Analysis of modern modeling methods in problems of stabilization of motions of mechatronic systems with differential constraints

  • Authors

    • Krasinskiy A.Ya
    • Krasinskaya E.M.
    2018-04-20
    https://doi.org/10.14419/ijet.v7i2.23.11873
  • differential constraints, stability, stabilization, vector-matrix equations.
  • Abstract

    The most important problem of controlling mechatronic systems is the development of methods for the fullest possible application of the properties of our own (without the application of controls) motions of the object for the optimal use of all available resources. The basis of this can be a non-linear mathematical model of the object, which allows to determine the degree of minimally necessary interference in the natural behavior of an object with the purpose of stable implementation of a given operating mode. The operating modes of the vast majority of modern mechatronic systems are realized due to the steady motions (equilibrium positions and stationary motions) of their mechanical components, and often these motions are constrained by connections of various kinds. The paper gives an analysis of methods for obtaining nonlinear mathematical models in stabilization problems of mechanical systems with differential holonomic and non-holonomic constraints.

     

  • References

    1. [1] Neimark Yu.I., Fufaev N.A. Dynamics of non-holonomic systems. M: Science, 1967.519 p.

      [2] Lurie A.I. Analytical mechanics. / A.I. Lurie. - Moscow: Gos. Izd-vo in fiz.-mat. Literature, 1961. - 824 s

      [3] Suslov G.K. Theoretical mechanics. Moscow-Leningrad: OGIZ. 1946. 656 p.

      [4] Shulgin M.F. On some differential equations of analytic dynamics and their integration. / M.F. Shulgin // Scientific works of the SAGU. - Tashkent. 1958, 183 p.

      [5] Okhotsimsky D.E., Martynenko Yu.G. New tasks of dynamics and traffic control of mobile wheeled robots. Advances in mechanics. 2003. â„– 1. Pp. 3-46

      [6] Krasinsky A.Ya., Kayumova D.R. On the influence of wheel deformability on the dynamics of a robot with a differential drive. // Nonlinear dynamics.2011.Т.7.№4, p.803-822.

      [7] Matyukhin V.I. Trajectory problems of control of wheel systems. M .: KRASAND, 2014.

      [8] Borisov A.V., Kilin A.A., Mamaev I.S. On the Hadamard-Hamel problem and the dynamics of wheeled vehicles, Nonlinear dynamics, 2016, vol. 12, No. 1, p. 145-163

      [9] Bolotin S.V. The problem of optimal control of ball rolling with rotors. Nonlinear dynamics. 2012, Vol. 8, No. 4, p. 837-852

      [10] Borisov A.V., Kilin AA, Mamaev IS How to control the Chaplygin ball using rotors, Nonlinear dynamics, 2012, v. 8, No. 2, p. 289-307

      [11] Karavaev Yu. L., Kilin AA Dynamics of the Spherical Robot with the Internal Omnipoles Platform, Nonlinear Dynamics. 2015, vol. 11, No. 1, p. 187-204

      [12] Borisov AV, Mamaev IS, Bizyaev IA Historical and critical review of the development of nonholonomic mechanics: the classical period. Nonlinear dynamics. 2016, Vol. 12, No. 3, p. 385-411

      [13] V.I. Kalenova, A. V. Karapetyan, V. M. Morozov, M. A. Salmina. Non-holonomic mechanical systems and stabilization of motion. Fundamental and applied mathematics. Center for New Information Technologies of Moscow State University. Publishing house "Open Systems". Vol. 11 (2005), no. 7, p. 117-158.

      [14] Zegzhda S.A., Soltakhanov Sh.H., Yushkov M.P. Equations of motion of nonholonomic systems and variational principles of mechanics. A new class of management tasks. M .: Fizmatlit. 2005. 272 p.

      [15] Zegzhda S.A., Soltakhanov Sh.H., Yushkov M.P. Non-head mechanics. Thorium and applications. Moscow: Fizmatlit, 2009.

      [16] Zobova A.A. Application of laconic forms of the equations of motion in the dynamics of nonholonomic mobile robots. Nonlinear dynamics. 2011, Vol. 7, No. 4, p. 771-783

      [17] Sumbatov A.S., On the Lagrange equations in nonholonomic mechanics. Nonlinear dynamics. 2013, v. 9, No. 1, p. 39-50

      [18] Borisov A.V., Mamaev I.S., Bizyaev I.A. The Jacobi integral in nonholonomic mechanics. Nonlinear dynamics. 2015, vol. 11, No. 2, p. 377-396

      [19] Karapetyan A.V., Rumyantsev V.V. Stability of conservative and dissipative systems // Itogi Nauki i Tekhniki. General mechanics. T.6. M.: VINITI. 1983. -129 with.

      [20] Yemelyanova I.S. To the definition of cyclic coordinates and stationary motions of mechanical systems. Q: Dynamics of systems. T. 3. Gorky 1974. P. 117-130.

      [21] Emelyanova I.S., Fufaev N.A. On the stability of stationary motions. In collection: Theory of oscillations, prikl. mat. and cybernet. Gory. 1974. pp. 3-9.

      [22] Sumbatov A.S. On linear integrals of non-holonomic systems. Vestnik Mosk. Un. Mat., Fur. 1972. â„– 6. from. 77-83.

      [23] Semenova L.N. On the Routh theorem for non-holonomic systems // PMM. 1965. Vol. 29. Issue. I. with. 156-157.

      [24] Karapetyan A.V. On the stability of stationary motions of systems of a certain type, Izv. AN SSSR. MTT.-1983.-No. 2.- S. 45-52.

      [25] Chetaev, N.G. Stability of motion. Works on analytical mechanics. - Moscow: Publishing House of the USSR Academy of Sciences, 1962.

      [26] Routh, E. J. The Advanced Part of the Treatise on the Dynamics of a System of Rigid Bodies. London: MacMillan and Co, 1884.

      [27] Poincare H. Sur l'equilibre d'une masse fluide animee d'un mouvement de rotation // Acta Math.-1885.-Vol. 7.-P. 259-380.

      [28] Lyapunov A.M. Collection op. T. 2.Izd. Academy of Sciences of the USSR, Moscow - Leningrad, 1956

      [29] Malkin I.G. Theory of stability of motion. Science Moscow, 1952.

      [30] Kamenkov G.V. Fav. Proceedings. T.2, Moscow, 1972.

      [31] Merkin D.R. Introduction to the theory of motion stability. - Moscow: Science, 1971.

      [32] Hagedorn P. Ueber die Instabilitaet konservativer Systeme mit gyroskopischen Kraeften // Arch. Rat. Mech.Anal, 1975.-58 (1) - pp. 1-9.

      [33] Huseyin K., Hagedorn P., Teschner W. On the stability of linear conservative gyroscopic systems, ZAMP, November 1983. - 34. - P. 807-815.

      [34] Krasinskaya-Tyumeneva E.M., Krasinsky A.Ya. On the influence of the structure of forces on the stability of equilibrium positions of nonholonomic systems. Questions subt. and prikl. mathematics. Вып.45, Ташкент, 1977, p. 172-186.

      [35] Krasinskaya E.M. To the stabilization of stationary motions of mechanical systems // PMM.1983.T.47.vyp.2.S.302-309.

      [36] Krasinsky A.Ya. On the stability and stabilization of equilibrium positions of non-holonomic systems // PMM.1988.Т.52.С.194-202.

      [37] Atazhanov B., Krasinskaya E.M. On the stabilization of stationary motions of non-holonomic mechanical systems. PMM, 1988, 52, issue 6. Pp. 902-908.

      [38] Krasinsky A.Ya. On stabilization of steady motions of systems with cyclic coordinates. //ПММ.1992.Т.56 .С. 939-950.

      [39] Martynenko Yu.G. On the matrix form of the equation of nonholonomic mechanics // Collection of scientific and methodical works on theoretical mechanics. - Moscow: Izd. University, 2000. - Вып. 23. - С. 9-21.

      [40] Kalenova V.I., Morozov V.M., Sheveleva E.V. Controllability and observability in the problem of stabilization of steady motions of nonholonomic mechanical systems with cyclic coordinates // PMM, 2001. - 65. - P. 915-924.

      [41] Martynenko Yu.G., Zatsepin M.F. Assignments of matrix methods for compiling the Magee and Euler-Lagrange equations of nonholonomic systems // A collection of scientific and methodological research on theoretical mechanics. - Moscow: Izd-vo Mosk. University, 2004. - Issue. 25.-S. 86-101.

      [42] Krasinsky A.Ya. On a method for studying the stability and stabilization of nonisolated steady motions of mechanical systems // Selected Works of the VIII International Conference "Stability and Oscillations of Nonlinear Control Systems". - Moscow, Institute for Control Sciences. V.A. Trapeznikova RAS, 2004. - Electronic publication. - P. 97-103. - http://www.ipu.ru/semin/arhiv/stab04.

      [43] Krasinsky A.Ya., Atazhanov B. The problem of stabilization of steady motions of non-holonomic systems SA Chaplygin // Problems of nonlinear analysis in engineering systems, 2007. - Issue. 2 (28), T. 13. - P. 74-96.

      [44] Krasinsky A. Ya., Khalikov A. A. Computer analysis of the problem of stabilization of stationary motions of mobile robots as non-holonomic systems // Bulletin of the Moscow Aviation Institute, 2008. - № 2, v. 15.-С. 66-76.

      [45] Krasinsky A.Ya., Krasinskaya E.M. On the stability and stabilization of the equilibrium of mechanical systems with redundant coordinates. Science and education. MSTU them. N.E. Bauman. Electron. Jour. 2013. No. 03. With. 347 - 376.DOI: 10.7463 / 0313.0541146.

      [46] Krasinsky A.Ya., Krasinskaya E.M. Modeling of the GBB 1005 BALL & BEAM stand dynamics as a controllable mechanical system with redundant coordinate. Science and education. MSTU them. N.E. Bauman. Electron. Jour. 2014. No. 01. pp. 282-297. DOI: 10.7463 / 0114.0646446.

      [47] Krasinsky, A.Ya., Krasinskaya, E.M, On One Method of Investigating the Stability and Stabilization of Steady Motions of Systems with Redundant Coordinates. XII All-Russian Meeting on the Management Problems of the VSPU-2014. MOSCOW, June 16-19, 2014 Proceedings [Electronic resource] Moscow: Institute for Control Sciences im. V.A. Trapeznikova RAS. 2014. Electron. text dan. (1074 file .: 537 MB). 1 elekt.opt. ROM (DVD-ROM). S. 1766-1778. ISBN 978-5-91450-151-5. Number of state registration: 0321401153.

      [48] Krasinsky A.Ya., Krasinskaya E.M. On the simulation of the dynamics of Ball & Beam stand and stabilization of its balance. XII All-Russian Meeting on the Management Problems of the VSPU-2014. MOSCOW, June 16-19, 2014 Proceedings [Electronic resource] Moscow: Institute for Control Sciences im. V.A. Trapeznikova RAS. 2014. Electron. text dan. (1074 file .: 537 MB). 1 elekt.opt. ROM (DVD-ROM). Pp. 2206-2218. ISBN 978-5-91450-151-5. Number of state registration: 0321401153.

      [49] Krasinsky, A.Ya., Krasinskaya, E.M., On the method of investigating a class of stabilization problems with incomplete information on the state. Proceedings of the conference dedicated to the 90th anniversary of the birth of Academician NN. Krasovskogo Publishing House: Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences. N.N. Krasovskogo (Yekaterinburg). 2015. P. 228-235. ISBN: 978-5-8295-0364-2.http: //elibrary.ru/item.asp? Id = 23795691.

      [50] Krasinsky, A.Ya., Krasinskaya, E.M., Asymptotic Stability in Stabilization Problems with Zero Roots in a Closed System. The All-Russian Congress on Fundamental Problems, Theor. and prikl. mechanics. Collection of Proceedings (Kazan, August 20-24, 2015) Kazan: Publishing House of the Kazan (Privolzhsky) Federal University .. 2015. С. 2055-2057.

      [51] Krasinsky A.Ya., Krasinskaya E.M., Ilina A.N. On modeling the dynamics of mechatronic systems with geometric constraints as systems with redundant coordinates. The Eighth All-Russian Multiconference on Management Problems // Proceedings of the 8th All-Russian Multiconference in 3 volumes T.2 - Rostov-on-Don: Publishing House of the Southern Federal University, 2015. P. 37-39. ISBN 978-5-9275-1633-9.

      [52] Krasinsky, A.Ya., Krasinskaya, E.M., On the admissibility of the linearization of the equations of geometric constraints in problems of stability and stabilization of equilibria. Collection of scientific and methodical articles. Theoretical mechanics. Issue 29 / under. Ed. prof. V.A. Samsonov. - Moscow: Publishing House of Moscow University, 2015. P. 54-65. ISBN 978-5-19-011085-2.

      [53] Krasinsky, A.Ya., Krasinskaya, E.M., On One Method of Stabilization of Steady Motions with Zero Roots in a Closed System. Automation and telemechanics (A & T), No. 8, 2016. P. 85-100. Krasinskii A.Y., Krasinskaya E.M. A stabilization method for steady motions with zero roots in the closed system. Automation and remote control. (2016) 77: 1386-1398. DOI: 10.1134 / S0005117916080051.

      [54] A. Ya. Krasinsky, A.N. Il'ina, "The mathematical modelling of the dynamics of systems with redundant coordinates in the neighborhood of steady motions", Vestn. SUSU. Ser. Mat. Modeling and programming, 10: 2 (2017), 38-50 https://doi.org/10.14529/mmp170203

      [55] Alexandr Krasinsky, Krasinskaya E.M., Ilyina A.N. About Mathematical Models of System Dynamics with Geometric Constraints in Problems of Stability and Stabilization by Incomplete State Information Int Rob Auto J 2 (1): 00007. DOI: 10.15406 / iratj.2017.02.00007.

      [56] Gabasov R., Kirillova F.M. Qualitative theory of optimal processes. M:., Science, 1971.

      [57] Zenkevich S.L. Yushchenko A.S. Fundamentals of manipulation robots. M .: Izd. MSTU. NE Bauman,. 2004.

      [58] Vukobratovich M., Stoich D., Kirchansky N. Nead adaptive and adaptive control of manipulative robots. Moscow: Mir. 1989.

      [59] Matyukhin V.I. Management of mechanical systems. M.: Fizmatlit, 2009.

      [60] Novozhilov I.V., Zatsepin M.F. Equations of motion of mechanical systems in an excessive set of variables. Collection of scientific and methodological articles on theoretical mechanics. M., 1987. Issue 18. Pp. 62-66.

      [61] Lyapunov A.M. Lectures on theoretical mechanics. Kiev: Naukova Dumka. 1982.

      [62] Krasinsky A.Ya., Krasinskaya E.M., Ilyina A.N. On the modeling of dynamics of the Ball and Beam system as a nonlinear mechatronic system with a geometric constraint. Bulletin of the Udmurt University. Mathematics. Mechanics. Computer Science, 2017, Vol. 27, no. 3, p. 414-430. DOI: 10.20537 / vm170310.

      [63] Rahmat M.F., Wahid H., Wahab N.A. Application of an intelligent controller in a Ball and Beam control system. International Journal on smart sensing and intelligent systems vol. 3, no. 1 March 2010. P. 45-60.

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  • How to Cite

    A.Ya, K., & E.M., K. (2018). Analysis of modern modeling methods in problems of stabilization of motions of mechatronic systems with differential constraints. International Journal of Engineering & Technology, 7(2.23), 9-13. https://doi.org/10.14419/ijet.v7i2.23.11873

    Received date: 2018-04-22

    Accepted date: 2018-04-22

    Published date: 2018-04-20