Geodesic
Abnormal extremals in the sub-Riemannian problem for a general model of a robot with a trailer
Sbornik. Mathematics, Tome 214 (2023) no. 10, pp. 1351-1372 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A symmetric mathematical model of a wheeled robot with a trailer is considered for various types of coupling between the robot and the trailer. It is shown that for fixed coupling parameters and fixed initial position of the robot with trailer there are two symmetric abnormal extremals. In motion along these trajectories the robot and the trailer traverse normal extremal trajectories for the sub-Riemannian problem on the group of motions of the plane; the coupling point always draws an inflectional elastica or a straight line. Bibliography: 33 titles.
Keywords: robot with trailer, kinematic model, Pontryagin maximum principle, abnormal trajectories, sub-Riemannian geometry. 
@article{SM_2023_214_10_a0,
     author = {A. A. Ardentov and E. M. Artemova},
     title = {Abnormal extremals in the {sub-Riemannian} problem for a~general model of a~robot with a~trailer},
     journal = {Sbornik. Mathematics},
     pages = {1351--1372},
     year = {2023},
     volume = {214},
     number = {10},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2023_214_10_a0/}
}
TY  - JOUR
AU  - A. A. Ardentov
AU  - E. M. Artemova
TI  - Abnormal extremals in the sub-Riemannian problem for a general model of a robot with a trailer
JO  - Sbornik. Mathematics
PY  - 2023
SP  - 1351
EP  - 1372
VL  - 214
IS  - 10
UR  - http://geodesic.mathdoc.fr/item/SM_2023_214_10_a0/
LA  - en
ID  - SM_2023_214_10_a0
ER  - 
%0 Journal Article
%A A. A. Ardentov
%A E. M. Artemova
%T Abnormal extremals in the sub-Riemannian problem for a general model of a robot with a trailer
%J Sbornik. Mathematics
%D 2023
%P 1351-1372
%V 214
%N 10
%U http://geodesic.mathdoc.fr/item/SM_2023_214_10_a0/
%G en
%F SM_2023_214_10_a0
A. A. Ardentov; E. M. Artemova. Abnormal extremals in the sub-Riemannian problem for a general model of a robot with a trailer. Sbornik. Mathematics, Tome 214 (2023) no. 10, pp. 1351-1372. http://geodesic.mathdoc.fr/item/SM_2023_214_10_a0/

[1] R. Montgomery, “Abnormal optimal controls and open problems in nonholonomic steering”, IFAC Proc. Vol., 25:13 (1992), 121–126 | DOI

[2] D. Barilari, U. Boscain, E. Le Donne and M. Sigalotti, “Sub-Finsler structures from the time-optimal control viewpoint for some nilpotent distributions”, J. Dyn. Control Syst., 23:3 (2017), 547–575 | DOI | MR | Zbl

[3] V. N. Berestovskii and I. A. Zubareva, “Abnormal extremals of left-invariant sub-Finsler quasimetrics on four-dimensional Lie groups”, Siberian Math. J., 62:3 (2021), 383–399 | DOI | MR | Zbl

[4] F. Pelletier and L. Bouche, “Abnormality of trajectory in sub-Riemannian structure”, Geometry in nonlinear control and differential inclusions (Warsaw 1993), Banach Center Publ., 32, Polish Acad. Sci., Inst. Math., Warsaw, 1995, 301–317 | DOI | MR | Zbl

[5] H. J. Sussmann, “A cornucopia of four-dimensional abnormal sub-Riemannian minimizers”, Sub-Riemannian geometry, Progr. Math., 144, Birkhäuser Verlag, Basel, 1996, 341–364 | DOI | MR | Zbl

[6] Yu. L. Sachkov and E. F. Sachkova, “The structure of abnormal extremals in a sub-Riemannian problem with growth vector $(2, 3, 5, 8)$”, Sb. Math., 211:10 (2020), 1460–1485 | DOI | MR | Zbl

[7] B. Bonnard and E. Trélat, “On the role of abnormal minimizers in sub-Riemannian geometry”, Ann. Fac. Sci. Toulouse Math. (6), 10:3 (2001), 405–491 | DOI | MR | Zbl

[8] A. A. Agrachev and A. V. Sarychev, “Abnormal sub-Riemannian geodesics: Morse index and rigidity”, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 13:6 (1996), 635–690 | DOI | MR | Zbl

[9] A. A. Agrachev and A. V. Sarychev, “Sub-Riemannian metrics: minimality of abnormal geodesics versus subanalyticity”, ESAIM Control Optim. Calc. Var., 4 (1999), 377–403 | DOI | MR | Zbl

[10] A. Agrachev and J.-P. Gauthier, “On the subanalyticity of Carnot-Caratheodory distances”, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 18:3 (2001), 359–382 | DOI | MR | Zbl

[11] Yu. L. Sachkov and A. Yu. Popov, “Sub-Riemannian Engel sphere”, Dokl. Math., 104:2 (2021), 301–305 | DOI | MR | Zbl

[12] A. Ardentov and E. Hakavuori, “Cut time in the sub-Riemannian problem on the Cartan group”, ESAIM Control Optim. Calc. Var., 28 (2022), 12, 19 pp. | DOI | MR | Zbl

[13] A. Agrachev, B. Bonnard, M. Chyba and I. Kupka, “Sub-Riemannian sphere in Martinet flat case”, ESAIM Control Optim. Calc. Var., 2 (1997), 377–448 | DOI | MR | Zbl

[14] R. S. Strichartz, “Sub-Riemannian geometry”, J. Differential Geom., 24:2 (1986), 221–263 | DOI | MR | Zbl

[15] R. Montgomery, A tour of subriemannian geometries, their geodesics and applications, Math. Surveys Monogr., 91, Amer. Math. Soc., Providence, RI, 2002, xx+259 pp. | DOI | MR | Zbl

[16] R. Monti, “The regularity problem for sub-Riemannian geodesics”, Geometric control theory and sub-Riemannian geometry, Springer INdAM Ser., 5, Springer, Cham, 2014, 313–332 | DOI | MR | Zbl

[17] E. Hakavuori and E. Le Donne, “Non-minimality of corners in subriemannian geometry”, Invent. Math., 206:3 (2016), 693–704 | DOI | MR | Zbl

[18] A. A. Agrachev and Yu. L. Sachkov, Control theory from the geometric viewpoint, Encyclopaedia Math. Sci., 87, Control Theory Optim., II, Springer-Verlag, Berlin, 2004, xiv+412 pp. | DOI | MR | Zbl

[19] M. Chyba and S. Sekhavat, “Time optimal paths for a mobile robot with one trailer”, Proceedings 1999 IEEE/RSJ International conference on intelligent robots and systems. Human and environment friendly robots with high intelligence and emotional quotients, v. 3, IEEE, 1999, 1669–1674 | DOI

[20] H. Chitsaz, “On time-optimal trajectories for a car-like robot with one trailer”, 2013 Proceedings of the conference on control and its applications (CT), SIAM, 2013, 114–120 | DOI

[21] I. Moiseev and Yu. L. Sachkov, “Maxwell strata in sub-Riemannian problem on the group of motions of a plane”, ESAIM Control Optim. Calc. Var., 16:2 (2010), 380–399 | DOI | MR | Zbl

[22] Yu. L. Sachkov, “Conjugate and cut time in the sub-Riemannian problem on the group of motions of a plane”, ESAIM Control Optim. Calc. Var., 16:4 (2010), 1018–1039 | DOI | MR | Zbl

[23] Yu. L. Sachkov, “Cut locus and optimal synthesis in the sub-Riemannian problem on the group of motions of a plane”, ESAIM Control Optim. Calc. Var., 17:2 (2011), 293–321 | DOI | MR | Zbl

[24] A. A. Ardentov and Yu. L. Sachkov, “Cut time in sub-Riemannian problem on Engel group”, ESAIM Control Optim. Calc. Var., 21:4 (2015), 958–988 | DOI | MR | Zbl

[25] A. A. Ardentov, “Controlling of a mobile robot with a trailer and its nilpotent approximation”, Regul. Chaotic Dyn., 21:7–8 (2016), 775–791 | DOI | MR | Zbl

[26] A. A. Ardentov and A. P. Mashtakov, “Control of a mobile robot with a trailer based on nilpotent approximation”, Autom. Remote Control, 82:1 (2021), 73–92 | DOI | MR | Zbl

[27] A. V. Borisov, A. A. Kilin and I. S. Mamaev, “On the Hadamard-Hamel problem and the dynamics of wheeled vehicles”, Regul. Chaotic Dyn., 20:6 (2015), 752–766 | DOI | MR | Zbl

[28] F. Lamiraux, S. Sekhavat and J.-P. Laumond, “Motion planning and control for Hilare pulling a trailer”, IEEE Trans. Robot. Autom., 15:4 (1999), 640–652 | DOI

[29] P. K. Rashevskii, “Connectivity of two arbitrary points in a totally unholonomic space by an admissible line”, Uchenye Zapiski Mos. Gos. Ped. Inst. Ser. Fiz. Mat. Nauk, 3:2 (1938), 83–94 (Russian)

[30] A. Ardentov, G. Bor, E. Le Donne, R. Montgomery and Yu. Sachkov, “Bicycle paths, elasticae and sub-Riemannian geometry”, Nonlinearity, 34:7 (2021), 4661–4683 | DOI | MR | Zbl

[31] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The mathematical theory of optimal processes, Intersci. Publ. John Wiley Sons, Inc., New York–London, 1962, viii+360 pp. | MR | MR | Zbl | Zbl

[32] A. Agrachev, D. Barilari and U. Boscain, A comprehensive introduction to sub-Riemannian geometry, Cambridge Stud. Adv. Math., 181, Cambridge Univ. Press, Cambridge, 2020, xviii+745 pp. | DOI | MR | Zbl

[33] E. Jahnke, F. Emde and F. Lösch, Tafeln höherer Funktionen. (Tables of higher functions.), 6. Aufl., B. G. Teubner, Stuttgart; McGraw-Hill Book Co., Inc., New York–Toronto–London, 1960, xiv+318 pp. | MR | Zbl