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Robust adaptive finite-time attitude tracking control of a 3D pendulum with external disturbance: numerical simulations and hardware experiments

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Abstract

The three-degree-of-freedom (3D) pendulum has been used as a benchmark in the field of nonlinear dynamics and control. Nonetheless, the attitude control of 3D pendulum is still an open problem presently since some issues remain not well addressed. In this paper, a robust adaptive finite-time attitude control method is proposed for the attitude tracking control of a 3D pendulum with external disturbance. First, a baseline finite-time attitude controller is designed based on the adding a power integrator technique. Then, a finite-time disturbance observer is designed to exactly estimate the unknown external disturbance. Finally, a robust adaptive finite-time attitude controller is constructed by integrating the baseline finite-time attitude controller with the finite-time disturbance observer. The proposed robust adaptive finite-time attitude controller can guarantee the global finite-time stability of the whole closed-loop system even in the presence of external disturbance owing to the feedforward dynamic compensation. Numerical simulations and hardware experiments are both performed to illustrate the effectiveness and superiority of the proposed control method.

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References

  1. Chung, C.C., Hauser, J.: Nonlinear control of a swinging pendulum. Automatica 31(6), 851–862 (1995)

    MathSciNet  MATH  Google Scholar 

  2. Åström, K.J., Furuta, K.: Swinging up a pendulum by energy control. Automatica 36(2), 287–295 (2000)

    MathSciNet  MATH  Google Scholar 

  3. Lozano, R., Fantoni, I., Block, D.J.: Stabilization of the inverted pendulum around its homoclinic orbit. Syst. Control Lett. 40(3), 197–204 (2000)

    MathSciNet  MATH  Google Scholar 

  4. Shiriaev, A., Pogromsky, A., Ludvigsen, H., Egeland, O.: On global properties of passivity-based control of an inverted pendulum. Int. J. Robust Nonlinear Control 10(4), 283–300 (2000)

    MathSciNet  MATH  Google Scholar 

  5. Shiriaev, A.S., Egeland, O., Ludvigsen, H., Fradkov, A.L.: VSS-version of energy-based control for swinging up a pendulum. Syst. Control Lett. 44(1), 45–56 (2001)

    MathSciNet  MATH  Google Scholar 

  6. Angeli, D.: Almost global stabilization of the inverted pendulum via continuous state feedback. Automatica 37(7), 1103–1108 (2001)

    MATH  Google Scholar 

  7. Shiriaev, A.S., Ludvigsen, H., Egeland, O.: Swinging up the spherical pendulum via stabilization of its first integrals. Automatica 40(1), 73–85 (2004)

    MathSciNet  MATH  Google Scholar 

  8. Chernousko, F.L., Reshmin, S.A.: Time-optimal swing-up feedback control of a pendulum. Nonlinear Dyn. 47(1–3), 65–73 (2005)

    MathSciNet  MATH  Google Scholar 

  9. Ibáñez, C.A., Frias, O.J.: Controlling the inverted pendulum by means of a nested saturation function. Nonlinear Dyn. 53(4), 273–280 (2008)

    MathSciNet  MATH  Google Scholar 

  10. Åström, K.J., Aracil, J., Gordillo, F.: A family of smooth controllers for swinging up a pendulum. Automatica 44(7), 1841–1848 (2008)

    MathSciNet  MATH  Google Scholar 

  11. Consolini, L., Tosques, M.: On the exact tracking of the spherical inverted pendulum via an homotopy method. Control Syst. Lett. 58(1), 1–6 (2009)

    MathSciNet  MATH  Google Scholar 

  12. Park, M.-S., Chwa, D.: Swing-up and stabilization control of inverted-pendulum systems via coupled sliding-mode control method. IEEE Trans. Ind. Electron. 56(9), 3541–3555 (2009)

    Google Scholar 

  13. Spong, M.W.: The swing up control problem for the Acrobot. IEEE Control Syst. Mag. 15(1), 49–55 (1995)

    Google Scholar 

  14. Fantoni, I., Lozano, R., Spong, M.W.: Energy based control of the pendubot. IEEE Trans. Autom. Control 45(4), 725–729 (2000)

    MathSciNet  MATH  Google Scholar 

  15. Spong, M.W., Corke, P., Lozano, R.: Nonlinear control of the reaction wheel pendulum. Automatica 37(11), 1845–1851 (2001)

    MATH  Google Scholar 

  16. Zhao, J., Spong, M.W.: Hybrid control for global stabilization of the cart-pendulum system. Automatica 37(12), 1941–1951 (2001)

    MathSciNet  MATH  Google Scholar 

  17. Chatterjee, D., Patra, A., Joglekar, H.K.: Swing-up and stabilization of a cart-pendulum system under restricted cart track length. Syst. Control Lett. 47(4), 355–364 (2002)

    MathSciNet  MATH  Google Scholar 

  18. Ibañez, C.A., Frias, O.G., Castañón, M.S.: Lyapunov-based controller for the inverted pendulum cart system. Nonlinear Dyn. 40(4), 367–374 (2005)

    MathSciNet  MATH  Google Scholar 

  19. Ibañez, C.A., Azuela, J.H.S.: Stabilization of the Furuta pendulum based on a Lyapunov function. Nonlinear Dyn. 49(1–2), 1–8 (2007)

    MathSciNet  MATH  Google Scholar 

  20. Xin, X., Kaneda, M.: Analysis of the energy-based swing-up control of the Acrobot. Int. J. Robust Nonlinear Control 17(16), 1503–1524 (2007)

    MathSciNet  MATH  Google Scholar 

  21. Xin, X., Tanaka, S., She, J., Yamasaki, T.: New analytical results of energy-based swing-up control for the Pendubot. Int. J. Non-Linear Mech. 52, 110–118 (2013)

    Google Scholar 

  22. Chen, Y.-F., Huang, A.-C.: Adaptive control of rotary inverted pendulum system with time-varying uncertainties. Nonlinear Dyn. 76(1), 95–102 (2014)

    MathSciNet  MATH  Google Scholar 

  23. Yue, M., Wei, X., Li, Z.: Adaptive sliding mode control for two-wheeled inverted pendulum vehicle based on zero-dynamics theory. Nonlinear Dyn. 76(1), 459–471 (2014)

    MathSciNet  MATH  Google Scholar 

  24. Cui, R., Guo, J., Mao, Z.: Adaptive backstepping control of wheeled inverted pendulums models. Nonlinear Dyn. 79(1), 501–511 (2015)

    MathSciNet  MATH  Google Scholar 

  25. Zhang, X.-L., Fan, H.-M., Zang, J.-Y., Zhao, L., Hao, S.: Nonlinear control of triple inverted pendulum based on GA-PIDNN. Nonlinear Dyn. 79(2), 1185–1194 (2015)

    Google Scholar 

  26. Udwadia, F.E., Koganti, P.B.: Dynamics and control of a multi-body planar pendulum. Nonlinear Dyn. 81(1–2), 845–866 (2015)

    MathSciNet  MATH  Google Scholar 

  27. Zhang, A., Yang, C., Gong, S., Qiu, J.: Nonlinear stabilizing control of underactuated inertia wheel pendulum based on coordinate transformation and time-reverse strategy. Nonlinear Dyn. 84(4), 2467–2476 (2016)

    MathSciNet  MATH  Google Scholar 

  28. Zhou, Y., Wang, Z.: Robust motion control of a two-wheeled inverted pendulum with an input delay based on optimal integral sliding mode manifold. Nonlinear Dyn. 85(3), 2065–2074 (2016)

    MathSciNet  Google Scholar 

  29. Zhang, Y., Qiu, B., Liao, B., Yang, Z.: Control of pendulum tracking (including swinging up) of IPC system using zeroing-gradient method. Nonlinear Dyn. 89(1), 1–25 (2017)

    MATH  Google Scholar 

  30. Gritli, H., Khraief, N., Chemori, A., Belghith, S.: Self-generated limit cycle tracking of the underactuated inertia wheel inverted pendulum under IDA-PBC. Nonlinear Dyn. 89(3), 2195–2226 (2017)

    MathSciNet  Google Scholar 

  31. Cho, S., Shen, J., McClamroch, N.H.: Mathematical models for the triaxial attitude control testbed. Math. Comput. Model. Dyn. Syst. 9(2), 165–192 (2003)

    MATH  Google Scholar 

  32. Chaturvedi, N.A., McClamroch, N.H.: Asymptotic stabilization of the hanging equilibrium manifold of the 3D pendulum. Int. J. Robust Nonlinear Control 17(16), 1435–1454 (2007)

    MathSciNet  MATH  Google Scholar 

  33. Chaturvedi, N.A., McClamroch, N.H., Bernstein, D.S.: Stabilization of a 3D axially symmetric pendulum. Automatica 44(9), 2258–2265 (2008)

    MathSciNet  MATH  Google Scholar 

  34. Chaturvedi, N.A., McClamroch, N.H., Bernstein, D.S.: Asymptotic smooth stabilization of the inverted 3-D pendulum. IEEE Trans. Autom. Control 54(6), 1204–1215 (2009)

    MathSciNet  MATH  Google Scholar 

  35. Chaturvedi, N., McClamroch, H.: Asymptotic stabilization of the inverted equilibrium manifold of the 3-D pendulum using non-smooth feedback. IEEE Trans. Autom. Control 54(11), 2658–2662 (2009)

    MathSciNet  MATH  Google Scholar 

  36. Chaturvedi, N.A., Lee, T., Leok, M., McClamroch, N.H.: Nonlinear dynamics of the 3D pendulum. J. Nonlinear Sci. 21(1), 3–32 (2011)

    MathSciNet  MATH  Google Scholar 

  37. Lee, T., Leok, M., McClamroch, N.H.: Computational dynamics of a 3D elastic string pendulum attached to a rigid body and an inertially fixed reel mechanism. Nonlinear Dyn. 64(1–2), 97–115 (2011)

    MathSciNet  MATH  Google Scholar 

  38. Muehlebach, M., D’Andrea, R.: Nonlinear analysis and control of a reaction-wheel-based 3-D inverted pendulum. IEEE Trans. Control Syst. Technol. 25(1), 235–246 (2017)

    Google Scholar 

  39. Zou, K., Ge, X.: Neural-network-based fuzzy logic control of a 3D rigid pendulum. Int. J. Control Autom. Syst. 15(5), 2425–2435 (2017)

    Google Scholar 

  40. Wang, S., Hoagg, J.B., Seigler, T.M.: Orientation control on SO(3) with piecewise sinusoids. Automatica 100, 114–122 (2019)

    MathSciNet  MATH  Google Scholar 

  41. Bhat, S.P., Bernstein, D.S.: Finite-time stability of continuous autonomous systems. SIAM J. Control Optim. 38(3), 751–766 (2000)

    MathSciNet  MATH  Google Scholar 

  42. Hong, Y., Xu, Y., Huang, J.: Finite-time control for robot manipulators. Syst. Control Lett. 46(4), 243–253 (2002)

    MathSciNet  MATH  Google Scholar 

  43. Bhat, S.P., Bernstein, D.S.: Geometric homogeneity with applications to finite-time stability. Math. Control Signals Syst. 17(2), 101–127 (2005)

    MathSciNet  MATH  Google Scholar 

  44. Man, Z., Paplinski, A.P., Wu, H.R.: A robust MIMO terminal sliding mode control scheme for rigid robotic manipulators. IEEE Trans. Autom. Control 39(12), 2464–2469 (1994)

    MathSciNet  MATH  Google Scholar 

  45. Feng, Y., Yu, X., Man, Z.: Non-singular terminal sliding mode control of robot manipulators. Automatica 38(12), 2159–2167 (2002)

    MathSciNet  MATH  Google Scholar 

  46. Yu, S., Yu, X., Shirinzadeh, B., Man, Z.: Continuous finite-time control for robotic manipulators with terminal sliding mode. Automatica 41(11), 1957–1964 (2005)

    MathSciNet  MATH  Google Scholar 

  47. Lin, W., Qian, C.: Adding one power integrator: a tool for global stabilization of high-order lower-triangular systems. Syst. Control Lett. 39(5), 339–351 (2000)

    MathSciNet  MATH  Google Scholar 

  48. Qian, C., Lin, W.: A continuous feedback approach to global strong stabilization of nonlinear systems. IEEE Trans. Autom. Control 46(7), 1061–1079 (2001)

    MathSciNet  MATH  Google Scholar 

  49. Du, H., Li, S., Qian, C.: Finite-time attitude tracking control of spacecraft with application to attitude synchronization. IEEE Trans. Autom. Control 56(11), 2711–2717 (2011)

    MathSciNet  MATH  Google Scholar 

  50. Jiang, B., Li, C., Ma, G.: Finite-time output feedback attitude control for spacecraft using “Adding a power integrator” technique. Aerosp. Sci. Technol. 66, 342–354 (2017)

    Google Scholar 

  51. Zou, A.-M., Kumar, K.D.: Finite-time attitude control for rigid spacecraft subject to actuator saturation. Nonlinear Dyn. 96(2), 1017–1035 (2019)

    MATH  Google Scholar 

  52. Li, S., Wang, X., Zhang, L.: Finite-time output feedback tracking control for autonomous underwater vehicles. IEEE J. Ocean. Eng. 40(3), 727–751 (2015)

    Google Scholar 

  53. Wang, N., Qian, C., Sun, J.-C., Liu, Y.-C.: Adaptive robust finite-time trajectory tracking control of fully actuated marine surface vehicles. IEEE Trans. Control Syst. Technol. 24(4), 1454–1462 (2016)

    Google Scholar 

  54. Jiang, T., Lin, D., Song, T.: Finite-time control for small-scale unmanned helicopter with disturbances. Nonlinear Dyn. 96(3), 1747–1763 (2019)

    MATH  Google Scholar 

  55. Shuster, M.D.: A survey of attitude representations. J. Astronaut. Sci. 41(4), 439–517 (1993)

    MathSciNet  Google Scholar 

  56. Schaub, H., Akella, M.R., Junkins, J.L.: Adaptive control of nonlinear attitude motions realizing linear closed loop dynamics. J. Guid. Control Dyn. 24(1), 95–100 (2001)

    Google Scholar 

  57. Levant, A.: High-order sliding modes, differentiation and output-feedback control. Int. J. Control 76(9–10), 924–941 (2003)

    MATH  Google Scholar 

  58. Shtessel, Y.B., Shkolnikov, I.A., Levant, A.: Smooth second-order sliding modes: missile guidance application. Automatica 43(8), 1470–1476 (2007)

    MathSciNet  MATH  Google Scholar 

  59. Levant, A.: Homogeneity approach to high-order sliding mode design. Automatica 41(5), 823–830 (2005)

    MathSciNet  MATH  Google Scholar 

  60. Hardy, G.H., Littlewood, J.E., Polya, G.: Inequalities. Cambridge University Press, Cambridge (1952)

    MATH  Google Scholar 

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  1. School of Aerospace Engineering, Beijing Institute of Technology, Beijing, 100081, China

    Qijia Yao

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Yao, Q. Robust adaptive finite-time attitude tracking control of a 3D pendulum with external disturbance: numerical simulations and hardware experiments. Nonlinear Dyn 102, 223–239 (2020). https://doi.org/10.1007/s11071-020-05932-9

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