Mathematical and computer modeling of rover with elastic suspension longitudinal dynamics

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Аuthors

Krasinsky A. Y.1*, Ilyina A. N.2**, Krasinskaya E. M.3*, Rukavishnikova A. S.4***

1. Moscow State University of Food Production, 11, Volokolamskoe shosse, Moscow, 125080, Russia
2. Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia
3. Bauman Moscow State Technical University, MSTU, 5, bldg. 1, 2-nd Baumanskaya str., Moscow, 105005, Russia
4. Moscow Institute of Physics and Technology, 9, Institutskiy per., Dolgoprudny, Moscow region, 141701, Russia

*e-mail: krasinsk@mail.ru
**e-mail: happyday@list.ru
***e-mail: nasty.ruka@mail.ru

Abstract

In the paper we provide a method of constructing an accurate mathematical model and solving a steady motion stabilization problem by the example of a simple rover − a four-wheeled mobile single-link manipulator with elastic suspension. It is an actual problem [16] because Lagrange equations of the second kind are not applicable for such systems. Redundant coordinates and equations of motion in M.F. Shul’gin’s form [1-3] are used to model the mechanical part of the manipulator dynamics. Vector-matrix form of the equations is convenient for analysis and control theory and the theory of critical cases [6-8] applying. The algorithm of solving stabilization problem was developed in [10-11].

The manipulator has to move rectilinearly with a constant speed and keep its clamp (with a camera or a scanner, for example) on a specified height. The system has a geometric constraint as the link is connected with a DC motor gear wheel by an inextensible arm AB. As a control action additional tension at the anchor engine is accepted. The wheels are simulated as two springs.

Solving the problem consists of a few steps:

  1. Mathematical modeling using F.M. Shul’gin’s equations.

  2. Calculating parameters of a steady motion.

  3. The first approximation of the disturbed motion equations around the steady motion is found.

  4. The linear substitution [10-11] is used for the system to have a special form of the theory of critical cases [6-8].

  5. Manipulator controllability [9] is confirmed.

  6. Coefficients of the optimal linear control law are determined uniquely by solving linear-quadratic problem using N.N. Krasovsky’s method [9, 15].

  7. System stability is analyzed.

Results of numerical computations showed that the control law provides asymptotic stability with respect to all variables. This algorithm was programmed in MATLAB.

Keywords:

geometric constraints, redundant coordinates, Shul’gin’s equations, stabilization, steady motion, manipulator

References

  1. Shul’gin M.F. Nauchnye trudy SAGU, 1958, no. 144, pp. 183.

  2. Krasinskaya E.M., Krasinskii A.Ya., Obnosov K.B. O razvitii nauchnykh metodov shkoly M.F. Shul’gina v primenenii k zadacham ustoichivosti i stabilizatsii ravnovesii mekhatronnykh sistem s izbytochnymi koordinatami. Sbornik statei. Teoreticheskaya mekhanika. Moscow, MGU, 2012, no. 28, pp. 169-184.

  3. Krasinskaya E.M., Krasinskii A.Ya. Nauka i obrazovanie, 2013, no. 3, pp. 347-376.

  4. Krasinskii A.Ya., Krasinskaya E.M., Il’ina A.N. Materialy 8-i Vserossiiskoi mul’tikonferentsii po problemam upravleniya. Gelendzhik, Rossiya, 28 sentyabrya — 3 oktyabrya 2015, vol. 2, pp. 37-39.

  5. Krasinskiy A.Ya., Krasinskaya E.M. O dopustimosti linearizacii uravnenij geometricheskih svyazej v zadachah ustojchivosti i stabilizacii ravnovesij. Sbornik statei. Teoreticheskaya mekhanika, Moscow, MGU, 2015, no. 29, pp. 54 — 65.

  6. Lyapunov A.M. Sobranie sochinenii (Collected Works), Moscow, Izd-vo AN SSSR, 1956, vol. 2, 481 p.

  7. Malkin I.G. Teoriya ustojchivosti dvizheniya (The Theory of Dynamic Stability), Moscow, Nauka, 1966, 532 p.

  8. Kamenkov G.V. Ustoichivost' i kolebaniya nelineinykh sistem (The Stability and Oscillation of Nonlinear Systems), Moscow, Nauka, 1972, vol. 2, 215 p.

  9. Gal’perin E.A., Krasovskii N.N. Prikladnaya matematika i mekhanika, 1963, vol. 27, no. 6. pp. 988-1004.

  10. Krasinskaya E.M., Krasinskiy A.Ya. Materialy XII Vserossijskogo soveshchaniya po problemam upravleniya VSPU-2014, Moscow, 2014, pp. 1766-1778.

  11. Krasinskiy A.Ya., Krasinskaya E.M. Avtomatika i telemekhanika, 2016, no. 8, pp. 85-100.

  12. Rumyancev V.V. Ob ustoichivosti statsionarnykh dvizhenii sputnikov (About Stability of Satellites Steady Motion), Moscow−Izhevsk, NITs «Regulyarnaya i khaoticheskaya dinamika», 2010, 156 p.

  13. Klokov A.S., Samsonov V.A. Prikladnaya matematika i mekhanika, 1985, vol. 49, no. 2, pp. 199-202.

  14. Zenkevich S.L., Yushchenko A.S. Osnovy upravleniya manipulyatsionnymi robotami (Basics of Manipulating Robots control), Moscow, Izd-vo MGTU im. N.E. Baumana, 2004, 480 p.

  15. Repin YU.M., Tretyakov V.E. Avtomatika i telemekhanika, 1963, vol. 24, no. 6, pp. 738-743.

  16. Kayumova D R Trudy MAI, 2012, no. 53, available at: http://trudymai.ru/eng/published.php?ID=29487


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