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Regul. Chaotic Dyn., 2012, Volume 17, Issue 3-4, Pages 258–272 (Mi rcd337)  

This article is cited in 53 scientific papers (total in 53 papers)

How to Control Chaplygin’s Sphere Using Rotors

Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev

Institute of Computer Science, Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034 Russia

Abstract: In the paper we study the control of a balanced dynamically non-symmetric sphere with rotors. The no-slip condition at the point of contact is assumed. The algebraic controllability is shown and the control inputs that steer the ball along a given trajectory on the plane are found. For some simple trajectories explicit tracking algorithms are proposed.

Keywords: non-holonomic constraint, non-holonomic distribution, control, Chow–Rashevsky theorem, drift

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation 11.G34.31.0039
1.1248.2011
02.740.11.0195
MK-8428.2010.1
This research was done at the Udmurt State University and was supported by the Grant Program of the Government of the Russian Federation for state support of scientific research conducted under the supervision of leading scientists at Russian institutions of higher professional education (Contract No11.G34.31.0039), by the Federal Target Program “Development of Scientific Potential of Higher Schools” (2012–2014, 1.1248.2011 “Nonholonomic Dynamical Systems and Control Problems”) and the Federal Target Program “Scientific and Scientific-Pedagogical Personnel of Innovative Russia” (2009—2013, “Scientific-Educational Center “Regular and Chaotic Dynamics” State Contract No 02.740.11.0195). The work of A. A.Kilin was subsidized by the Presidential Grant of the Russian Federation for Support of young Candidates of Science MK-8428.2010.1.


DOI: https://doi.org/10.1134/S1560354712030045


Bibliographic databases:

Document Type: Article
MSC: 37J60, 37J35, 70E18, 70F25, 70H45
Received: 10.03.2012
Language: English

Citation: Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev, “How to Control Chaplygin’s Sphere Using Rotors”, Regul. Chaotic Dyn., 17:3-4 (2012), 258–272

Citation in format AMSBIB
\Bibitem{BorKilMam12}
\by Alexey V.~Borisov, Alexander A.~Kilin, Ivan S.~Mamaev
\paper How to Control Chaplygin’s Sphere Using Rotors
\jour Regul. Chaotic Dyn.
\yr 2012
\vol 17
\issue 3-4
\pages 258--272
\mathnet{http://mi.mathnet.ru/rcd337}
\crossref{https://doi.org/10.1134/S1560354712030045}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2956222}
\zmath{https://zbmath.org/?q=an:1264.37016}
\adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2012RCD....17..258B}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84865571827}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. V. Bolsinov, A. V. Borisov, I. S. Mamaev, “Kachenie bez vercheniya shara po ploskosti: otsutstvie invariantnoi mery v sisteme s polnym naborom integralov”, Nelineinaya dinam., 8:3 (2012), 605–616  mathnet
    2. Alexey V. Borisov, Ivan S. Mamaev, Dmitrii V. Treschev, “Rolling of a rigid body without slipping and spinning: kinematics and dynamics”, J. Appl. Nonlinear Dyn., 2:2 (2013), 161–173  mathnet  crossref
    3. Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev, “The Problem of Drift and Recurrence for the Rolling Chaplygin Ball”, Regul. Chaotic Dyn., 18:6 (2013), 832–859  mathnet  crossref  mathscinet  zmath
    4. Alexey V. Borisov, Ivan S. Mamaev, Ivan A. Bizyaev, “The Hierarchy of Dynamics of a Rigid Body Rolling without Slipping and Spinning on a Plane and a Sphere”, Regul. Chaotic Dyn., 18:3 (2013), 277–328  mathnet  crossref  mathscinet  zmath
    5. Mikhail Svinin, Akihiro Morinaga, Motoji Yamamoto, “On the Dynamic Model and Motion Planning for a Spherical Rolling Robot Actuated by Orthogonal Internal Rotors”, Regul. Chaotic Dyn., 18:1-2 (2013), 126–143  mathnet  crossref  mathscinet  zmath
    6. Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev, “How to Control the Chaplygin Ball Using Rotors. II”, Regul. Chaotic Dyn., 18:1-2 (2013), 144–158  mathnet  crossref  mathscinet  zmath
    7. A. V. Borisov, A. A. Kilin, I. S. Mamaev, “Kak upravlyat sharom Chaplygina pri pomoschi rotorov. II”, Nelineinaya dinam., 9:1 (2013), 59–76  mathnet
    8. A. V. Borisov, I. S. Mamaev, I. A. Bizyaev, “Ierarkhiya dinamiki pri kachenii tverdogo tela bez proskalzyvaniya i vercheniya po ploskosti i sfere”, Nelineinaya dinam., 9:2 (2013), 141–202  mathnet
    9. I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “Dinamika negolonomnykh sistem, sostoyaschikh iz sfericheskoi obolochki s podvizhnym tverdym telom vnutri”, Nelineinaya dinam., 9:3 (2013), 547–566  mathnet
    10. A. V. Borisov, A. A. Kilin, I. S. Mamaev, “Problema dreifa i vozvraschaemosti pri kachenii shara Chaplygina”, Nelineinaya dinam., 9:4 (2013), 721–754  mathnet
    11. Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “The Dynamics of Nonholonomic Systems Consisting of a Spherical Shell with a Moving Rigid Body Inside”, Regul. Chaotic Dyn., 19:2 (2014), 198–213  mathnet  crossref  mathscinet  zmath
    12. Tatiana B. Ivanova, Elena N. Pivovarova, “Comments on the Paper by A.V. Borisov, A.A. Kilin, I.S. Mamaev “How to Control the Chaplygin Ball Using Rotors. II””, Regul. Chaotic Dyn., 19:1 (2014), 140–143  mathnet  crossref  mathscinet  zmath
    13. A. A. Kilin, Yu. L. Karavaev, A. V. Klekovkin, “Kinematicheskaya model upravleniya vysokomanevrennym mobilnym sferorobotom s vnutrennei omnikolesnoi platformoi”, Nelineinaya dinam., 10:1 (2014), 113–126  mathnet
    14. T. B. Ivanova, E. N. Pivovarova, “Kommentarii k state A. V. Borisova, A. A. Kilina, I. S. Mamaeva «Kak upravlyat sharom Chaplygina pri pomoschi rotorov. II»”, Nelineinaya dinam., 10:1 (2014), 127–132  mathnet
    15. A. A. Kilin, A. D. Bobykin, “Upravlenie telezhkoi s omnikolesami na ploskosti”, Nelineinaya dinam., 10:4 (2014), 473–481  mathnet
    16. A. A. Kilin, Yu. L. Karavaev, “Kinematicheskaya model upravleniya sferorobotom s neuravnoveshennoi omnikolesnoi platformoi”, Nelineinaya dinam., 10:4 (2014), 497–511  mathnet
    17. Alexander A. Kilin, Elena N. Pivovarova, Tatyana B. Ivanova, “Spherical Robot of Combined Type: Dynamics and Control”, Regul. Chaotic Dyn., 20:6 (2015), 716–728  mathnet  crossref  mathscinet  adsnasa
    18. Alexey V. Borisov, Ivan S. Mamaev, Alexander A. Kilin, Ivan A. Bizyaev, “Qualitative Analysis of the Dynamics of a Wheeled Vehicle”, Regul. Chaotic Dyn., 20:6 (2015), 739–751  mathnet  crossref  mathscinet  adsnasa
    19. A. V. Borisov, Yu. L. Karavaev, I. S. Mamaev, N. N. Erdakova, T. B. Ivanova, V. V. Tarasov, “Experimental Investigation of the Motion of a Body with an Axisymmetric Base Sliding on a Rough Plane”, Regul. Chaotic Dyn., 20:5 (2015), 518–541  mathnet  crossref
    20. V. V. Kozlov, “The dynamics of systems with servoconstraints. II”, Regular and Chaotic Dinamics, 20:4 (2015), 401–427  mathnet  crossref
    21. Alexander P. Ivanov, “On the Control of a Robot Ball Using Two Omniwheels”, Regul. Chaotic Dyn., 20:4 (2015), 441–448  mathnet  crossref  mathscinet  zmath  adsnasa  elib
    22. Ferraro S. Jimenez F. Martin de Diego D., “New Developments on the Geometric Nonholonomic Integrator”, Nonlinearity, 28:4 (2015), 871–900  crossref  mathscinet  zmath  isi  scopus
    23. Yury L. Karavaev, Alexander A. Kilin, “The Dynamics and Control of a Spherical Robot with an Internal Omniwheel Platform”, Regul. Chaotic Dyn., 20:2 (2015), 134–152  mathnet  crossref  mathscinet  zmath  adsnasa  elib
    24. Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev, “Dynamics and Control of an Omniwheel Vehicle”, Regul. Chaotic Dyn., 20:2 (2015), 153–172  mathnet  crossref  mathscinet  zmath  adsnasa
    25. E. V. Vetchanin, Yu. L. Karavaev, A. A. Kalinkin, A. V. Klekovkin, E. N. Pivovarova, “Model bezvintovogo podvodnogo robota”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 25:4 (2015), 544–553  mathnet  elib
    26. E. N. Pivovarova, A. V. Klekovkin, “Vliyanie treniya kacheniya na upravlyaemoe dvizhenie robota-kolesa”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 25:4 (2015), 583–592  mathnet  elib
    27. Yu. L. Karavaev, A. A. Kilin, “Dinamika sferorobota s vnutrennei omnikolesnoi platformoi”, Nelineinaya dinam., 11:1 (2015), 187–204  mathnet  elib
    28. A. A. Kilin, E. V. Vetchanin, “Upravlenie dvizheniem tverdogo tela v zhidkosti s pomoschyu dvukh podvizhnykh mass”, Nelineinaya dinam., 11:4 (2015), 633–645  mathnet
    29. A. A. Kilin, Yu. L. Karavaev, “Eksperimentalnye issledovaniya dinamiki sfericheskogo robota kombinirovannogo tipa”, Nelineinaya dinam., 11:4 (2015), 721–734  mathnet
    30. Alexander P. Ivanov, “On Final Motions of a Chaplygin Ball on a Rough Plane”, Regul. Chaotic Dyn., 21:7-8 (2016), 804–810  mathnet  crossref
    31. Evgeny V. Vetchanin, Alexander A. Kilin, Ivan S. Mamaev, “Control of the Motion of a Helical Body in a Fluid Using Rotors”, Regul. Chaotic Dyn., 21:7-8 (2016), 874–884  mathnet  crossref
    32. Alexey V. Borisov, Alexey O. Kazakov, Elena N. Pivovarova, “Regular and Chaotic Dynamics in the Rubber Model of a Chaplygin Top”, Regul. Chaotic Dyn., 21:7-8 (2016), 885–901  mathnet  crossref
    33. Yury L. Karavaev, Alexander A. Kilin, Anton V. Klekovkin, “Experimental Investigations of the Controlled Motion of a Screwless Underwater Robot”, Regul. Chaotic Dyn., 21:7-8 (2016), 918–926  mathnet  crossref
    34. Alexey V. Borisov, Alexey O. Kazakov, Igor R. Sataev, “Spiral Chaos in the Nonholonomic Model of a Chaplygin Top”, Regul. Chaotic Dyn., 21:7-8 (2016), 939–954  mathnet  crossref
    35. Ivanov A.P., “The Anais Billiard Experiment”, Dokl. Phys., 61:6 (2016), 285–287  crossref  mathscinet  isi  scopus
    36. Alexey V. Borisov, Ivan S. Mamaev, “Adiabatic Invariants, Diffusion and Acceleration in Rigid Body Dynamics”, Regul. Chaotic Dyn., 21:2 (2016), 232–248  mathnet  crossref  mathscinet  zmath  elib
    37. I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “Dynamics of the Chaplygin sleigh on a cylinder”, Regular and Chaotic Dynamics, 21:1 (2016), 136–146  mathnet  crossref  crossref  mathscinet  zmath  elib
    38. Vetchanin E.V., Kilin A.A., “Free and Controlled Motion of a Body With a Moving Internal Mass Through a Fluid in the Presence of Circulation Around the Body”, Dokl. Phys., 61:1 (2016), 32–36  mathnet  crossref  mathscinet  isi  scopus
    39. E. V. Vetchanin, A. A. Kilin, “Upravlenie dvizheniem neuravnoveshennogo tyazhelogo ellipsoida v zhidkosti s pomoschyu rotorov”, Nelineinaya dinam., 12:4 (2016), 663–674  mathnet  crossref  mathscinet  elib
    40. Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “The Chaplygin Sleigh with Parametric Excitation: Chaotic Dynamics and Nonholonomic Acceleration”, Regul. Chaotic Dyn., 22:8 (2017), 955–975  mathnet  crossref
    41. 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, 85 (2017), 327–338  crossref  mathscinet  zmath  isi  scopus
    42. Bizyaev I.A. Borisov A.V. Kuznetsov S.P., “Chaplygin Sleigh With Periodically Oscillating Internal Mass”, EPL, 119:6 (2017), 60008  crossref  isi
    43. Vetchanin E.V. Mamaev I.S., “Optimal Control of the Motion of a Helical Body in a Liquid Using Rotors”, Russ. J. Math. Phys., 24:3 (2017), 399–411  crossref  mathscinet  zmath  isi  scopus
    44. Alexey V. Borisov, Ivan S. Mamaev, “An Inhomogeneous Chaplygin Sleigh”, Regul. Chaotic Dyn., 22:4 (2017), 435–447  mathnet  crossref  mathscinet
    45. Alexander A. Kilin, Elena N. Pivovarova, “The Rolling Motion of a Truncated Ball Without Slipping and Spinning on a Plane”, Regul. Chaotic Dyn., 22:3 (2017), 298–317  mathnet  crossref  mathscinet
    46. Vetchanin E.V. Kilin A.A., “Control of Body Motion in An Ideal Fluid Using the Internal Mass and the Rotor in the Presence of Circulation Around the Body”, J. Dyn. Control Syst., 23:2 (2017), 435–458  crossref  mathscinet  zmath  isi  scopus
    47. A. V. Borisov, I. S. Mamaev, I. A. Bizyaev, “Dynamical systems with non-integrable constraints, vakonomic mechanics, sub-Riemannian geometry, and non-holonomic mechanics”, Russian Math. Surveys, 72:5 (2017), 783–840  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    48. E. V. Vetchanin, V. A. Tenenev, A. A. Kilin, “Optimalnoe upravlenie dvizheniem v idealnoi zhidkosti tela s vintovoi simmetriei s vnutrennimi rotorami”, Kompyuternye issledovaniya i modelirovanie, 9:5 (2017), 741–759  mathnet  crossref
    49. A. V. Borisov, E. V. Vetchanin, A. A. Kilin, “Control of the Motion of a Triaxial Ellipsoid in a Fluid Using Rotors”, Math. Notes, 102:4 (2017), 455–464  mathnet  crossref  crossref  mathscinet  isi  elib
    50. E. N. Pivovarova, “Issledovanie ustoichivosti statsionarnykh dvizhenii sferorobota kombinirovannogo tipa”, Nelineinaya dinam., 13:4 (2017), 611–623  mathnet  crossref  elib
    51. A. V. Borisov, I. S. Mamaev, “Neodnorodnye sani Chaplygina”, Nelineinaya dinam., 13:4 (2017), 625–639  mathnet  crossref  elib
    52. Ivanova T.B. Kilin A.A. Pivovarova E.N., “Controlled Motion of a Spherical Robot With Feedback. i”, J. Dyn. Control Syst., 24:3 (2018), 497–510  crossref  mathscinet  zmath  isi  scopus
    53. Yang Bai, Mikhail Svinin, Motoji Yamamoto, “Dynamics-Based Motion Planning for a Pendulum-Actuated Spherical Rolling Robot”, Regul. Chaotic Dyn., 23:4 (2018), 372–388  mathnet  crossref  mathscinet
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