Figure 1.
The model of soft crawler
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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
| 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) |
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 and Continuous Dynamical Systems,
2022, 42
(2)
: 737-757.
doi: 10.3934/dcds.2021135
![]()
References:
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J. Andres,
Nonlinear rotations, Nonlinear Analysis: Theory, Methods & Applications, 30 (1997), 495-503.
doi: 10.1016/S0362-546X(96)00208-8. |
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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. |
| [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. |
| [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. |
| [5] |
G. Colombo and L. Thibault, Prox-regular sets and applications, Handbook of Nonconvex Analysis and Applications, (2010), 99–182. |
| [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. |
| [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. |
| [8] |
P. Gidoni,
Rate-independent soft crawlers, Quart. J. Mech. Appl. Math., 71 (2018), 369-409.
doi: 10.1093/qjmam/hby010. |
| [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. |
| [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. |
| [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. |
| [12] |
I. Gudoshnikov, O. Makarenkov and D. Rachinskiy, Finite-time stability of polyhedral sweeping processes with application to elastoplastic systems, preprint, arXiv: 2011.07744. |
| [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. |
| [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. |
| [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. |
| [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. |
| [17] |
P. Krejčí, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Gattotoscho, 1996. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [24] |
A. Mielke and T. Roubíček, Rate-Independent Systems. Theory and Application, Applied Mathematical Sciences, 193. Springer, New York, 2015. |
| [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. |
| [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. |
show all references
References:
| [1] |
J. Andres,
Nonlinear rotations, Nonlinear Analysis: Theory, Methods & Applications, 30 (1997), 495-503.
doi: 10.1016/S0362-546X(96)00208-8. |
| [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. |
| [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. |
| [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. |
| [5] |
G. Colombo and L. Thibault, Prox-regular sets and applications, Handbook of Nonconvex Analysis and Applications, (2010), 99–182. |
| [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. |
| [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. |
| [8] |
P. Gidoni,
Rate-independent soft crawlers, Quart. J. Mech. Appl. Math., 71 (2018), 369-409.
doi: 10.1093/qjmam/hby010. |
| [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. |
| [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. |
| [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. |
| [12] |
I. Gudoshnikov, O. Makarenkov and D. Rachinskiy, Finite-time stability of polyhedral sweeping processes with application to elastoplastic systems, preprint, arXiv: 2011.07744. |
| [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. |
| [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. |
| [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. |
| [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. |
| [17] |
P. Krejčí, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Gattotoscho, 1996. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [24] |
A. Mielke and T. Roubíček, Rate-Independent Systems. Theory and Application, Applied Mathematical Sciences, 193. Springer, New York, 2015. |
| [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. |
| [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. |
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
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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