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Modal characterization with nonlinear behaviors of a two-link flexible manipulator

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Abstract

This present work has made a noteworthy attempt to demonstrate brief modeling of N-link manipulator and subsequent modal characterization along with the determination of static deflection of a two-link flexible manipulator with a payload. In addition, investigation of nonlinear phenomena of dynamic responses under 3:1 internal resonance has also been accomplished considering geometric nonlinearities. An appropriate and realistic dynamic modeling of the two-link manipulator taking into account of inertia coupling and geometry compatibility between equations of motion and boundary conditions has been derived using the extended Hamilton’s principle. The effect of parametric variation on system eigenfrequencies is well tabulated, and the corresponding eigenspectrums are illustrated graphically. Further, the nonlinear phenomena of dynamic solutions have been demonstrated by using MMS of second order for its statutory effect onto the system instability for the existence of S-N bifurcations. The effect of nonlinearities and various design parameters on the dynamic responses and subsequent bifurcations for 3:1 internal resonance has also been demonstrated. The outcome of the present work enables new understanding into the design criterion and performance limitation of multi-link flexible robots.

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Abbreviations

A :

Area of cross section of link \(({\hbox {m}}^{2})\)

b :

Width of link (m)

E :

Young’s modulus of material of link \(({\hbox {N}}/{\hbox {m}}^{2})\)

g :

Acceleration due to gravity \(({\hbox {m}}/{\hbox {s}}^{2})\)

h :

Thickness of link (m)

I :

Moment of inertia of link \(({\hbox {m}}^{4})\)

L :

Length of link (m)

\({m}_1 \) :

Mass at the end of link 1 (kg)

\({m}_2 \) :

Mass at the end of link 2 called payload (kg)

R :

Position vector of the end point on flexible link

s :

Position vector of general point on the flexible link

w(x, t):

Transverse displacement of link

\({w}_{\mathrm{L}} \) :

Transverse defection at the end of link

\({\rho } \) :

Density of material of link \(({\hbox {kg}}/{\hbox {m}}^{3})\)

\({\theta } \) :

Angular rotation of motor

\(\bar{{\beta }}\) :

Nondimensional eigenfrequency

\({\alpha } _{\mathrm{L}} \) :

Nondimensional length parameter

\({\alpha } _{\mathrm{m}_1 } \) :

Nondimensional mass parameter

\({\alpha } _{\mathrm{m}_2 } \) :

Nondimensional tip mass parameter

\({\alpha } _{\mathrm{M}} \) :

Nondimensional beam mass density parameter

\(\varOmega \) :

Eigenfrequency of the system

\({\chi } \) :

Flexural rigidity ratio

\(\bar{{x}} \) :

Nondimensional position coordinate.

References

  1. Low, H.: A systematic formulation of dynamic equations for robot manipulators with elastic links. J. Field Robot. 4(3), 435–456 (1987)

    Google Scholar 

  2. Coleman, M.P.: Vibration eigenfrequency analysis of a single-link flexible manipulator. J. Sound Vib. 212, 107–120 (1998)

    Article  Google Scholar 

  3. Hwang, Y.L.: A new approach for dynamic analysis of flexible manipulator systems. Int. J. Non Linear Mech. 40, 925–938 (2005)

    Article  MATH  Google Scholar 

  4. Yuan, K.: Regulation of single-link flexible manipulator involving large elastic deflections. J. Guid. Control Dyn. 18(3), 635–637 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  5. Poppelwell, N., Chang, D.: Influence of an offset payload on a flexible manipulator. J. Sound Vib. 190, 721–725 (1996)

    Article  Google Scholar 

  6. Low, K.H., Vidyasagar, M.: A Lagrangian formulation of the dynamic model for flexible manipulator systems. J. Dyn. Syst. Meas. Control 110, 175–181 (1998)

    Article  MATH  Google Scholar 

  7. Ower, J.C., Van De Vegte, J.: Classical control design for a flexible manipulator: modeling and control system design. IEEE J. Robot. Autom. RA 3(5), 485–489 (1987)

    Article  Google Scholar 

  8. Benati, M., Morro, A.: Formulation of equations of motion for a chain flexible links using Hamilton’s principle. J. Dyn. Syst. Meas. Control 116, 81–88 (1994)

    Article  MATH  Google Scholar 

  9. Matsuno, F., Asano, T., Sakawa, Y.: Modeling and quasi-static hybrid position-force control of constrained planar two-link flexible manipulators. IEEE Trans. Robot. Autom. 10, 287–297 (1994)

    Article  Google Scholar 

  10. Zhang, X., Xu, W., Nair, S., Chellaboina, V.: PDE modeling and control of a flexible two-link manipulator. IEEE Trans. Control Syst. Technol. 13(2), 301–312 (2005)

    Article  Google Scholar 

  11. Oakley, C.M., Cannon Jr., R.H.: Theory and experiments in selecting mode shapes for two-link flexible manipulators. In: Proceedings of the First International Symposium on Experimental Robotics, Montreal, Canada (1989)

  12. Chiou, C., Shahinpoor, M.: Dynamic stability analysis of a two-link force-controlled flexible manipulator. J. Dyn. Syst. Meas. Control 112(6), 661–666 (1990)

    Article  Google Scholar 

  13. Fung, R.F., Chang, C.: Dynamic modeling of a non-linearly constrained flexible manipulator with a tip mass by Hamilton’s principle. J. Sound Vib. 216(5), 751–769 (1998)

    Article  Google Scholar 

  14. Zhang, L., Liu, J.: Adaptive boundary control for flexible two-link manipulator based on partial differential equation dynamic model. IET Control Theory Appl. 7(1), 43–51 (2013)

    Article  MathSciNet  Google Scholar 

  15. Ahmed, M.A., Mohamed, Z., Hambali, N.: Dynamic modeling of a two-link flexible manipulator system incorporating payload. In: 3rd IEEE Conference on Industrial Electronics and Applications, 3–5, June (2008)

  16. Sato, O., Sato, A., Takahashi, N., Yokomichi, M.: Analysis of the two–link manipulator in consideration of the horizontal motion about object. Artif. Life Robot. 21(1), 43–48 (2016)

    Article  Google Scholar 

  17. Sato, A., Sato, O., Takahashi, N., Yokomichi, M.: Experimental analysis of the two-link-manipulator in consideration of the relative motion between link and object. In: Proceedings of the 19th ISAROB, pp. 748–751 (2013)

  18. Ata, A.A., Fares, W.F., Sa’adeh, M.Y.: Dynamic analysis of a two-link flexible manipulator subject to different sets of conditions. Proc. Eng. 41, 1253–1260 (2012)

    Article  Google Scholar 

  19. Abe, A., Hashimobo, K.: A novel feedforward control technique for a flexible dual manipulator. Robot. Comput. Integr. Manuf. 35, 169–177 (2015)

    Article  Google Scholar 

  20. Lochan, K., Roy, B.K., Subudhi, B.: A review on two-link flexible manipulators. Annu. Rev. Control 42, 346–367 (2016)

    Article  Google Scholar 

  21. Yang, H., Yu, Y., Yuan, Y., Fan, X.: Back-stepping control of two-link flexible manipulator based on an extended state observer. Adv. Space Res. 56, 2312–2322 (2015)

    Article  Google Scholar 

  22. Lochan, K., Roy, B.K., Subudhi, B.: Robust tip trajectory synchronisation between assumed modes modelled two-link flexible manipulators using second-order PID terminal SMC. Rob. Auton. Syst. 97, 108–124 (2017)

    Article  Google Scholar 

  23. Lochan, K., Roy, B.K., Subudhi, B.: SMC controlled chaotic trajectory tracking of two-link flexible manipulator with PID sliding surface. IFAC Pap. OnLine 49–1, 219–224 (2016)

    Article  Google Scholar 

  24. Pedro, J.O., Tshabalala, T.: Hybrid NNMPC/PID control of a two-link flexible manipulator with actuator dynamics. In: 10th Asian Control Conference (ASCC), Kota Kinabalu, pp. 1-6 (2015)

  25. Ding, W., Shen, Y.: Analysis of transient deformation response for flexible robotic manipulator using assumed mode method. In: 2nd Asia-Pacific Conference on Intelligent Robot Systems (2017)

  26. Pratiher, B., Dwivedy, S.K.: Nonlinear vibration of magnetoelastic cantilever beam with tip mass. J. Vib. Acoust. 131, 1–9 (2009)

    Article  Google Scholar 

  27. Pratiher, B.: Non-linear response of a magneto-elastic translating beam with prismatic joint for higher resonance conditions. Int. J. Non Linear Mech. 46(5), 685–692 (2011)

    Article  Google Scholar 

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  1. Indian Institute of Technology Jodhpur, Rajasthan, 342037, India

    Pravesh Kumar & Barun Pratiher

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  1. Pravesh Kumar
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  2. Barun Pratiher
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Correspondence to Barun Pratiher.

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Subscripts 1 and 2 represent link 1 and link 2, respectively. Also, ()’ and \((\cdot )\) in the following discussion denote the differentiation with respect to the space and time, respectively.

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Kumar, P., Pratiher, B. Modal characterization with nonlinear behaviors of a two-link flexible manipulator. Arch Appl Mech 89, 1201–1220 (2019). https://doi.org/10.1007/s00419-018-1472-9

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