Skip to main content
SpringerLink
Log in

Evolutionary design of swing-up controllers for stabilization task of underactuated inverted pendulums

  • Research
  • Published:
Genetic Programming and Evolvable Machines Aims and scope Submit manuscript

Abstract

The development of control laws for underactuated mechanical systems with pendulum-like behaviors is of paramount importance due to their use in the modeling of more complex systems and other challenging tasks. The underactuated feature describes constraints in the maneuverability and capabilities of a mechanical system with the advantage of offering less energy consumption. In this work, a novel methodology for solving the automation of evolved nonlinear controllers for the swing-up phase of switching control laws for underactuated inverted pendulums is proposed. Automatic synthesis of linear controllers with optimal performance applied to linear systems modeled as transfer functions is a forward leap proposed by Koza in 2003. Our proposed approach introduces the nonlinear nature within the automated construction of a set of swing-up controllers integrating an evolutionary process based on Genetic Programming (GP). The presented framework is based on an analytic behaviorist setup that merges Control Theory (CT) with GP. CT is applied to formulate the mathematical description of the problem and the design of the fitness function that guides the automated synthesis; GP is implemented as an evolutionary engine for the construction of the solutions. The advantage is that the symbolic feature of GP is exploited to develop large sets of nonlinear controllers that can be further studied with analytic tools from the CT approach. The proposed framework is applied to an underactuated two-link inverted pendulum giving a set of 13,590 evolved nonlinear swing-up controllers with the same and better fitness value than a state-of-the-art human-made design.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. M.W. Spong, Underactuated mechanical systems, in Control Problems in Robotics and Automation. ed. by B. Siciliano, K.P. Valavanis (Springer, Berlin, Heidelberg, 1998), pp.135–150

    Chapter  Google Scholar 

  2. I. Fantoni, R. Lozano, Non-Linear Control for Underactuated Mechanical Systems (Springer, Berlin, Heidelberg, 2001)

    MATH  Google Scholar 

  3. S. Kajita, F. Kanehiro, K. Kaneko, K. Fujiwara, K. Yokoi, H. Hirukawa, Biped walking pattern generation by a simple three-dimensional inverted pendulum model. Adv. Robotics 17(2), 131–147 (2003). https://doi.org/10.1163/156855303321165097

    Article  Google Scholar 

  4. Y. Yoshida, K. Takeuchi, Y. Miyamoto, D. Sato, D. Nenchev, Postural balance strategies in response to disturbances in the frontal plane and their implementation with a humanoid robot. IEEE Trans. Syst. Man Cybern.: Syst. 44(6), 692–704 (2014). https://doi.org/10.1109/TSMC.2013.2272612

    Article  Google Scholar 

  5. M. Yue, C. An, Z. Li, Constrained adaptive robust trajectory tracking for WIP vehicles using model predictive control and extended state observer. IEEE Trans. Syst. Man Cybern.: Syst. 48(5), 733–742 (2018). https://doi.org/10.1109/TSMC.2016.2621181

    Article  Google Scholar 

  6. A. Elhasairi, A. Pechev, Humanoid robot balance control using the spherical inverted pendulum mode. Front. Robotics AI 2, 21 (2015). https://doi.org/10.3389/frobt.2015.00021

    Article  Google Scholar 

  7. O. Boubaker, R. Iriarte, The Inverted Pendulum in Control Theory and Robotics: From Theory to New Innovations (Series Control, Robotics and Sensors) (Institution of Engineering and Technology, London, England, 2017)

    MATH  Google Scholar 

  8. I. Izadgoshasb, Y.Y. Lim, L. Tang, R.V. Padilla, Z.S. Tang, M. Sedighi, Improving efficiency of piezoelectric based energy harvesting from human motions using double pendulum system. Energy Convers. Manag. 184, 559–570 (2019). https://doi.org/10.1016/j.enconman.2019.02.001

    Article  Google Scholar 

  9. T. Biswas, S. Rao, V. Bhandawat, A simple extension of inverted pendulum template to explain features of slow walking. J. Theor. Biol. 457, 112–123 (2018). https://doi.org/10.1016/j.jtbi.2018.08.027

    Article  MATH  Google Scholar 

  10. Z.B. Hazem, M.J. Fotuhi, Z. Bingül, Development of a Fuzzy-LQR and Fuzzy-LQG stability control for a double link rotary inverted pendulum. J. Franklin Inst. 357(15), 10529–10556 (2020). https://doi.org/10.1016/j.jfranklin.2020.08.030

    Article  MathSciNet  MATH  Google Scholar 

  11. B. Sánchez, P. Ordaz, O. Santos, Swing-stabilization up for a rotatory-elastic pendulum via nonlinear sub-optimal control. Asian J. Control 22(1), 34–48 (2020). https://doi.org/10.1002/asjc.1925

    Article  MathSciNet  Google Scholar 

  12. A. Nayak, R.N. Banavar, D.H.S. Maithripala, Stabilizing a spherical pendulum on a quadrotor. Asian J. Control (2021). https://doi.org/10.1002/asjc.2577

    Article  Google Scholar 

  13. I. Chawla, V. Chopra, A. Singla, Robust stabilization control of a spatial inverted pendulum using integral sliding mode controller. Int. J. Nonlinear Sci. Numer. Simul. 22(2), 183–195 (2021). https://doi.org/10.1515/ijnsns-2018-0029

    Article  MathSciNet  MATH  Google Scholar 

  14. L. Herrera, Md.C. Rodríguez-Liñán, M. Meza-Sánchez, E. Clemente, Orbital synchronization of homogeneous mechanical systems with one degree of underactuation. Int. J. Robust. Nonlinear Control (2022). https://doi.org/10.1002/rnc.6052

    Article  MathSciNet  Google Scholar 

  15. D. Liberzon, Switching in Systems and Control (Birkhäuser, Boston, 2003)

    Book  MATH  Google Scholar 

  16. Y. Orlov, L.T. Aguilar, L. Acho, A. Ortiz, Robust orbital stabilization of pendubot: algorithm synthesis, experimental verification, and application to swing up and balancing control, in Modern Sliding Mode Control Theory: New Perspectives and Applications. ed. by G. Bartolini, L. Fridman, A. Pisano, E. Usai (Springer, Berlin, Heidelberg, 2008), pp.383–400. https://doi.org/10.1007/978-3-540-79016-7_18

    Chapter  MATH  Google Scholar 

  17. I. Fantoni, R. Lozano, M.W. Spong, Energy based control of the pendubot. IEEE Trans. Automat. Control 45(4), 725–729 (2000). https://doi.org/10.1109/9.847110

    Article  MathSciNet  MATH  Google Scholar 

  18. M. Zhang, T.-J. Tarn, Hybrid control of the pendubot. IEEE/ASME Trans. Mechatron. 7(1), 79–86 (2002). https://doi.org/10.1109/3516.990890

    Article  Google Scholar 

  19. O. Saleem, K. Mahmood-ul-Hasan, Robust stabilisation of rotary inverted pendulum using intelligently optimised nonlinear self-adaptive dual fractional-order PD controllers. Int. J. Syst. Sci. 50(7), 1399–1414 (2019). https://doi.org/10.1080/00207721.2019.1615575

    Article  MathSciNet  MATH  Google Scholar 

  20. N. Sun, T. Yang, Y. Fang, Y. Wu, H. Chen, Transportation control of double-pendulum cranes with a nonlinear quasi-PID scheme: design and experiments. IEEE Trans. Syst. Man Cybern.: Syst. 49(7), 1408–1418 (2019). https://doi.org/10.1109/TSMC.2018.2871627

    Article  Google Scholar 

  21. D. Gutiérrez-Oribio, J.A. Mercado-Uribe, J.A. Moreno, L. Fridman, Robust global stabilization of a class of underactuated mechanical systems of two degrees of freedom. Int. J. Robust Nonlinear Control 31(9), 3908–3928 (2020). https://doi.org/10.1002/rnc.5176

    Article  MathSciNet  MATH  Google Scholar 

  22. M.W. Spong, The swing up control problem for the acrobot. IEEE Control Syst. Mag. 15(1), 49–55 (1995). https://doi.org/10.1109/37.341864

    Article  Google Scholar 

  23. R. Lozano, I. Fantoni, D.J. Block, Stabilization of the inverted pendulum around its homoclinic orbit. Syst. Control Lett. 40(3), 197–204 (2000). https://doi.org/10.1016/S0167-6911(00)00025-6

    Article  MathSciNet  MATH  Google Scholar 

  24. H. Bui, M. Pham, T.S. Nguyen, Swing-up control of an inverted pendulum cart system using the approach of hedge-algebras theory. Soft. Comput. (2022). https://doi.org/10.1007/s00500-022-06968-2

    Article  Google Scholar 

  25. A.S. Shiriaev, H. Ludvigsen, O. Egeland, Swinging up the spherical pendulum via stabilization of its first integrals. Automatica 40(1), 73–85 (2004). https://doi.org/10.1016/j.automatica.2003.07.009

    Article  MathSciNet  MATH  Google Scholar 

  26. T. Albahkali, R. Mukherjee, T. Das, Swing-up control of the pendubot: An impulse-momentum approach. IEEE Trans. Robot. 25(4), 975–982 (2009). https://doi.org/10.1109/TRO.2009.2022427

    Article  Google Scholar 

  27. P.X. La Hera, L.B. Freidovich, A.S. Shiriaev, U. Mettin, New approach for swinging up the furuta pendulum: theory and experiments. Mechatronics 19(8), 1240–1250 (2009). https://doi.org/10.1016/j.mechatronics.2009.07.005

    Article  Google Scholar 

  28. K. Flaßkamp, J. Timmermann, S. Ober-Blöbaum, A. Trächtler, Control strategies on stable manifolds for energy-efficient swing-ups of double pendula. Int. J. Control 87(9), 1886–1905 (2014). https://doi.org/10.1080/00207179.2014.893450

    Article  MathSciNet  MATH  Google Scholar 

  29. M. Hesse, J. Timmermann, E. Hüllermeier, A. Trächtler, A reinforcement learning strategy for the swing-up of the double pendulum on a cart. Procedia Manuf. 24, 15–20 (2018). https://doi.org/10.1016/j.promfg.2018.06.004

    Article  Google Scholar 

  30. J.F.S. Trentin, S. da Silva, JMd.S. Ribeiro, H. Schaub, An experimental study to swing up and control a pendulum with two reaction wheels. Meccanica 56, 981–990 (2021). https://doi.org/10.1007/s11012-021-01311-9

    Article  MathSciNet  Google Scholar 

  31. J.R. Koza, Genetic Programming: On the Programming of Computers by Means of Natural Selection (MIT Press, Cambridge, 1992)

    MATH  Google Scholar 

  32. J.R. Koza, M.A. Keane, M.J. Streeter, W. Mydlowec, J. Yu, G. Lanza, Automatic Synthesis of Controllers (Springer, Boston, 2003)

    Google Scholar 

  33. J.R. Koza, M.A. Keane, J. Yu, F.H. Bennett, W. Mydlowec, Method and apparatus for automatic synthesis controllers (U.S. Patent US6564194B1, May. 2003)

  34. K.A. Seeler, Transfer Functions Block Diagrams and the s-Plane (Springer, New York, 2014), pp.519–576. https://doi.org/10.1007/978-1-4614-9152-1_9

    Book  Google Scholar 

  35. A. Soltoggio, A comparison of genetic programming and genetic algorithms in the design of a robust, saturated control system, in Genetic and Evolutionary Computation - GECCO 2004. ed. by K. Deb (Springer, Berlin, Heidelberg, 2004), pp.174–185

    Chapter  Google Scholar 

  36. E. Clemente, M. Meza-Sánchez, E. Bugarin, A.Y. Aguilar-Bustos, Adaptive behaviors in autonomous navigation with collision avoidance and bounded velocity of an omnidirectional mobile robot. J. Intell. Robotic Syst. 92(2), 359–380 (2018). https://doi.org/10.1007/s10846-017-0751-y

    Article  Google Scholar 

  37. O. Peñaloza-Mejía, E. Clemente, M. Meza-Sánchez, C.B. Pérez, Evolving behaviors for bounded-flow tracking control of second-order dynamical systems. Eng. Appl. Artif. Intell. 78, 12–27 (2019). https://doi.org/10.1016/j.engappai.2018.10.001

    Article  Google Scholar 

  38. M. Meza-Sánchez, E. Clemente, M.C. Rodríguez-Liñán, G. Olague, Synthetic-analytic behavior-based control framework: Constraining velocity in tracking for nonholonomic wheeled mobile robots. Inf. Sci. 501, 436–459 (2019). https://doi.org/10.1016/j.ins.2019.06.025

    Article  MathSciNet  MATH  Google Scholar 

  39. L. Herrera, M.C. Rodríguez-Liñán, E. Clemente, M. Meza-Sánchez, L. Monay-Arredondo, Evolved extended Kalman filter for first-order dynamical systems with unknown measurements noise covariance. Appl. Soft Comput. 115, 108174 (2022)

    Article  Google Scholar 

  40. B. Siciliano, O. Khatib, Springer Handbook of Robotics (Springer, Secaucus, NJ, USA, 2007)

    MATH  Google Scholar 

  41. J. Nakamura, Applied Numerical Methods with Software, 1st edn. (Prentice Hall PTR, Upper Saddle River, NJ, USA, 1990)

    MATH  Google Scholar 

  42. R.W. Hamming, Numerical Methods for Scientists and Engineers (McGraw-Hill Inc, New York, 1973)

    MATH  Google Scholar 

  43. M. Meza-Sánchez, M.C. Rodríguez-Liñán, E. Clemente, Family of controllers based on sector non-linear functions: an application for first-order dynamical systems. IET Control Theory A 14(10), 1387–1392 (2020). https://doi.org/10.1049/iet-cta.2019.0680

    Article  Google Scholar 

Download references

Acknowledgements

This research was supported by Tecnológico Nacional de México through the project 18344.23-P “Desarrollo de sistemas evolutivos jerárquicos para su aplicación en sistemas adaptativos complejos”.

Funding

The present work was supported by Tecnológico Nacional de México (TecNM) under Grant 18344.23-P.

Author information

Authors and Affiliations

  1. Department of Industrial Engineering, Tecnológico Nacional de México/Instituto Tecnológico de Tijuana, Calzada Del Tecnológico S/N, Fracc. Tomas Aquino, 22414, Tijuana, BC, Mexico

    Marlen Meza-Sánchez

  2. Crater Labs, 365 Bloor Street East, Suite 1003, Toronto, ON, M4W3L4, Canada

    M. C. Rodríguez-Liñán

  3. Department of Computing and Systems, TecNM/I.T. Ensenada, Blvd. Tecnológico 150, Ex-ejido Chapultepec, 22780, Ensenada, BC, Mexico

    Eddie Clemente

  4. Mechanical and Aerospace Engineering Department, Naval Postgraduate School, 1 University Circle, Monterey, CA, 93943, USA

    Leonardo Herrera

Authors
  1. Marlen Meza-Sánchez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. C. Rodríguez-Liñán
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Eddie Clemente
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Leonardo Herrera
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed equally to this work.

Corresponding author

Correspondence to Eddie Clemente.

Ethics declarations

Conflict of interest

The submitted work is original and it not have been published elsewhere in any form or language. The authors declare no conflict of interest.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Meza-Sánchez, M., Rodríguez-Liñán, M.C., Clemente, E. et al. Evolutionary design of swing-up controllers for stabilization task of underactuated inverted pendulums. Genet Program Evolvable Mach 24, 9 (2023). https://doi.org/10.1007/s10710-023-09457-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10710-023-09457-z

Keywords

Search

Navigation