Skip to main content
SpringerLink
Log in

A vanishing-inertia analysis for finite-dimensional rate-independent systems with nonautonomous dissipation and an application to soft crawlers

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

We study the approximation of finite-dimensional rate-independent quasistatic systems, via a vanishing-inertia asymptotic analysis of dynamic evolutions. We prove the uniform convergence of dynamic solutions to a rate-independent one, employing the variational concept of energetic solution. Motivated by applications in soft locomotion, we allow time-dependence of the dissipation potential, and translation invariance of the potential energy.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Notes

  1. Sometimes Froude’s number for crawlers is defined as

    $$\begin{aligned} \text {Froude's number}:=\frac{\text {inertial forces}}{\text {gravitational forces}}=\frac{v_\mathrm {char}^2}{gL_\mathrm {char}}\quad \left( =\mu _\mathrm {char}\, \frac{m_\mathrm {char}\, L_\mathrm {char}}{T_\mathrm {char}^2\, F_\mathrm {char}} \right) , \end{aligned}$$

    which is the same expression used in legged locomotion. The validity of this second definition is based on the assumption that the normal load proportional to dry friction forces is caused by gravity, so that \(F_\mathrm {char}= m_\mathrm {char}g\mu _\mathrm {char}\). The two notions are thus related by setting the characteristic speed as \(v_\mathrm {char}=L_\mathrm {char}/T_\mathrm {char}\). We prefer definition (1.4) for two reasons. Firstly, it provides a direct measure of the relevance of inertia in the gait, without the need to compare it with the characteristic friction coefficient \(\mu _\mathrm {char}\). Secondly, not necessarily the normal load is produced by gravity: consider for instance a crawler underground or in a pipe.

References

  1. Accoto, D., Castrataro, P., Dario, P.: Biomechanical analysis of oligochaeta crawling. J. Theor. Biol. 230, 49–55 (2004)

    MathSciNet  MATH  Google Scholar 

  2. Adly, S., Le, B.K.: Unbounded second-order state-dependent Moreau’s sweeping processes in Hilbert spaces. J. Optim. Theory Appl. 169, 407–423 (2016)

  3. Agostiniani, V.: Second order approximations of quasistatic evolution problems in finite dimension. Disc. Contin. Dyn. Syst. 32, 1125–1167 (2012)

    MathSciNet  MATH  Google Scholar 

  4. Agostiniani, V., Rossi, R.: Singular vanishing-viscosity limits of gradient flows: the finite-dimensional case. J. Differ. Equ. 263, 7815–7855 (2017)

    MathSciNet  MATH  Google Scholar 

  5. Alexander, R.M.: Principles of Animal Locomotion. Princeton University Press (2003)

  6. Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Birkhäuser Verlag, Basel-Boston-Berlin (2005)

    MATH  Google Scholar 

  7. Arroud, ChE, Colombo, G.: A maximum principle for the controlled sweeping process. Set-Valued Var. Anal. 26, 607–629 (2018)

    MathSciNet  MATH  Google Scholar 

  8. Artina, M., Cagnetti, F., Fornasier, M., Solombrino, F.: Linearly constrained evolutions of critical points and an application to cohesive fractures. Math. Models Methods Appl. Sci. 27, 231–290 (2017)

    MathSciNet  MATH  Google Scholar 

  9. Aubin, J.P., Cellina, A.: Differential Inclusions: Set-Valued Maps and Viability Theory. Springer (2012)

  10. Bastien, J., Bernardin, F., Lamarque, C.-H.: Non Smooth Deterministic or Stochastic Discrete Dynamical Systems: Applications to Models with Friction or Impact. Wiley (2013)

  11. Bastien, J., Schatzman, M., Lamarque, C.-H.: Study of some rheological models with a finite number of degrees of freedom. Eur. J. Mech. A Solids 19, 277–307 (2000)

    MathSciNet  MATH  Google Scholar 

  12. Behn, C., Schale, F., Zeidis, I., Zimmermann, K., Bolotnik, N.: Dynamics and motion control of a chain of particles on a rough surface. Mech. Syst. Sig. Process. 89, 3–13 (2017)

    Google Scholar 

  13. Berthé, R.A., Westhoff, G., Bleckmann, H., Gorb, S.N.: Surface structure and frictional properties of the skin of the Amazon tree boa Corallus hortulanus (Squamata, Boidae). J. Comp. Physiol. A 195, 311–318 (2009)

    Google Scholar 

  14. Brezis, H.: Operateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, North–Holland Publishing Company Amsterdam (1973)

  15. Brogliato, B., Tanwani, A.: Dynamical systems coupled with monotone set-valued operators: formalisms, applications, well-posedness, and stability. SIAM Rev. 62, 3–129 (2020)

    MathSciNet  MATH  Google Scholar 

  16. Brokate, M., Krejčí, P.: Optimal control of ODE systems involving a rate independent variational inequality. Disc. Cont. Dyn. Syst. Ser. B 18, 331–348 (2013)

    MathSciNet  MATH  Google Scholar 

  17. Calisti, M., Picardi, G., Laschi, C.: Fundamentals of soft robot locomotion. J. Roy. Soc. Interf. 14, 20170101 (2017)

    Google Scholar 

  18. Cicconofri, G., DeSimone, A.: A study of snake-like locomotion through the analysis of a flexible robot model. Proc. Roy. Soc. A Math. Phys. Eng. Sci. 471, 20150054 (2015)

    MathSciNet  MATH  Google Scholar 

  19. Colombo, G., Gidoni, P.: On the optimal control of rate-independent soft crawlers. J. Mathématiques Pures et Appliquées 146, 127–157 (2021)

    MathSciNet  MATH  Google Scholar 

  20. Colombo, G., Gidoni, P., Vilches, E.: Stabilization of periodic sweeping processes and asymptotic average velocity for soft locomotors with dry friction. Disc. Contin. Dyn. Syst. (in press)

  21. Crismale, V.: Globally stable quasistatic evolution for a coupled elastoplastic-damage model. ESAIM Control Optim. Calc. Var. 22, 883–912 (2016)

    MathSciNet  MATH  Google Scholar 

  22. Dal Maso, G., Scala, R.: Quasistatic evolution in perfect plasticity as limit of dynamic processes. J. Dyn. Differ. Equ. 26, 915–954 (2014)

    MathSciNet  MATH  Google Scholar 

  23. DeSimone, A., Guarnieri, F., Noselli, G., Tatone, A.: Crawlers in viscous environments: linear vs non-linear rheology. Int. J. Non-Linear Mech. 56, 142–147 (2013)

    Google Scholar 

  24. Efendiev, M.A., Mielke, A.: On the rate-independent limit of systems with dry friction and small viscosity. J. Conv. Anal. 13, 151–167 (2006)

    MathSciNet  MATH  Google Scholar 

  25. Francfort, G., Mielke, A.: Existence results for a class of rate-independent material models with nonconvex elastic energies. J. Reine Angew. Math. 595, 55–91 (2006)

    MathSciNet  MATH  Google Scholar 

  26. Gamus, B., Salem, L., Gat, A.D., Or, Y.: Understanding inchworm crawling for soft-robotics. IEEE Robotics Autom. Lett. 5, 1397–1404 (2020)

    Google Scholar 

  27. Gidoni, P.: Rate-independent soft crawlers. Quart. J. Mech. Appl. Math. 71, 369–409 (2018)

    MathSciNet  MATH  Google Scholar 

  28. Gidoni, P., DeSimone, A.: On the genesis of directional friction through bristle-like mediating elements. ESAIM Control Optim. Calc. Var. 23, 1023–1046 (2017)

    MathSciNet  MATH  Google Scholar 

  29. Heida, M., Mielke, A.: Averaging of time-periodic dissipation potentials in rate-independent processes. Disc. Contin. Dyn. Syst. Ser. S 10(6), 1303–1327 (2017)

    MathSciNet  MATH  Google Scholar 

  30. Hu, D.L., Nirody, J., Scott, T., Shelley, M.J.: The mechanics of slithering locomotion. Proc. Natl. Acad. Sci. USA 106, 10081–10085 (2009)

    Google Scholar 

  31. Jung, K., Koo, J.C., Nam, J., Lee, Y.K., Choi, H.R.: Artificial annelid robot driven by soft actuators. Bioinsp, Biomim. 2, S42–S49 (2007)

    Google Scholar 

  32. Kim, S., Laschi, C., Trimmer, B.: Soft robotics: a bioinspired evolution in robotics. Trends Biotechnol. 31, 287–294 (2013)

    Google Scholar 

  33. Krejcí, P.: Hysteresis, Convexity and Dissipation in Hyperbolic Equations. Gattötoscho (1996)

  34. Krejcí, P., Monteiro, G.A.: What is the Best Viscous Approximation to a Rate-Independent Process? J. Convex Anal. 27, (2020)

  35. Kunze, M., Monteiro Marques, M.D.P.: An introduction to Moreau’s sweeping process. Impacts Mech. Syst. 551, 1–60 (1999)

  36. Lazzaroni, G., Nardini, L.: On the quasistatic limit of dynamic evolutions for a peeling test in dimension one. J. Nonlinear Sci. 28, 269–304 (2018)

    MathSciNet  MATH  Google Scholar 

  37. Lazzaroni, G., Rossi, R., Thomas, M., Toader, R.: Rate-independent damage in thermoviscoelastic materials with inertia. J. Dynam. Differ. Equ. 30, 1311–1364 (2018)

    MATH  Google Scholar 

  38. Manoonpong, P., Petersen, D., Kovalev, A., Wörgötter, F., Gorb, S.N., Spinner, M., Heepe, L.: Enhanced locomotion efficiency of a bio-inspired walking robot using contact surfaces with frictional anisotropy. Sci. Rep. 6, 39455 (2016)

    Google Scholar 

  39. Manwell, T., Guo, B., Back, J., Liu, H.: Bioinspired setae for soft worm robot locomotion. In: 2018 IEEE International Conference on Soft Robotics (Robosoft), pp. 54–59 (2018)

  40. Martins, J.A.C., Monteiro Marques, M.D.P., Petrov, A.: On the stability of quasi-static paths for finite dimensional elastic-plastic systems with hardening. ZAMM Z. Angew. Math. Mech. 87, 303–313 (2007)

    MathSciNet  MATH  Google Scholar 

  41. Marvi, H., Bridges, J., Hu, D.L.: Snakes mimic earthworms: propulsion using rectilinear travelling waves. J. Roy. Soc. Interf. 10, 20130188 (2013)

    Google Scholar 

  42. Marvi, H., Meyers, G., Russell, G., Hu, D.L.: A snake-inspired robot with active control of friction. In: Proceedings of the ASME Dynamic Systems and Control Conference and BATH/ASME Symposium on Fluid Power and Motion Control, pp. 443–450 (2012)

  43. Menciassi, A., Accoto, D., Gorini, S., Dario, P.: Development of a biomimetic miniature robotic crawler. Auton. Robots 21, 155–163 (2006)

    Google Scholar 

  44. Mielke, A.: Evolution of rate-independent systems, Evolutionary equations. In: Handbook of Differential Equations, vol. II, pp. 461–559. Elsevier/North-Holland, Amsterdam (2005)

  45. Mielke, A.: Three examples concerning the interaction of dry friction and oscillations. In: Trends in Applications of Mathematics to Mechanics, pp. 159–177. Springer, Cham (2018)

  46. Mielke, A., Petrov, A., Martins, J.A.C.: Convergence of solutions of kinetic variational inequalities in the rate-independent quasistatic limit. J. Math. Anal. Appl. 348, 1012–1020 (2008)

    MathSciNet  MATH  Google Scholar 

  47. Mielke, A., Rossi, R., Savaré, G.: Modeling solutions with jumps for rate-independent systems on metric spaces. Disc. Cont. Dyn. Syst. 25, 585–615 (2009)

    MathSciNet  MATH  Google Scholar 

  48. Mielke, A., Rossi, R., Savaré, G.: BV solutions and viscosity approximations of rate-independent systems. ESAIM Control Optim. Calc. Var. 18, 36–80 (2012)

    MathSciNet  MATH  Google Scholar 

  49. Mielke, A., Rossi, R., Savaré, G.: Balanced viscosity (BV) solutions to infinite-dimensional rate-independent systems. J. Eur. Math. Soc. 18, 2107–2165 (2016)

    MathSciNet  MATH  Google Scholar 

  50. Mielke, A., Roubíček, T.: Rate-Independent Systems: Theory and Application. Springer-Verlag, New York (2015)

    MATH  Google Scholar 

  51. Mielke, A., Theil, F.: On rate-independent hysteresis models. NoDEA Nonlinear Differ. Equ. Appl. 11, 151–189 (2004)

    MathSciNet  MATH  Google Scholar 

  52. Mielke, A., Thomas, M.: Damage of nonlinearly elastic materials at small strain - Existence and regularity results -. Z. Angew. Math. Mech. 90, 88–112 (2010)

    MathSciNet  MATH  Google Scholar 

  53. Moreau, J.J.: Bounded Variation in Time. Topics in nonsmooth mechanics, Birkhäuser, Basel (1988)

    MATH  Google Scholar 

  54. Nardini, L.: A note on the convergence of singularly perturbed second order potential-type equations. J. Dyn. Differ. Equ. 29, 783–797 (2017)

    MathSciNet  MATH  Google Scholar 

  55. Negri, M.: Quasi-static rate-independent evolutions: characterization, existence, approximation and application to fracture mechanics. ESAIM Control Optim. Calc. Var. 20, 983–1008 (2014)

    MathSciNet  MATH  Google Scholar 

  56. Quillin, K.: Kinematic scaling of locomotion by hydrostatic animals: ontogeny of peristaltic crawling by the earthworm lumbricus terrestris. J. Exp. Biol. 202, 661–674 (1999)

    Google Scholar 

  57. Rafsanjani, A., Zhang, Y., Liu, B., Rubinstein, S.M., Bertoldi, K.: Kirigami skins make a simple soft actuator crawl. Sci. Robotics 3, eaar7555 (2018)

    Google Scholar 

  58. Riva, F.: On the approximation of quasistatic evolutions for the debonding of a thin film via vanishing inertia and viscosity. J. Nonlinear Sci. 30, 903–951 (2020)

    MathSciNet  MATH  Google Scholar 

  59. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer-Verlag, Berlin (1998)

  60. Rossi, R.: From visco to perfect plasticity in thermoviscoelastic materials. ZAMM Z. Angew. Math. Mech. 98, 1123–1189 (2018)

    MathSciNet  Google Scholar 

  61. Roubíček, T.: Adhesive contact of visco-elastic bodies and defect measures arising by vanishing viscosity. SIAM J. Math. Anal. 45, 101–126 (2013)

    MathSciNet  MATH  Google Scholar 

  62. Roubíček, T.: Rate-independent processes in viscous solids at small strains. Math. Methods Appl. Sci. 32, 825–862 (2009)

    MathSciNet  MATH  Google Scholar 

  63. Scala, R.: Limit of viscous dynamic processes in delamination as the viscosity and inertia vanish, ESAIM: Control Optim. Calc. Var. 23, 593–625 (2017)

    MathSciNet  MATH  Google Scholar 

  64. Scilla, G., Solombrino, F.: A variational approach to the quasistatic limit of viscous dynamic evolutions in finite dimension. J. Differ. Equ. 267, 6216–6264 (2019)

    MathSciNet  MATH  Google Scholar 

  65. Sellers, W.I., Manning, P.L.: Estimating dinosaur maximum running speeds using evolutionary robotics. Proc. Roy. Soc. B Biol. Sci. 274, 2711–2716 (2007)

    Google Scholar 

  66. Seok, S., Onal, C.D., Cho, K.J., Wood, R.J., Rus, D., Kim, S.: Meshworm: a peristaltic soft robot with antagonistic nickel titanium coil actuators. IEEE/ASME Trans. Mechatr. 18, 1485–1497 (2013)

    Google Scholar 

  67. Vaughan, C.L., O’Malley, M.J.: Froude and the contribution of naval architecture to our understanding of bipedal locomotion. Gait Post. 21, 350–362 (2005)

  68. Vikas, V., Cohen, E., Grassi, R., Sözer, C., Trimmer, B.: Design and locomotion control of a soft robot using friction manipulation and motor-tendon actuation. IEEE Trans. Robot 32, 949–959 (2016)

    Google Scholar 

  69. Wagner, G.L., Lauga, E.: Crawling scallop: friction-based locomotion with one degree of freedom. J. Theor. Biol. 324, 42–51 (2013)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are grateful to Professors Gianni Dal Maso and Giovanni Colombo for many helpful discussions on the topic. A special thank goes also to an anonymous Referee for the careful reading of the manuscript and for several valuable suggestions on the presentation of our results.

Part of the work was carried out while F. R. was a Ph.D. student in SISSA (International School for Advanced Studies, Via Bonomea, 265, 34136, Trieste, Italy).

P. G. was partially supported by the GAČR–FWF grant 19-29646L and the MŠMTČR grant 8J19AT013. F. R. is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

Author information

Authors and Affiliations

  1. Institute for Information Theory and Automation (ÚTIA), Czech Academy of Sciences, Pod vodárenskou veží- 4, 182 08, Prague 8, Czech Republic

    Paolo Gidoni

  2. Dipartimento di Matematica “Felice Casorati”, Università degli Studi di Pavia, Via Ferrata, 5, 27100, Pavia, Italy

    Filippo Riva

Authors
  1. Paolo Gidoni
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Filippo Riva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paolo Gidoni.

Additional information

Communicated by J. M. Ball.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gidoni, P., Riva, F. A vanishing-inertia analysis for finite-dimensional rate-independent systems with nonautonomous dissipation and an application to soft crawlers. Calc. Var. 60, 191 (2021). https://doi.org/10.1007/s00526-021-02067-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-021-02067-6

Mathematics Subject Classification

Search

Navigation