Mathematical and computer modeling of rover with elastic suspension longitudinal dynamics

Mathematical support and software for computers, complexes and networks


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



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.


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


  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:

Download — informational site MAI

Copyright © 2000-2021 by MAI