Kybernetika 54 no. 3, 476-495, 2018

Smooth super twisting sliding mode based steering control for nonholonomic systems transformable into chained form

Waseem Abbasi, Fazal ur Rehman and Ibrahim ShahDOI: 10.14736/kyb-2018-3-0476


In this article, a new solution to the steering control problem of nonholonomic systems, which are transformable into chained form is investigated. A smooth super twisting sliding mode control technique is used to steer nonholonomic systems. Firstly, the nonholonomic system is transformed into a chained form system, which is further decomposed into two subsystems. Secondly, the second subsystem is steered to the origin by using smooth super twisting sliding mode control. Finally, the first subsystem is steered to zero using signum function. The proposed method is tested on three nonholonomic systems, which are transformable into chained form; a two-wheel car model, a model of front-wheel car, and a fire truck model. Numerical computer simulations show the effectiveness of the proposed method when applied to chained form nonholonomic systems.


nonholonomic mechanical systems, chained form, steering control, smooth super twisting sliding mode control and lyapunov function.


93C85, 70Q05


  1. W. Abbassi and F. Rehman: Adaptive integral sliding mode stabilization of nonholonomic drift-free systems. Math. Problems Engrg. 2016 (2016), 1-11.   DOI:10.1155/2016/9617283
  2. S. sam Ge, J. Wang, T. heng Lee and GY. Zhou: Adaptive robust stabilization of dynamic nonholonomic chained systems. J. Field Robotics 18 (2001), 3, 119-133.   DOI:10.1002/rob.1010.abs
  3. I. Kolmanovsky and N. H. McClamroch: Developments in nonholonomic control problems. IEEE Control Systems 15 (1995), 6, 20-36.   DOI:10.1109/37.476384
  4. A. Levant: Higher-order sliding modes, differentiation and output-feedback control. Int, J. Control 76 (2003), 9-10, 924-941.   DOI:10.1080/0020717031000099029
  5. L. Zhen-Ying and C-Li. Wang: Robust stabilization of nonholonomic chained form systems with uncertainties. Acta Automat. Sinica 37 (2011), 2, 129-142.   DOI:10.3724/sp.j.1004.2011.00129
  6. P. Li and Z. Zheng: Global finite-time stabilization of planar nonlinear systems with disturbance. Asian J. Control 14 (2012), 3, 851-858.   DOI:10.1002/asjc.377
  7. Z. Li, H. Xiao, C. Yang and Y. Zhao: Model predictive control of nonholonomic chained systems using general projection neural networks optimization. IEEE Trans. Systems Man Cybernetics: Systems 45 (2015), 10, 1313-1321.   DOI:10.1109/tsmc.2015.2398833
  8. L. Luque-Vega, B. Castillo-Toledo and A. G. Loukianov: Robust block second order sliding mode control for a quadrotor. J. Franklin Inst. 349 (2012), 2, 719-739.   DOI:10.1016/j.jfranklin.2011.10.017
  9. S. Mobayen: Fast terminal sliding mode tracking of non-holonomic systems with exponential decay rate. IET Control Theory Appl. 9 (2015), 8, 1294-1301.   DOI:10.1049/iet-cta.2014.1118
  10. S. Mobayen: Finite-time tracking control of chained-form nonholonomic systems with external disturbances based on recursive terminal sliding mode method. Nonlinear Dynamics 80 (2015), 1-2, 669-683.   DOI:10.1007/s11071-015-1897-4
  11. S. Mobayen and D. Baleanu: Linear matrix inequalities design approach for robust stabilization of uncertain nonlinear systems with perturbation based on optimally-tuned global sliding mode control. J. Vibration Control 23, (2017), 8, 1285-1295.   DOI:10.1177/1077546315592516
  12. J. A. Moreno and M. Osorio: A Lyapunov approach to second-order sliding mode controllers and observers. In: Proc. 47th IEEE Conference on Decision and Control 2008, pp. 2856-2861   CrossRef
  13. J. A. Moreno and M. Osorio: Strict Lyapunov functions for the super-twisting algorithm. IEEE Trans. Automat. Control 57 (2012), 4, 1035-1040.   DOI:10.1109/tac.2012.2186179
  14. R. M. Murray and S. S. Sastry: Steering nonholonomic systems in chained form. In: Proc. 30th IEEE Conference on Decision and Control 2 (1991), pp. 1121-1126.   DOI:10.1109/cdc.1991.261508
  15. I. Nagesh and C. Edwards: A multivariable super-twisting sliding mode approachtor: A field-oriented control approach. Automatica 50 (2014), 3, 984-988.   DOI:10.1016/j.automatica.2013.12.032
  16. J. Picó, E. Picó-Marco, A. Vignoni and H. De Battista: Stability preserving maps for finite-time convergence: super-twisting sliding-mode algorithm. Automatica 49 (2013), 2, 534-539.   DOI:10.1016/j.automatica.2012.11.022
  17. F. Rehman: Feedback stabilization of nonholonomic control systems using model decomposition. Asian J. Control 7, (2005), 3, 256-265.   DOI:10.1111/j.1934-6093.2005.tb00235.x
  18. Y. B. Shtessel, I. A. Shkolnikov and A. Levant: Smooth second-order sliding modes: Missile guidance application. Automatica 43 (2007), 8, 1470-1476.   DOI:10.1016/j.automatica.2007.01.008
  19. O. J. Sordalen and O. Egeland: Exponential stabilization of nonholonomic chained systems. IEEE Trans. Automat. Control 40 (1995), 1, 35-49.   DOI:10.1109/9.362901
  20. Y. Wang, Z. Miao, H. Zhong and Qi. Pan: Simultaneous stabilization and tracking of nonholonomic mobile robots: A Lyapunov-based approach. IEEE Trans. Control Systems Technol. 23 (2015), 4, 1440-1450.   DOI:10.1109/tcst.2014.2375812
  21. V. Utkin, J. Guldner, J. Shi, S. Ge and F. Lewis: Sliding Mode Control in Electro-mechanical Systems. Second Edition. Boca Raton: CRC Press, 2009.   DOI:10.1201/9781420065619