Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
167 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
42 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Nonlinear Modes as a Tool for Comparing the Mathematical Structure of Dynamic Models of Soft Robots (2402.07038v1)

Published 10 Feb 2024 in cs.RO

Abstract: Continuum soft robots are nonlinear mechanical systems with theoretically infinite degrees of freedom (DoFs) that exhibit complex behaviors. Achieving motor intelligence under dynamic conditions necessitates the development of control-oriented reduced-order models (ROMs), which employ as few DoFs as possible while still accurately capturing the core characteristics of the theoretically infinite-dimensional dynamics. However, there is no quantitative way to measure if the ROM of a soft robot has succeeded in this task. In other fields, like structural dynamics or flexible link robotics, linear normal modes are routinely used to this end. Yet, this theory is not applicable to soft robots due to their nonlinearities. In this work, we propose to use the recent nonlinear extension in modal theory -- called eigenmanifolds -- as a means to evaluate control-oriented models for soft robots and compare them. To achieve this, we propose three similarity metrics relying on the projection of the nonlinear modes of the system into a task space of interest. We use this approach to compare quantitatively, for the first time, ROMs of increasing order generated under the piecewise constant curvature (PCC) hypothesis with a high-dimensional finite element (FE)-like model of a soft arm. Results show that by increasing the order of the discretization, the eigenmanifolds of the PCC model converge to those of the FE model.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (22)
  1. Y. Mikhlin and K. V. Avramov, “Nonlinear normal modes of vibrating mechanical systems: 10 years of progress,” Applied Mechanics Reviews, pp. 1–57, 2023.
  2. G. Kerschen, M. Peeters, J.-C. Golinval, and C. Stéphan, “Nonlinear modal analysis of a full-scale aircraft,” Journal of Aircraft, vol. 50, no. 5, pp. 1409–1419, 2013.
  3. E. Gavassoni, P. B. Gonçalves, and D. de Mesquita Roehl, “Nonlinear vibration modes of an offshore articulated tower,” Ocean Engineering, vol. 109, pp. 226–242, 2015.
  4. M. Matheny, L. Villanueva, R. Karabalin, J. E. Sader, and M. Roukes, “Nonlinear mode-coupling in nanomechanical systems,” Nano Letters, vol. 13, no. 4, pp. 1622–1626, 2013.
  5. 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 Robotics and Automation Letters, vol. 7, no. 2, pp. 1689–1696, 2022.
  6. A. Coelho, A. Albu-Schaeffer, A. Sachtler, H. Mishra, D. Bicego, C. Ott, and A. Franchi, “EigenMPC: An eigenmanifold-inspired model-predictive control framework for exciting efficient oscillations in mechanical systems,” in 61st IEEE Conference on Decision and Control, 2022, pp. 2437–2442.
  7. Y. P. Wotte, S. Dummer, N. Botteghi, C. Brune, S. Stramigioli, and F. Califano, “Discovering efficient periodic behaviors in mechanical systems via neural approximators,” Optimal Control Applications and Methods, 2023.
  8. T. Simpson, G. Tsialiamanis, N. Dervilis, K. Worden, and E. Chatzi, “On the use of variational autoencoders for nonlinear modal analysis,” in Nonlinear Structures & Systems, Volume 1, M. Brake, L. Renson, R. Kuether, and P. Tiso, Eds., 2023, pp. 297–300.
  9. E. Pesheck, C. Pierre, and S. W. Shaw, “Modal reduction of a nonlinear rotating beam through nonlinear normal modes,” Journal of Vibration and Acoustics, vol. 124, no. 2, pp. 229–236, 2002.
  10. C. Touzé, M. Amabili, and O. Thomas, “Reduced-order models for large-amplitude vibrations of shells including in-plane inertia,” Computer Methods in Applied Mechanics and Engineering, vol. 197, no. 21-24, pp. 2030–2045, 2008.
  11. R. J. Kuether, M. R. Brake, and M. S. Allen, “Evaluating convergence of reduced order models using nonlinear normal modes,” in Model Validation and Uncertainty Quantification, Volume 3, H. Atamturktur, B. Moaveni, C. Papadimitriou, and T. Schoenherr, Eds., 2014, pp. 287–300.
  12. R. J. Kuether, B. J. Deaner, J. J. Hollkamp, and M. S. Allen, “Evaluation of geometrically nonlinear reduced-order models with nonlinear normal modes,” AIAA Journal, vol. 53, no. 11, pp. 3273–3285, 2015.
  13. C. Della Santina, M. G. Catalano, and A. Bicchi, “Soft robots,” in Encyclopedia of Robotics, M. Ang, O. Khatib, and B. Siciliano, Eds.   Springer, 2021.
  14. A. Albu-Schäffer and C. Della Santina, “A review on nonlinear modes in conservative mechanical systems,” Annual Reviews in Control, vol. 50, pp. 49–71, 2020.
  15. S. H. Sadati, S. E. Naghibi, I. D. Walker, K. Althoefer, and T. Nanayakkara, “Control space reduction and real-time accurate modeling of continuum manipulators using Ritz and Ritz–Galerkin methods,” IEEE Robotics and Automation Letters, vol. 3, no. 1, pp. 328–335, 2017.
  16. S. Grazioso, G. Di Gironimo, and B. Siciliano, “A geometrically exact model for soft continuum robots: The finite element deformation space formulation,” Soft Robotics, vol. 6, no. 6, pp. 790–811, 2019.
  17. F. Renda, C. Armanini, V. Lebastard, F. Candelier, and F. Boyer, “A geometric variable-strain approach for static modeling of soft manipulators with tendon and fluidic actuation,” IEEE Robotics and Automation Letters, vol. 5, no. 3, pp. 4006–4013, 2020.
  18. B. Caasenbrood, A. Pogromsky, and H. Nijmeijer, “Control-oriented models for hyperelastic soft robots through differential geometry of curves,” Soft Robotics, vol. 10, no. 1, pp. 129–148, 2023.
  19. C. Della Santina, C. Duriez, and D. Rus, “Model-based control of soft robots: A survey of the state of the art and open challenges,” IEEE Control Systems Magazine, vol. 43, no. 3, pp. 30–65, 2023.
  20. G. Kerschen, M. Peeters, J.-C. Golinval, and A. F. Vakakis, “Nonlinear normal modes, part I: A useful framework for the structural dynamicist,” Mechanical Systems and Signal Processing, vol. 23, no. 1, pp. 170–194, 2009.
  21. M. Peeters, R. Viguié, G. Sérandour, G. Kerschen, and J.-C. Golinval, “Nonlinear normal modes, Part II: Toward a practical computation using numerical continuation techniques,” Mechanical Systems and Signal Processing, vol. 23, no. 1, pp. 195–216, 2009.
  22. Y. Tao, A. Both, R. I. Silveira, K. Buchin, S. Sijben, R. S. Purves, P. Laube, D. Peng, K. Toohey, and M. Duckham, “A comparative analysis of trajectory similarity measures,” GIScience & Remote Sensing, vol. 58, no. 5, pp. 643–669, 2021.

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now