Multibody System Dynamics

, Volume 42, Issue 3, pp 283–315 | Cite as

Feedback control of multibody systems with joint clearance and dynamic backlash: a tutorial



The problem of feedback control of mechanisms with joint clearance is analysed. Various control strategies are reviewed: impactless trajectories with persistent contact, control through collisions, the stabilization of equilibrium points, and trajectory tracking control. This article sets a general control framework, brings some preliminary answers and leaves some problems open, which are mentioned throughout the article and in the conclusions.


Joint clearance Multi-body system Lyapunov stability Feedback control Trajectory tracking Impact Moreau’s sweeping process Contact linear complementarity problem Stabilization Non-smooth Lagrangian system Juggling system 


  1. 1.
    Acary, V.: Projected event-capturing time-stepping schemes for nonsmooth mechanical systems with unilateral contact and Coulomb’s friction. Comput. Methods Appl. Mech. Eng. 256, 224–250 (2013) MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Acary, V., Brogliato, B.: Numerical Methods for Nonsmooth Dynamical Systems. Applications in Mechanics and Electronics. Lecture Notes in Applied and Computational Mechanics, vol. 35. Springer, Berlin (2008) MATHGoogle Scholar
  3. 3.
    Addi, K., Brogliato, B., Goeleven, D.: A qualitative mathematical analysis of a class of linear variational inequalities via semi-complementarity problems: applications in electronics. Math. Program. 126(1), 31–67 (2011) MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Akhadkar, N., Acary, V., Brogliato, B.: Analysis of collocated feedback controllers for four-bar planar mechanisms with joint clearances. Multibody Syst. Dyn. 38(2), 101–136 (2016) MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Ames, A.D., Galloway, K., Sreenath, K., Grizzle, J.W.: Rapidly exponentially stabilizing control Lyapunov functions and hybrid zero dynamics. IEEE Trans. Autom. Control 59(4), 876–891 (2014) MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Ballard, P.: Formulation and well-posedness of the dynamics of rigid body systems with perfect unilateral constraints. Philos. Trans. R. Soc. Lond. A 359(1789), 2327–2346 (2001) MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Baumann, M., Leine, R.I.: A synchronization-based state observer for impact oscillators using only collision time information. Int. J. Robust Nonlinear Control 26, 2542–2563 (2016) MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bernstein, D.S.: Matrix Mathematics. Theory, Facts, and Formulas with Application to Linear Systems Theory. Princeton University Press, Princeton (2005) MATHGoogle Scholar
  9. 9.
    Blumentals, A., Brogliato, B., Bertails-Descoubes, F.: The contact problem in Lagrangian systems subject to bilateral and unilateral constraints,with or without sliding Coulomb’s friction: a tutorial. Multibody Syst. Dyn. 38, 43–76 (2016) MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Bourgeot, J.M., Brogliato, B.: Tracking control of Lagrangian complementarity systems. Int. J. Bifurc. Chaos 15(6), 1839–1866 (2005) CrossRefMATHGoogle Scholar
  11. 11.
    Brogliato, B.: Absolute stability and the Lagrange–Dirichlet theorem with monotone multivalued mappings. Syst. Control Lett. 51(5), 343–353 (2004) MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Brogliato, B.: Inertial couplings between unilateral and bilateral holonomic constraints in frictionless Lagrangian systems. Multibody Syst. Dyn. 29(3), 289–325 (2013) MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Brogliato, B.: Kinetic quasi-velocities in unilaterally constrained Lagrangian mechanics with impacts and friction. Multibody Syst. Dyn. 32(2), 175–216 (2014) MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Brogliato, B.: Nonsmooth Mechanics. Models, Dynamics and Control, third edn. Communications and Control Engineering. Springer, Berlin (2016) MATHGoogle Scholar
  15. 15.
    Brogliato, B., Zavala-Rio, A.: On the control of complementary-slackness mechanical juggling systems. IEEE Trans. Autom. Control 45(2), 235–246 (2000) CrossRefMATHGoogle Scholar
  16. 16.
    Brogliato, B., Niculescu, S.I., Orhant, P.: On the control of finite-dimensional mechanical systems with unilateral constraints. IEEE Trans. Autom. Control 42(2), 200–215 (1997) MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Brogliato, B., Mabrouk, M., Zavala-Rio, A.: On the controllability of linear juggling mechanical systems. Syst. Control Lett. 55, 350–367 (2006) MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Brogliato, B., Lozano, R., Maschke, B., Egeland, O.: Dissipative Systems Analysis and Control. Theory and Applications, 2nd edn. Communications and Control Engineering. Springer, London (2007) MATHGoogle Scholar
  19. 19.
    Byrnes, C.I., Isidori, A.: Asymptotic stabilization of minimum phase nonlinear systems. IEEE Trans. Autom. Control 36(10), 1122–1337 (1991) MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Chang, S.L., Tsai, L.W.: On the redundant-drive backlash-free robotic mechanisms. ASME J. Mech. Des. 115, 247–254 (1993) CrossRefGoogle Scholar
  21. 21.
    Changqing, B., Qingyu, X.: Dynamic model of ball bearings with internal clearance and waviness. J. Sound Vib. 294, 23–48 (2006) CrossRefGoogle Scholar
  22. 22.
    Chevallereau, C., Westervelt, E.R., Grizzle, J.W.: Asymptotically stable running for a five-link, four-actuator, planar bipedal robot. Int. J. Robot. Res. 24(6), 431–464 (2005) CrossRefGoogle Scholar
  23. 23.
    Duarte, F., Machado, J.T.: Describing function of two masses with backlash. Nonlinear Dyn. 56, 409–413 (2009) CrossRefMATHGoogle Scholar
  24. 24.
    Facchinei, F., Pang, J.S.: Finite-Dimensional Inequalities and Complementarity Problems, vol. I. Springer Series in Operations Research. Springer, Berlin (2003) MATHGoogle Scholar
  25. 25.
    Flores, P., Ambrosio, J., Claro, J.P., Lankarani, H.: Kinematics and Dynamics of Multibody Systems with Imperfect Joints. Lecture Notes in Applied and Computational Mechanics, vol. 34. Springer, Berlin (2008) MATHGoogle Scholar
  26. 26.
    Flores, P., Koshy, C., Lankarani, H., Ambrosio, J., Claro, J.: Numerical and experimental investigation on multibody systems with revolute joint clearance joints. Nonlinear Dyn. 65(4), 383–398 (2011) CrossRefMATHGoogle Scholar
  27. 27.
    Galeani, S., Menini, L., Potini, A., Tornambè, A.: Trajectory tracking for a particle in elliptical billiards. Int. J. Control 81(2), 189–213 (2008) MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Glocker, C.: Set-Valued Force Laws. Lecture Notes in Applied Mechanics, vol. 1. Springer, Berlin (2001) CrossRefMATHGoogle Scholar
  29. 29.
    Glocker, C.: An introduction to impacts. In: Haslinger, J., Stavroulakis, G. (eds.) Nonsmooth Mechanics of Solids. CISM Courses and Lectures, vol. 485 pp. 45–102. Springer, New York (2006) CrossRefGoogle Scholar
  30. 30.
    Guan, Y., Li, M., Lim, T., Jr, W.S.: Comparative analysis of actuator concepts for active gear pair vibration control. J. Sound Vib. 269, 273–294 (2004) CrossRefGoogle Scholar
  31. 31.
    Hiriat-Urruty, J.B., Lemaréchal, C.: Fundamentals of Convex Analysis. Grundlehren text Editions. Springer, Berlin (2000) Google Scholar
  32. 32.
    Jang, G., Jeong, S.: Vibration analysis of a rotating system due to the effect of ball bearing waviness. J. Sound Vib. 269, 709–726 (2004) CrossRefGoogle Scholar
  33. 33.
    Koshy, C., Flores, P., Lankarani, H.: Study of the effect of contact force model on the dynamic response of mechanical systems with dry clearance joints: computational and experimental approaches. Nonlinear Dyn. 73(1), 325–338 (2013) CrossRefGoogle Scholar
  34. 34.
    Krinner, A., Thümmel, T.: Non-smooth behaviour of a linkage mechanism with revolute clearance joints. In: New Advances in Mechanisms, Transmissions and Applications. Proceedings of the Second Conference MeTrApp 2013. Mechanisms and Machine Science, vol. 17, pp. 233–241 (2014) Google Scholar
  35. 35.
    Lagerberg, A., Egardt, B.: Backlash estimation with application to automotive powertrains. IEEE Trans. Control Syst. Technol. 15(3), 483–493 (2007) CrossRefGoogle Scholar
  36. 36.
    Lancaster, P., Tismenetsky, M.: The Theory of Matrices, 2nd edn. Academic Press, Orlando (1985) MATHGoogle Scholar
  37. 37.
    Leine, R.I., van de Wouw, N.: Stability and Convergence of Mechanical Systems with Unilateral Constraints. Lecture Notes in Applied and Computational Mechanics, vol. 36. Springer, Berlin (2008) MATHGoogle Scholar
  38. 38.
    Mata-Jimenez, M., Brogliato, B.: Analysis of PD and nonlinear control of mechanical systems with dynamic backlash. J. Vib. Control 9(1), 119–156 (2003) MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    McClamroch, N., Wang, D.: Feedback stabilization and tracking of constrained robots. IEEE Trans. Autom. Control 33(5), 419–426 (1988) MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Menini, L., Tornambé, A.: Velocity observers for non-linear mechanical systems subject to non-smooth impacts. Automatica 38, 2169–2175 (2002) MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Menini, L., Tornambè, A.: Control of (otherwise) uncontrollable linear mechanical systems through non-smooth impacts. Syst. Control Lett. 49(4), 311–322 (2003) MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Menini, L., Possieri, C., Tornambé, A.: On the computation of the continuous-time reference trajectory for mechanical juggling systems. In: Proc. IEEE 54th Annual Conference on Decision and Control, Osaka, Japan, pp. 145–150 (2015) Google Scholar
  43. 43.
    Morarescu, C.I., Brogliato, B.: Trajectory tracking control of multiconstraint complementarity Lagrangian systems. IEEE Trans. Autom. Control 55(6), 1300–1313 (2010) MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Moreau, J.J.: Unilateral contact and dry friction in finite freedom dynamics. In: Moreau, J.J., Panagiotopoulos, P. (eds.) Nonsmooth Mechanics and Applications. CISM Courses and Lectures, pp. 1–82. Springer, New York (1988) CrossRefGoogle Scholar
  45. 45.
    Müller, A.: Internal preload control of redundantly actuated parallel manipulators–its application to backlash avoiding control. IEEE Trans. Robot. 21(4), 668–677 (2005) CrossRefGoogle Scholar
  46. 46.
    Nguyen, N., Brogliato, B.: Multiple Impacts in Dissipative Granular Chains. Lecture Notes in Applied and Computational Mechanics, vol. 72. Springer, Berlin (2014) Google Scholar
  47. 47.
    Nordin, M., Gutman, P.: Controlling mechanical systems with backlash—a survey. Automatica 38, 1633–1649 (2002) MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Nordin, M., Galic, J., Gutman, P.: New models for backlash and gear play. Int. J. Adapt. Control Signal Process. 11, 49–63 (1997) MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Paden, B., Panja, R.: Globally asymptotically stable ‘PD+’ controller for robot manipulators. Int. J. Control 47(6), 1697–1712 (1988) CrossRefMATHGoogle Scholar
  50. 50.
    Paoli, L., Schatzman, M.: Penalty approximation for dynamical systems submitted to multiple non-smooth constraints. Multibody Syst. Dyn. 8(3), 347–366 (2002) MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Pereira, C., Ramalho, A., Ambrosio, J.: A critical overview of internal and external cylinder contact force models. Nonlinear Dyn. 63(4), 681–697 (2011) CrossRefGoogle Scholar
  52. 52.
    Pfeiffer, F., Glocker, C.: Multibody Dynamics with Unilateral Contacts. Nonlinear Science. Wiley, New York (1996) CrossRefMATHGoogle Scholar
  53. 53.
    Reyhanoglu, M., van der Schaft, A., McClamroch, N., Kolmanovsky, I.: Dynamics and control of a class of underactuated mechanical systems. IEEE Trans. Autom. Control 44(9), 1663–1671 (1999) MathSciNetCrossRefMATHGoogle Scholar
  54. 54.
    Saperstone, S., Yorke, J.: Controllability of linear oscillatory systems using positive controls. SIAM J. Control 9(2), 253–262 (1971) MathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    Spong, M.W., Kelly, R., Ortega, R.: Comments on “Adaptive manipulator control: a case study”. IEEE Trans. Autom. Control 35(6), 761–762 (1990) CrossRefMATHGoogle Scholar
  56. 56.
    Tanwani, A., Brogliato, B., Prieur, C.: Observer design for unilaterally constrained Lagrangian systems: a passivity-based approach. IEEE Trans. Autom. Control 61(9), 2386–2401 (2016) MathSciNetCrossRefMATHGoogle Scholar
  57. 57.
    Thuemmel, T., Rossner, M.: Introduction to modelling and parameter identification methodology of linkages by measurements and simulation. In: Proceedings of 13rd World Congress in Mechanisms and Machine Science, Guanajuato, Mexico, 19–25 June, pp. IMD-123 (2011) Google Scholar
  58. 58.
    Thümmel, T., Ginzinger, L.: Measurements and simulations of a crank and rocker mechanism including friction, clearance and impacts. In: Proceedings of IXth Intern. Conf. on the Theory of Machines and Mechanisms, August 31–September 02, Liberec, Czech Republic (2004) Google Scholar
  59. 59.
    van de Wouw, N., Leine, R.I.: Attractivity of equilibrium sets of systems with dry friction. Nonlinear Dyn. 35(1), 19–39 (2008) MathSciNetCrossRefMATHGoogle Scholar
  60. 60.
    Varedi, S., Daniali, H., Dardel, M., Fathi, A.: Optimal dynamic design of a planar slider–crank mechanism with a joint clearance. Mech. Mach. Theory 86, 191–200 (2015) CrossRefGoogle Scholar
  61. 61.
    Xiao, H., Shao, Y., Brennan, M.: On the contact stiffness and nonlinear vibration of an elastic body with a rough surface in contact with a rigid surface. Eur. J. Mech. A, Solids 49, 315–328 (2015) CrossRefGoogle Scholar
  62. 62.
    Yan, S., Xiang, W., Zhang, L.: A comprehensive model for 3D revolute joints with clearances in mechanical systems. Nonlinear Dyn. 80(1), 309–328 (2015) CrossRefGoogle Scholar
  63. 63.
    Zavala-Rio, A., Brogliato, B.: On the control of a one degree-of-freedom juggling robot. Dyn. Control 9, 67–90 (1999) MathSciNetCrossRefMATHGoogle Scholar
  64. 64.
    Zavala-Rio, A., Brogliato, B.: Direct adaptive control design for one-degree-of-freedom complementary-slackness jugglers. Automatica 37, 1117–1123 (2001) CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.INRIA Grenoble Rhône-Alpes and Laboratoire Jean KuntzmanUniversity Grenoble-AlpesSaint-IsmierFrance

Personalised recommendations