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Contribution to the determination of the global minimum time for the brachistochronic motion of a holonomic mechanical system

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Abstract

The problem of the brachistochronic motion of a holonomic scleronomic mechanical system is analyzed. The system moves in an arbitrary field of known potential forces. The problem is formulated as an optimal control task, where generalized speeds are taken as control variables. The problem considered is reduced to solving the corresponding two-point boundary-value problem (TPBVP). In order to determine the global minimal solution of the TPBVP, an appropriate numerical procedure based on the shooting method is presented. The global minimal solution represents the solution with the minimum time of motion. The procedure is illustrated by an example of determining the brachistochronic motion of a disk that performs plane motion in a vertical plane in a homogeneous field of gravity.

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References

  1. Djukic Dj S, Atanackovic TM (1976) A note on the classical brachistochrone. Z Angew Math Phys 27:677–681

    Article  MathSciNet  MATH  Google Scholar 

  2. McConnell AJ (1930) The brachistochronic motion of a dynamical system. Proc R Irish Acad 39A:31–48

    Google Scholar 

  3. Bertolazzi E, Biral F, Da Lio M (2006) Symbolic numeric efficient solution of optimal control problems for multibody systems. J Comput Appl Math 185:404–421

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Stoer J, Bulirsch J (1993) Introduction to numerical analysis. Springer, New York

    Book  MATH  Google Scholar 

  5. Dixon LCW, Biggs MC (1972) The advantages of adjoint control transformations when determining optimal trajectories by Pontryagin’s maximum principle. Areonaut J 76:169–174

    Google Scholar 

  6. Seywald H, Kumar RR (1996) Method for automatic costate calculation. J Guid Control Dyn 19:1252–1261

    Article  ADS  MATH  Google Scholar 

  7. Fahroo F, Ross IM (2001) Costate estimation by a Legendre pseudospectral method. J Guid Control Dyn 24:270–277

    Article  ADS  Google Scholar 

  8. Hull DG (2008) Initial Lagrange multipliers for the shooting method. J Guid Control Dyn 31:1490–1492

    Article  ADS  Google Scholar 

  9. Graichen K, Petit N (2008) A continuation approach to state and adjoint calculation in optimal control applied to the reentry problem. In: Proceedings of the 17th World congress the international federation of automatic control Seoul, Korea, July 6–11, pp 14307–14312

  10. Jiang F, Baoyin H, Li J (2012) Practical techniques for low-thrust trajectory optimization with homotopic approach. J Guid Control Dyn 35:245–258

    Article  ADS  Google Scholar 

  11. Park Ch, Guibout V, Scheeres D (2006) Solving optimal continuous thrust rendezvous problems with generating functions. J Guid Control Dyn 29:321–331

    Article  ADS  Google Scholar 

  12. Obradović A, Šalinić S, Jeremić O et al (2014) On the brachistochronic motion of a variable-mass mechanical system in general force fields. Math Mech Solids 19:398–410

    Article  MathSciNet  MATH  Google Scholar 

  13. Mehrpouya MA, Shamsi M (2015) Gauss pseudospectral and continuation methods for solving two-point boundary value problems in optimal control theory. Appl Math Model 39:5047–5057

    Article  MathSciNet  Google Scholar 

  14. Lurie AI (2002) Analytical mechanics. Springer, Berlin Heidelberg

    Book  MATH  Google Scholar 

  15. Papastavridis JG (2002) Analytical mechanics. Oxford University Press, New York

    MATH  Google Scholar 

  16. Kelley H, Kopp RE, Moyer GH (1967) Singular extremals. In: Leitmann G (ed) Topics in optimization (mathematics in science and engineering, 31). Academic Press, New York, pp 63–101

    Google Scholar 

  17. Gabasov R, Kirillova FM (1972) High order necessary conditions for optimality. SIAM J Control 10:127–168

    Article  MathSciNet  MATH  Google Scholar 

  18. Bryson AE, Ho YC (1975) Applied optimal control. Hemisphere, New York

    Google Scholar 

  19. Hull D (2003) Optimal control theory for applications. Springer, New York

    Book  MATH  Google Scholar 

  20. Hull D (1990) On the variational process in optimal control theory. J Optimiz Theory App 67:447–462

    Article  MathSciNet  MATH  Google Scholar 

  21. Strang G (1988) Linear algebra and its applications, 3rd edn. Harcourt Brace Jovanovich, San Diego

    MATH  Google Scholar 

  22. Ruskeepaa H (2009) Mathematica ®Navigator: mathematics, statistics, and graphics, 3rd edn. Academic Press, Burlington

    MATH  Google Scholar 

  23. Hirsch MJ, Pardalos PM, Resende MGC (2009) Solving systems of nonlinear equations with continuous GRASP. Nonlinear Anal Real 10:2000–2006

    Article  MathSciNet  MATH  Google Scholar 

  24. Song W, Wang Y, Li H-X et al (2015) Locating multiple optimal solutions of nonlinear equation systems based on multiobjective optimization. IEEE T Evolut Comput 19:414–431

    Article  Google Scholar 

  25. Antunes ACB, Siguad C (2010) Controling nonholonomic Chaplygin systems. Braz J Phys 40:131–140

    Article  ADS  Google Scholar 

  26. Čović V, Vesković M (2009) Brachistochronic motion of a multibody system with Coulomb friction. Eur J Mech A Solid 28:882–890

    Article  MathSciNet  MATH  Google Scholar 

  27. Šalinić S, Obradović A, Mitrović Z (2012) On the brachistochronic motion of mechanical systems with unilateral constraints. Mech Res Commun 45:1–6

    Article  Google Scholar 

  28. Šalinić S, Obradović A, Mitrović Z et al (2013) On the brachistochronic motion of the Chaplygin sleigh. Acta Mech 224:2127–2141

    Article  MathSciNet  MATH  Google Scholar 

  29. Yablonskii AA, Nikiforova VM (1966) Course of theoretical mechanics, vol 1, 3rd edn. Vischaya Shkola, Moscow (in Russian)

    Google Scholar 

  30. Scheiber E, Lupu M (1997) Numerical solutions to inverse problems of general planar motion. Comput Methods Appl Mech Eng 146:197–214

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Zypman FR (2007) Instantaneous center of rotation and centrodes: background and new examples. Int J Mech Eng Educ 35:79–90

    Article  Google Scholar 

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Acknowledgments

This research was supported under Grants Nos. ON17400 and TR35006 by the Ministry of Education, Science and Technological Development of the Republic of Serbia. This support is gratefully acknowledged.

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Authors and Affiliations

  1. Faculty of Mechanical Engineering, University of Belgrade, Belgrade, Serbia

    Radoslav Radulović & Aleksandar Obradović

  2. Faculty of Mechanical and Civil Engineering in Kraljevo, University of Kragujevac, Kragujevac, Serbia

    Slaviša Šalinić

Authors
  1. Radoslav Radulović
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  2. Aleksandar Obradović
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  3. Slaviša Šalinić
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Correspondence to Slaviša Šalinić.

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Radulović, R., Obradović, A. & Šalinić, S. Contribution to the determination of the global minimum time for the brachistochronic motion of a holonomic mechanical system. Meccanica 52, 795–805 (2017). https://doi.org/10.1007/s11012-016-0425-z

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  • DOI: https://doi.org/10.1007/s11012-016-0425-z

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