| Title:
|
The CR geometry of the three-segment snake (English) |
| Author:
|
Frelik, Tymon |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
61 |
| Issue:
|
3 |
| Year:
|
2025 |
| Pages:
|
93-99 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We study the geometry associated with the kinematics of a planar robot known as the three-segment snake, whose velocity distribution belongs to a class of $(2,3,5)$ distributions. We discover that, under certain assumptions on its construction parameters, the snake may be endowed with a CR structure of CR dimension 1 and real codimension 3. We solve the associated Cartan equivalence problem and find the invariants of the snake’s CR structure. (English) |
| Keyword:
|
snake robot |
| Keyword:
|
CR structure |
| Keyword:
|
Cartan equivalence method |
| MSC:
|
32V05 |
| MSC:
|
53A17 |
| MSC:
|
53A55 |
| MSC:
|
58A15 |
| DOI:
|
10.5817/AM2025-3-93 |
| . |
| Date available:
|
2025-11-07T11:56:50Z |
| Last updated:
|
2025-11-14 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153130 |
| . |
| Reference:
|
[1] Agrachev, A., Nurowski, P.: Ants and bracket generating distributions in dimension 5 and 6.Automatica 147 (2023), 10pp. DOI: http://dx.doi.org/10.1016/j.automatica.2022.110693 10.1016/j.automatica.2022.110693 |
| Reference:
|
[2] Agrachev, A.A., Sachkov, Y.L.: Control theory from the geometric viewpoint.Encyclopaedia of Mathematical Sciences, vol. 87, Springer-Verlag, Berlin, 2004, Control Theory and Optimization, II. MR 2062547 DOI: http://dx.doi.org/10.1007/978-3-662-06404-7 10.1007/978-3-662-06404-7 |
| Reference:
|
[3] Čap, A., Slovák, J.: Parabolic geometries I. Background and general theory.Math. Surveys Monogr., vol. 154, Amer. Math. Soc., Providence, RI, 2009. 10.1090/surv/154 |
| Reference:
|
[4] Cartan, É.: Les systèmes de Pfaff à cinq variables et les équations aux dérivées partielles du seconde ordre.Ann. Sc. Norm. Super. 27 (1910), 109–192. |
| Reference:
|
[5] Doležal, M.: Lie algebra structure in the model of 3-link snake robot.Arch. Math. (Brno) 60 (4) (2024), 221–229. 10.5817/AM2024-4-221 |
| Reference:
|
[6] Hill, C.D., Nurowski, P.: A car as parabolic geometry.Geometry, Lie theory and applications—the Abel Symposium 2019, Abel Symp., vol. 16, Springer, 2022, pp. 93–130. DOI: http://dx.doi.org/10.1007/978-3-030-81296-6_6 10.1007/978-3-030-81296-6_6 |
| Reference:
|
[7] Ishikawa, M.: Iterative feedback control of snake-like robot based on principal fibre bundle modelling.Int. J. of Adv. Mech. Sys. 1 (2009), 175–182. DOI: http://dx.doi.org/10.1504/IJAMECHS.2009.023200 10.1504/IJAMECHS.2009.023200 |
| Reference:
|
[8] Jean, F.: The car with $n$ trailers: characterisation of the singular configurations.ESAIM Controle Optim. Calc. Var. 1 (1995/96), 241–266. 10.1051/cocv:1996108 |
| Reference:
|
[9] Merker, J., Pocchiola, S., Sabzevari, M.: Canonical Cartan Connections on Maximally Minimal Generic Submanifolds $M^5\subset \mathbb{C}^4$.Electron. Res. Ann. Math. Sci. (ERA-MS) 21 (2014), 16pp. |
| Reference:
|
[10] Merker, J., Sabzevari, M.: Cartan Equivalence Problem for 5-Dimensional Bracket-Generating CR Manifolds in $\mathbb{C}^4$.J. Geom. Anal. 26 (4) (2015), 3194–3251. DOI: http://dx.doi.org/10.1007/s12220-015-9667-6 10.1007/s12220-015-9667-6 |
| Reference:
|
[11] Nurowski, P.: Differential equations and conformal structures.J. Geom. Phys. 55 (1) (2005), 19–49. DOI: http://dx.doi.org/10.1016/j.geomphys.2004.11.006 Zbl 1082.53024, 10.1016/j.geomphys.2004.11.006 |
| Reference:
|
[12] Nurowski, P.: Hunting for a $G_2$ snake.International Meeting on Lorentzian and Conformal Geometry, Alfried Krupp Wissenschaftskolleg Greifswald, March 18 2014, https://www.fuw.edu.pl/~nurowski/prace/hunting.pdf. |
| . |
