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Archive of Mechanical Engineering

The Journal of Committee on Machine Building of Polish Academy of Sciences

4 Issues per year

CiteScore 2016: 0.44

SCImago Journal Rank (SJR) 2016: 0.162
Source Normalized Impact per Paper (SNIP) 2016: 0.459

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Volume 61, Issue 2 (Aug 2014)

Tracking Control of A Balancing Robot – A Model-Based Approach

Sterowanie Śledzące Robota Zachowującego Równowagę – Podejście Oparte Na Modelu

Tobias Zaiczek / Matthias Franke
Published Online: 2014-08-15 | DOI: https://doi.org/10.2478/meceng-2014-0019


This paper presents a control concept for a single-axle mobile robot moving on the horizontal plane. A mathematical model of the nonholonomic mechanical system is derived using Hamel's equations of motion. Subsequently, a concept for a tracking controller is described in detail. This controller keeps the mobile robot on a given reference trajectory while maintaining it in an upright position. The control objective is reached by a cascade control structure. By an appropriate input transformation, we are able to utilize an input-output linearization of a subsystem. For the remaining dynamics a linear set-point control law is presented. Finally, the performance of the implemented control law is illustrated by simulation results.


W artykule przedstawiono koncepcję sterowania ruchem jednoosiowego robota poruszajĄcego się po płaszczyźnie poziomej. Model matematyczny nieholonomicznego systemu mechanicznego wyprowadzono korzystajĄc z równań ruchu Hamela. Opisano następnie szczegółowo koncepcję sterownika śledzĄcego. Sterownik prowadzi poruszajĄcy się robot po zadanej trajektorii utrzymujĄc go jednocześnie w pozycji pionowej. Cel sterowania jest osiĄgnięty przy zastosowaniu kaskadowej struktury sterowania. Dzięki odpowiedniej transformacji danych wejściowych, w podsystemie istnieje moŻliwość linearyzacji wejście-wyjście. Dla dynamiki pozostałej części zaprezentowano prawo liniowego sterowania stałowartościowego. Ostatecznie, jakość działania zastosowanych praw sterowania zilustrowano wynikami symulacji.

Key words:: single-axle mobile robot; nonholonomic constraints; Hamel's equation; cascade control structure; input-output linearization; tracking control; nonlinear control


  • [1] Hamel G.: Über die virtuellen Verschiebungen in der Mechanik. Mathematische Annalen, 59, 416-434, 1904.Google Scholar

  • [2] Kim B. M., and Tsiotras P.: Controllers for unicycle-type wheeled robots: Theoretical results and experimental validation. IEEE Trans. on Robotics and Automation, 18(3), 294-307, June 2002.Google Scholar

  • [3] Lee T.-Ch., Song K.-T., Lee Ch.-H., and Teng Ch.-Ch.: Tracking control of unicycle-modeled mobile robots using a saturation feedback controller. IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, 9(2), 305-328, March 2001.Google Scholar

  • [4] M'Closkey R.T., and Murray R.M.: Exponential stabilization of driftless nonlinear control systems using homogeneous feedback. TAC, 42(5), 614-628, 1997.Google Scholar

  • [5] Sharp R. S.: On the stability and control of unicycles. In Proc. R. Soc. A, 2010.Google Scholar

  • [6] Franke M., Rudolph J., and Woittennek F.: Motion planning and feedback control of a planar robotic unicycle model. In Proc. Int. Conf. on Methods and Models in Automation and Robotics, volume 14, 2009.Google Scholar

  • [7] Franke M., Zaiczek T., and Röbenack K.: Simulation of nonholonomic mechanical systems using algorithmic differentiation. In Proc. 7th Vienna International Conference on Mathematical Modelling (MATHMOD), 2012.Google Scholar

  • [8] Goldstein H.: Classical Mechanics. Addison-Wesley series in physics. Addison-Wesley Pub. Co., 1980.Google Scholar

  • [9] Neimark I. I., and Fufaev N. A.: Dynamics of Nonholonomic Systems. American Mathemtical Society, 1972.Google Scholar

  • [10] Griepentrog E., and März R.: Differential-Algebraic Equations and Their Numerical Treatment, volume 88. Teubner Verlagsgesellschaft, Leipzig, 1986.Google Scholar

  • [11] Brenan K. E., Campbell S. L., and Petzold L. R.: Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. SIAM, Philadelphia, 2nd edition, 1996.Google Scholar

  • [12] Röbenack K., Bausch T., and Uhlig A.: Struktureller Index von Deskriptorvariablen. In Klaus Panreck and Frank Dörrscheidt, editors, Simulationstechnik, 15. Symposium in Paderborn, ASIM 2001, Frontiers in Simulation, pages 569-574. SCS – The Society for Modeling and Simulation International, 2001.Google Scholar

  • [13] Kunkel P., and Mehrmann V.: Differential-Algebraic Equations: Analysis and Numerical Solution. EMS Publishing House, Zürich, 2006.Google Scholar

  • [14] Cameron J.M., and Book W.J.: Modeling mechanisms with nonholonomic joints using the Boltzmann-Hamel equations. International Journal of Robotics Research, 16(1), 47-59, February 1997.CrossrefGoogle Scholar

  • [15] Jarzebowska E., and Lewandowski R.: Modeling and control design using the Boltzmann-Hamel equations: A roller-racer example. In Proc. 8th IFAC Symposium on Robot Control, pages 236-241, 2006.Google Scholar

  • [16] Isidori A.: Nonlinear Control Systems. Springer, 3. edition, 1995.Google Scholar

About the article

Published Online: 2014-08-15

Citation Information: Archive of Mechanical Engineering, ISSN (Online) 2300-1895, DOI: https://doi.org/10.2478/meceng-2014-0019.

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© 2014 Tobias Zaiczek and Matthias Franke. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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