On local classification of Goursat structures

Cheaito, Mohamad H.; Mormul, Piotr; Pasillas-Lépine, William; Respondek, Witold

doi:10.1016/s0764-4442(99)80030-5

Cited by 7 publications

(5 citation statements)

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“…Secondly, we consider the models S(κ 3 ) and R c (κ 3 ), which are equivalent to R 0 (κ 3 ), as being Kumpera-Ruiz normal forms. The following result of Kumpera and Ruiz (1982) (see also Cheaito and Mormul, 1999;Cheaito et al, 1998;Montgomery and Zhitomirskiȋ, 1999;Pasillas-Lépine and Respondek, 1999b) shows clearly the interest of their normal forms.…”

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confidence: 79%

“…But Kumpera-Ruiz normal forms have also an other interesting property: they are "triangular" -in the sense of Murray and Sastry (1993), see also (Marigo, 1999). Indeed, it follows directly from their construction that they give a control system ẋ = κ n 1 (x) u 1 + κ n 2 (x) u 2 that can be written -see (Cheaito and Mormul, 1999), (Cheaito et al, 1998), (Kumpera and Ruiz, 1982), and (Pasillas-Lépine and Respondek, 1999b), after a permutation of the x i 's, in the following form:…”

Section: Motion Planningmentioning

confidence: 99%

“…Note that our definition differs slightly from that of Cheaito et al (1998). Firstly, we do not ask the coordinates to satisfy x(p) = 0, where p is the point around which we work.…”

mentioning

confidence: 99%

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Nilpotentization of the kinematics of the n-trailer system at singular points and motion planning through the singular locus

Pasillas-Lépine

Respondek

2001

International Journal of Control

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We propose in this paper a constructive procedure that transforms locally, even at singular con® gurations, the kinematics of a car towing trailers into Kumpera± Ruiz normal form. This construction converts the non-holonomic motion planning problem into an algebraic problem (the resolution of a system of polynomial equations), which we illustrate by steering the two-trailer system in a neighbourhood of singular con® gurations. We show also that the n-trailer system is a universal local model for all Goursat structures and that all Goursat structures are locally nilpotentizable.

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confidence: 79%

Section: Motion Planningmentioning

confidence: 99%

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Nilpotentization of the kinematics of the n-trailer system at singular points and motion planning through the singular locus

Pasillas-Lépine

Respondek

2001

International Journal of Control

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“…The equation ( 29) reduces at the point x to y 9 y 7 7 = −y 8 7 + x 9 y 8 8 and, by (28), y 8 7 = 0 hence, for x 9 = 1 and y 9 = −1, y 7 7 = −y 8 8 at the point x. Differentiating, with respect to x 8 , the last equation resulting from the congruence requirement in the (3.1.3.3.) model, namely the equation y 8 y 7 7 = x 8 φ, we obtain y 7 7 y 8 8 = φ + x 8 ∂φ/∂x 8 hence, at the point x, y 7 7 y 8 8 = φ < 0. However, by the second equation in ( 27), y 6 6 = φ < 0 and, for x 5 = 0, the expression (26) reduces to y 6 y 4 4 = x 6 y 5 5 hence to y 4 4 = y 5 5 = 0 at the point x (x 6 = y 6 = 1).…”

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confidence: 86%

Automorphisms of flag systems

Kumpera

2019

Journal of Differential Equations

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We first discuss the problems in the theory of ordinary differential equations that gave rise to the concept of a flag system and illustrate these with the Cartan criterion for Monge equations (1st order) as well as the Cartan statement concerning the local equivalence of Monge-Ampère type equations (2nd order). Next, we describe a prolongation functor operating on the infinitesimal symmetries (automorphisms) of the Darboux flag and extending these, isomorphically, to all the symmetries of any other flag. Hence, flag systems cannot be distinguished by their symmetry algebras and the local classification of these objects is approached by considering higher order isotropies of these algebras as well as the groupoids of k − th order formal equivalences since the differential equations defining the latter provide precious information for the application of flag systems to differential equations (e.g., Cartan's criterion for non-linear Monge equations). In examining the behaviour of the isotropy algebras, that can either diminish or remain the same, when passing from a derived system Sν to the previous system S ν−1 , we obtain a full set of numerical invariants for the elementary flag systems that moreover specify the local models. flag systems and models and infinitesimal automorphisms and isotropy and Darboux prolongations and k − th order equivalence groupoids.

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“…The basic theory as well as an initial attempt towards a local classification of flag systems can be found in [15]. Further classification can also be found in [5], [6], [8] and [21]. in one independent variable x and two dependent variables y and z, by means of parametrized expressions:…”

Section: Ie Do = Xa ~ a W I W I 9 F(s)mentioning

confidence: 99%

Flag systems and ordinary differential equations

Kumpera¹

1999

Annali di Matematica pura ed applicata

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During many decades, it was a general belief that a flag system could always be represented locally by the von Weber model. However, this is not the case and, in 1978, Giaro, Ruiz and the author brought to evidence a non-homogeneous model ([9]). More recently, the same model was found independently by Mormul ([20]). The presence of such non-homogeneous models sheds some doubts on Cartan's criterion since statements such as in [4], p. 89, [11], p. 328, and [22], p. 227, become inaccurate. It is our purpose here to prove the Cartan criterion with the added hypothesis of homogeneity and show as well that this criterion can be applied directly to higher order ordinary differential equations without the necessity of reducing them to first order equations. Finally, we examine first and second order single equations, these involving flag systems of length two and three respectively.(*) Entrata in Redazione il 1 agosto 1998. Indirizzo dell'A.: Rua Campos Sales 225,

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On local classification of Goursat structures

Cited by 7 publications

References 1 publication

Nilpotentization of the kinematics of the n-trailer system at singular points and motion planning through the singular locus

Nilpotentization of the kinematics of the n-trailer system at singular points and motion planning through the singular locus

Automorphisms of flag systems

Flag systems and ordinary differential equations

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