 49, issue 18, pp. 820-825. https://doi.org/10.1016/j.ifacol.2016.10.267
  15. Ishikawa M., Kitayoshi R., Sugie T. Volvot: A spherical mobile robot with eccentric twin rotors, 2011 IEEE International Conference on Robotics and Biomimetics, 2011, pp. 1462-1467. https://doi.org/10.1109/ROBIO.2011.6181496
  16. Joshi V.A., Banavar R.N. Motion analysis of a spherical mobile robot, Robotica, 2009, vol. 27, issue 3, pp. 343-353. https://doi.org/10.1017/S0263574708004748
  17. Joshi V.A., Banavar R.N., Hippalgaonkar R. Design and analysis of a spherical mobile robot, Mechanism and Machine Theory, 2010, vol. 45, issue 2, pp. 130-136. https://doi.org/10.1016/j.mechmachtheory.2009.04.003
  18. Kilin A.A. The dynamics of Chaplygin ball: the qualitative and computer analysis, Regular and Chaotic Dynamics, 2001, vol. 6, no. 3, pp. 291-306. https://doi.org/10.1070/RD2001v006n03ABEH000178
  19. Kilin A.A., Pivovarova E.N. Chaplygin top with a periodic gyrostatic moment, Russian Journal of Mathematical Physics, 2018, vol. 25, no. 4, pp. 509-524. https://doi.org/10.1134/S1061920818040088
  20. Kilin A.A., Pivovarova E.N., Ivanova T.B. Spherical robot of combined type: dynamics and control, Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 716-728. https://doi.org/10.1134/S1560354715060076
  21. Madhushani T.W.U., Maithripala D.H.S., Wijayakulasooriya J.V., Berg J.M. Semi-globally exponential trajectory tracking for a class of spherical robots, Automatica, 2017, vol. 85, pp. 327-338. https://doi.org/10.1016/j.automatica.2017.07.060
  22. Morinaga A., Svinin M., Yamamoto M. A motion planning strategy for a spherical rolling robot driven by two internal rotors, IEEE Transactions on Robotics, 2014, vol. 30, no. 4, pp. 993-1002. https://doi.org/10.1109/TRO.2014.2307112
  23. Muralidharan V., Mahindrakar A.D. Geometric controllability and stabilization of spherical robot dynamics, IEEE Transactions on Automatic Control, 2015, vol. 60, no. 10, pp. 2762-2767. https://doi.org/10.1109/TAC.2015.2404512
  24. Ohsawa T. Geometric kinematic control of a spherical rolling robot, Journal of Nonlinear Science, 2019, pp. 1-25. https://doi.org/10.1007/s00332-019-09568-x
  25. Otani T., Urakubo T., Maekawa S., Tamaki H., Tada Y. Position and attitude control of a spherical rolling robot equipped with a gyro, 9th IEEE International Workshop on Advanced Motion Control, 2006, pp. 416-421. https://doi.org/10.1109/AMC.2006.1631695
  26. Svinin M., Hosoe Sh. Motion planning algorithms for a rolling sphere with limited contact area, IEEE Transactions on Robotics, 2008, vol. 24, no. 3, pp. 612-625. https://doi.org/10.1109/TRO.2008.921568
  27. Svinin M., Morinaga A., Yamamoto M. An analysis of the motion planning problem for a spherical rolling robot driven by internal rotors, 2012 IEEE/RSJ International Conference on Intelligent Robots and Systems, 2012, pp. 414-419. https://doi.org/10.1109/IROS.2012.6386077
  28. Svinin M., Morinaga A., Yamamoto M. On the dynamic model and motion planning for a class of spherical rolling robots, 2012 IEEE International Conference on Robotics and Automation, 2012, pp. 3226-3231. https://doi.org/10.1109/ICRA.2012.6224795
  29. Svinin M., Morinaga A., Yamamoto M. On the dynamic model and motion planning for a spherical rolling robot actuated by orthogonal internal rotors, Regular and Chaotic Dynamics, 2013, vol. 18, nos. 1-2, pp. 126-143. https://doi.org/10.1134/S1560354713010097
  30. Sugiyama Y., Hirai S. Crawling and jumping by a deformable robot, Experimental Robotics IX, Eds.: M.H. Ang, O. Khatib, vol. 21, Berlin: Springer, 2006, pp. 281-291. https://doi.org/10.1007/11552246_27
  31. Tao Y., Hanxu S., Qingxuan J., Wei Zh. Path following control of a spherical robot rolling on an inclined plane, Sensors and Transducers, 2013, vol. 21, pp. 42-47.
  32. Movement of the ball with three internal flywheel engines in a straight line, Mathematical Modeling in UrFU. https://youtu.be/kUta7xlhB-o
  33. Movement of the ball with three internal engines around the circle, Mathematical Modeling in UrFU. https://youtu.be/I8BvP84_YcI
  34. Movement of the ball with three internal engines on the trajectory of “slalom”, Mathematical Modeling in UrFU. https://youtu.be/aHbboXhw1ZY
  35. Movement of the ball with three internal engines along the clothoid, Mathematical Modeling in UrFU. https://youtu.be/-r566MX3keg
  36. Movement of the ball with three internal engines along the astroid, Mathematical Modeling in UrFU. https://youtu.be/7mo-XR0QebI
  37. Animation of a ball moving on a plane without slipping, Mathematical Modeling in UrFU. https://youtu.be/9PVqnH1Ap9A
Full text
/doc/issues-2019/19-03-10.pdf
<< Previous article
Next article >>