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Adaptive sliding mode control of \(n\) flexible-joint robot manipulators in the presence of structured and unstructured uncertainties

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This study investigates a voltage-based adaptive sliding mode control (VB-ASMC) to tracking the position of an \(n\) rigid-link flexible-joint (RLFJ) robot manipulator under the presence of uncertainties and external disturbances. First, the dynamic equations of the \(n\)-RLFJ robot manipulator have been divided into \(n\) subsystems, and for each of them a voltage-based sliding mode control (VB-SMC) is designed simultaneously. The mathematical proof shows that the closed-loop system under VB-SMC has global asymptotic stability. Second, due to the use of the sign function in the VB-SMC structure, the occurrence of chattering is inevitable. Therefore, to overcome this problem, an adaptive estimator is designed to estimate the boundary of uncertainties. Since the adaptive estimator part in the VB-ASMC has only one law, the proposed control has a very low computational volume. The Lyapunov stability theorem shows that the controlled closed-loop system under the VB-ASMC has global asymptotic stability. Finally, extensive simulations on the single and 2-RLFJ robot manipulator and practical implementation on the single-RLFJ robot manipulator are presented to demonstrate the effectiveness and improved performance of the proposed control scheme.

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  1. 1.

    Zhang, Y., Jin, L.: Robot Manipulator Redundancy Resolution. Wiley, New York (2017)

  2. 2.

    Sharifnia, M., Akbarzadeh, A.: A constrained assumed modes method for dynamics of a flexible planar serial robot with prismatic joints. Multibody Syst. Dyn. 40(3), 261–285 (2017)

  3. 3.

    Dwivedy, S.K., Eberhard, P.: Dynamic analysis of flexible manipulators, a literature review. Mech. Mach. Theory 41(7), 749–777 (2006)

  4. 4.

    Kiang, C.T., Spowage, A., Yoong, C.K.: Review of control and sensor system of flexible manipulator. J. Intell. Robot. Syst. 77(1), 187–213 (2015)

  5. 5.

    Subudhi, B., Morris, A.S.: Dynamic modelling, simulation and control of a manipulator with flexible links and joints. Robot. Auton. Syst. 41(4), 257–270 (2002)

  6. 6.

    Spong, M.W.: Modeling and control of elastic joint robots. J. Dyn. Syst. Meas. Control 109, 310–318 (1987).

  7. 7.

    Siciliano, B., Khatib, O.: Springer Handbook of Robotics. Springer, Berlin (2016)

  8. 8.

    Olfati-Saber, R.: Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles. Massachusetts Inst. Technol. Press, Cambridge (2001)

  9. 9.

    Meng, D., She, Y., Xu, W., Lu, W., Liang, B.: Dynamic modeling and vibration characteristics analysis of flexible-link and flexible-joint space manipulator. Multibody Syst. Dyn. 43(4), 321–347 (2018)

  10. 10.

    Jin, L., Li, S., Yu, J., He, J.: Robot manipulator control using neural networks: a survey. Neurocomputing 285, 23–34 (2018)

  11. 11.

    Gattringer, H., Oberhuber, B., Mayr, J., Bremer, H.: Recursive methods in control of flexible joint manipulators. Multibody Syst. Dyn. 32(1), 117–131 (2014)

  12. 12.

    Miranda-Colorado, R., Moreno-Valenzuela, J.: Experimental parameter identification of flexible joint robot manipulators. Robotica 36(3), 313–332 (2018)

  13. 13.

    Ginoya, D., Shendge, P., Phadke, S.: Delta-operator-based extended disturbance observer and its applications. IEEE Trans. Ind. Electron. 62(9), 5817–5828 (2015)

  14. 14.

    Liu, H.-S., Huang, Y.: Bounded adaptive output feedback tracking control for flexible-joint robot manipulators. J. Zhejiang Univ. Sci. A 19(7), 557–578 (2018)

  15. 15.

    Korayem, M.H., Nekoo, S.R.: Finite-time state-dependent Riccati equation for time-varying nonaffine systems: rigid and flexible joint manipulator control. ISA Trans. 54, 125–144 (2015).

  16. 16.

    Ulrich, S., Sasiadek, J.Z., Barkana, I.: Nonlinear adaptive output feedback control of flexible-joint space manipulators with joint stiffness uncertainties. J. Guid. Control Dyn. 37(6), 1961–1975 (2014)

  17. 17.

    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)

  18. 18.

    Subudhi, B., Morris, A.S.: Singular perturbation based neuro-H\(\infty \) control scheme for a manipulator with flexible links and joints. Robotica 24(2), 151–161 (2006)

  19. 19.

    Abe, A.: Trajectory planning for flexible Cartesian robot manipulator by using artificial neural network: numerical simulation and experimental verification. Robotica 29(5), 797–804 (2011)

  20. 20.

    Chaoui, H., Gueaieb, W., Biglarbegian, M., Yagoub, M.C.: Computationally efficient adaptive type-2 fuzzy control of flexible-joint manipulators. Robotics 2(2), 66–91 (2013)

  21. 21.

    Chaoui, H., Gueaieb, W.: Type-2 fuzzy logic control of a flexible-joint manipulator. J. Intell. Robot. Syst. 51(2), 159–186 (2008)

  22. 22.

    Lightcap, C.A., Banks, S.A.: An extended Kalman filter for real-time estimation and control of a rigid-link flexible-joint manipulator. IEEE Trans. Control Syst. Technol. 18(1), 91–103 (2010)

  23. 23.

    Ulrich, S., Sasiadek, J.Z.: Extended Kalman filtering for flexible joint space robot control. In: American Control Conference, ACC, 2011, pp. 1021–1026. IEEE Press, New York (2011)

  24. 24.

    Nikdel, P., Hosseinpour, M., Badamchizadeh, M.A., Akbari, M.A.: Improved Takagi–Sugeno fuzzy model-based control of flexible joint robot via hybrid-Taguchi genetic algorithm. Eng. Appl. Artif. Intell. 33, 12–20 (2014)

  25. 25.

    Fateh, M.M.: On the voltage-based control of robot manipulators. Int. J. Control. Autom. Syst. 6(5), 702–712 (2008)

  26. 26.

    Fateh, M.M.: Robust control of flexible-joint robots using voltage control strategy. Nonlinear Dyn. 67(2), 1525–1537 (2012)

  27. 27.

    Cheah, C.C., Liu, C., Slotine, J.J.E.: Adaptive Jacobian vision based control for robots with uncertain depth information. Automatica 46(7), 1228–1233 (2010)

  28. 28.

    Izadbakhsh, A.: A note on the nonlinear control of electrical flexible-joint robots. Nonlinear Dyn. 89, 2753–2767 (2017)

  29. 29.

    Fateh, M.M.: Nonlinear control of electrical flexible-joint robots. Nonlinear Dyn. 67(4), 2549–2559 (2012)

  30. 30.

    Izadbakhsh, A.: Robust control design for rigid-link flexible-joint electrically driven robot subjected to constraint: theory and experimental verification. Nonlinear Dyn. 85(2), 751–765 (2016)

  31. 31.

    Zirkohi, M.M., Fateh, M.M., Shoorehdeli, M.A.: Type-2 fuzzy control for a flexible-joint robot using voltage control strategy. Int. J. Autom. Comput. 10(3), 242–255 (2013)

  32. 32.

    Khooban, M.H., Niknam, T., Blaabjerg, F., Dehghani, M.: Free chattering hybrid sliding mode control for a class of non-linear systems: electric vehicles as a case study. IET Sci. Meas. Technol. 10(7), 776–785 (2016)

  33. 33.

    Shokoohinia, M.R., Fateh, M.M.: Robust dynamic sliding mode control of robot manipulators using the Fourier series expansion. Trans. Inst. Meas. Control 1–8 (2018).

  34. 34.

    Soltanpour, M.R., Zolfaghari, B., Soltani, M., Khooban, M.H.: Fuzzy sliding mode control design for a class of nonlinear systems with structured and unstructured uncertainties. Int. J. Innov. Comput. Inf. Control 9(7), 2713–2726 (2013)

  35. 35.

    Adhikary, N., Mahanta, C.: Integral backstepping sliding mode control for underactuated systems: swing-up and stabilization of the Cart–Pendulum System. ISA Trans. 52(6), 870–880 (2013)

  36. 36.

    Orlov, Y.V., Utkin, V.: Sliding mode control in indefinite-dimensional systems. Automatica 23(6), 753–757 (1987)

  37. 37.

    Huang, A.-C., Chen, Y.-C.: Adaptive sliding control for single-link flexible-joint robot with mismatched uncertainties. IEEE Trans. Control Syst. Technol. 12(5), 770–775 (2004)

  38. 38.

    Zhang, L., Liu, L., Wang, Z., Xia, Y.: Continuous finite-time control for uncertain robot manipulators with integral sliding mode. IET Control Theory Appl. (2018).

  39. 39.

    Soltanpour, M.R., Khooban, M.H.: A particle swarm optimization approach for fuzzy sliding mode control for tracking the robot manipulator. Nonlinear Dyn. 74(1–2), 467–478 (2013)

  40. 40.

    Soltanpour, M.R., Otadolajam, P., Khooban, M.H.: Robust control strategy for electrically driven robot manipulators: adaptive fuzzy sliding mode. IET Sci. Meas. Technol. 9(3), 322–334 (2014)

  41. 41.

    Soltanpour, M.R., Khooban, M.H., Soltani, M.: Robust fuzzy sliding mode control for tracking the robot manipulator in joint space and in presence of uncertainties. Robotica 32(3), 433–446 (2014)

  42. 42.

    Zouari, L., Abid, H., Abid, M.: Sliding mode and PI controllers for uncertain flexible joint manipulator. Int. J. Autom. Comput. 12(2), 117–124 (2015)

  43. 43.

    Zhang, B., Yang, X., Zhao, D., Spurgeon, S.K., Yan, X.: Sliding mode control for nonlinear manipulator systems. IFAC-PapersOnLine 50(1), 5127–5132 (2017)

  44. 44.

    Suryawanshi, P.V., Shendge, P.D., Phadke, S.B.: A boundary layer sliding mode control design for chatter reduction using uncertainty and disturbance estimator. Int. J. Dyn. Control 4(4), 456–465 (2016)

  45. 45.

    Feng, Y., Han, F., Yu, X.: Chattering free full-order sliding-mode control. Automatica 50(4), 1310–1314 (2014)

  46. 46.

    Ebrahimi, M.M., Piltan, F., Bazregar, M., Nabaee, A.: Artificial chattering free on-line modified sliding mode algorithm: applied in continuum robot manipulator. Int. J. Inf. Eng. Electron. Bus. 5(5), 57 (2013)

  47. 47.

    Lewis, F.L.: Neural network control of robot manipulators. IEEE Expert 11(3), 64–75 (1996)

  48. 48.

    Moberg, S.: Modeling and Control of Flexible Manipulators. Linköping University Electronic Press, Linköping (2010)

  49. 49.

    Spong, M.W., Hutchinson, S., Vidyasagar, M.: Robot Modeling and Control, vol. 3. Wiley, New York (2006)

  50. 50.

    Khalil, H.K.: Nonlinear Systems. Prentice Hall, New York (2002)

  51. 51.

    Quanser Inc: 2 DOF Serial Flexible Joint Robot, User Manual (2018). Accessed June 2018

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Correspondence to Mohammad Reza Soltanpour.

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Dynamic equations of Quanser 2-RLFJ serial robot manipulator according to (14) can be defined as [51]:


where the vector of the state variables is considered as

$$ x= [ \theta _{m_{1}}, \theta _{m_{2}},\alpha _{1}, \alpha _{2}, \dot{\theta }_{m_{1}}, \dot{\theta }_{m_{2}}, \dot{\alpha }_{1}, \dot{\alpha }_{2} ]^{T}. $$

The other coefficients expressed in (72) and (73) are as bellow:

$$\begin{aligned} f_{1} &= \left . \frac{R_{m_{1}} K_{s_{1}} x_{3} - ( K_{g_{1}} ^{2} K_{m_{1}} K_{t_{1}} \eta _{g_{1}} \eta _{m_{1}} + B_{r_{1}} R_{m _{1}} ) x_{5}}{R_{m_{1}} J_{eq_{1}}} \right \vert _{x=e+ x_{d}}, \end{aligned}$$
$$\begin{aligned} f_{2} &= \left .\frac{R_{m_{2}} K_{s_{2}} x_{4} - ( K_{g_{2}} ^{2} K_{m_{2}} K_{t_{2}} \eta _{g_{2}} \eta _{m_{2}} + B_{r_{2}} R_{m _{2}} ) x_{6}}{R_{m_{2}} J_{eq_{2}}} \right \vert _{x=e+ x_{d}}, \end{aligned}$$
$$\begin{aligned} f_{3} &= \frac{1}{R_{m_{1}} J_{eq_{1}} J_{l_{1}}} \bigl( g J _{eq_{1}} R_{m_{1}} L_{2} m_{1} \sin ( x_{1} + x_{3} ) - R_{m_{1}} K_{s_{1}} ( J_{eq_{1}} + J_{l_{1}} ) x_{3} \\ &\quad {}+ \bigl( ( - B_{l_{1}} J_{eq_{1}} + B_{r_{1}} J_{l_{1}} ) R_{m_{1}} + K_{g_{1}}^{2} K_{m_{1}} K_{t_{1}} \eta _{g_{1}} \eta _{m _{1}} J_{l_{1}} \bigr) x_{5} - B_{l_{1}} R_{m_{1}} J_{eq_{1}} x_{7} \bigr) \Biggr\vert _{x=e+ x_{d}}, \end{aligned}$$
$$\begin{aligned} f_{4} &= \frac{1}{R_{m_{2}} J_{eq_{2}} J_{l_{2}}} \bigl( g J _{eq_{2}} R_{m_{2}} L_{2} m_{2} \sin ( x_{2} + x_{4} ) - R_{m_{2}} K_{s_{2}} ( J_{eq_{2}} + J_{l_{2}} ) x_{4} \\ &\quad {}+ \bigl( ( - B_{l_{2}} J_{eq_{2}} + B_{r_{2}} J_{l_{2}} ) R_{m_{2}} + K_{g_{2}}^{2} K_{m_{2}} K_{t_{2}} \eta _{g_{2}} \eta _{m _{2}} J_{l_{2}} \bigr) x_{6} - B_{l_{2}} R_{m_{2}} J_{eq_{2}} x_{8} \bigr) \Biggr\vert _{x=e+ x_{d}}, \end{aligned}$$
$$\begin{aligned} &\left \{ \textstyle\begin{array}{l} b_{1} = \frac{\eta _{g_{1}} K_{g_{1}} \eta _{m_{1}} K_{t_{1}}}{R_{m_{1}} J_{eq_{1}}}, \\ b_{2} = \frac{\eta _{g_{2}} K_{g_{2}} \eta _{m_{2}} K_{t_{2}}}{R_{m_{2}} J_{eq_{2}}}, \end{array}\displaystyle \right . \end{aligned}$$
$$\begin{aligned} &\left \{ \textstyle\begin{array}{l} b_{3} =- \frac{\eta _{g_{1}} K_{g_{1}} \eta _{m_{1}} K_{t_{1}}}{R_{m _{1}} J_{eq_{1}}}, \\ b_{4} =- \frac{\eta _{g_{2}} K_{g_{2}} \eta _{m_{2}} K_{t_{2}}}{R_{m _{2}} J_{eq_{2}}}. \end{array}\displaystyle \right . \end{aligned}$$

The parameters, as well as their values, for both subsystems are shown in Table 4.

Table 4 Motor and manipulator parameters of Quanser 2-RLFJ robot manipulator [51]

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Zaare, S., Soltanpour, M.R. & Moattari, M. Adaptive sliding mode control of \(n\) flexible-joint robot manipulators in the presence of structured and unstructured uncertainties. Multibody Syst Dyn 47, 397–434 (2019).

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  • Robot manipulator
  • Joint flexibility
  • Uncertainty
  • Chattering
  • Adaptive estimator
  • Adaptive sliding mode control