A constraint-following control for uncertain mechanical systems: given force coupled with constraint force

  • Ruiying Zhao
  • Ye-Hwa Chen
  • Shengjie Jiao
  • Xiyong Ma
Original Paper


A novel constraint-following control for uncertain mechanical systems is proposed. In mechanical systems, certain given forces may arise due to the constraint forces, which means the given forces are coupled with the constraint forces. By using the second-order form of the constraints, the given forces are decoupled explicitly. The uncertainty of the mechanical system is time-varying and bounded. But its bound is unknown. A series of adaptive parameters are invoked to estimate the bound information of the uncertainty in virtue of state feedback. Based on the estimated bound information, a robust control is designed to render the mechanical system an approximate constraint-following. The system performance under the control is guaranteed as uniform boundedness and uniform ultimate boundedness.


Mechanical system Constraint-following Robust control Given force Constraint force 



Ruiying Zhao is supported by the National Natural Science Foundation of China (Grant No. 51605038) and Natural Science Basic Research Plan in Shaanxi Province of China (No. 2017JQ5034). Shengjie Jiao is supported by the National Science and Technology Pillar Program (No. 2015BAF07B08).


  1. 1.
    Chen, Y.H.: Constraint-following servo control design for mechanical systems. J. Vib. Control 15(3), 369–389 (2009)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Pars, L.A.: A Treatise on Analytical Dynamics. OX Bow Press, Connecticut (1979)MATHGoogle Scholar
  3. 3.
    Rosenberg, R.M.: Analytical Dynamics of Discrete Systems. Plenum, New York (1977)CrossRefMATHGoogle Scholar
  4. 4.
    Whittaker, E.T.: A Treatise on The Analytical Dynamics of Particles and Rigid Bodies. Cambridge University Press, Cambridge (1993)Google Scholar
  5. 5.
    Papastavridis, J.G.: Analytical Mechanics: A Comprehensive Treatise on The Dynamics of Constrained Systems; for Engineers, Physicists, and Mathematicians. Oxford University Press, New York (2002)MATHGoogle Scholar
  6. 6.
    Fantoni, I., Lozano, R.: Nonlinear Control for Underactuated Mechanical Systems. Springer, New York (2001)MATHGoogle Scholar
  7. 7.
    Khalil, H.K.: Nonlinear Systems. Prentice-Hall, Upper Saddle River (2002)MATHGoogle Scholar
  8. 8.
    Udwadia, F.E.: A new perspective on the tracking control of nonlinear structural and mechanical systems. Proc. Math. Phys. Eng. Sci. 459(2035), 1783–1800 (2003)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Jin, X., Wang, S., Yang, G., et al.: Robust adaptive hierarchical insensitive tracking control of a class of leader-follower agents. Inf. Sci. 406–407, 234–247 (2017)CrossRefGoogle Scholar
  10. 10.
    Jin, X., He, Y.: Finite-time robust fault-tolerant control against actuator faults and saturations. IET Control Theory Appl. 11(4), 550–556 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Shen, H., Huang, X., Zhou, J., et al.: Global exponential estimates for uncertain Markovian jump neural networks with reaction–diffusion terms. Nonlinear Dyn. 69(1–2), 473–486 (2011)MathSciNetMATHGoogle Scholar
  12. 12.
    Shen, H., Ju, H., Wu, Z.: Finite-time synchronization control for uncertain Markov jump neural networks with input constraints. Nonlinear Dyn. 77(4), 1709–1720 (2014)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Chen, Y.H., Zhang, X.: Adaptive robust approximate constraint-following control for mechanical systems. J. Franklin Inst. 347(1), 69–86 (2010)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Sun, H., Zhao, H., Zhen, S., et al.: Application of the Udwadia–Kalaba approach to tracking control of mobile robots. Nonlinear Dyn. 83(1–2), 1–12 (2015)MathSciNetGoogle Scholar
  15. 15.
    Goldstein, H.: Classical Mechanics, 2nd edn. Addison-Wesley, Reading (1982)MATHGoogle Scholar
  16. 16.
    Gantmacher, L.: Lectures in Analytical Mechanics. Mir Publishing, Mosocow (1970)Google Scholar
  17. 17.
    Greenwood, D.T.: Classical Dynamics. Dover, New York (1997)Google Scholar
  18. 18.
    Shang, W., Cong, S.: Adaptive compensation of dynamics and friction for a planar parallel manipulator with redundant actuation. In: Proceedings of the 2010 IEEE International Conference on Robotics and Biomimetics, pp. 507–512 (2010)Google Scholar
  19. 19.
    Hao, R., Wang, J., Zhao, J., et al.: Observer-based robust control of 6-DOF parallel electrical manipulator with fast friction estimation. IEEE Trans. Autom. Sci. Eng. 13(3), 1399–1408 (2016)CrossRefGoogle Scholar
  20. 20.
    Chen, Y.H.: Second order constraints for equations of motion of constrained systems. IEEE/ASME Trans. Mechatron. 3(4), 240–248 (1998)CrossRefGoogle Scholar
  21. 21.
    Chen, Y.H., Leitmann, G., Chen, J.S.: Robust control for rigid serial manipulators: a general setting. In: Proceedings of the 1998 American Control Conference, vol. 2, pp. 912–916 (1998)Google Scholar
  22. 22.
    Udwadia, F.E., Kalaba, R.E.: Analytical Dynamics: A New Approach. Cambridge University Press, New York (1996)CrossRefMATHGoogle Scholar
  23. 23.
    Kalaba, R.E., Udwadia, F.E.: Analytical dynamics with constraint forces that do work in virtual displacements. Appl. Math. Comput. 121, 211–217 (2001)MathSciNetMATHGoogle Scholar
  24. 24.
    Udwadia, F.E., Kalaba, R.E.: Nonideal constraints and lagrangian dynamics. J. Aerosp. Eng. 13(1), 17–22 (2000)CrossRefGoogle Scholar
  25. 25.
    Chen, Y.H.: On the deterministic performance of uncertain dynamical systems. Int. J. Control 43(5), 1557–1579 (1986)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Koivo, A.J., Thoma, M., Kocaoglan, E., Andradecetto, J.: Modeling and control of excavator dynamics during digging operation. J. Aerosp. Eng. 9(1), 10–18 (1996)CrossRefGoogle Scholar
  27. 27.
    Jin, X., Qin, J., Shi, Y., et al.: Auxiliary fault tolerant control with actuator amplitude saturation and limited rate. IEEE Trans. Syst. Man Cybern. Syst. 99, 1–10 (2017)Google Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.National Engineering Laboratory for Highway Maintenance EquipmentChang’an UniversityXi’anPeople’s Republic of China
  2. 2.The George W. Woodruff School of Mechanical EngineeringGeorgia Institute of TechnologyAtlantaUSA

Personalised recommendations