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Control of the Dynamics of a System with Differential Constraints

  • CONTROL IN DETERMINISTIC SYSTEMS
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Abstract

We propose a method for solving the control problem of a system with allowance for the dynamics of actuation mechanisms. The aim of the control and kinematic properties of the system are determined by the holonomic and nonholonomic constraints imposed on the phase coordinates of the control plant. Control actions are generated with allowance for conditions for stabilizing the constraints in the numerical solution of the equations of the dynamics of a closed system.

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REFERENCES

  1. H. F. Olson, Dynamical Analogies (Van Nostrand, New York, 1943).

    Google Scholar 

  2. R. A. Layton, Principles of Analytical System Dynamics (Springer, New York, 1998).

    Book  MATH  Google Scholar 

  3. H. Béghin, Étude théorique des compas gyrostatiques Anschütz et Sperry (C. R. Acad. Sci., Paris, 1922) [in French].

    MATH  Google Scholar 

  4. G. V. Korenev, Purpose and Adaptability of Movement (Nauka, Moscow, 1974) [in Russian].

    Google Scholar 

  5. A. S. Galiullin, I. A. Mukhametzyanov, R. G. Mukharlyamov, and V. D. Furasov, Building Software Systems Movement (Nauka, Moscow, 1971) [in Russian].

    MATH  Google Scholar 

  6. R. G. Mukharlyamov, “To inverse problems of the qualitative theory of differential equations,” Differ. Uravn. 3, 1673–1681 (1967).

    MathSciNet  MATH  Google Scholar 

  7. O. V. Matukhina, “On the problem of simulating kinematics and dynamics of controllable systems with software links,” Inzhen. Zh.: Nauka Innov., No. 4 (2018). https://doi.org/10.18698/2308-6033-2018-4-1753

  8. J. Baumgarte, “Stabilization of constraints and integrals of motion in dynamical systems,” Comput. Methods Appl. Mech. Eng., No. 1, 1–16 (1972).

  9. J. Baumgarte, “Stabilized Kepler motion connected with analytic step adaptation,” Celest. Mech. 13, 105–109 (1976).

    Article  MATH  Google Scholar 

  10. V. A. Avdyushev, Numerical Orbit Modeling (NTL, Tomsk, 2010) [in Russian].

    Google Scholar 

  11. A. A. Burov and I. I. Kosenko, “The Lagrange differential-algebraic equations,” J. Appl. Math. Mech. 78, 587–598 (2014).

    Article  MathSciNet  Google Scholar 

  12. J. Wittenburg, Dynamics of Systems of Rigid Bodies (Vieweg Teubner, Berlin, 1977).

    Book  MATH  Google Scholar 

  13. F. Amirouche, Fundamentals of Multibody Dynamics: Theory and Applications (Birkhäuser, Boston, 2006).

    MATH  Google Scholar 

  14. Shih-Tin Lin and Jiann-Nan Huang, “Numerical integration of multibody mechanical systems using Baumgarte’s constraint stabilization method,” J. Chin. Inst. Eng. 25, 243–252 (2002).

    Article  Google Scholar 

  15. U. M. Ascher, Hongsheng Chin, L. R. Petzold, and S. Reich, “Stabilization of constrained mechanical systems with DAEs and invariant manifolds,” J. Mech. Struct. Mach. 23, 135–158 (1995).

    Article  MathSciNet  Google Scholar 

  16. U. M. Ascher, “Stabilization of invariant of discretized differential systems,” Numer. Algorithms 14, 1–24 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  17. U. M. Ascher, Hongsheng Chin, and S. Reich, “Stabilization of DAEs and invariant manifolds,” Numer. Math. 67, 131–149 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  18. Shih-Tin Lin and Ming-Chong Hong, “Stabilization method for the numerical integration of controlled multibody mechanical system: a hybrid integration approach,” JSME Int. J., Ser. C 44, 79–88 (2001).

  19. G. K. Suslov, On the Force Function, Admitting Given Integrals (Kievsk. Univ., Kiev, 1890) [in Russian].

    Google Scholar 

  20. N. E. Zhukovskii, “Determination of the force function for this family of trajectories,” in Complete Collection of Works (ONTI NKTP SSSR, Moscow, Leningrad, 1937), Vol. 1, pp. 293–308 [in Russian].

    Google Scholar 

  21. T. Levi Chivita and U. Amaldi, Lezioni di meccanica razionale (Compomat, Italy, 2013), Vol. 2, Part 2 [in Italian].

    Google Scholar 

  22. A. S. Galiullin, Solving Methods for Inverse Dynamics Problems (Nauka, Moscow, 1986) [in Russian].

    MATH  Google Scholar 

  23. G. Bozis and S. Ichtiaroglou, “Existence and construction of dynamical systems having a prescribed integral of motion–an inverse problem,” Inverse Probl. 3, 213–227 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  24. N. P. Erugin, “Construction of the entire set of systems of differential equations with a given integral curve,” Prikl. Mat. Mekh. 21, 659–670 (1952).

    MathSciNet  Google Scholar 

  25. R. G. Mukharlyamov, “On the construction of differential equations of optimal motion for a given variety,” Differ. Uravn. 7, 1825–1834 (1971).

    MathSciNet  Google Scholar 

  26. R. G. Mukharlyamov, “On the construction of the set of systems of differential equations of stable motion on an integral manifold,” Differ. Uravn. 5, 688–699 (1969).

    Google Scholar 

  27. R. G. Mukharlyamov, “On solving systems of nonlinear equations,” Zh. Vychisl. Mat. Mat. Fiz. 11, 829–836 (1971).

    Google Scholar 

  28. R. G. Mukharlyamov, “On equations of motion of mechanical systems,” Differ. Uravn. 19, 2048–2056 (1983).

    MathSciNet  MATH  Google Scholar 

  29. R. G. Mukharlyamov and Deressa Chernet Tuge, “Stabilization of redundantly constrained dynamic system,” Vestn. RUDN, Ser.: Mat. Inform. Fiz., No. 1, 60–72 (2015).

  30. R. G. Mukharlyamov, “Control of dynamics of systems with positional relations,” in Analytical Mechanics, Stability and Control, Proceedings of the 11th International Chetaev Conference, Sect. 3: Control (KNITU-KAI, Kazan’, 2017), Vol. 3, Part 2, pp. 140–146.

  31. V. V. Kozlov, “The dynamics of systems with servoconstraints. I,” Regular Chaot. Dyn. 20, 205–224 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  32. V. V. Kozlov, “The dynamics of systems with servoconstraints. II,” Regular Chaot. Dyn. 20, 401–427 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  33. A. W. Beshaw, “Dynamic equation of constrained mechanical system,” Vestn. RUDN. Ser.: Mat. Inform. Fiz., No. 3, 115–124 (2014).

  34. Adaptive Optics: Collection of Articles, Ed. by E. A. Vitrichenko (Mir, Moscow, 1980) [in Russian].

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ACKNOWLEDGMENTS

This work was financially supported by the Russian Foundation for Basic Research (project no. 19-08-00261 a).

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Correspondence to R. G. Mukharlyamov.

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Translated by L. Kartvelishvili

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Mukharlyamov, R.G. Control of the Dynamics of a System with Differential Constraints. J. Comput. Syst. Sci. Int. 58, 515–527 (2019). https://doi.org/10.1134/S1064230719030134

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  • DOI: https://doi.org/10.1134/S1064230719030134

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