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On the Geometry of Goursat Structures

Published online by Cambridge University Press:  15 August 2002

William Pasillas-Lépine  and
Witold Respondek
William Pasillas-Lépine
Affiliation:
Laboratoire des Signaux et Systèmes, CNRS – Supélec – Université Paris-Sud XI, Plateau du Moulon, 91192 Gif-sur-Yvette, France; [email protected].
Witold Respondek
Affiliation:
Laboratoire de Mathématiques de l'INSA, Institut National des Sciences Appliquées de Rouen, Place E. Blondel, 76130 Mont-Saint-Aignan, France; [email protected].
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Abstract

A Goursat structure on a manifold of dimension n is a rank two distribution Ɗ such that dim Ɗ(i) = i + 2, for 0 ≤ i ≤ n-2, where Ɗ(i) denote the elements of the derived flag of Ɗ, defined by Ɗ(0) = Ɗ and Ɗ(i+1) = Ɗ(i) + [Ɗ(i),Ɗ(i)] . Goursat structures appeared first in the work of von Weber and Cartan, who have shown that on an open and dense subset they can be converted into the so-called Goursat normal form. Later, Goursat structures have been studied by Kumpera and Ruiz. In the paper, we introduce a new local invariant for Goursat structures, called the singularity type, and prove that the growth vector and the abnormal curves of all elements of the derived flag are determined by this invariant. We provide a detailed analysis of all abnormal and rigid curves of Goursat structures. We show that neither abnormal curves, if n ≥ 6, nor abnormal curves of all elements of the derived flag, if n ≥ 9, determine the local equivalence class of a Goursat structure. The latter observation is deduced from a generalized version of Bäcklund's theorem. We also propose a new proof of a classical theorem of Kumpera and Ruiz. All results are illustrated by the n-trailer system, which, as we show, turns out to be a universal model for all local Goursat structures.

Keywords

Goursat structuresKumpera-Ruiz normal formsabnormal curvesnonholonomic control systemstrailer systems.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 6 , 2001 , pp. 119 - 181
Copyright
© EDP Sciences, SMAI, 2001

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References

Agrachev, A. and Sarychev, A., On abnormal extremals for Lagrange variational problems. J. Math. Systems Estim. Control 8 (1998) 87-118.
Bäcklund, A., Über Flachentransformationen. Math. Ann. 9 (1876) 297-320. CrossRef
B. Bonnard and I. Kupka, Théorie des singularités de l'application entrée/sortie et optimalité des trajectoires singulières dans le problème du temps minimal. Forum Math. (5) (1993) 111-159.
R. Brockett, Control theory and singular Riemannian geometry, edited by P. Hilton and G. Young, New Directions in Applied Mathematics. Springer-Verlag, New York (1981) 11-27.
R. Brockett, Asymptotic stability and feedback stabilization, edited by R. Brockett, R. Millman and H. Sussmann, Differential Geometric Control Theory. Birkhäuser, Boston (1983) 181-191.
R. Bryant, S.-S. Chern, R. Gardner, H. Goldschmidt and P. Griffiths, Exterior Differential Systems. Mathematical Sciences Research Institute Publications. Springer-Verlag, New York (1991).
R. Bryant and L. Hsu, Rigidity of integral curves of rank 2 distributions. Invent. Math. (114) (1993) 435-461.
M. Ca nadas-Pinedo and C. Ruiz, Pfaffian systems with derived length one. The class of flag systems. Preprint, University of Granada.
Cartan, E., Sur l'intégration de certains systèmes de Pfaff de caractère deux. Bull. Soc. Math. France 29 (1901) 233-302. œuvres complètes, Part. II, Vol. 1, Gauthiers-Villars, Paris. CrossRef
Cartan, E., Les systèmes de Pfaff à cinq variables et les équations aux dérivées partielles du second ordre. Ann. École Norm. Sup. 27 (1910) 108-192. œuvres complètes, Part. II, Vol. 2, Gauthiers-Villars, Paris.
Cartan, E., Sur l'équivalence absolue de certains systèmes d'équations différentielles et sur certaines familles de courbes. Bull. Soc. Math. France 42 (1914) 12-48. œuvres complètes, Part. II, Vol. 2, Gauthiers-Villars, Paris. CrossRef
Cheaito, M. and Mormul, P., Rank-2 distributions satisfying the Goursat condition: All their local models in dimension 7 and 8. ESAIM: COCV 4 (1999) 137-158. CrossRef
Cheaito, M., Mormul, P., Pasillas-Lépine, W. and Respondek, W., On local classification of Goursat structures. C. R. Acad. Sci. Paris Sér. I Math. 327 (1998) 503-508. CrossRef
Coron, J.-M., Global asymptotic stabilization for controllable systems without drift. Math. Control Signals Systems 5 (1991) 295-312. CrossRef
Darboux, G., Sur le problème de Pfaff. Bull. Sci. Math. 2 (1882) 14-36, 49-68.
F. Engel, Zur Invariantentheorie der Systeme Pfaff'scher Gleichungen. Ber. Verhandlungen der Koniglich Sachsischen Gesellshaft der Wissenshaften Mathematisch-Physikalische Klasse, Leipzig 41 (1889, 1890) 157-176, 192-207.
Fliess, M., Lévine, J., Martin, P. and Rouchon, P., Flatness and defect of nonlinear systems: Introductory theory and examples. Int. J. Control 61 (1995) 1327-1361. CrossRef
Frobenius, G., Über das Pfaff'sche problem. J. Reine Angew. Math. 82 (1877) 230-315.
M. Gaspar, Sobre la clasificacion de sistemas de Pfaff en bandera, in Proc. of the Spanish-Portuguese Conference on Mathematics. Murcia, Spain (1985) 67-74.
Giaro, A., Kumpera, A. and Ruiz, C., Sur la lecture correcte d'un resultat d'Élie Cartan. C. R. Acad. Sci. Paris Sér. I Math. 287 (1978) 241-244.
E. Goursat, Sur le problème de Monge. Bull. Soc. Math. France (33) (1905) 201-210.
E. Goursat, Leçons sur le problème de Pfaff. Hermann, Paris (1923).
Hilbert, D., Über den Begriff der Klasse von Differentialgleichungen. Math. Ann. 73 (1912) 95-108. CrossRef
B. Jacquard, Le problème de la voiture à deux, trois et quatre remorques. Preprint, DMI-ENS Paris (1993).
B. Jakubczyk, Invariants of dynamic feedback and free systems, in Proc. of the European Control Conference. Groningen, The Netherlands (1993) 1510-1513.
B. Jakubczyk, Characteristic varieties of distributions and abnormal curves. Preprint (1999).
B. Jakubczyk and F. Przytycki, Singularities of k-tuples of vector fields. Diss. Math. (213) (1984) 1-64.
Jakubczyk, B. and Zhitomirskiĭ, M., Odd-dimensional Pfaffian equations: Reduction to the hypersurface of singular points. C. R. Acad. Sci. Paris Sér. I Math. 325 (1997) 423-428. CrossRef
Jean, F., The car with n trailers: Characterization of the singular configurations. ESAIM: COCV 1 (1996) 241-266. CrossRef
Jiang, Z.-P. and Nijmeijer, H., A recursive technique for tracking control of nonholonomic systems in chained form. IEEE Trans. Automat. Control 44 (1999) 265-279. CrossRef
Kazarian, M., Montgomery, R. and Shapiro, B., Characteristic classes for the degenerations of two-plane fields in four dimensions. Pacific J. Math. 179 (1997) 355-370. CrossRef
A. Kumpera and C. Ruiz, Sur l'équivalence locale des systèmes de Pfaff en drapeau, edited by F. Gherardelli, Monge-Ampère equations and related topics. Instituto Nazionale di Alta Matematica Francesco Severi, Rome (1982) 201-247.
G. Lafferriere and H. Sussmann, A differential geometric approach to motion planning, Nonholonomic motion planning, edited by Z. Li and J. F. Canny, International Series in Engineering and Computer Sciences. Kluwer, Dordrecht (1992) 235-270.
Laumond, J.-P., Controllability of a multibody mobile robot. IEEE Trans. Robotics and Automation 9 (1991) 755-763. CrossRef
J.-P. Laumond, Singularities and topological aspects in nonholonomic motion planning, edited by Z. Li and J.F. Canny, Nonholonomic motion planning, International Series in Engineering and Computer Sciences. Kluwer, Dordrecht (1992) 755-763.
J.-P. Laumond, Robot Motion Planning and Control. Springer-Verlag, Berlin, Lecture Notes on Control and Information Sciences (1997).
Laumond, J.-P., Jacobs, P., Taïx, M. and Murray, R., A motion planner for nonholonomic mobile robots. IEEE Trans. Robotics and Automation 10 (1994) 577-593. CrossRef
Z. Li and J.-F. Canny, Nonholonomic Motion Planning, International Series in Engineering and Computer Sciences. Kluwer, Dordrecht (1992).
Libermann, P., Sur le problème d'équivalence des systèmes de Pfaff non complètement intégrables. Publ. Paris VII 3 (1977) 73-110.
S. Lie and G. Scheffers, Geometrie of Berührungstransformationen. B. G. Teubners, Leipzig (1896).
Liu, W., An approximation algorithm for non-holonomic systems. SIAM J. Control Optim. 35 (1997) 1328-1365. CrossRef
F. Luca and J.-J. Risler, The maximum degree of nonholonomy for the car with n trailers, in Proc. of the IFAC Symposium on Robot Control. Capri, Italy (1994) 165-170.
Martin, P. and Rouchon, P., Feedback linearization and driftless systems. Math. Control Signals Systems 7 (1994) 235-254. CrossRef
M'Closkey, R. and Murray, R., Exponential stabilization of driftless nonlinear control systems using homogeneous feedback. IEEE Trans. Automat. Control 42 (1997) 614-628. CrossRef
R. Montgomery, A survey of singular curves in sub-Riemannian geometry. J. Dynam. Control Systems (1995) 49-90.
R. Montgomery and M. Zhitomirskiĭ, Geometric approach to Goursat flags. Preprint, University of California Santa Cruz (1999).
Morin, P. and Samson, C., Exponential stabilization of nonlinear driftless systems with robustness to unmodeled dynamics. ESAIM: COCV 4 (1999) 1-35. CrossRef
P. Mormul, Contact hamiltonians distinguishing locally certain Goursat systems. Preprint, Warsaw (1998).
P. Mormul, Local models of 2-distributions in 5 dimensions everywhere fulfilling the Goursat condition. Research report, Rouen (1994).
P. Mormul, Rank-2 distributions satisfying the Goursat condition: All their local models in dimension 9. Preprint, Institute of Mathematics, Polish Academy of Sciences (1997).
P. Mormul, Goursat distributions with one singular hypersurface - constants important in their Kumpera-Ruiz pseudo-normal forms. Preprint, Université de Bourgogne (1999).
P. Mormul, Goursat flags: Classification of codimension-one singularities. Preprint, Warsaw University (1999).
Murray, R., Nilpotent bases for a class of nonintegrable distributions with applications to trajectory generation for nonholonomic systems. Math. Control Signals Systems 7 (1994) 58-75. CrossRef
Murray, R. and Sastry, S., Nonholonomic motion planning: Steering using sinusoids. IEEE Trans. Automat. Control 38 (1993) 700-716. CrossRef
P. Olver, Equivalence, Invariants, and Symmetry. Cambridge University Press (1995).
W. Pasillas-Lépine and W. Respondek, Applications of the geometry of Goursat structures to nonholonomic control systems, in Proc. of the IFAC Nonlinear Control Systems Design Symposium. Enschede, The Netherlands (1998) 789-794.
W. Pasillas-Lépine and W. Respondek, Conversion of the n-trailer into Kumpera-Ruiz normal form and motion planning through the singular locus, in Proc. of the IEEE Conference on Decision and Control. Phoenix, Arizona (1999) 2914-2919.
Pomet, J.-B., Explicit design of time-varying stabilizing control laws for a class of controllable systems without drift. Systems Control Lett. 18 (1992) 147-158. CrossRef
L. Pontryagin, V. Boltyanskiĭ, R. Gamkrelidze and E. Mischenko, The Mathematical Theory of Optimal Processes. Wiley, New York (1962).
P. Rouchon, M. Fliess, J. Lévine and P. Martin, Flatness and motion planning: The car with n trailers, in Proc. of the European Control Conference. Groningen (1993) 1518-1522.
Samson, C., Control of chained systems: Application to path following and time-varying point-stabilization of mobile robots. IEEE Trans. Automat. Control 40 (1995) 64-77. CrossRef
O. Sørdalen, Conversion of the kinematics of a car with n trailers into a chained form, in Proc. of the IEEE Conference on Robotics and Automation. Atlanta, Georgia (1993) 382-387.
O. Sørdalen, On the global degree of nonholonomy of a car with n trailers, in Proc. of the IFAC Symposium on Robot Control. Capri, Italy (1994) 343-348.
Sørdalen, O. and Egeland, O., Exponential stabilization of nonholonomic chained systems. IEEE Trans. Automat. Control 40 (1995) 35-49. CrossRef
O. Sørdalen, Y. Nakamura and W. Chung, Design and control of a nonholonomic manipulator, in École d'été d'automatique de l'ENSIEG. Grenoble, France (1996).
O. Sørdalen and K. Wichlund, Exponential stabilization of a car with n trailers, in Proc. of the IEEE Conference on Decision and Control. San Antonio, Texas (1993) 978-983.
H. Sussmann and W. Liu, Shortest paths for sub-Riemannian metrics of rank-2 distributions. Mem. Amer. Math. Soc. 192 (1995).
Teel, A., Murray, R. and Walsh, G., Nonholonomic control systems: From steering to stabilization with sinusoids. Int. J. Control 62 (1995) 849-870. CrossRef
Tilbury, D., Murray, R. and Sastry, S., Trajectory generation for the n-trailer problem using Goursat normal form. IEEE Trans. Automat. Control 40 (1995) 802-819. CrossRef
A. Vershik and V. Gershkovich, Nonholonomic dynamical systems, geometry of distributions and variational problems, edited by V. Arnol'd and S. Novikov, Dynamical systems VII, Encyclopaedia of Mathematical Sciences. Springer-Verlag, New-York (1991).
Walsh, G., Tilbury, D., Sastry, S., Murray, R. and Laumond, J.-P., Stabilization of trajectories for systems with nonholonomic constraints. IEEE Trans. Automat. Control 39 (1994) 216-222. CrossRef
E. von Weber, Zur Invariantentheorie der Systeme Pfaff'scher Gleichungen. Ber. Verhandlungen der Koniglich Sachsischen Gesellshaft der Wissenshaften Mathematisch-Physikalische Klasse, Leipzig 50 (1898) 207-229.
Zelenko, I. and Zhitomirskiĭ, M., Rigid paths of generic 2-distributions on 3-manifolds. Duke Math. J. 79 (1995) 281-307. CrossRef
P. Zervos, Le problème de Monge. Mémorial des Sciences Mathématiques. Gauthier-Villars, Paris (1932).
Zhitomirskiĭ, M., Normal forms of germs of 2-dimensional distributions on R 4. Funct. Analys. Appl. 24 (1990) 150-152. CrossRef
M. Zhitomirskiĭ, Normal forms of germs of distributions with a fixed segment of growth vector. Leningrad Math. J. (2) (1991) 1043-1065 (English translation).
M. Zhitomirskiĭ, Rigid and abnormal line subdistributions of 2-distributions. J. Dynam. Control Systems (1) (1995) 253-294.
M. Zhitomirskiĭ, Singularities and normal forms of smooth distributions, edited by B. Jakubczyk, W. Respondek and T. Rzezuchowski, Geometry in Nonlinear Control and Differential Inclusions. Banach Center Publications, Warszawa (1995) 395-409.