Stabilization of periodic sweeping processes and asymptotic average velocity for soft locomotors with dry friction

Giovanni Colombo; Paolo Gidoni; Emilio Vilches

doi:10.3934/dcds.2021135

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February 2022, 42(2): 737-757. doi: 10.3934/dcds.2021135

Stabilization of periodic sweeping processes and asymptotic average velocity for soft locomotors with dry friction

Giovanni Colombo ^1, , Paolo Gidoni ^2,, and Emilio Vilches ^3,

1.	Department of Mathematics "Tullio Levi-Civita", University of Padua, via Trieste 63, 35121 Padova (Italy)
2.	Institute of Information Theory and Automation, Czech Academy of Sciences, Pod vodárenskou věží 4, CZ-182 08 Praha 8 (Czech Republic)
3.	Instituto de Ciencias de la Ingeniería, Universidad de O'Higgins, Av. Libertador Bernardo O'Higgins 611, 2841959 Rancagua (Chile)

^*Corresponding author

Received March 2021 Revised July 2021 Published February 2022 Early access September 2021

Fund Project: G.C. was partially supported by Padua University grant SID 2018 "Controllability, stabilizability and infimum gaps for control systems", prot. BIRD 187147, and by GNAMPA of INdAM. P.G. was partially supported by by the GAČR–FWF grant 19-29646L. E.V. was partially supported by ANID-Chile under grant Fondecyt de Iniciación N$ ^{\circ} $ 11180098

We study the asymptotic stability of periodic solutions for sweeping processes defined by a polyhedron with translationally moving faces. Previous results are improved by obtaining a stronger $ W^{1,2} $ convergence. Then we present an application to a model of crawling locomotion. Our stronger convergence allows us to prove the stabilization of the system to a running-periodic (or derivo-periodic, or relative-periodic) solution and the well-posedness of an average asymptotic velocity depending only on the gait adopted by the crawler. Finally, we discuss some examples of finite-time versus asymptotic-only convergence.

Keywords: Sweeping process, running-periodic solutions, relative-periodic solutions, asymptotic stability, crawling locomotion.

Mathematics Subject Classification: Primary: 70K42, 34A60; Secondary: 34D45, 37C25.

Citation: Giovanni Colombo, Paolo Gidoni, Emilio Vilches. Stabilization of periodic sweeping processes and asymptotic average velocity for soft locomotors with dry friction. Discrete & Continuous Dynamical Systems, 2022, 42 (2) : 737-757. doi: 10.3934/dcds.2021135

References:

[1]	J. Andres, Nonlinear rotations, Nonlinear Analysis: Theory, Methods & Applications, 30 (1997), 495-503. doi: 10.1016/S0362-546X(96)00208-8. Google Scholar
[2]	J. Andres, D. Bednařík and K. Pastor, On the notion of derivo-periodicity, J. Math. Anal. Appl., 303 (2005), 405-417. doi: 10.1016/j.jmaa.2004.08.020. Google Scholar
[3]	T. H. Cao, G. Colombo, B. S. Mordukhovich and D. Nguyen, Optimization of fully controlled sweeping processes, J. Differ. Eq., 295 (2021), 138-186. doi: 10.1016/j.jde.2021.05.042. Google Scholar
[4]	G. Colombo and P. Gidoni, On the optimal control of rate-independent soft crawlers, J. Math. Pures Appl., 146 (2021), 127-157. doi: 10.1016/j.matpur.2020.11.005. Google Scholar
[5]	G. Colombo and L. Thibault, Prox-regular sets and applications, Handbook of Nonconvex Analysis and Applications, (2010), 99–182. Google Scholar
[6]	J. Eldering and H. O. Jacobs, The role of symmetry and dissipation in biolocomotion, SIAM J. Appl. Dyn. Syst., 15 (2016), 24-59. doi: 10.1137/140970914. Google Scholar
[7]	F. Fassò, S. Passarella and M. Zoppello, Control of locomotion systems and dynamics in relative periodic orbits, J. Geome. Mech., 12 (2020), 395-420. doi: 10.3934/jgm.2020022. Google Scholar
[8]	P. Gidoni, Rate-independent soft crawlers, Quart. J. Mech. Appl. Math., 71 (2018), 369-409. doi: 10.1093/qjmam/hby010. Google Scholar
[9]	P. Gidoni and A. DeSimone, Stasis domains and slip surfaces in the locomotion of a bio-inspired two-segment crawler, Meccanica, 52 (2017), 587-601. doi: 10.1007/s11012-016-0408-0. Google Scholar
[10]	P. Gidoni and A. DeSimone, On the genesis of directional friction through bristle-like mediating elements, ESAIM Control Optim. Calc. Var., 23 (2017), 1023-1046. doi: 10.1051/cocv/2017030. Google Scholar
[11]	P. Gidoni and F. Riva, A vanishing inertia analysis for finite dimensional rate-independent systems with nonautonomous dissipation and an application to soft crawlers, Calc. Var. Partial Differential Equations, 60 (2021), 54pp. doi: 10.1007/s00526-021-02067-6. Google Scholar
[12]	I. Gudoshnikov, O. Makarenkov and D. Rachinskiy, Finite-time stability of polyhedral sweeping processes with application to elastoplastic systems, preprint, arXiv: 2011.07744. Google Scholar
[13]	I. Gudoshnikov and O. Makarenkov, Stabilization of the response of cyclically loaded lattice spring models with plasticity, ESAIM Control Optim. Calc. Var., 27 (2021), 43pp. doi: 10.1051/cocv/2020043. Google Scholar
[14]	I. Gudoshnikov, M. Kamenskii, O. Makarenkov and N. Voskovskaia., One-period stability analysis of polygonal sweeping processes with application to an elastoplastic model, Math. Model. Nat. Phenom., 15 (2020), 18pp. doi: 10.1051/mmnp/2019030. Google Scholar
[15]	D. G. E. Hobbelen and M. Wisse, Limit Cycle Walking, Humanoid Robots, Human-like Machines, edited by M. Hackel, I-Tech Education and Publishing, 2007. Google Scholar
[16]	S. D. Kelly and R. M. Murray, Geometric phases and robotic locomotion, Journal of Robotic Systems, 12 (1995), 417-431. doi: 10.1002/rob.4620120607. Google Scholar
[17]	P. Krejčí, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Gattotoscho, 1996. Google Scholar
[18]	P. Krejčí and V. Recupero, BV solutions of rate independent differential inclusions, Math. Bohem., 139 (2014), 607-619. doi: 10.21136/MB.2014.144138. Google Scholar
[19]	P. Krejčí and A. Vladimirov, Polyhedral sweeping processes with oblique reflection in the space of regulated functions, Set-Valued Anal., 11 (2003), 91-110. doi: 10.1023/A:1021980201718. Google Scholar
[20]	M. Levi, F. C. Hoppensteadt and W. L. Miranker, Dynamics of the Josephson junction, Quart. Appl. Math., 36 (1978/79), 167-198. doi: 10.1090/qam/484023. Google Scholar
[21]	O. Makarenkov, Existence and stability of limit cycles in the model of a planar passive biped walking down a slope, Proc. R. Soc. A., 476 (2020), 20190450. doi: 10.1098/rspa.2019.0450. Google Scholar
[22]	R. Martins, The attractor of an equation of Tricomi's type, J. Math. Anal. Appl., 342 (2008), 1265-1270. doi: 10.1016/j.jmaa.2008.01.017. Google Scholar
[23]	A. Mielke and F. Theil, On rate-independent hysteresis models, NoDEA. Nonlinear Differential Equations Appl., 11 (2004), 151-189. doi: 10.1007/s00030-003-1052-7. Google Scholar
[24]	A. Mielke and T. Roubíček, Rate-Independent Systems. Theory and Application, Applied Mathematical Sciences, 193. Springer, New York, 2015. Google Scholar
[25]	B. Pollard, V. Fedonyuk and P. Tallapragada, Swimming on limit cycles with nonholonomic constraints, Nonlinear Dyn., 97 (2019), 2453-2468. doi: 10.1007/s11071-019-05141-z. Google Scholar
[26]	A. A. Tolstonogov, Polyhedral set-valued maps: Properties and applications, Sibirsk. Mat. Zh., 61 (2020), 428-452. doi: 10.33048/smzh.2020.61.216. Google Scholar

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References:

[1]	J. Andres, Nonlinear rotations, Nonlinear Analysis: Theory, Methods & Applications, 30 (1997), 495-503. doi: 10.1016/S0362-546X(96)00208-8. Google Scholar
[2]	J. Andres, D. Bednařík and K. Pastor, On the notion of derivo-periodicity, J. Math. Anal. Appl., 303 (2005), 405-417. doi: 10.1016/j.jmaa.2004.08.020. Google Scholar
[3]	T. H. Cao, G. Colombo, B. S. Mordukhovich and D. Nguyen, Optimization of fully controlled sweeping processes, J. Differ. Eq., 295 (2021), 138-186. doi: 10.1016/j.jde.2021.05.042. Google Scholar
[4]	G. Colombo and P. Gidoni, On the optimal control of rate-independent soft crawlers, J. Math. Pures Appl., 146 (2021), 127-157. doi: 10.1016/j.matpur.2020.11.005. Google Scholar
[5]	G. Colombo and L. Thibault, Prox-regular sets and applications, Handbook of Nonconvex Analysis and Applications, (2010), 99–182. Google Scholar
[6]	J. Eldering and H. O. Jacobs, The role of symmetry and dissipation in biolocomotion, SIAM J. Appl. Dyn. Syst., 15 (2016), 24-59. doi: 10.1137/140970914. Google Scholar
[7]	F. Fassò, S. Passarella and M. Zoppello, Control of locomotion systems and dynamics in relative periodic orbits, J. Geome. Mech., 12 (2020), 395-420. doi: 10.3934/jgm.2020022. Google Scholar
[8]	P. Gidoni, Rate-independent soft crawlers, Quart. J. Mech. Appl. Math., 71 (2018), 369-409. doi: 10.1093/qjmam/hby010. Google Scholar
[9]	P. Gidoni and A. DeSimone, Stasis domains and slip surfaces in the locomotion of a bio-inspired two-segment crawler, Meccanica, 52 (2017), 587-601. doi: 10.1007/s11012-016-0408-0. Google Scholar
[10]	P. Gidoni and A. DeSimone, On the genesis of directional friction through bristle-like mediating elements, ESAIM Control Optim. Calc. Var., 23 (2017), 1023-1046. doi: 10.1051/cocv/2017030. Google Scholar
[11]	P. Gidoni and F. Riva, A vanishing inertia analysis for finite dimensional rate-independent systems with nonautonomous dissipation and an application to soft crawlers, Calc. Var. Partial Differential Equations, 60 (2021), 54pp. doi: 10.1007/s00526-021-02067-6. Google Scholar
[12]	I. Gudoshnikov, O. Makarenkov and D. Rachinskiy, Finite-time stability of polyhedral sweeping processes with application to elastoplastic systems, preprint, arXiv: 2011.07744. Google Scholar
[13]	I. Gudoshnikov and O. Makarenkov, Stabilization of the response of cyclically loaded lattice spring models with plasticity, ESAIM Control Optim. Calc. Var., 27 (2021), 43pp. doi: 10.1051/cocv/2020043. Google Scholar
[14]	I. Gudoshnikov, M. Kamenskii, O. Makarenkov and N. Voskovskaia., One-period stability analysis of polygonal sweeping processes with application to an elastoplastic model, Math. Model. Nat. Phenom., 15 (2020), 18pp. doi: 10.1051/mmnp/2019030. Google Scholar
[15]	D. G. E. Hobbelen and M. Wisse, Limit Cycle Walking, Humanoid Robots, Human-like Machines, edited by M. Hackel, I-Tech Education and Publishing, 2007. Google Scholar
[16]	S. D. Kelly and R. M. Murray, Geometric phases and robotic locomotion, Journal of Robotic Systems, 12 (1995), 417-431. doi: 10.1002/rob.4620120607. Google Scholar
[17]	P. Krejčí, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Gattotoscho, 1996. Google Scholar
[18]	P. Krejčí and V. Recupero, BV solutions of rate independent differential inclusions, Math. Bohem., 139 (2014), 607-619. doi: 10.21136/MB.2014.144138. Google Scholar
[19]	P. Krejčí and A. Vladimirov, Polyhedral sweeping processes with oblique reflection in the space of regulated functions, Set-Valued Anal., 11 (2003), 91-110. doi: 10.1023/A:1021980201718. Google Scholar
[20]	M. Levi, F. C. Hoppensteadt and W. L. Miranker, Dynamics of the Josephson junction, Quart. Appl. Math., 36 (1978/79), 167-198. doi: 10.1090/qam/484023. Google Scholar
[21]	O. Makarenkov, Existence and stability of limit cycles in the model of a planar passive biped walking down a slope, Proc. R. Soc. A., 476 (2020), 20190450. doi: 10.1098/rspa.2019.0450. Google Scholar
[22]	R. Martins, The attractor of an equation of Tricomi's type, J. Math. Anal. Appl., 342 (2008), 1265-1270. doi: 10.1016/j.jmaa.2008.01.017. Google Scholar
[23]	A. Mielke and F. Theil, On rate-independent hysteresis models, NoDEA. Nonlinear Differential Equations Appl., 11 (2004), 151-189. doi: 10.1007/s00030-003-1052-7. Google Scholar
[24]	A. Mielke and T. Roubíček, Rate-Independent Systems. Theory and Application, Applied Mathematical Sciences, 193. Springer, New York, 2015. Google Scholar
[25]	B. Pollard, V. Fedonyuk and P. Tallapragada, Swimming on limit cycles with nonholonomic constraints, Nonlinear Dyn., 97 (2019), 2453-2468. doi: 10.1007/s11071-019-05141-z. Google Scholar
[26]	A. A. Tolstonogov, Polyhedral set-valued maps: Properties and applications, Sibirsk. Mat. Zh., 61 (2020), 428-452. doi: 10.33048/smzh.2020.61.216. Google Scholar

Figure 1. The model of soft crawler

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Figure 2. The orbit (red) of a solution $ z(t) $ of counterexample 5.2, converging only asymptotically to a constant solution $ \bar z $. The set $ C(t) $ is periodically sliding vertically. Notice that the solution will stop for progressively longer intervals at the edges of the orbit. The set $ \mathcal{Z} $ of periodic solutions is the set of constant functions with value in the light-gray triangle

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Figure 3. The orbit (red) of a solution $ z(t) $ of counterexample 5.3, converging only asymptotically to a periodic solution

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Stabilization of periodic sweeping processes and asymptotic average velocity for soft locomotors with dry friction

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Stabilization of periodic sweeping processes and asymptotic average velocity for soft locomotors with dry friction

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