Dynamic Model and Intelligent Optimal Controller of Flexible Link Manipulator System with Payload Uncertainty


There is a high interest in research for using flexible link manipulators in industrial robots as flexible link manipulators are more advantageous than heavy and rigid link manipulators. However, flexible link manipulators still have a critical problem of less accuracy due to their tip vibration. Thus, this research contributes to this topic by obtaining a mathematical model and proposing an intelligent optimal controller for a single-flexible link manipulator with variable payload. The study developed the mathematical model of the single-flexible link manipulator using finite element method and Lagrange’s equation, the mathematical model of the flexible link manipulator has been validated with a SimMechamics model. The novel intelligent optimal controller is an integration of a fuzzy logic controller and an optimal linear quadratic regulator controller, the proposed intelligent optimal controller has the advantages of simplicity and effectiveness for position tracking and vibration suppression. The concept of integrating the fuzzy and linear quadratic regulator solves the problem of rules’ explosion of fuzzy control as only uses the minimum and active rules. The proposed controller has shown better position tracking performance and has demonstrated better effectiveness for vibration suppression of the flexible link manipulator than the linear quadratic regulator controller. Furthermore, the proposed controller is more robust than the linear quadratic regulator controller for dealing with uncertainties.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15


  1. 1.

    Xiao, B.; Yin, S.; Kaynak, O.: Tracking control of robotic manipulators with uncertain kinematics and dynamics. IEEE Trans. Industr. Electron. 63(10), 6439–6449 (2016). https://doi.org/10.1109/TIE.2016.2569068

    Article  Google Scholar 

  2. 2.

    Lee, T.S.; Alandoli, E.A.: A critical review of modelling methods for flexible and rigid link manipulators. J. Braz. Soc. Mech. Sci. Eng. 42(10), 508 (2020). https://doi.org/10.1007/s40430-020-02602-0

    Article  Google Scholar 

  3. 3.

    Sun, C.; Gao, H.; He, W.; Yu, Y.: Fuzzy neural network control of a flexible robotic manipulator using assumed mode method. IEEE Trans. Neural Netw. Learn. Syst. 29(11), 5214–5227 (2018). https://doi.org/10.1109/TNNLS.2017.2743103

    MathSciNet  Article  Google Scholar 

  4. 4.

    My, C.A.; Bien, D.X.; Le, C.H.; Packianather, M.: An efficient finite element formulation of dynamics for a flexible robot with different type of joints. Mech. Mach. Theory 134, 267–288 (2019). https://doi.org/10.1016/j.mechmachtheory.2018.12.026

    Article  Google Scholar 

  5. 5.

    Alandoli, E.A.; Shah, H.N.; Sulaiman, M.; Rashid, M.Z.A.; Aras, S.M.: PD/H-∞ integrated controller for position tracking and vibration suppression of flexible link manipulator system. Int. J. Mech. Mechatron. Eng. 18(3), 54–61 (2018)

    Google Scholar 

  6. 6.

    Tavasoli, A.; Mohammadpour, O.: Dynamic modeling and adaptive robust boundary control of a flexible robotic arm with 2-dimensional rigid body rotation. Int. J. Adapt. Control Signal Process. 32(6), 891–907 (2018). https://doi.org/10.1002/acs.2874

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Alandoli, E.A.; Sulaiman, M.; Rashid, M.Z.: Robustness and disturbance rejection of PD/H-∞ integrated controller for flexible link manipulator system. J. Eng. Sci. Technol. Rev. 21(1), 27–36 (2019). https://doi.org/10.25103/jestr.121.04

    Article  Google Scholar 

  8. 8.

    Shitole, C.; Sumathi, P.: Sliding DFT-based vibration mode estimator for single-link flexible manipulator. IEEE/ASME Trans. Mechatron. 20(6), 3249–3256 (2015). https://doi.org/10.1109/TMECH.2015.2391132

    Article  Google Scholar 

  9. 9.

    San-Millan, A.; Feliu, V.; Garcia, A.: A two-stage control scheme of single-link flexible manipulators. In: 23rd Mediterranean Conference on Control and Automation, Spain, pp. 1098–1105 (2015). https://doi.org/10.1109/MED.2015.7158903

  10. 10.

    My, C.A.; Bien, D.X.: New development of the dynamic modeling and the inverse dynamic analysis for flexible robot. Int. J. Adv. Rob. Syst. 17(4), 1–12 (2020). https://doi.org/10.1177/1729881420943341

    Article  Google Scholar 

  11. 11.

    Wang, F.Y.; Gao, Y.: On frequency sensitivity and mode orthogonality of flexible robotic manipulators. IEEE/CAA J. Autom. Sin. 3(4), 394–397 (2016). https://doi.org/10.1109/JAS.2016.7510112

    MathSciNet  Article  Google Scholar 

  12. 12.

    Gao, H.; He, W.; Zhou, C.; Sun, C.: Neural network control of a two-link flexible robotic manipulator using assumed mode method. IEEE Trans. Ind. Inform. 15(2), 755–765 (2018). https://doi.org/10.1109/TII.2018.2818120

    Article  Google Scholar 

  13. 13.

    Lochan, K.; Roy, B.K.; Subudhi, B.: A review on two-link flexible manipulators. Ann. Rev. Control 42, 346–367 (2016). https://doi.org/10.1016/j.arcontrol.2016.09.019

    Article  Google Scholar 

  14. 14.

    Korayem, M.H.; Rahimi, H.N.: Nonlinear dynamic analysis for elastic robotic arms. Front. Mech. Eng. 6(2), 219–228 (2011). https://doi.org/10.1007/s11465-011-0218-y

    Article  Google Scholar 

  15. 15.

    Peza-Solis, J.F.; Silva-Navarro, G.; Castro-Linares, N.R.: Trajectory tracking control in a single flexible-link robot using finite differences and sliding modes. J. Appl. Res. Technol. 13(1), 70–78 (2015). https://doi.org/10.1016/S1665-6423(15)30006-7

    Article  Google Scholar 

  16. 16.

    Kim, S.M.: Lumped element modeling of a flexible manipulator system. IEEE/ASME Trans. Mechatron. 20(2), 967–974 (2014). https://doi.org/10.1109/TMECH.2014.2327070

    Article  Google Scholar 

  17. 17.

    Fayazi, A.; Pariz, N.; Karimpour, A.; Hosseinnia, S.H.: Robust position-based impedance control of lightweight single-link flexible robots interacting with the unknown environment via a fractional-order sliding mode controller. Robotica 36(12), 1920–1942 (2018). https://doi.org/10.1017/S0263574718000802

    Article  Google Scholar 

  18. 18.

    Rahimi, H.N.; Nazemizadeh, M.: Dynamic analysis and intelligent control techniques for flexible manipulators: a review. Adv. Robot. 28(2), 63–76 (2014). https://doi.org/10.1080/01691864.2013.839079

    Article  Google Scholar 

  19. 19.

    Quan, Q.Q.; Chen, C.B.; Deng, Z.Q.; Tang, J.Y.; Tang, D.W.: On modeling drilling load in lunar regolith simulant. Chin. J. Mech. Eng. 31(1), 1–12 (2018). https://doi.org/10.1186/s10033-018-0207-8

    Article  Google Scholar 

  20. 20.

    Mortazavi, B.; Baniassadi, M.; Bardon, J.; Ahzi, S.: Modeling of two phase random composite materials by finite element, Mori-Tanaka and strong contrast methods. Compos. B Eng. 45(1), 1117–1125 (2013). https://doi.org/10.1016/j.compositesb.2012.05.015

    Article  Google Scholar 

  21. 21.

    My, C.A.; Nguyen, C.D.; Duong, X.B.; Nguyen, A.V.; Nguyen, T.A.; Le, C.H.; Packianather, M.: A novel mathematical approach for finite element formulation of flexible robot dynamics. Mech. Based Des. Struct. Mach. (2020). https://doi.org/10.1080/15397734.2020.1820874

    Article  Google Scholar 

  22. 22.

    Pedro, J.O.; Smith, R.V.: Real-time hybrid PID/ILC control of two-link flexible manipulators. IFAC-PapersOnLine 50(2), 145–150 (2017). https://doi.org/10.1016/j.ifacol.2017.12.027

    Article  Google Scholar 

  23. 23.

    Vakil, M.; Fotouhi, R.; Nikiforuk, P.N.: A new method for dynamic modeling of flexible-link flexible-joint manipulators. J. Vib. Acoust. 134(1), 1–11 (2012). https://doi.org/10.1115/1.4004677

    Article  Google Scholar 

  24. 24.

    Fareh, R.; Saad, M.; Saad, M.: Distributed control strategy for flexible link manipulators. Robotica 33(4), 768–786 (2015). https://doi.org/10.1017/S0263574714000459

    Article  MATH  Google Scholar 

  25. 25.

    Akyüz, I.H.; Kizir, S.; Bingül, Z.: Fuzzy logic control of single-link flexible joint manipulator. In: IEEE International Conference on Industrial Technology, USA, pp. 306–311 (2011). https://doi.org/10.1109/ICIT.2011.5754392

  26. 26.

    Dixit, R.; Kumar, R.P.: Working and limitations of cable stiffening in flexible link manipulators. Adv. Acoust. Vib. (2016). https://doi.org/10.1155/2016/4503696

    Article  Google Scholar 

  27. 27.

    Alandoli, E.A.; Rashid, M.Z.; Sulaiman, M.: A comparison of PID and LQR controllers for position tracking and vibration suppression of flexible link manipulator. J. Theor. Appl. Inf. Technol. 95(13), 2949–2955 (2017)

    Google Scholar 

  28. 28.

    Li, Y.; Tong, S.; Li, T.: Adaptive fuzzy output feedback control for a single-link flexible robot manipulator driven DC motor via backstepping. Nonlinear Anal. Real World Appl. 14(1), 483–494 (2013). https://doi.org/10.1016/j.nonrwa.2012.07.010

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Xu, B.; Zhang, P.: Composite learning sliding mode control of flexible-link manipulator. Complexity (2017). https://doi.org/10.1155/2017/9430259

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Mohamed, Z.; Faudzi, A.A.; Supriyanto, E., et al.: Hybrid vibration and rest-to-rest control of a two-link flexible robotic arm using H∞ loop-shaping control design. Eng. .ations 33(2), 395–409 (2016). https://doi.org/10.1108/EC-11-2014-0228

    Article  Google Scholar 

  31. 31.

    Hu, J.; Xu, G.: Vibration control of piezoelectric flexible structure using robust control methodology. J. Theor. Appl. Inf. Technol. 51(2), 264–274 (2013)

    Google Scholar 

  32. 32.

    Alandoli, E.A.; Lee, T.S.: A critical review of control techniques for flexible and rigid link manipulators. Robotica 38(12), 2239–2265 (2020). https://doi.org/10.1017/S0263574720000223

    Article  Google Scholar 

  33. 33.

    He, W.; Ouyang, Y.; Hong, J.: Vibration control of a flexible robotic manipulator in the presence of input deadzone. IEEE Trans. Ind. Inf. 13(1), 48–59 (2017). https://doi.org/10.1109/TII.2016.2608739

    Article  Google Scholar 

  34. 34.

    Shawky, A.; Zydek, D.; Elhalwagy, Y.Z.; Ordys, A.: Modeling and nonlinear control of a flexible-link manipulator. Appl. Math. Model. 37(23), 9591–9602 (2013). https://doi.org/10.1016/j.apm.2013.05.003

    MathSciNet  Article  MATH  Google Scholar 

  35. 35.

    Suklabaidya, S.; Lochan, K.; Roy, B.K.: Control of rotational base single link flexible manipulator using different SMC techniques for variable payloads. In: International Conference on Energy, Power and Environment: Towards Sustainable Growth, India, pp. 1–6 (2015). https://doi.org/10.1109/EPETSG.2015.7510095

  36. 36.

    Zebin, T., Alam, M.S.: Dynamic modeling and fuzzy logic control of a two-link flexible manipulator using genetic optimization techniques. In: 13th International Conference on Computer and Information Technology (ICCIT), Bangladesh, pp. 418–423 (2010). https://doi.org/10.1109/ICCITECHN.2010.5723894

  37. 37.

    Martins, J.M.; Mohamed, Z.; Tokhi, M.O.; Costa, J.S.D.; Botto, M.A.: Approaches for dynamic modelling of flexible manipulator systems. IEE Proc. Control Theory Appl. 150(4), 401–411 (2003). https://doi.org/10.1049/ip-cta:20030496

    Article  Google Scholar 

  38. 38.

    Tokhi, M.O.; Mohamed, Z.; Shaheed, M.H.: Dynamic characterisation of a flexible manipulator system. Robotica 19(5), 571–580 (2001). https://doi.org/10.1017/S0263574700003209

    Article  Google Scholar 

  39. 39.

    Hussein, M.T., Nemah, M.N.: Control of a two-link (rigid-flexible) manipulator. In: 3rd RSI International Conference on Robotics and Mechatronics, Iran, pp. 720–724 (2015). https://doi.org/10.1109/ICRoM.2015.7367871

  40. 40.

    Kumar, E.V., Jerome, J., Srikanth, K.: Algebraic approach for selecting the weighting matrices of linear quadratic regulator. In: International Conference on Green Computing Communication and Electrical Engineering, Coimbatore, India, pp. 1–6 (2014). https://doi.org/10.1109/ICGCCEE.2014.6922382

  41. 41.

    Tarbosh, Q.A.; Aydoğdu, Ö.; Farah, N.; Talib, M.H.N.; Salh, A.; Çankaya, N.; Omar, F.A.: Review and investigation of simplified rules fuzzy logic speed controller of high performance induction motor drives. IEEE Access 8, 49377–49394 (2020). https://doi.org/10.1109/ACCESS.2020.2977115

    Article  Google Scholar 

  42. 42.

    Nise, N.S.: Control Systems Engineering. Wiley, New York (2011)

    Google Scholar 

Download references


The authors would like to express their thanks to Multimedia University (MMU) for supporting this research through MMU GRA Scheme (MMUI/180265) and to thank the reviewers for their constructive comments.

Author information



Corresponding author

Correspondence to Esmail Ali Alandoli.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Alandoli, E.A., Lee, T.S., Lin, Y.J. et al. Dynamic Model and Intelligent Optimal Controller of Flexible Link Manipulator System with Payload Uncertainty. Arab J Sci Eng (2021). https://doi.org/10.1007/s13369-021-05436-7

Download citation


  • Flexible link manipulator
  • Finite element method
  • Fuzzy logic control
  • Linear quadratic regulator
  • Intelligent optimal controller