Configurations of an Articulated Arm and Singularities of Special Multi-Flags

P. Mormul has classified the singularities of special multi-flags in terms of"EKR class"encoded by sequences $j_1,\dots, j_k$ of integers (see [Singularity Theory Seminar, Warsaw University of Technology, Vol. 8, 2003, 87-100] and [Banach Center Publ., Vol. 65, Polish Acad. Sci., Warsaw, 2004, 157-178]). However, A.L. Castro and R. Montgomery have proposed in [Israel J. Math. 192 (2012), 381-427] a codification of singularities of multi-flags by RC and RVT codes. The main results of this paper describe a decomposition of each"EKR"set of depth $1$ in terms of RVT codes as well as characterize such a set in terms of configurations of an articulated arm. Indeed, an analogue description for some"EKR"sets of depth $2$ is provided. All these results give rise to a complete characterization of all"EKR"sets for $1\leq k\leq 4$.


Introduction and Results
A special multi-flag of step m ≥ 1 and length k ≥ 1 is a sequence (see [7]): (see section 2.1 for a more precise definition) The notion of special multi-flags is described in some ways in [13] and [8]. Furthermore, for m ≥ 2, it is proved in [2] and [16] that the existence of a completely integrable subdistribution F of D1 implies property (iii), and when such a distribution F exists, it is than unique ( see Theorem 2.1). When m = 1 a special multi-flag is a Goursat flag, and, in this case, the conditions (iii) and (iv) are automatically satisfied but for such a distribution F is not unique. One fundamental result on Goursat flags is the existence of locally universal Goursat distributions proved by R. Montgomery and M. Zhitomirskii in [10]: the "monster Goursat manifold" which is constructed by applying Cartan prolongations k times. On the other hand, the kinematic system of a car with k − 1 trailers can be described by an appropriate Goursat distribution ∆ k on R 2 × (S 1 ) k and moreover, this configuration space is diffeomorphic to the Cartan prolongation of the distribution ∆ k−1 on R 2 × (S 1 ) k−1 (see Appendix D of [10]).
Theorem 1 of this paper (which is announced in [12]) is a generalization of this last result for special multi-flags of step m ≥ 2. As all the arguments used to show this results are basic arguments for the poof of Theorem 2 and Theorem 3 of this paper, we give here a complete this result More precisely, a special multi-flag can be considered as a generalization of the notion of Goursat flags and the fundamental result of [2] and [16] is again obtained by Cartan prolongation (see also [8]). So, in this situation, we can also defined a "monster tower" by sucessive Cartan prolongations of T R m+1 (see for instance [2], [16], [4] or [5] ). On the other hand, we can construct a kinematic system, called articulated arm in [15], and also called system of rigid bars in [6]. The configuration space C k (m) of such a kinematic system is diffeomorphic to R m+1 × (S m ) k , and and this system is characterized by a distribution D k which generates a special multi-flags of length k (see section 3.1).
Note that we have a canonical 2-fold coverinĝ P k (m) → P k (m) for any k ≥ 1 and m ≥ 2.
In this context, we have the following result announced in [12]: Let be∆ k the canonical distribution obtained onP k (m) after k-fold "spherical prolongation". Than we have: For each k ≥ 1 and m ≥ 2, there exists a diffeomorphism F k fromP k (m) on C k (m) such that: (i) ifπ k :P k (m) →P k−1 (m) and ρ k : C k (m) → C k−1 (m) are the canonical projections, we have: In particular, this result gives a positive answer to a conjecture proposed in section 6 of [5] .
The singularities of special multi-flags were firstly described by P Mormul in [7] and [8]. This classification was founded on one hand, on a generalization of Cartan prolongation and, on the other hand, on some "operation" denoted j which produce a new (m + 1)-distributions from the older ones. So P. Mormul constructs a coding system for labeling singularity classes of germs of special multi-flags which he called "Extended Kumpera-Ruiz classes" and "EKR classes" in short (for more details see section 5.1). An EKR class is coded by a sequence j1 · · · j k so that j l+1 ≤ 1 + sup{j1, · · · j l }. The integer sup{j1, · · · , j k } − 1 is called the depth of the EKR class.
More recently, in [4], A-L. Castro and R. Montgomery have proposed a codification of singularities of multi-flags founded on the tower of projective bundles (1). They use RC and RVT codes. So they get the classification of singularities of special multi-flags in terms of RC or RVT classes The essential results of this paper is to give a decomposition of EKR class of depth 1 in terms of RVT classes and also to give an interpretation of EKR class of depth 1 and RVT classes in terms of configurations of an articulated arm More precisely, in the tower (1) we can define sub-towers by taking the tower of Cartan prolongation of any fiber of P j (m) → P j−1 (m). We get the so called " baby monsters" in [4], so that, in each vector space ∆ k (p) ⊂ TpP k (m) we have a family of "critical" hyperplanes coming from these "baby monsters". One of them is the vertical space VpP k (m) i.e. the tangent space to a fiber of P k (m) → P k−1 (m). A point p ∈ P k (m) can be written p = (p k−1 , z) where p k−1 ∈ P k−1 (m) and z is a line in ∆ k−1 (p k−1 ). The point p is called vertical if z is tangent to the fiber at p k−1 , tangency if z is not vertical but belongs to one critical hyperplane and otherwise p is called regular. So we can affect to p a word in letters {R, V, T } so that the letters of rank l is R or V or T according to the fact that the projection of p onto P l (m) is regular or vertical or tangency respectively.
In a word ω, a sub-word of type R h or T l means a sequence of h (resp. l) consecutive letters R (resp. T ) if h > 0 (resp. l > 0), and no letter R (resp. T ) if h = 0 (resp. l = 0). For any EKR class of 1-depth we will denote by {i1, · · · , iν } the set {i such that ji = 2}. The following result gives a a complete description of EKR classes of depth 1 in terms of class of word in RVT codes.
(2) Denote by C R h 0 V R h 1 ···V R hν a class of point whose RVT code is R h 0 V R h 1 · · · V R hν . Then any class Cω of point p whose RVT code is ω is contained in an EKR class Σj 1 ···j k if an only if ω is of type R h 0 V R h 1 · · · V R hν where each letter V is exactly at rank i1, · · · , iν . Such a class C R h 0 V R h 1 ···V R hν is an analytic submanifold of Σj 1 ···j k of codimension l1 + · · · + lν (3) The EKR class Σj 1 ···j k is the union of all class C R h 0 V R h 1 ···V R hν which satisfy property (2).

Theorem 3.
Let be (M0, · · · , M k ) an articulated arm in R m+1 (1) a configuration q ∈ C k (m) of the articulated arm belongs to the EKR class Σj 1 ···j k of depth 1 if and only if, for this configuration, the segments [Mi−2, Mi−1] and [Mi−1, Mi] are orthogonal in Mi−1 for all i = i1, · · · , iν (2) There exists a family for l = 0, · · · l λ and λ = 1, · · · , ν; • We finally give the same type of results for EKR classes of depth at most 2 for 1 ≤ k ≤ 4 in the last section (section 5.4

).
This paper is self-contained and organized as follows. In Section 2 we recall at first all the context and essential results about special multi-flags which are used in this paper. In a second subsection, We present a summary on Cartan prolongation and tower of projective bundle. Spherical prolongations, tower of sphere bundles and their properties are developed in the last subsection.
Section 3 is devoted to the context of the configurations of an articulated arm of length k ≥ 1 in R m+1 . The space C k (m) of such configuration is presented in the first subsection. The relation between tower of sphere bundles and C k (m) is given in the second subsection in which the reader can find of proof of Theorem 1. Finally we present the hyperspherical coordinates on C k (m) in the last subsection.
In the first subsection of section 4, according to [4], we present a summary on the RC and RVT codes and adapt these codes to the context of tower of sphere bundles. The following subsection gives some interpretations of the property of verticality in terms of configurations of an articulated arm. In the same way, some interpretations of the property of tangency are given in the last subsection.
Section 5 is devoted to the relation between EKR classes of depth at most 1 and RVT classes. In the first subsection, we summary the definition and procedures about EKR classes according to [7] and [8]. The following subsection gives a global description of EKR classes in terms of RVT classes. In particular, it contains a proof of theorem 2. The following subsection presents an interpretation in terms of configurations of an articulated arm for EKR classes (of depth at most 1) and RVT classes. In particular it contains a proof of Theorem 3. Finally, the last subsection develops, for 1 ≤ k ≤ 4, the decomposition of EKR classes of depth at most 2 in RVT classes and the corresponding interpretation in terms of configurations of an articulated arm. We end this paper by some commentaries about these results and the results contained in [4], [5] and [9].

Special multi-flags.
A distribution D on a manifold M is an assignement D : x → Dx ⊂ T M of subspace Dx of the tangent space TxM . A local vector field X on M is tangent to D if for any X(x) belongs to Dx for all x in the open set on which X is defined. A distribution is called a smooth distribution if there exists a set X of of local vector fields such that Dx is generated by the set {X(x), X ∈ X }, we then say that D is generated by X .
In this paper any distribution will be smooth and we denote by Γ(D) the set of all local vector fields which are tangent to D. Such a distribution will be called a distribution of constant rank if D defines is a subbbundle of T M . According to [2] and [16], any pair (M, D) of a distribution of constant rank on a smooth manifold M will be called a differential system. Given two differential systems (M, D) and (N, ∆), we will say that (M, The Lie square of a distribution D is the distribution denoted D 2 which is generated by the sets Γ(D) and {[X, Y ], X, Y ∈ Γ(D)}. The Cauchy characteristic distribution L(D) of a distribution D is the distribution generated by the set vector fields is a distribution of constant rank, then it is an integrable distribution.
A special multi-flag of step m ≥ 2 and length k ≥ 1 is a sequence (see [7]): of distributions of constant rank on a manifold M of dimension (k + 1)m + 1 which satisfies the following conditions: (iii) Each Cauchy characteristic subdistribution L(Dj ) of Dj is a subdistribution of constant corank one in each Dj+1, for j = 1, · · · , k − 1, and L(D k ) = 0. (iv) there exists a completely integrable subdistribution F ⊂ D1 of corank one in D1.
In the following, a flag F which satisfies conditions (i), (ii) but not conditions (iii) and (iv) will be just called a multi-flag of step m or a m-flag and we say that F is generated by D.
The necessary and sufficient condition of a multi-flag to be a special multi-flag is given by the following result (see [2] Proposition 1.3 and [16] Theorem 6.2) ) Theorem 2. 1. [2], [16] for k ≥ 2, a m ≥ 1 consider a multi-flag of step m : F is a special multi-flag if and only if there exists a completely integrable subbundle F of D1 of corank 1. Moreover, if such a subbundle F exists, F is unique.
According to the previous the definition of a special multi-flag, we obtain the following sandwich flag: All vertical inclusions in this diagram are of codimension one, while all horizontal inclusions are of codimension k. The squares built by these inclusions can be perceived as certain sandwiches, i.e. each subdiagram" number j indexed by the upper left vertices Dj: In a sandwich number j, at each point x ∈ M , in the (m+1) dimensional vector space Dj−1/L(Dj−1)(x) we can look for the relative position of the m dimensional subspace L(Dj−2)/L(Dj−1)(x) and the 1 dimensional . We say that x ∈ M is a Cartan point if the first situation is true in each sandwich number j, for j = 2, · · · , k. Otherwise x is called a singular point.

Cartan prolongation and tower of projective bundles.
Let be D a distribution of constant rank m + 1 on a manifold M of dimension n. Classically the Grassmannian bundle G(D, 1) on M is the set where P (D(x), 1) is the projective space of the vector space D(x). So we have a bundle π : G(D, 1) → M whose fiber π −1 (x) is diffeomorphic to the projective space RP m . The rank one Cartan prolongation is the distribution D (1) defined in the following way: given a point (x, λ) ∈ G(D, 1) then where λ is a direction of D(x). Then D (1) is a distribution on G(D, 1, M ) of constant rank m + 1. Let be M a manifold of dimension m + 1 As in [16], for any m ≥ 2 and k ≥ 1 starting with D = T M , we obtain inductively a tower of bundles: where, for any j = 0, · · · , k, P j (M ) is a manifold of dimension (j + 1)m + 1, and on each P j (M ), ∆j is a distribution and these data are defined inductively by : P j (M ) = G(∆j−1, 1, P j−1 (M )) and ∆j = (∆j−1) (1) for j = 1, · · · , k and ∆0 = T M .
In the particular case of M = R m+1 , for j = 0, · · · , k, we denote simply by P j (m) the manifold P j (R m+1 ) and we have the following tower of bundles: We have then the following result : Theorem 2.2. [16] (1) On P k (m), the distribution ∆ k generates a special multi-flag of step m and length k.
(2) let be F : D = D k ⊂ D k−1 ⊂ · · · ⊂ Dj ⊂ · · · ⊂ D1 ⊂ D0 = T M a special multi-flag of step m ≥ 2 and length k ≥ 1. Then, for any x ∈ M , there exists y ∈ P k (m) for which the differential system (P k (m), ∆s, y) is locally equivalent to the differential system (M, D, x).
The part 2 of Theorem 2.2 can be found precisely in [16] called "Drapeau Theorem". However, according to the definition of a special multi-flag, we can easily deduce this result from the following Theorem of [8]: Suppose that D is a (m + 1)-dimensional distribution on a manifold M s+m such that: In particular τ is a local diffeomorphism. On S(D, M, g) we consider the distribution D [1] defined in the following way where ν is a norm one vector in D(x).
The distribution D [1] is called the rank one spherical prolongation of (M, D, g) In fact, the unit sphere bundle of S(D, M ) is defined as soon as we fix some riemannian metric on D. In this case, the distribution D [1] is also well defined.
There exists a canonical riemannian metricĝ on S(D, M, g) which is uniquely defined from the riemannian metric g on M . Proof.
At first we show part (i) locally. Choose a chart domain U over which D is trivial. We choose an orthonormal frame {e0, · · · em} of D over U . Without loss of generality we can assume that D |U ≡ R n × R m+1 so, the bundle S(D, M, g) |U is isomorphic to to R n × S m and G(D, 1, M ) |U is isomorphic to R n × RP m . So locally, is the line bundle generated by ν. From the definition of D [1] (x,ν) and D (1) µ(x,[ν]) we have τ * (D [1] (x,ν) ) = D (1) τ (x,[ν]) . As τ is a local diffeomorphism we get the part (i) locally. On the other hand, the mapα : S(D, M, g) → S(D, M, g) given byα(x, ν) = (x, −ν) is a diffeomorphism which commutes with τ . From the definition of of D [1] , we get This ends the proof of part (i). For part (ii), denote byḡ the canonical riemannian metric on T M associated to g. As, S(D, M, g) can be considered as a submanifold on T M we get an natural induced riemannian metricĝ on S(D, M, g).
Let be g0 and g1 two riemannian metrics on M . We denote by Si(D, M ) the sphere bundle of D associated to the metric gi, and D It is easy to see that Ψ is a diffeomorphism from D o into itself which commutes with Π and which sends S0(D, M ) to S1(D, M ). So the restriction ψ of Ψ to S0(D, M ) is a diffeomorphism onto S1(D, M ). Moreover, from (9), dΨ map the the linear span Ru into itself, for any u in the fiber D o x over x. So we have ψ * (D [1] 0 ) = D  for any x ∈ M . Given any riemanian metric g ′ on M ′ , we get an induced riemannian metric g on M and we can consider the associated spherical prolongation then we have: Thenφ is a bundle morphism over φ which is an injective immersion and such that (i)φ(S(D, M, g)) = S(φ * (D), φ(M ), g ′ ) (ii)φ * (D [1] ) = (φ * (D)) [1] ⊂ (D ′ ) [1] . Moreover, if φ is a diffeomorphism such that φ * (D) = D ′ , thenφ is also a diffeomorphism and we havê φ * (D [1] ) = (D ′ ) [1] and the riemannian metricφ * ĝ ′ is nothing but the canonical metricĝ naturally associated to g on M .

Proof.
As in Lemma, consider the mapφ(x, ν) = (φ(x), dxφ(ν)). From our assumptions , we get a smooth map from S(D, M, g) to S(D ′ , M ′ , g ′ ) and from its definition, clearlyφ is a bundle morphism over φ. As φ is an injective immersion, it follows that at firstφ is injective.
Note that the tangent space T (x,ν) Sx of the fiber Sx over x of S(D, M, g) can be identified with {v ∈ Dx such that g(ν, v) = 0}. Now any V ∈ T (x,ν) S(D, M, g) can be written V = (u, v) with u ∈ TxM and v ∈ T (x,ν) Sx. So we have then: So,φ is an immersion, from (10).
On the other hand, as φ * g ′ = g, dxφ is an isometry on its range, and then, dxφ(Sx) is the fiber over φ(x) of S(φ * (D), φ(M ), g ′ ) and we get (i).
So, as in the context of Cartan prolongation, for any m ≥ 2 and k ≥ 1 we can define, inductively, a tower of sphere bundles (for a fixed choice of the metric g on a manifold M ): where, for any j = 0, · · · , k,P j (M ) is a manifold of dimension (j + 1)m + 1, and, on eachP j (M ) we have a canonical distribution∆j and a riemannian metric gj onP j (M ), all these data are defined inductively in the following way : Note that if g ′ is another riemannian metric on M , according to Lemma 2.2 and Lemma 2.3, by induction we get a family of diffeomorphisms ψ j such that, if : is the tower of sphere bundles associated to the choice g ′ on M we have, for all j = 0, · · · , k : So the properties which characterize the tower (11) are independent of the choice of the riemannian metric g on M.
For simplicity we writeP j (m) :=P j (R m+1 ) for any j ∈ N. From Theorem 2.2, and Lemma 2.1 we get the following result : the tower of sphere bundles associated to the canonical metric on R m+1 .
(1) we have a canonical two-fold covering τ k :P k (m) → P k (m) such that (2) On each manifoldP k (m), the distribution∆ k generates a special multi-flag of step m and length k.
and length k ≥ 1. Then, for any x ∈ M , there exists y ∈P k (m) for which the differential system (P k (m),∆s, y) is locally equivalent to the differential system (M, D, x).
The tower (12) will be called the spherical tower of special multi-flags of step k.

3.
Tower of sphere bundles associated to a kinematic system 3.1. A kinematic system for special multi-flags. We locate us in the context of [6] and [15]. Consider a set of k segments [Mi; Mi+1], i = 0, · · · , k − 1, in R m+1 , with m ≥ 2, keeping a constant length li = 1 between Mi and Mi+1, and the articulation occurs at points Mi, for i = 1, · · · , k − 1.
Such a system is called a "k-bar system" in [6] and an "articulated arm of length k" in [15]. The kinematic evolution of the extremity M0, under the constraint that the velocity of each point Mi, i = 0, · · · , k − 1, is collinear with the segment [Mi, Mi+1] is completely described in terms of hypersperical coordinate in [15] and result of flatness and controllability for such a system are proved in [6]. We can associate to this problem a special multi-flag of step m ≥ 2 and length k ≥ 1 as explained in the following: ) be the canonical coordinates on the space R m+1 i which is equipped with its canonical scalar product < , >. (R m+1 ) k+1 is then equipped with its canonical scalar product too.
Denote by E k the distribution generated by the vector fields The properties of D k are summarized in the following result. (see [15] also [6]) On C k (m), the distribution D k satisfies the following properties: (1) D k is a distribution of rank m + 1.
(2) The distribution D k is a special multi-flag on C k (m) of step m and length k.

Articulated arm and spherical prolongation.
To an articulated arm on R m+1 (m ≥ 2) of length k ≥ 1 we can associate the following canonical tower of sphere bundles: According to Theorem 3.1, and Theorem 2.4, we know that the differential system (C k (m), D k )) associated to an articulated arm of length k on R m+1 is locally isomorphic to the canonical differential system (P k (m),∆ k ) at some appropriate points. In fact, we have more: (see Theorem 1 in the introduction) are the canonical projections, we have: Then, for any x ∈ M , there exists y ∈ C k (m) for wich the differential system (C k (m), D k , y) is locally equivalent to the differential system (M, D, x) .
The end of this subsection is devoted to the proof of Theorem 3.2.
Before proving these results, we need some auxiliary results.

Lemma 3.2.
For k ≥ 1, consider, on C k (m), the natural decomposition: On the other hand, let L k be the involutive distribution whose leafs are the fibers of the natural fibration of C k (m) onto C k−1 (m).
(2) The distribution D k , is generated by L k and the vector field (3) The distribution D k is also generated by the family of vector fields Let be H k the subdistribution of E k generated by the family vector fields { ∂ ∂x r k , r = 1, · · · m + 1}. So In fact, we have So L k is generated by the family vector fields {Π k ( ∂ ∂x r k ), r = 1, · · · m + 1}. On the other hand, as H k is the vertical bundle of the canonical projection it follows that L k is also the vertical bundle of the induced projection of C k (m) onto C k−1 (m). On the other hand, the fiber over q ∈ C k−1 (m) of the previous fibration is the unit sphere Sq = {(q, x k ) : Ψ k−1 (q, x k ) = 0} which proves (1).

The vector field
is vertical for the projection (18) and is orthogonal to each Sq.
As ||V k || = 1 we thus have: ] is tangent to D k but clearly this vector fields is not tangent to L k . As D k is a distribution of constant rank m+1 and L k is an (integrable ) subdistribution of rank m, it follows that D k is generated by L k and X k which proves (2) .
On the other hand, according to (19), each vector field ( According to Lemma 3.1 we get (3).
(2) Let be γ k a riemmaniann metric on the bundle D k so that the morphismΨ is an isometry between D k and C k (m) × R m+1 (for the canonical euclidian product on the fiber R m+1 ). ThenΨ induces a diffeomorphism Ψ :
So, in the global chart of S(D k , C k (m), γ k ) defined by Ψ, according to (20), the spherical prolongation (D k ) [1] of D k is generated by the tangent space to the sphere centered at (x0, · · · , x k ) and the vector field: According to (19) and the value of X k , this vector field can be written and the distribution D k+1 is generated by Again from Lemma 3.2 part (2), at level k + 1 we get: Proof of Theorem 3.2 Note that on the tangent bundle T R m+1 , we can put the global chart defined by the map (x1, x2) → (x1, x2 − x1). On the other hand, the riemannian metric g1 on T R m+1 induces by the canonical metic g is again the canonical metric on the product In the previous coordinates, any tangent vector of R m+1 at a point x1 can be written (x1, x2 − x1). So, Z0 defines a global section of the unit bundle associated to T R m+1 According to Lemma 3.2, the distribution D1 is generated by [1] and we get the result for k = 1.
Assume that we have a diffeomorphism F k :P k (m) → C k (m) which satisfies the properties (i), and (ii) of Theorem 2. 4. [1] ] = D k+1 and which commutes with the natural projections According to previous induction, we can put onP k (m) , the riemannian metric [1] and which commutes with the natural projections Finally, according to Lemma 2.2, when we put onP k (m) the riemannain metric induces by induction on the tower bundle (11), we also have a diffeomorphism Φ :P k+1 (m) → S(∆ k ,P k (m),γ k ) which commutes with the canonical projectionŝ and such that Φ * (∆ k+1 ) =∆ [1] k .
Remark 3.2. According to Theorem 2.4 and Theorem 3.2, from towers (12) and (17) we get the following diagram each vertical map is a 2-fold covering for k ≥ 1:

Hyperspherical coordinates.
Consider the natural global diffeomorphism F k : C k (m) → R m+1 × (S m ) k given by Note that, according to Theorem 3.2, the map F k • F k is also a global diffeomorphism fromP k (m) to S k (m) = R m+1 ×(S m ) k and, if ̺ k : S k → S k−1 is the canonical projection, we have the following commutative diagram: Via this global chart, each point q = (x0, x1, · · · , xi, · · · , x k ) ∈ C k (m) could be identified with We will put on each factor S m charts given by hyperspherical coordinates. We first recall some basic facts about this type of coordinates.
(1) For any ζ ∈ S k (m) any map of type H k around ζ is called a hyperpsherical chart on S k (m) (2) For any q = (F k ) −1 (ζ) in C k (m), any map of type H k • F k around q is called a hyperspherical chart on C k (m). (3) for any p = (F k •F k ) −1 (ζ) onP k (m), any map of type H k •F k •F k around p is called a hyperspherical chart onP k (m). Now, we introduce the following notations in hyperpsherical coordinates: Aj, for i = 0, · · · , l − 1, 0 ≤ l ≤ k − 1 and f l l = 1.
Remark 3. 3. In Lemma 3.1 we already defined a function Aj (q) = − < Nj (q), Nj−1(q) >. It is clear that that we have: Theorem 3. 4. For any k ≥ 1 we have (1) In hyperspherical coordinates, on each manifold S k (m), C k (m) andP k (m), the corresponding distributions F k * (D k ), D k and∆ k respectively, is generated by {X 0 k−1 , X 1 k−1 , · · · , X m k−1 }. (2) we have a tower of commutative diagrams: where, for all l = 1, · · · , k, • at each horizontal level the horizontal map between column number l to column number l − 1 is a fibration in sphere (resp. projective space) for the three first line (resp. for the last line) • Each vertical map sends each corresponding distribution of index l onto the corresponding distribution l on the lower line . Moreover the three first vertical maps are diffeomorphisms and the last vertical ones are two-fold coverings.

Remark 3.4.
All manifolds which appear in the towers (24) are analytic manifolds and all maps in these towers are also analytic. 4. RC -RVT codes and configuration of articulated arms 4. 1. RC and RVT codes according to [4]. In this subsection we expose the theory of RC and RVT codes introduced in [4] and we also adapt it to the context of spherical prolongation.
Let be D a distribution of constant rank on a manifold M fitted with a Riemannian metric. We will denote indistinctly by P(M, D) the sphere sphere bundle S(M, D, g) or the projective bundle P (M, D) and by D {1} the spherical prolongation or the Cartan prolongation of D on P(M, D). By induction the corresponding tower of bundle ( cf (7) and (11)) fitted at each level j with a distribution denoted by D k = (D k−1 ) {1} .
When M = R m+1 we simply denote P k (m) any one spaceP k (m) or P k (m).
For any k ≥ 1 we denote by Π k the natural projection of P k (m) onto P k−1 (m). The tangent bundle to the fiber of Π k is the vertical bundle denoted V k and by construction we have V k (p) ⊂ D k (p).
For any p ∈ P k−1 (m), the fiber (Π k ) −1 (p) is denoted S k (p). So for such a point p, from (25), we get a tower of fiber bundles · · · → P l (S k (p)) → P l−1 (S k (p)) → · · · → P 1 (S k (p)) → P 0 (S k (p)) = S k (p) Coming back to our general context, on each P j (S k (p)), we have a distribution d k j again inductively defined by d k j = [δ k j−1 ] {1} . Such a tower will be called a fiber prolongation tower. Note that, in such a tower, the differential system (P j (S k (p)), d k j , q) is always equivalent to (P l (m − 1), D l , p) for some adequate point p ∈ P l (m − 1). Of course we have P l (S k (p)) ⊂ P k+l (m) and d k j (q) is an hyperplane in D k+j (q) for any q ∈ P j (S k (p)). In particular V k (q) is nothing but that d k 0 (q) for any q ∈ S k (p) On the other hand, for any k > l ≥ 0, let be Π kl the natural projection of P k (m) onto P l (m) which is the composition Π k •Π k−1 •· · ·•Π l+1 . If p is a point in P k (M ), we denote by p l its projection p l = Π kl (p) ∈ P l (m) and we say that p l is under p k = p. With these notations, for k ≥ 1, each point p k ∈ P k (m) can be written (p k−1 , z) for some z ∈ S k (p k−1 ).
So, at each level k ≥ 1, inside the space D k (p), we have the family of hyperplanes d i j (p), with i + j = k, coming from a fiber prolongation of order j of the tangent space of the fiber S i (pi−1) for i = 1, · · · , k.
Recall that a family {Ei}i∈I of hyperplanes of R N is in general position if, for every subset J ⊂ I of indices whose cardinality |J| ≤ N we have that the codimension of the intersection i∈J Ej is exactly |J|. Proof. This result is proved in [4] for Cartan prolongation (Theorem 6.1). Ifp is a point inP k (m) we denote by p its projection τ k (p) in P k (m) (cf Theorem 2.4). According to Theorem 2.4, each hyperplane d i j (p) in D k (p) project via τ k onto any hyperplane d j i (p) in ∆ k (p) corresponding to the previous process of fiber prolongation in the tower of bundles (11). By applying result of [4] we get the proof in the context of spherical prolongations.
According to [4] and [5] we have the following Definitions (1) any hyperplane d i j (p), with i + j = k in the space D k (p) is called a critical hyperplane at p. A direction l or a vector v in D k (p) is called critical if it lies in one critical hyperplane. Otherwise l (resp. v) is called regular. Moreover a critical direction l or a vector v in D k (p) is called vertical (resp. tangency) if the singular hyperplane which contains it is V k (p) = d k 0 (p) (resp. d i j (p) for i > 0).
(1) let bep ∈P k (m) and p = τ k (p) ∈ P k (m). It follows from Theorem 2.4 thatp is regular, critical vertical, tangency if and only if p is respectively regular, critical vertical, tangency. In the converse, for any p ∈ P k (m), each points in τ k (p) ⊂P k (m) have the same previous qualification as p (2) Inside any fiber prolongations tower (26) we can consider a fiber prolongation tower from some fiber of the projection P l (S k (p)) → P l−1 (S k (p)) and look for the corresponding critical hyperplane in d k l (q). Then such a critical hyperplane is in fact an intersection of type d k l (q) ∩ d i j (q) with i > k and k + l = i + j (see Proposition 6.2

of [4] ).
(3) if a point p = (p k−1 , z) ∈ P k (m) is critical, than z can belongs to the intersection of many critical hyperplanes and not only one critical hyperplane.
The RC code of a point p ∈ P k (m) is a word σ = σ1 · · · σ l · · · σ k whose letter σ l is R (resp. C) if the point p l under p is regular (resp. critical). Note that, by convention, the first letter is always R.
The RVT code of a point p ∈ P k (m) is a word ω = ω1 · · · ω l · · · ω k obtained fron its RC σ(p) such that:  (1) According to Remark 4.1, the RC (resp. RVT) code of any pointp ∈P k (m) is the same as the RC (resp. RVT) code of its projection p = τ k (p) ∈ P k (m). (2) The RC code gives rise to a partition of P k (m) into 2 k−1 set of points which have the same RC code σ. Let beĈσ (resp. Cσ) the set of pointp ∈P k (m) (resp. p ∈ P k (m)) whose RC code is σ.
Then τ k (Ĉσ) = Cσ and (τ k ) −1 (Cσ) =Ĉσ. (3) Remark 4.1 part (2) imposes that each pi which is tangency, pi must lies in a fiber tower prolongations for some pj under pi. So the first letter C which appears in the RC codes, says level i, imposes that pi must vertical. (4) Each RC code σ generates, a priori, 2 nσ RVT codes ω, if nσ is the number of letters C in σ.
However, from part (3), a letter T cannot follows immediately a letter R in such a code because each tangency point must lies in a prolongation tower of some point pj under pi. So, after a letter R the first eventually letter T which appear, we must at most one letter V between these letters. (5) According to Remark 4.1 (3), for a critical point p = (p k−1 , z) ∈ P k (m), when z can belongs to the intersection of many critical hyperplanes the RVT code generated by the RC code can be not well defined. In this case, when z belongs is not vertical and belongs to one (and only one) of them, we need much precision in the code about the possible letters "T" for instance T1, T2 · · · , Tν. Moreover, much more complicated codification is needed if z belongs to a the intersection of many critical hyperplanes. For instance, if this intersection is a line we can use a codification by letters L1, L2 · · · as it is proposed in [4] and [5].
In the RVT code, according to Remark 4.2 (5), and [4] and [5] • if z to belongs to only one critical hyperplane we will use letters V, T1, T2 · · · , in the RVT code; • if z belongs to the intersection of exactly two critical hyperplanes referenced Ti and Tj we will use letters of type Tij in the RVT code. Moreover we adopt the following convention : T0 is always relative to a vertical hyperplanes and T i , for i > 0 to a critical hyperplanes which is not vertical.
• more generally, if z belongs to the intersection of exactly n critical hyperplanes referenced T0 = V and Ti 1 , · · · Ti n−1 or Ti 1 , · · · Ti n with ii · · · in = 0 we will use letters of type T0i 1 ···in or Ti 1 ···in in the RVT code.
More generally we can transpose the characterization of points ofP k (m) onto points of C k (m). More precisely we have: (1) q is called regular, critical ,vertical, tangency ifp is regular, critical ,vertical, tangency respectively (2) the code of q will the code of the corresponding pointp At first, we have the following characterization of vertical points in C k (m) (1) For all 2 ≤ l ≤ k − 1, We have the following equivalent properties: (a) let be   (1) The set CS of singular points of C k (m) is a subanalytic set of codimension 1. In particular, the set of Cartan points C k C (m) = C k (m) \ CS is an open dense set. (2) Let be ω a word of length k in in letters R and V and denote by {l1, · · · , lr} the set of index {l ∈ {1, · · · , k}} such that ω l = V . We have the following properties: (i) The set C(ω) of points q ∈ C k (m) whose RVT code is ω is an analytic submanifold of C k (m) whose codimension is r.  (1) Our original definition of Cartan points (see the end of section 2.1) is somewhat different from [4] or [5]. However part (2) of Proposition 4.1 proves the equivalence of these definitions. (2) The result of part (1) of Theorem 4.2 is well known (see [4], [5], [6], [7], [14]). (2) is also proved in [14] with an another notation for this set.
The proof of the Proposition 4.1 needs the following Lemma: Then in hyperpsherical coordinates, Proof. We place ourselves in the context of Notations 3. 1. By a simple calculation in hyperpsherical coordinates, we obtain that the member [D k ] l of the multi-flag associate to D k is generated by On the other hand, we also have On the other hand, as {X 0 l−2 , X 1 l−2 , · · · , X m+1 l−2 } is a basis of D l−1 at q l−1 and Z l−1 is a linear combination {X 1 l−2 , · · · , X m+1 l−2 } we must have X 0 l−2 (q) = 0. So we get: Remark 3.3, this ends the proof of part (1).
As {Π l−1 ( ∂ ∂x r l−1 ), r = 1, · · · , m + 1} generates the tangent space to each fiber of the projection Note that at, a point q, we have So, if A l (q) = 0 we must have ∂A l ∂x r l+1 (q) = 0 for some 1 ≤ r ≤ m + 1. According to Remark 3.4, it follows that CS is a subanalytic subset of C k (m) of codimension 1, which ends the proof of part (1).
Note that q belongs to C(ω) if and only if each points q l 1 , · · · , q lr under q are vertical. So, from property (b) of Proposition 4.1, the equations of C(ω) are: Ai(q) = 0 : for i + 1 = l1, · · · , lr (28) On the other hand each A l depends only of the variables x l−1 , x l , x l+1 . So using the previous argument " ∂A l ∂x r l+1 (q) = 0 for some 1 ≤ r ≤ m + 1", the equations in (28) are independent. According again to Remark 3.4, it follows that C(ω) is an analytic submanifold of C k (m) of codimension r.
Part (2) of Theorem 4.2 is a direct consequence of property (c) of Proposition 4.1.

Tangency points and configurations of articulated arms.
We will prove the fundamental following results for tangency points q ∈ C k (m) : (1) Let be q ∈ C k (m) a tangency point. Then, there exists 2 ≤ i ≤ k − 1 such that the point qi under q is vertical. Moreover, let be Then if l < k, for any l < j ≤ k the point qj under q is tangency.
letters equal to R and T l means l consecutive letters equal to T . Then the set C R h V T l of points q ∈ C h+l+1 (m) whose RVT code is R h V T l is an analytic submanifold of C h+l+1 (m) of codimension l + 1. Moreover, the fiber of the projection of C R h V T l onto (C h (m))C over q h ∈ (C h (m))C is the set F h+l+1 (P l (Ŝ(q h )) (3) Consider the the flag (3) associated to D h+l+1 . The set C R h V T l can be defined inductively in the following way: (4) To a word R h V T l we associate a family {K0, · · · K l } of directions in R m+1 in the following way: Ki(q) is the direction generated by x h+i − x h−1 for i = 0, · · · , l Let be (M0, · · · , M h+l+1 ) an articulated arm in R m+1 . Its configuration q = (x0, · · · , x h+l+1 ) belongs to C R h V T l if and only if, in configuration q ∈ C k (m), each segment [Mi−1, Mi] and [Mi,Mi+1] are not orthogonal at point Mi for each i = 1, · · · h − 1 and each segment [M h+i , M h+i+1 ] is contained in the affine hyperplane though x h+i which is orthogonal to the direction Ki for all i = 0, · · · , l Remark 4.5.
(1) The result of part (3) condition (iii) can be seen as a justification of term "tangency" as in the context [11] for m = 1.
(2) For m = 1, in the context of the "car with n trailers", in [3], F. Jean has build a family of "critical angulars" . In our situation (m ≥ 2) the family {K0, · · · K l } of directions in R m+1 , is nothing but a generalization of Jean's result.  (4) can be find in [14] (with quite different assumptions), by using hyperspherical coordinates.
Proof. fix some tangency point q ∈ C k (m). From Remark 4.2 part (3), there must exist a vertical point qi under q. Let q l be the last such tangency point under q. Then we have A l−1 (q) = 0. In fact , for q l−1 = (x0, · · · , x l−1 ) given, this relation characterizes such points q l = (x0, · · · , x l−1 , x l ) ∈ C l (m) which are vertical. In other words, the set F l (Ŝ l (q l−1 )) are exactly the set of points (q l−1 , x l ) ∈ C l (m) such that Now, from the definition of l if l + 1 ≤ k, q l+1 cannot be vertical. The point q l+1 is no more regular otherwise between q l+1 and q we must have a vertical point which contradicts the definition of l. So q l+1 must be a tangency point. Set q l+1 = (x0, · · · , x l , x l+1 ). Taking in account the proof of Proposition 3.1, in the trivializationΨ l : is a tangency point, this vector field must project on C l (m) onto a vector which is tangent to F l−1 (Ŝ l (q l−1 )) at point q l . So we must have: Taking in account that A l−1 (q l ) = 0, and Remark 3.1 we get Y l = Z l−1 on A l−1 = 0. According to the value of Z l−1 and Z l , modulo A l−1 = 0, equation (29) gives rise to the relation Note that, for q l−1 = (x0, · · · , x l ) fixed, the equation A l−1 (q l ) = 0 associated to (30) are exactly the equations of the set F l+1 (P 1 (Ŝ l (q l−1 )). Moreover, it is easy to see that these equations are independent on C h+2 (m). Assume that for l ≤ i < k the point qi is tangency. By same arguments as previously, qi+1 must also a tangency point. So, by induction we get part (1).
We now look for part (2). Note that, from the previous proof we have already shown that C R h V T is an analytic submanifold of C h+2 (m) of codimension 2 and that the fiber of the projection of The global result will be obtained by induction on l.
To precise the induction hypothesis we need the following notations and Lemma.
We set Ai,j (q) =< xi+1 − xi , xj+1 − xj > for i, j = 0, · · · , k − 1. The family of functions {Ai,j } have the immediate following properties: Proof of Lemma. (1) is then a consequence of the definition of Ai,j . The other parts is a direct calculation.

At first note that Z h is a vector field with components only on
Induction hypothesis (Hi): Assume that for any 1 ≤ i < l we have: • each set C R h V T i of point q ∈ C h+i+1 (m) whose RVT code is R h V T i is a subset of C h+i+1 (m) which is defined by the system of independent equations of type • for j = 0, · · · , n modulo the previous equations φ j ′ = 0 for 0 ≤ j ′ < j, the equation φj = 0 can be reduced to • the set F h+i+1 (P i (Ŝ(q h )) ∩ C h+i+1 (m) is defined by system of equations (31), but for each fixed q h ∈ C h (m)C • we have Dφj (Y h+i+1 ) = 0 for all 0 ≤ j < i Note that we have already shown that this hypothesis is true for i = 1. Referring to the previous proof for point q l+1 to obtain the equation (29), from the same argument applied for q h+i+1 = (q h+i , x h+i+1 ), we get that, when q h+i ∈ C R h V T i , the point q h+i+1 belongs to C R h V T i+1 if and only if Dq h+i φj(Y h+i+2 ) = 0 for all 0 ≤ j < i + 1.
According to (32), these conditions are equivalent to On one hand, the vector field Y h+i+2 can be written and so, from our induction hypothesis, on On the other hand, from (32), eachφj, depends only on variables (x h−1 , · · · , x h+j+1 ), so we always have Dφj(Z h+i+1 ) = 0 for all 0 ≤ j < i. So, the only equation we must add to the system (31) for j = i is: In fact, it is clear that to the system (31) , this equation is of course equivalent to φi+1 = Dq h+i φi(Y h+i+2 ) = 0 At first, note that, for q h ∈ C h C (m) given, the system (31) with adding the equation φi+1 = 0 characterizes any point q h+i+1 = (q h+i , x h+i+1 ) which belongs to F h+i+2 (P i+1 (Ŝ(q h )). We must show now that φi+1 has a reduction of type (32), modulo the previous equation From our hypothesis that A h = 0 and Remark 3.1: we have Using the decomposition (32) and Lemma 4.2, we comput DA k+j,i ′′ (Z i ′ ) for i ′′ = h − 1, · · · ; h + j − 1 and i ′ = h · · · , h + j + 1: We must also show that {φ0, φ1, · · · , φi, φi+1} is a set of independent functions. Using the reductions of equation (31), it is sufficient to proved that {φ0,φ1, · · · ,φi,φi+1} is a set of independent functions From the expression ofφj, j = 0, · · · , i + 1 ( of type (32)), we havē It follows that {φ0,φ1, · · · ,φi,φi+1} is a set of independent functions on C h+i+2 (m). This ends the proof of part (2) Now, in part (3), the condition (i) and (ii) are a direct consequence of Proposition 4.1. We will look for condition (iii).
For simplicity, we denote only by D the distribution D h+l+1 and consider the flag associated to D: For each 0 ≤ i < l, we can identify C h+i+1 (m) with the submanifold of R h+l+1 of equations: x h+j+2 = 0 for j = i + 1, · · · l ||xj − xj−1|| = 1 for j = 1, · · · h + i + 1 With these identification, we have: Proof of Lemma 4. 3. For j = h + l + 1 the result is trivial. Assume that the result is true for Dj for some 0 < j ≤ h + l + 1. We know that Dj−1 is a distribution of rank (h + l − j)m + 1 which contained Dj and is generated by Lie brackets [X, Y ] for all vector fields X and Y tangents to Dj . We first prove that Dj +Dj−1 is contained in Dj−1. We know that Dj is generated by the family{(x r j −x r j−1 )yj + ∂ ∂x r j : r = 1, · · · , m+1} and Yj = Aj−1Yj−1 + Zj−1 also belongs to Dj (Remark 3.1). As Yj−1 depends only on variables (x0, · · · , xj−1) and is tangent to C h+j (m), we have then in C h+j (m), we get that Dj + Dj−1 is a distribution of constant rank (h + l − j)m + 1 which ends the proof.
We come back to the proof of condition (iii) of part (2). According to the fact that (i), (ii) are already proved, assume that condition (iii) is true for all 0 ≤ i ≤ j < l.
From this assumption we already know that D h+j+1 is tangent to C R i V T j at each point of this manifold. So, according to Lemma 4.3, the set of points q = (x0, · · · , x h+l+1 ) ∈ C h+l+1 (m) such that D h+j+2 is tangent to C R i V T j is the set of points where D h+j+2 is tangent to C R i V T j . But, we have already shown that C R i V T j is defined by the system (31), for 0 ≤ j ≤ l. Moreover this system of equations depends only on variables }, this last condition is reduced to Dφi(Y h+j+2 ) = 0 for all equations φi = 0 in (31).
From the value of Y h−j+2 and the properties of theses equations, the previous condition is equivalent to: So finally, the set of points q = (x0, · · · , x h+l+1 ) ∈ C h+l+1 (m) such that D h+j+2 is tangent to C R i V T j is the set defined by the system (31). This ends the proof of condition (iii) of part (3).
The part (4) we have already noted that So part (4) is a simple geometrical interpretation of reduced system {φj = 0, j = 0, · · · l} of the set C R i V T l Remark 4. 6. Let be q ∈ C k (m) and assume that its projection q h+1 is vertical.
In [7] and [8], P. Mormul has constructed a coding system for labeling singularity classes of germs of special multi-flag which he called "Extended Kumpera-Ruiz classes" ( "EKR classes" in short). Mormul's codes are finite sequences in N. For each such a sequence (under some constraints), using Theorem 2.3, he build an associated polynomially germ at 0 ∈ R N , for N large enough, of (k + 1)-dimensional distributions. These germs are called "Extended Kumpera-Ruiz pseudo-normal forms ".
In the following construction, any external variables x will denoted by a capital letter X = x + c, when such a variable can be shifted by a constant c , not excluding, the value c = 0 and only by x when such a shift is excluded.
For each j ∈ {1, 2, · · · , m + 1}, we will define an operation denoted j producing new (m + 1)-distributions from the older ones. More precisely, given a distribution generated on a neighbourhood of 0 ∈ R s (y1, · · · , ys) the operation j, as operation at level l, gives rise to a new (m + 1)-distribution defined in the neighbourhood of 0 some space R s+m (y1, · · · , ys, x l 1 , · · · , x l m ), generated by the vector fields Z ′ 1 = x l 1 Z1 + · · · + x l j−1 Zj−1 + Zj + X l j Zj+1 + · · · + X l m Zm+1 and by Z It is important that these new local generators are written precisely in this order, thus producing a new system of generators (Z ′ 1 , · · · , Z ′ m+1 ).
Note, however, that possible constants in such an EKR representing a given germ D are not, in general, uniquely defined . On the other hand, for a given germ of distribution D, the sequence of operations j1j2 · · · jr associated to it, is unique and satisfy the rule of least upward jumps. Such a sequence is called a EKR class of multi-flags and the integer d = sup{j1, · · · , j k } − 1 is called the depth of this EKR class. 5. 2. Stratification of EKR classes of at most 1-depth by RVT codes. We denote by Σj 1 ···j k the set of points q ∈ C k (m) such that germ of distribution at q belongs to the EKR class j1, · · · , j k . Recall in a word ω, a sub-word of type R h or T l means a sequence of h (resp. l) consecutive letters R (resp. T ) if h > 0 (resp. l > 0, and no letter R (resp. T ) if h = 0 (resp. l = 0). For any EKR class of 1-depth we will denote by {i1, · · · , iν } the set {i such that ji = 2}.
The following result gives a complete description of EKR classes of at most 1-depth in terms of RVT classes. In particular, it contains the results announced in Theorem 2 and part (1) of Theorem 3: (1) Σ1···1 is of Cartan points. So Σ1···1 is an open dense set whose complementary is a subanalytic set of C k (m) of codimension 1 (2) for any EKR class j1, · · · , j k of 1-depth, the set Σj 1 ···j k is an analytic submanifold of C k (m) of codimension equal to ν (cardinal of index i such that ji = 2). Moreover, q belongs to Σj 1 ···j k if and only if the articulated arm M0, · · · , M k the segments [Mi−2, Mi−1] and [Mi−1, Mi] are orthogonal in Mi−1 for all i = i1, · · · , iν and any other pair of consecutive segments are not orthogonal (3) in the previous situation we have: (i) a point q belongs to Σj 1 ···j k if and only if its RVT code is a word such that the only letters V are at rank i1, · · · , iν.
(iii) Σj 1 ···j k is the union of all classes of type C R h 0 V T l 1 R h 1 ···V T lν R hν which satisfies the previous properties (ii) Remark 5. 1. The decomposition of Σj 1 ···j k given in (iii) into RVT classes is in agreement with the decomposition of such EKR classes for k = 3 described by Howard in appendix of [4]. So the description in (iii) can be seen as a generalization of Howard's result.
For the proof of this Theorem we need the following Lemma: We now look for property (ii). If ω is of type R h 0 V T l 1 R h 1 · · · V T lν R hν and each letter V are exactly at rank i1, · · · , iν , from part (i), C R h 0 V T l 1 R h 1 ···V T lν R hν must be contained in Σj 1 ···j k . On the other hand consider any word ω in RVT code such that each letters of rank r1, · · · rν is V and take any q ∈ Cω ⊂ C k (m). If i2 − i1 > 0, consider some q l under q with i1 < l < i2. Assume that q l is critical. According to the RVT code of q, the point q l must be tangency from Theorem 4.3 part (1), and, moreover, for i1 < l ′ ≤ l, the point q l ′ must be also tangency. Otherwise q l must be regular, and then, according to the RVT code of q and Theorem 4.3 part (1), each point q l ′ must be also regular for l ≤ l ′ < i2. It follows that the RVT code of qi 2 −1 is of type R h 0 V T l 1 R h 1 . By induction on 1 ≤ i ≤ ν, using the same arguments we get that ω must be of type R h 0 V T l 1 R h 1 · · · V T lν R hν . Finally, from the proof Theorem 4.2 part (2), it follows that the equations of C R h 0 V T l 1 R h 1 ···V T lν R hν is the union of ν systems of type (31) and so gives rise to ν + l1 + · · · + lν independent equations. So C R h 0 V T l 1 R h 1 ···V T lν R hν is an analytic submanifold of C k (m) of codimension ν + l1 + · · · + lν. On the other hand, the equations of Σj 1 ···j k are Ai λ −1 = 0 for λ = 1, · · · ν. These equations are exactly the first equations of the ν systems of type (31) which define C R h 0 V T l 1 R h 1 ···V T lν R hν . This ends the proof of the property (ii).
The property (iii) is a direct consequence of properties (i) and (ii). 5. 3. EKR classes of 1-depth , RVT codes and configurations of articulated arms.
We will now give a complete interpretation of the previous result in terms of configuration of an articulated arm such is announced in Theorem 3 part (2): Let be Σj 1 ···j k a EKR class of depth 1 and {i1, · · · , iν } the set {i such that ji = 2}. There exists a family for l = 0, · · · l λ and λ = 1, · · · , ν;  For the proof of this result, we need the notion of "induced articulated arm" : Given two integers r and s such that 0 ≤ r < s ≤ k, we can look for the motion of an induced articulated arm , which consists of segments of the original articulated arm which joints Mr to Ms included. We can then study the induced articulated arm (Mr, · · · , Ms). We put κ = s − r, and we denote by C rs (m) the image of C k (m) by the canonical projection ̺ rs from R m+1 In fact, we have: C rs (m) = {qrs = (xr, xr+1, · · · xs)}, if q = (x0, · · · , x k )} Taking in account subsection 3.1, let be Ers is the distribution on (R m+1 ) κ+1 generated by Zr, · · · , Zs−1, ∂ ∂x 1 s , · · · , ∂ ∂x m+1 s and let be Drs the distribution induced by Ers on C rs (m).
In terms of Notations 3.1, the mechanical system of the evolution of an induced arm (Mr, · · · , Ms), is a controlled system on R m+1 × (S m ) κ ≡ C κ (m) naturally associated to the distribution Drs.
Consider a word R h 0 V T l 1 R h 1 · · · V T lν R hν of k letters in RVT code and we associate to this word the sequences r0, · · · , rν and s0, · · · , sν defined by: s0 = h0 and r0 = 0; si = si−1 + hi + li + 1 = h0 + h1 + l1 + 1 + · · · + hi + li + 1 and ri = si−1 − 1 for i = 1, · · · , ν We have the following characterization: The configuration q ∈ C k (m) of an articulated arm (M0, · · · , M k ) belongs to the class C R h 0 V T l 1 R h 1 ···V T lν R hν if and only if the induced articulated arm associated to integer (ri, si) satisfies ̺ r i s i (q) belongs to the class C R h 0 ⊂ C r 0 s 0 (m) = C s 0 (m) for i = 0 and to the class C RV T h i R h i ⊂ C r i s i (m) for all i = 1, · · · ν Proof. For any q ∈ C k (m), as usual we denote by q l any point of C l (m) under q. Fix a configuration q ∈ C k (m) of the articulated arm (M0, · · · M k ). At first for i = 0, the induced articulated arm associated (r0, s0) has the induced configuration qs 0 . The RVT code of qs 0 is the h0 first letters of the RVT code of q. So, these h0 first letters are R h 0 if and only if the RVT code of qs 0 is R h 0 .
Assume that we have proved that the si first letters of the RVT code of q are R h 0 V T l 1 R h 1 · · · V T l i R h i if and only if RVT code of the configuration qr i s i = ̺ r i s i (q) of the associated induced articulated arm is R h 0 for i = 0 and RV T l i R h i for all 1 ≤ i ≤ µ − 1 < ν.
Proof of Theorem 5.2. According to Lemma 5.2 and Theorem 4.3 property (ii), to each induced articulated arm associated to a pair (ri, si) there exists a family of directions {K i 0 , · · · , K i κ i } of directions in R m+1 such that qr i s i belongs to the class C R V T l i R h i ⊂ C r i s i (m) if and only if in the configuration qr i s i we have : • each segment [M r i +1+l , M r i +2+l ] is orthogonal in M r i +1+l to K i l , for l = 0, · · · li ; • each pair of consecutive segments [M l−2 , M l−1 ] and [M l−1 , M l ] are not orthogonal in M l−1 for all ri + 1 + li < l ≤ ri + 1 + li + hi = si − 1.
On the other hand, each index i λ is equal to r λ + 2 for λ = 1, · · · , ν. So we get the announced results 5.4. EKR classes of 2-depth , RVT codes and configurations of articulated arms for k ≤ 4.
The combination of all possible RVT codes of depth 2 has an exponential growth relatively to the length k of special multi-flag. So, in this subsection we only describe the relations between EKR classes of 2-depth, RVT codes and configurations of articulated arms for k = 4. In fact this situation recovers the results of [4], [5] and [9].
At first for for k = 3 we have only Σ123 which is a EKR class of depth 2 and k = 4 we have 14 EKR classes (of depth 2) whose numerical code are (see for instance [9]) :  1111, 1112, 1121, 1122, 1123, 1211, 1212, 1213, 1221, 1222, 1223, 1231, 1232, 1233. So, for k ≤ 4, the other EKR classes are of depth 1. On the other, at the end of section 4.3, we have seen that for k = 3, we have only one RVT class of depth 2 (i. e RT0T01) but we have that we have 10 RVT classes for k = 4 All other RVT classes are of depth 1.
For the decomposition of EKR classes of depth 1 into RVT classes are of depth 1 can be found in Theorem 5.2 and the corresponding interpretation in terms of configurations of an articulated arm can be found in Theorem 5.2. So we have only to give such result for EKR classes of depth 2 previously enumerated.
For this purpose we need the following Proposition: Proposition 5.1. Let be Σj 1 ···j k a EKR class of depth 2. Consider an index j l = 3 for l ≥ 3 and denote by {i1, · · · iν } the set of index such that {i : ji = 2, 1 < i < l} (1) q ∈ C k (m) belongs to Σj 1 ···j k , (2) the projection q l = (x0, · · · , x l ) of q onto C l (m) satisfies the following equations : A l−1 (q l ) = 0 < x l − x l−1 , x l−1 − xi λ −2 >= 0 for some 1 ≤ λ ≤ ν For k = 4, we get easily the decomposition of EKR class of at most depth 2 into RVT classes as given in the following table and the proof is left to the reader. For k ≤ 3 the results are essentially particular cases of Theorem 5.2.
On the other hand, as in [11] for k = 1, in [4] for k ≥ 1, Castro and Montgomery look for the definition of RVT code in terms of germs curves at 0 ∈ R m+1 whose all directional derivatives at order k is a point of P k (m). They used the classification of curve singularities, under "Right-Left" equivalence (see Definition 3.11 in [4]), due to Arnold ([1]), to obtain beautiful results about classification of RVT classes in terms of Arnold's classification. Also let us mention [5] for complete results of classification of RVT classes in P 4 (2).
In this paper this aspect is not approached. However, it would be interesting to give an interpretation of a such approach in terms of germs of trajectories of the motion of an articulated arm.