Nonlinear Dynamics

, Volume 97, Issue 4, pp 2271–2289 | Cite as

Dynamic analysis of pneumatic artificial muscle (PAM) actuator for rehabilitation with principal parametric resonance condition

  • Bhaben KalitaEmail author
  • S. K. Dwivedy
Original paper


In this present work, the dynamic behavior of a pneumatic artificial muscle (PAM) has been studied by varying different muscle parameters and air pressure into it. The governing equation of motion has been derived using Newton’s law of motion to study the various responses in the system at principal parametric resonance condition. The temporal equation of motion contains various nonlinear parameters with forced and nonlinear parametric excitation. Then, the second-order method of multiple scales is used to find the approximate solutions and to study the dynamic stability and bifurcations of the system. The results are found to be in good agreement with the solutions obtained by solving the temporal equation of motion numerically. The instability regions by varying different system parameters have been plotted. The time responses and phase portraits have been plotted to study the system behavior with the nonlinearity. The influences of the different system parameters in the amplitude for the muscle have also been studied with the help frequency responses. In order to verify the solution, the basin of attraction has also been plotted. The obtained results will be very useful for designing the desired PAM use in different rehabilitation robotics and exoskeleton system.


Pneumatic artificial muscle (PAM) Method of multiple scales Principal parametric resonance Frequency response Basin of attraction 


Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interest.


  1. 1.
    Chou, C.P., Hannaford, B.: Measurement and modeling of McKibben pneumatic artificial muscles. IEEE Trans. Robot. Autom. 12, 90–102 (1996)CrossRefGoogle Scholar
  2. 2.
    Daerden, F., Lefeber, D.: Pneumatic artificial muscles: actuators for robotics and automation. Eur. J. Mech. Environ. Eng. 47, 11–21 (2002)Google Scholar
  3. 3.
    Klute, G.K., Czerniecki, J.M., Hannaford, B.: McKibben artificial muscles: pneumatic actuators with biomechanical intelligence. In: 1999 IEEE/ASME International Conference on Advanced Intelligent Mechatronics, pp. 221–226 (1999)Google Scholar
  4. 4.
    Tondu, B., Lopez, P.: Modeling and control of McKibben artificial muscle robot actuators. IEEE Control Syst. Mag. 20, 15–38 (2000)Google Scholar
  5. 5.
    Singh, M.D., Liem, K., Leontjievs, V., Kecskeméthy, A.: A fluidic-muscle driven force-controlled parallel platform for physical simulation of virtual spatial force-displacement laws. Meccanica 46, 171–182 (2011)CrossRefzbMATHGoogle Scholar
  6. 6.
    Davis, S., Caldwell, D.G.: Pneumatic muscle actuators for humanoid applications-sensor and valve integration. In: 2006 IEEE-RAS International Conference on Humanoid Robots, pp. 456–461 (2006)Google Scholar
  7. 7.
    Daerden, F., Lefeber, D.: The concept and design of pleated pneumatic artificial muscles. Int. J. Fluid Power 2, 41–50 (2001)CrossRefGoogle Scholar
  8. 8.
    Nakamura, T., Shinohara, H.: Position and force control based on mathematical models of pneumatic artificial muscles reinforced by straight glass fibers. In: 2007 IEEE International Conference on Robotics and Automation, pp. 4361–4366 (2007)Google Scholar
  9. 9.
    Veale, A.J., Xie, S.Q., Anderson, I.A.: Modeling the Peano fluidic muscle and the effects of its material properties on its static and dynamic behavior. Smart Mater. Struct. 25, 065014 (2016)CrossRefGoogle Scholar
  10. 10.
    Yarlott, J.M.: Fluid actuator. US Patent No. 3,645,173 (1972)Google Scholar
  11. 11.
    Immega, G., Kukolj, M.: Axially contractable actuator. US Patent No. 4,939,982 (1990)Google Scholar
  12. 12.
    Kukolj, M.: Axially contractable actuautor. US Patent No. 4,733,603 (1988)Google Scholar
  13. 13.
    Morin, A.H.: Elastic diaphragm. US Patent No. 2,642,091 (1953)Google Scholar
  14. 14.
    Baldwin, H.A.: Realizable models of muscle function. In: Bootzin, D., Muffley, H.C. (eds.) Biomechanics, pp. 139–147. Springer, Boston, MA (1969)CrossRefGoogle Scholar
  15. 15.
    Paynter, H.M.: High pressure fluid-driven tension actuators and methods for constructing them. US Patent No. 4,751,869 (1988a)Google Scholar
  16. 16.
    Paynter, H.M.: Hyperboloid of revolution fluid-driven tension actuators and methods of making. US Patent No. 4,721,030 (1988b)Google Scholar
  17. 17.
    Kleinwachter, H., Geerk, J.: Device with a pressurizable variable capacity chamber for transforming a fluid pressure into a moment. US Patent No. 3,638,536 (1972)Google Scholar
  18. 18.
    Beullens, T.: Hydraulic or pneumatic drive device. US Patent No. 4, 841,845 (1989)Google Scholar
  19. 19.
    Festo AG & Co.: Fluidic Muscle MAS. Accessed 19 June 2018
  20. 20.
    Žilić, T., Pavković, D., Zorc, D.: Modeling and control of a pneumatically actuated inverted pendulum. ISA Trans. 48, 327–335 (2009)CrossRefGoogle Scholar
  21. 21.
    Lu, C.H., Hwang, Y.R.: Hybrid sliding mode position control for a piston air motor ball screw table. ISA Trans. 51, 373–385 (2012)CrossRefGoogle Scholar
  22. 22.
    Van Kien, C., Son, N.N., Anh, H.P.H.: Identification of 2-DOF pneumatic artificial muscle system with multilayer fuzzy logic and differential evolution algorithm. In: 2017 IEEE Conference on Industrial Electronics and Applications (ICIEA), pp. 1264–1269 (2017)Google Scholar
  23. 23.
    Liu, Y., Zang, X., Lin, Z., Liu, X., Zhao, J.: Modelling length/pressure hysteresis of a pneumatic artificial muscle using a modified Prandtl–Ishlinskii model. Stroj. Vestn. J. Mech E 63, 56–64 (2017)CrossRefGoogle Scholar
  24. 24.
    Xie, S., Mei, J., Liu, H., Wang, Y.: Hysteresis modeling and trajectory tracking control of the pneumatic muscle actuator using modified Prandtl–Ishlinskii model. Mech Mach Theory. 120, 213–224 (2018)CrossRefGoogle Scholar
  25. 25.
    Wickramatunge, K.C., Leephakpreeda, T.: Empirical modeling of dynamic behaviors of pneumatic artificial muscle actuators. ISA Trans. 52, 825–834 (2013)CrossRefGoogle Scholar
  26. 26.
    Sárosi, J., Biro, I., Nemeth, J., Cveticanin, L.: Dynamic modeling of a pneumatic muscle actuator with two-direction motion. Mech Mach Theory. 85, 25–34 (2015)CrossRefGoogle Scholar
  27. 27.
    Palomares, E., Nieto, A.J., Morales, A.L., Chicharro, J.M., Pintado, P.: Dynamic behaviour of pneumatic linear actuators. Mechatronics 45, 37–48 (2017)CrossRefGoogle Scholar
  28. 28.
    Al-Fahaam, H., Davis, S., Nefti-Meziani, S.: The design and mathematical modelling of novel extensor bending pneumatic artificial muscles (EBPAMs) for soft exoskeletons. Robot. Auton. Syst. 99, 63–74 (2018)CrossRefGoogle Scholar
  29. 29.
    Rimár, M., Šmeringai, P., Fedak, M., Hatala, M., Kulikov, A.: Analysis of step responses in nonlinear dynamic systems consisting of antagonistic involvement of pneumatic artificial muscles. Adv. Mater. Sci. Eng. (2017). Google Scholar
  30. 30.
    Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (2008)zbMATHGoogle Scholar
  31. 31.
    Butikov, E.I.: A physically meaningful new approach to parametric excitation and attenuation of oscillations in nonlinear systems. Nonlinear Dyn. 88, 2609–2627 (2017)CrossRefGoogle Scholar
  32. 32.
    Lacarbonara, W., Antman, S.S.: Parametric instabilities of the radial motions of non-linearly viscoelastic shells under pulsating pressures. Int. J. Nonlinear Mech. 47, 461–472 (2012)CrossRefGoogle Scholar
  33. 33.
    Dwivedy, S.K., Kar, R.C.: Nonlinear dynamics of a cantilever beam carrying an attached mass with 1: 3: 9 internal resonances. Nonlinear Dyn. 31, 49–72 (2003)CrossRefzbMATHGoogle Scholar
  34. 34.
    Araumi, N., Yabuno, H.: Cubic-quintic nonlinear parametric resonance of a simply supported beam. Nonlinear Dyn. 90, 549–560 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Shaker, M.C., Ghosal, A.: Nonlinear modeling of flexible manipulators using nondimensional variables. ASME J. Comput. Nonlinear Dyn. 1, 123–134 (2006)CrossRefGoogle Scholar
  36. 36.
    Pratiher, B., Dwivedy, S.K.: Nonlinear response of a flexible Cartesian manipulator with payload and pulsating axial force. Nonlinear Dyn. 57, 177–195 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Daqaq, M.F., Masana, R., Erturk, A., Quinn, D.D.: On the role of nonlinearities in vibratory energy harvesting: a critical review and discussion. Appl. Mech. Rev. 66, 040801 (2014)CrossRefGoogle Scholar
  38. 38.
    Dwivedy, S.K., Reddy, A.K., Garg, A.: Dynamic analysis of parametrically excited piezoelectric bimorph beam for energy harvesting. In: Sinha, J. (ed.) Vibration Engineering and Technology of Machinery, pp. 363–371. Springer, Cham (2015)CrossRefGoogle Scholar
  39. 39.
    Wilcox, B., Dankowicz, H., Lacarbonara, W.: Response of electrostatically actuated flexible MEMS structures to the onset of low-velocity contact. In: 2009 ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, pp. 1777–1786 (2009) Google Scholar
  40. 40.
    Lacarbonara, W. (eds.): Concepts, methods, and paradigms. In: Nonlinear Structural Mechanics, pp. 1–66. Springer, Boston, MA (2013)Google Scholar
  41. 41.
    Kalita, B., Dwivedy, S.K.: Nonlinear dynamics of a parametrically excited pneumatic artificial muscle (PAM) actuator with simultaneous resonance condition. Mech. Mach. Theory 135, 281–297 (2019)CrossRefGoogle Scholar
  42. 42.
    Li, H., Ganguly, S., Nakano, S., Tadano, K., Kawashima, K.: Development of a light-weight forceps manipulator using pneumatic artificial rubber muscle for sensor-free haptic feedback. In: 2010 International Conference on Applied Bionics and Biomechanics (2010)Google Scholar
  43. 43.
    Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods. Wiley, New York (2008)zbMATHGoogle Scholar
  44. 44.
    Lacarbonara, W.: Nonlinear Structural Mechanics: Theory, Dynamical Phenomena and Modeling. Springer, Berlin (2013)CrossRefzbMATHGoogle Scholar
  45. 45.
    Wakimoto, S., Suzumori, K., Kanda, T.: Development of intelligent McKibben actuator. In: 2005 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 487–492 (2005)Google Scholar
  46. 46.
    De Volder, M., Moers, A.J.M., Reynaerts, D.: Fabrication and control of miniature McKibben actuators. Sens. Actuators A Phys. 166, 111–116 (2011)CrossRefGoogle Scholar
  47. 47.
    Das, A.S., Dutt, J.K., Ray, K.: Active vibration control of unbalanced flexible rotor-shaft systems parametrically excited due to base motion. Appl. Math. Model. 34, 2353–2369 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Azadi, M., Behzadipour, S., Faulkner, G.: Performance analysis of a semi-active mount made by a new variable stiffness spring. J. Sound Vib. 330, 2733–2746 (2011)CrossRefGoogle Scholar
  49. 49.
    Luongo, A., Paolone, A.: On the reconstitution problem in the multiple time-scale method. Nonlinear Dyn. 19, 135–158 (1999)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Mechanical Engineering DepartmentIndian Institute of Technology GuwahatiGuwahatiIndia

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