2000 character limit reached
Swing-Up of a Weakly Actuated Double Pendulum via Nonlinear Normal Modes (2404.08478v1)
Published 12 Apr 2024 in eess.SY, cs.RO, and cs.SY
Abstract: We identify the nonlinear normal modes spawning from the stable equilibrium of a double pendulum under gravity, and we establish their connection to homoclinic orbits through the unstable upright position as energy increases. This result is exploited to devise an efficient swing-up strategy for a double pendulum with weak, saturating actuators. Our approach involves stabilizing the system onto periodic orbits associated with the nonlinear modes while gradually injecting energy. Since these modes are autonomous system evolutions, the required control effort for stabilization is minimal. Even with actuator limitations of less than 1% of the maximum gravitational torque, the proposed method accomplishes the swing-up of the double pendulum by allowing sufficient time.
- K. Graichen, M. Treuer, and M. Zeitz, “Swing-up of the double pendulum on a cart by feedforward and feedback control with experimental validation,” Automatica, vol. 43, no. 1, pp. 63–71, Jan. 2007.
- M. Yamakita, M. Iwashiro, Y. Sugahara, and K. Furuta, “Robust swing up control of double pendulum,” Proc. of 1995 Am. Control Conf., vol. 1, pp. 290–295, 1995.
- K. Flaßkamp, J. Timmermann, S. Ober-Blöbaum, and A. Trächtler, “Control strategies on stable manifolds for energy-efficient swing-ups of double pendula,” Int. J. Control, vol. 87, no. 9, 2014.
- I. Fantoni, R. Lozano, and M. Spong, “Energy based control of the pendubot,” IEEE Trans. on Autom. Control, vol. 45, no. 4, pp. 725–729, 2000.
- X. Xin and T. Yamasaki, “Energy-based swing-up control for a remotely driven acrobot: Theoretical and experimental results,” IEEE Trans. on Control Syst. Techn., vol. 20, no. 4, pp. 1048–1056, 2012.
- M. Spong, “The swing up control problem for the acrobot,” IEEE Control Syst. Magazine, vol. 15, no. 1, pp. 49–55, 1995.
- K. Flaßkamp, A. R. Ansari, and T. D. Murphey, “Hybrid control for tracking of invariant manifolds,” Nonlinear Analysis: Hybrid Syst., vol. 25, pp. 298–311, 2017.
- K. Åström and K. Furuta, “Swinging up a pendulum by energy control,” Automatica, vol. 36, no. 2, pp. 287–295, 2000.
- T. Shinbrot, C. Grebogi, J. Wisdom, and J. A. Yorke, “Chaos in a double pendulum,” Am. J. of Phys., vol. 60, no. 6, pp. 491–499, 1992.
- A. Albu-Schäffer and A. Sachtler, “What can algebraic topology and differential geometry teach us about intrinsic dynamics and global behavior of robots?” in Robotics Research, A. Billard, T. Asfour, and O. Khatib, Eds. Springer Nature Switzerland, 2023, pp. 468–484.
- G. Kerschen, M. Peeters, J. Golinval, and A. Vakakis, “Nonlinear normal modes, part i: A useful framework for the structural dynamicist,” Mech. Syst. Signal Process., vol. 23, no. 1, pp. 170–194, 2009.
- A. Albu-Schäffer and C. Della Santina, “A review on nonlinear modes in conservative mechanical systems,” Annu. Rev. Control, vol. 50, pp. 49 – 71, 2020.
- C. Della Santina and A. Albu-Schäffer, “Exciting Efficient Oscillations in Nonlinear Mechanical Systems Through Eigenmanifold Stabilization,” IEEE Control Syst. Lett., vol. 5, no. 6, pp. 1916–1921, 2021.
- F. Bjelonic, A. Sachtler, A. Albu-Schäffer, and C. Della Santina, “Experimental closed-loop excitation of nonlinear normal modes on an elastic industrial robot,” IEEE Robot. Autom. Lett., vol. 7, no. 2, pp. 1689–1696, Apr. 2022.
- D. Calzolari, C. D. Santina, A. M. Giordano, A. Schmidt, and A. Albu-Schäffer, “Embodying quasi-passive modal trotting and pronking in a sagittal elastic quadruped,” IEEE Robot. Autom. Lett., vol. 8, no. 4, pp. 2285–2292, 2023.
- A. Sesselmann, F. Loeffl, C. D. Santina, M. A. Roa, and A. Albu-Schäffer, “Embedding a nonlinear strict oscillatory mode into a segmented leg,” in Proc. IEEE/RSJ Int. Conf. Intell. Robots and Syst., 2021, pp. 1370–1377.
- A. Sachtler and A. Albu-Schäffer, “Strict modes everywhere - bringing order into dynamics of mechanical systems by a potential compatible with the geodesic flow,” IEEE Robot. Autom. Lett., vol. 7, no. 2, pp. 2337–2344, 2022.
- R. P. Dickinson and R. J. Gelinas, “Sensitivity analysis of ordinary differential equation systems—a direct method,” J Comp. Phys., vol. 21, no. 2, pp. 123–143, 1976.
- Y. P. Wotte, A. Sachtler, A. Albu-Schäffer, and C. Della Santina, “Sufficient conditions for an eigenmanifold to be of the extended Rosenberg type,” Research Square Preprint, 2022.
- R. Rosenberg, “On nonlinear vibrations of systems with many degrees of freedom,” in Adv. in Appl. Mech., 1966, vol. 9, pp. 155 – 242.
- C. B. Barber, D. P. Dobkin, and H. Huhdanpaa, “The quickhull algorithm for convex hulls,” ACM Trans. Math. Softw., vol. 22, no. 4, p. 469–483, 1996.
- M. S. Floater, “Generalized barycentric coordinates and applications,” Acta Numerica, vol. 24, p. 161–214, 2015.
- E. Todorov, T. Erez, and Y. Tassa, “MuJoCo: A physics engine for model-based control,” in Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst., 2012, pp. 5026–5033.
Collections
Sign up for free to add this paper to one or more collections.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.