Control design and comprehensive stability analysis of acrobots based on Lyapunov functions

  • Lai Xu-zhi Email author
  • Wu Yun-xin 
  • She Jin-hua 
  • Wu Min 
Electro-Mechanical Engineering And Information Science


A design method for controllers and a comprehensive stability analysis for an acrobat based on Lyapunov functions are presented. Three control laws based on three Lyapunov functions are designed to increase the energy so as to move the acrobot into the unstable inverted equilibrium position, and solve the problem of posture and energy. The concept of a non-smooth Lyapunov function is employed to analyze the stability of the whole system. The validity of this strategy is demonstrated by simulations.

Key words

stability Lyapunov function acrobot fuzzy control 

CLC number



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Oriolo G, Nakamura Y. Control of mechanical systems with second-order nonholonomic constraints: underactuated manipulators[A]. Proc 30th IEEE Conf on Decision and Control[C]. 1991. 2398–2403.Google Scholar
  2. [2]
    Hauser J, Murray R M. Nonlinear controllers for non-integrable systems: the acrobot example[A]. Proc of American Control Conference[C]. 1990. 669–671.Google Scholar
  3. [3]
    Spong M W. The swing up control problem for the acrobot[J]. IEEE Trans Control Systems, 1995, 15: 49–55.Google Scholar
  4. [4]
    Spong M W. Energy based control of a class of underactuated mechanical systems [A]. Proc IFAC 13th World Congress[C]. 1996. 431–435.CrossRefGoogle Scholar
  5. [5]
    Brown S, Passin K. Intelligent control for acrobot[J]. Journal of Intelligent and Robotic Systems, 1997, 18: 209–248.CrossRefGoogle Scholar
  6. [6]
    Bortoff S A, Spong M W. Pseudolinearization of acrobot using spline functions[A]. Proc 31st IEEE Conf on Decision and Control[C]. 1992. 593–598.Google Scholar
  7. [7]
    She Jinhua, Ohyama Y, Hirano K, et al. Motion control of acrobot using time-state control form[A]. Proc IASTED International Conference on Control and Applications[C]. 1998. 171–174.Google Scholar
  8. [8]
    Smith M H, Lee M A, Uliea M, et al. Design limitation of PD versus fuzzy controllers for the acrobot[A]. Proc IEEE Int Conf on Robotics and Automation[C]. 1997. 1130–1135.Google Scholar
  9. [9]
    Miyazaki M, Sampei M, Koga M, et al. A control of underactuated hopping gait systems: acrobot example [A]. Proc 39th IEEE Conference on Decision and Control[C]. 2000. 4797–4802.Google Scholar
  10. [10]
    Malmborg J, Bernhardsson B, Astron J. A stabilizing switching scheme for multi controller systems [A]. IFAC 13th World Congress[C]. 1996. 229–234.CrossRefGoogle Scholar
  11. [11]
    LAI Xu-zhi, SHE Jin-hua, Ohyama Y, et al. A fuzzy control strategy for acrobot combining model-free and model-based control[A]. IEEE Proceedings in Control Theory and Applications[C]. 1999, 146: 505–511.CrossRefGoogle Scholar
  12. [12]
    Schultz D G, Melsa J L. State Functions and Linear Control Systems[M]. McGraw-Hill Book Company, 1967.Google Scholar
  13. [13]
    Yamada K, Yuzawa A. Approximate feedback linearization for nonlinear systems and its application to the acrobot[A]. Proc 2002 American Control Conference [C]. 2002. 1672–1677.Google Scholar
  14. [14]
    Liberzon D. Systems and Control: Foundations and Applications[M]. Boston: Birkhauser, 2003.zbMATHGoogle Scholar

Copyright information

© Central South University 2005

Authors and Affiliations

  • Lai Xu-zhi 
    • 1
    Email author
  • Wu Yun-xin 
    • 2
  • She Jin-hua 
    • 3
  • Wu Min 
    • 1
  1. 1.School of Information Science and EngineeringCentral South UniversityChangshaChina
  2. 2.School of Mechanical and EngineeringCentral South UniversityChangshaChina
  3. 3.School of BionicsTokyo University of TechnologyTokyoJapan

Personalised recommendations