Lyapunov Self-triggered Controller for Nonlinear Trajectory Tracking of Unicycle-type Robot


This paper focuses on the design and implementation of an aperiodic control of nonholonomic robots tracking nonlinear trajectories. The main objective of our controller is to reduce the number of updates while preserving control performance guarantees. To solve the problem in a more efficient way, we design two aperiodic control solutions, one to reach a target point and a second to track a predefined nonlinear trajectory. Unlike most previous work, our triggering condition only updates the controller when the time derivative of the Lyapunov function becomes non-negative, without taking into account the measurement error. Multiple simulated results with different initial conditions are included, showing how our control solution significantly reduces the need for communication in comparison with periodic and other aperiodic strategies while preserving a desired tracking performance. To validate the proposal experimental tests of each control technique with a P3-DX robot remotely controlled through an IEEE 802.11g wireless network are also carried out.

This is a preview of subscription content, log in to check access.


  1. [1]

    D. Liu and F. Hao, “Decentralized event-triggered control strategy in distributed networked systems with delays,” International Journal of Control, Automation and Systems, vol. 11, pp. 33–40, Feb. 2013.

    Article  Google Scholar 

  2. [2]

    R. Postoyan, M. C. Bragagnolo, E. Galbrun, J. Daafouz, D. Nesic, and E. B. Castelan, “Nonlinear event-triggered tracking control of a mobile robot: design, analysis and experimental results,” IFAC Proceedings Volumes, vol. 46, pp. 318–323, Jan. 2013.

    Article  Google Scholar 

  3. [3]

    C. Santos, M. Martinez-Rey, F. Espinosa, A. Gardel, and E. Santiso, “Event-based sensing and control for remote robot guidance: An experimental case,” Sensors, vol. 17, no. 9, 2017.

    Google Scholar 

  4. [4]

    X. Chen, F. Hao, and B. Ma, “Periodic event-triggered cooperative control of multiple non-holonomic wheeled mobile robots,” IET Control Theory Applications, vol. 11, no. 2, pp. 890–899, 2017.

    MathSciNet  Article  Google Scholar 

  5. [5]

    D. Zhao, T. Dong, and W. Hu, “Event-triggered consensus of discrete time second-order multi-agent network,” International Journal of Control, Automation and Systems, vol. 16, pp. 87–96, Feb. 2018.

    Article  Google Scholar 

  6. [6]

    Z. Tang, “Event-triggered consensus of linear discrete-time multi-agent systems with time-varying topology,” International Journal of Control, Automation and Systems, vol. 16, pp. 1179–1185, Jun 2018.

    Article  Google Scholar 

  7. [7]

    M. Mazo, A. Anta, and P. Tabuada, “On self-triggered control for linear systems: Guarantees and complexity,” in 2009 European Control Conference (ECC), pp. 3767–3772, Aug 2009.

    Google Scholar 

  8. [8]

    A. Eqtami, S. Heshmati-alamdari, D. V. Dimarogonas, and K. J. Kyriakopoulos, “Self-triggered model predictive control for nonholonomic systems,” Proc. of European Control Conference (ECC), pp. 638–643, July 2013.

    Google Scholar 

  9. [9]

    C. Santos, M. Mazo, and F. Espinosa, “Adaptive self-triggered control of a remotely operated p3-dx robot: Simulation and experimentation,” Robotics and Autonomous Systems, vol. 62, no. 2, pp. 847–854, 2014.

    Article  Google Scholar 

  10. [10]

    U. Tiberi and K. Johansson, “A simple self-triggered sampler for perturbed nonlinear systems,” Nonlinear Analysis: Hybrid Systems, vol. 10, pp. 126–140, 2013. Special Issue related to IFAC Conference on Analysis and Design of Hybrid Systems (ADHS 12).

    MathSciNet  MATH  Google Scholar 

  11. [11]

    M. D. D. Benedetto, S. D. Gennaro, and A. D’Innocenzo, “Digital self triggered robust control of nonlinear systems,” Proc. of 50th IEEE Conference on Decision and Control and European Control Conference, pp. 1674–1679, Dec. 2011.

    Google Scholar 

  12. [12]

    W. P. M. H. Heemels, K. H. Johansson, and P. Tabuada, “An introduction to event-triggered and self-triggered control,” Proc. of IEEE 51st IEEE Conference on Decision and Control (CDC), pp. 3270–3285, Dec 2012.

    Google Scholar 

  13. [13]

    C. Santos, F. Espinosa, E. Santiso, and M. Martinez-Rey, “A simplified event-triggering condition non-dependent on measurement error,” Proc. of 3rd International Conference on Event-Based Control, Communication and Signal Processing (EBCCSP), pp. 1–6, May 2017.

    Google Scholar 

  14. [14]

    Y. Su, Q. Wang, and C. Sun, “Self-triggered robust model predictive control for nonlinear systems with bounded disturbances,” IET Control Theory Applications, vol. 13, no. 2, pp. 1336–1343, 2019.

    MathSciNet  Article  Google Scholar 

  15. [15]

    N. Marchand, J. Martinez, S. Durand, and J. Guerrero-Castellanos, “Lyapunov event-triggered control: a new event strategy based on the control,” IFAC Proceedings Volumes, vol. 46, no. 2, pp. 324–328, 2013.

    Article  Google Scholar 

  16. [16]

    Q. Cao, Z. Sun, Y. Xia, and L. Dai, “Self-triggered mpc for trajectory tracking of unicycle-type robots with external disturbance,” Journal of the Franklin Institute, vol. 356, no. 2, pp. 5593–5610, 2019.

    MathSciNet  Article  Google Scholar 

  17. [17]

    F. Heidari and R. Fotouhi, “A human-inspired method for point-to-point and path-following navigation of mobile robots,” Journal of Mechanisms and Robotics, vol. 7, pp. 041025–041025-18, July 2015.

    Article  Google Scholar 

  18. [18]

    M. Aicardi, G. Casalino, A. Bicchi, and A. Balestrino, “Closed loop steering of unicycle like vehicles via lyapunov techniques,” Robotics Automation Magazine, IEEE, vol. 2, pp. 27–35, Mar 1995.

    Article  Google Scholar 

  19. [19]

    R. W. Brockett, “Asymptotic stability and feedback stabilization,” Differential Geometric Control Theory, pp. 181–191, Birkhauser, 1983.

    Google Scholar 

  20. [20]

    M. Amoozgar and Y. Zhang, “Trajectory tracking of wheeled mobile robots: A kinematical approach,” Proc. of IEEE/ASME International Conference on Mechatronics and Embedded Systems and Applications (MESA), pp. 275–280, July 2012.

    Google Scholar 

  21. [21]

    Z. Wang and Y. Liu, “Visual regulation of a nonholo-nomic wheeled mobile robot with two points using lyapunov functions,” Proc. of International Conference on Mechatronics and Automation (ICMA), pp. 1603–1608, Aug. 2010.

    Google Scholar 

  22. [22]

    D. P. Borgers and W. P. M. H. Heemels, “Event-separation properties of event-triggered control systems,” IEEE Transactions on Automatic Control, vol. 59, pp. 2644–2656, Oct. 2014.

    MathSciNet  Article  Google Scholar 

  23. [23]

    Y. Batmani, M. Davoodi, and N. Meskin, “On design of nonlinear event-triggerec suboptimal tracking controller,” Proc. of 4th International Conference on Control, Decision and Information Technologies (CoDIT), pp. 1048–1053, April 2017.

    Google Scholar 

  24. [24]

    H. Khalil, Nonlinear Systems, Prentice Hall, 2002.

    Google Scholar 

  25. [25]

    J. Kurzweil, “On the inversion of Lypaunov’s second theorem on stability of motion,” Czechoslovak Mathematical Journal, vol. 81, pp. 217–259, 1956.

    MATH  Google Scholar 

  26. [26]

    A. Chaillet and A. Loria, “Necessary and sufficient conditions for uniform semiglobal practical asymptotic stability: Application to cascaded systems,” Automatica, vol. 42, no. 2, pp. 1899–1906, 2006.

    MathSciNet  Article  Google Scholar 

  27. [27]

    F. Espinosa, M. Salazar, D. Pizarro, and F. Valdes, “Electronics proposal for telerobotics operation of p3-dx units”, Remote and Telerobotics (N. Mollet, ed.), ch. 1, InTech, Mar. 2010.

    Google Scholar 

  28. [28]

    C. Santos, F. Espinosa, E. Santiso, and M. Mazo, “Aperiodic linear networked control considering variable channel delays: Application to robots coordination,” Sensors, vol. 15, no. 6, p. 12454, 2015.

    Article  Google Scholar 

  29. [29]

    B. M. Kim and P. Tsiotras, “Controllers for unicycle-type wheeled robots: Theoretical results and experimental validation,” IEEE Transactions on Robotics and Automation, vol. 18, pp. 294–307, Jun 2002.

    Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Carlos Santos.

Additional information

Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Carlos Santos received his B.S. degree in Telecommunications Engineering in 2010 and an M.Sc. in Electrical Engineering in 2011, both from the University of Alcala, Spain, and a Ph.D. degree in Electronics in 2016. He is currently at the Department of Electronics of the University of Alcala D with a Postdoctoral Research Contract. His research interest focuses on the field of fusion algorithms, trajectory generation for navigation in mobile robotics and varying-time sampling control techniques.

Felipe Espinosa received his M.S. degree (Polytechnics University of Madrid, Spain) and a Ph.D. degree (University of Alcala, Spain) in telecommunication, in 1991 and 1999 respectively. He became an Associate Professor in 2000 and a Full Professor in 2016 in the Electronics De-partment with the University of Alcala, regularly involved in electronic control and automation subjects in the Post-Degree programme (Quality Awared of the Education and Science Spanish Ministry). His current research interests include control, communication and sensorial systems applied to intelligent transportation systems, industrial automation and smart cities.

Enrique Santiso is an assistant professor at the Electronics Department, University of Alcala. He received his M.Sc. degree (Polytechnic University of Valencia, Spain) in 1996, and a Ph.D. (University of Alcala, Spain) in Telecommunications in 2003. His main research interest focuses on sensorial systems, control and electronics instrumentation.

David Gualda received his B.S. degree in Electronics Systems and an M.Sc. degree in Advanced Electronics System, from the University of Alcala (Spain), in 2009 and 2011, respectively; and a Ph.D. degree in Electronics in 2016. He is currently at the Department of Electronics of the University of Alcala with a Postdoctoral Research Contract. His main research interests are in the areas of ultrasonic indoor location, signal processing and information fusion.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Santos, C., Espinosa, F., Santiso, E. et al. Lyapunov Self-triggered Controller for Nonlinear Trajectory Tracking of Unicycle-type Robot. Int. J. Control Autom. Syst. 18, 1829–1838 (2020).

Download citation


  • Lyapunov-based controller
  • nonlinear trajectory-tracking
  • self-triggered
  • semiglobal practical stability