nach oben

Automation and Remote Control

Erschienen in:

01.01.2021 | ROBUST, ADAPTIVE AND NETWORK CONTROL

Control of a Mobile Robot with a Trailer Based on Nilpotent Approximation

verfasst von: A. A. Ardentov, A. P. Mashtakov

Erschienen in: Automation and Remote Control | Ausgabe 1/2021

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

We consider a kinematic model of a mobile robot with a trailer moving on a homogeneous plane. The robot can move back and forth and make a pivot turn. For this model, we pose the following optimal control problem: transfer the “robot–trailer” system from an arbitrarily given initial configuration into an arbitrarily given final configuration so that the amount of maneuvering is minimal. By a maneuver we mean a functional that defines a trade-off between the linear and angular robot motion. Depending on the trailer–robot coupling, this problem corresponds to a two-parameter family of optimal control problems in the 4-dimensional space with a 2-dimensional control.

We propose a nilpotent approximation method for the approximate solution of the problem. A number of iterative algorithms and programs have been developed that successfully solve the posed problem in the ideal case, namely, with no state constraints. Based on these algorithms, we propose a dedicated reparking algorithm that solves a particular case of the problem where the initial and final robot position coincide and takes into account a state constraint on the trailer’s turning angle occurring in real systems.

Vorheriger Artikel A Criterion for the Asymptotic Stability of a Periodic Selector-Linear Differential Inclusion

Nächster Artikel Adaptive Approach to Solving a Two-Point Boundary Value Problem under Partial Uncertainty in the Disturbance Field

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Nur mit Berechtigung zugänglich

The multiplication law in the group can be found in [20, Ch. 15].

Laumond, J.-P., Nonholonomic Motion Planning for Mobile Robots. Tutorial Notes, Toulouse: LAAS-CNRS, 1998.

Ardentov, A.A., Controlling of a mobile robot with a trailer and its nilpotent approximation, Regular Chaot. Dyn., 2016, vol. 21, no. 7–8, pp. 775–791.MathSciNetCrossRef

Mashtakov, A.P., Algorithmic and software tools for solving constructive problem of control of nonholonomic five-dimensional systems, Program. Sist: Teoriya Prilozh., 2012, vol. 3, no. 1(10), pp. 3–29.

Krasovskii, N.N., Teoriya upravleniya dvizheniem (Motion Control Theory), Moscow: Nauka, 1968.

Chitour, Y., Jean, F., and Long, R., A global steering method for nonholonomic systems, J. Differ. Equat., 2013, vol. 254, pp. 1903–1956.MathSciNetCrossRef

Kushner, A.G., Lychagin, V.V., and Rubtsov, V.N., Contact Geometry and Nonlinear Differential Equations, Cambridge: Cambridge Univ. Press, 2007.MATH

Murray, R.M. and Sastry, S., Steering nonholonomic systems using sinusoids, IEEE Int. Conf. Decis. Control. (1990), pp. 2097–2101.

Murray, R.M., Robotic control and nonholonomic motion planning, PhD Thesis, Memo. no. UCB/ERL M90/117, Berkeley: Univ. California, 1990.

Tilbury, D., Murray, R., and Sastry, S., Trajectory generation for the \(n \)-trailer problem using Goursat normal form, IEEE TAC, 1995, vol. 40, no. 5, pp. 802–819.MathSciNetMATH

10.

Monaco, S. and Norman-Cyrot, D., On Carnot–Caratheodory metrics, J. Differ. Geom., 1985, vol. 21, pp. 35–45.MathSciNetCrossRef

11.

Murray, R.M., Nilpotent bases for a class on nonintegrable distributions with applications to trajectory generation for nonholonomic systems, in Math. Control Signal Syst., Berkeley: Univ. California, 1990.

12.

Venditelli, M., Oriolo, G., Jea, F., and Laumond, J.P., Nonhomogeneous nilpotent approximations for nonholonomic systems with singularities, Trans. Autom. Control, 2004, vol. 49, no. 2, pp. 261–266.MathSciNetCrossRef

13.

Fernandes, C., Gurvits, L., and Li, Z.X., A variational approach to optimal nonholonomic motion planning, IEEE ICRA (Sacramento, 1991), pp. 680–685.

14.

Agrachev, A.A. and Sachkov, Yu.L., Geometricheskaya teoriya upravleniya (Geometric Control Theory), Moscow: Fizmatlit, 2005.MATH

15.

Duits, R., Meesters, S.P.L., Mirebeau, J.M., and Portegies, J.M., Optimal paths for variants of the 2D and 3D Reeds-Shepp car with applications in image analysis, J. Math. Imaging Vision, 2018, vol. 60, no. 6, pp. 816–848.MathSciNetCrossRef

16.

Lafferriere, G. and Sussmann, H.J., A differential geometric approach to motion planning, in Nonholonomic Motion Planing, Zexiang Li and Canny, J.F., Eds., 1992.

17.

Bellaiche, A., Laumond, J.P., and Chyba, M., Canonical nilpotent approximation of control systems: application to nonholonomic motion planning, 32nd IEEE CDC (1993).

18.

Bellaiche, A., Laumond, J.P., and Riser, J.J., Nilpotent infinitesimal approximations to a control Lie algebra, IFAC NCSDS (Bordeaux, 1992), pp. 174–181.

19.

Ardentov, A.A. and Sachkov, Yu.L., Extremal trajectories in a nilpotent sub-Riemannian problem on the Engel group, Sb. Math., 2011, vol. 202, no. 11, pp. 1593–1615.MathSciNetCrossRef

20.

Sachkov, Yu.L., Upravlyaemost’ i simmetrii invariantnykh sistem na gruppakh Li i odnorodnykh prostranstvakh (Controllability and Symmetries of Invariant Systems on Lie Groups and Homogeneous Spaces), Moscow: Fizmatlit, 2007.

21.

Montgomery, R., A Tour of Sub-Riemannian Geometries, Their Geodesics and Applications. Vol. 91 of Math. Surv. Monogr., Providence: Am. Math. Soc., 2002.

22.

Stefani, G., Polynomial approximations to control systems and local controllability, Proc. 24th. IEEE Conf. Decis. Control (Ft. Lauderdale. Fla., 1985), pp. 33–38.

23.

Agrachev, A.A. and Sarychev, A.V., Filtration of the Lie algebra of vector fields and nilpotent approximation to control systems, Dokl. Akad. Nauk SSSR, 1987, vol. 295, pp. 777–781.

24.

Hermes, H., Nilpotent and high-order approximations of vector fields systems, SIAM, 1991, vol. 33, pp. 238–264.MathSciNetCrossRef

25.

Bellaiche, A., The tangent space in sub-Riemannian geometry, in Sub-Riemannian Geometry, Basel: Birkhäuser, 1996, pp. 1–78.

26.

Gromov, M., Lafontaine, J., and Pansu, P., Structures métriques pour les variétés riemanniennes, in Textes Mathématiques, Paris: CEDIC/Fernand Nathan, 1981.

27.

Jean, F., Control of Nonholonomic Systems: from Sub-Riemannian Geometry to Motion Planning, Berlin–Heidelberg: Springer, 2014.MATH

28.

Sachkov, Yu.L., Symmetries of flat rank two distributions and sub-Riemannian structures, Trans. Am. Math. Soc., 2004, vol. 356, pp. 457–494.MathSciNetCrossRef

29.

Ardentov, A.A. and Sachkov, Yu.L., Conjugate points in nilpotent sub-Riemannian problem on the Engel group, JMS, 2013, vol. 195, no. 3, pp. 369–390.MathSciNetMATH

30.

Ardentov, A.A. and Sachkov, Yu.L., Cut time in sub-Riemannian problem on Engel group, ESAIM: COCV., 2015, vol. 21, no. 4, pp. 958–988.MathSciNetMATH

31.

Ardentov, A.A. and Sachkov, Yu.L., Maxwell strata and cut locus in sub-Riemannian problem on Engel group, RCD, 2017, vol. 22, no. 8, pp. 909–936.MathSciNetMATH

32.

Ardentov, A.A. and Sachkov, Yu.L., Cut locus in the sub-Riemannian problem on Engel group, Dokl. Math., 2018, vol. 97, no. 1, pp. 82–85.MathSciNetCrossRef

33.

Whittacker, E.T. and Watson, J.N., A Course of Modern Analysis, Cambridge: Cambridge Univ. Press, 1996. Translated under the title: Kurs sovremennogo analiza, Moscow: URSS, 2002.CrossRef

34.

Moiseev, I. and Sachkov, Yu.L., Maxwell strata in sub-Riemannian problem on the group of motions of a plane, ESAIM: Control Optim. Calculus Var., 2010, vol. 16, pp. 380–399.MathSciNetMATH

35.

Sachkov, Yu.L., Conjugate and cut time in the sub-Riemannian problem on the group of motions of a plane, ESAIM: Control Optim. Calculus Var., 2010, vol. 16, pp. 1018–1039.MathSciNetMATH

36.

Sachkov, Yu.L., Cut locus and optimal synthesis in the sub-Riemannian problem on the group of motions of a plane, ESAIM: Control Optim. Calculus Var., 2011, vol. 17, pp. 293–321.MathSciNetMATH

37.

David, J. and Manivannan, P.V., Control of truck-trailer mobile robots: a survey, Intell. Serv. Rob., 2014, vol. 7, no. 4, pp. 245–258.CrossRef

38.

Lamiraux, F., Sekhavat, S., and Laumond, J.-P., Motion planning and control for Hilare pulling a trailer, IEEE Trans. Rob. Autom., 1999, vol. 15, no. 4, pp. 640–652.CrossRef

39.

Dubins, L.E., On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents, Am. J. Math., 1957, vol. 79, no. 3, pp. 497–516.MathSciNetCrossRef

40.

Ardentov, A.A., Karavaev, Y.L., and Yefremov, K.S., Euler elasticas for optimal control of the motion of mobile wheeled robots: the problem of experimental realization, RCD, 2019, vol. 24, no. 3, pp. 312–328.MathSciNetMATH

41.

Lokutsievskii, L.V., Convex trigonometry with applications to sub-Finsler geometry, Sb. Math., 2019, vol. 210, no. 8, pp. 1179–1205.MathSciNetCrossRef

Titel: Control of a Mobile Robot with a Trailer Based on Nilpotent Approximation
verfasst von: A. A. Ardentov
A. P. Mashtakov
Publikationsdatum: 01.01.2021
Verlag: Pleiades Publishing
Erschienen in: Automation and Remote Control / Ausgabe 1/2021
Print ISSN: 0005-1179
Elektronische ISSN: 1608-3032
DOI: https://doi.org/10.1134/S0005117921010057

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Weitere Artikel der Ausgabe 1/2021

Adaptive Approach to Solving a Two-Point Boundary Value Problem under Partial Uncertainty in the Disturbance Field

Lemma on Boundedness of Anisotropic Norm for Systems with Multiplicative Noises under a Noncentered Disturbance

Solutions of Cooperative Games with a Major Player and a Hierarchical Structure

Mathematical Modeling of Kinetics of Iodine-Containing Radiotracers in Nuclear Medicine Problems

A Modified Myerson Value for Determining the Centrality of Graph Vertices

A Criterion for the Asymptotic Stability of a Periodic Selector-Linear Differential Inclusion

Premium Partner