Wisama Khalil
email: wisama.khalil@lan.ec-nantes.fr

Denis Creusot

SYMORO+: A SYSTEM FOR THE SYMBOLIC MODELLING OF ROBOTS

Keywords: Symbolic calculation, robots, modelling, identification, kinematic, dynamic, simulation. Robot Geometric Kinematic New Transformation matrix... Jacobian matrix... Open... Fast geometric model... Inverse Jacobian & determinant... Save I

This paper presents the software package SYMORO+ for the automatic symbolic modelling of robots. This package permits to generate the direct geometric model, the inverse geometric model, the direct kinematic model, the inverse kinematic model, the dynamic model, and the inertial parameters identification models.

The structure of the robots can be serial, tree structure or containing closed loops. The package runs on Sun stations and PC computers, it has been developed under MATHEMATICA and C language. In this paper we give an overview of the algorithms used in the different models, the computational cost of the dynamic models of the PUMA robot are given.

INTRODUCTION

Many works have been devoted to the automatic generation of the symbolic modelling of robots. Most of these works were interested in the generation of some models or more especially in the dynamic model only. SYMORO+ generates almost all the symbolic models needed in the simulation, control, identification, and design of robots. This package is the result of the research work of the robotics team of the "Laboratoire d'Automatique de Nantes", in the field of modelling, control and identification of robots.

The programs generating the different models are developed using MATHEMATICA [START_REF] Wolfram | Mathematica , a system for doing Mathematics[END_REF]. A graphic interface, written in language C, is developed such that the user can define his robot or the desired model using recent stations environment. The main interface page is seen in figure 1, on which we distinguish the parameters of the robot and the following main menus : Robot, Geometric, Kinematic, Dynamic, Identification, Optimizer.

The functions and models corresponding to these menus are given in table 1.

The paper is organized as follows: Section 2 gives the parameters needed to define a robot, then sections 3,…, 7 present the different models which can be generated by SYMORO+, an idea will be given to describe the algorithms which are used in generating the different models.

2-ROBOT DESCRIPTION

The definition of a robot is carried out using the menu Robot. The software package SYMORO+ can treat serial, tree structure or closed loop robots. The description of the geometry of the robot is carried out using the notations of Khalil and Kleinfinger [START_REF] Khalil | A new geometric notation for open and closed-loop robots[END_REF]. At first the number of moving links (N L ), the number of joints (N j ) and the type of structure (serial, tree structure or closed) must be given. The following three sets of parameters (figure 1) are then defined either numerically or symbolically :

The geometric parameters

These parameters define the kinematic of the structure, the type of joints and the location of the link frames with respect to its antecedent [START_REF] Khalil | A new geometric notation for open and closed-loop robots[END_REF]. The coordinate frame j is assigned fixed with respect to link j. The z j axis is along the axis of joint j, the x j axis is along the common perpendicular of z j and one of the following axis on the same link. The geometry of the robot is defined by the following parameters (for j=1,…,N f ):

-γ j , b j , α j , d j , θ j , r j defining frame j with respect to its antecedent frame a(j), figure 2, in serial robots γ j , b j are equal to zero. It is to be noted that the joint variable q j is equal to θ j if j is rotational and equal to r j for j translational, -σ j defining the type of joint j. σ j = 0 for j rotational, σ j = 1 for j translational, -the antecedent frames a(j), -µ j indicates if the joint j is motorized (active) or not (passive). In open loop robots (serial or tree structure robots) all the joints are supposed active. It is to be noted that the number of frames N f in serial or tree structure robot is equal to the number of moving links N L . In the case of closed loop robot we suppose each closed loop opened in one of its passive joint to construct an equivalent tree structure, then we add two frames on one of the links surrounding the opened joints. If the number of the opened joint is k the number of the corresponding additional frames will be k and k+B, these two frames are aligned but their antecedent frames are not the same (figure 3). Thus in the case of closed

loop robots N f = N L + 2B
where B = N j -N L = number of closed loops.

Dynamic parameters

These parameters are composed of inertial and friction parameters. For each link the following parameters have to be defined: [ XX j , XY j , XZ j , YY j ,YZ j , ZZ j , MX j , MY j , MZ j , Mj, Ia j ], and [ Fv j , Fsj]

where:

-(XX j , ...,ZZ j ) are the elements of the inertia matrix j J j , defining the inertia of link j, around the origin of frame j, -(MX j ,MY j , MZ j ) are the elements of j MS j defining the first moments of link j, -M j the mass of link j, -Ia j inertia of motor j referred to the joint side.

Fv j , Fs j are the viscous and static friction coefficients of joint j.

General parameters

The following parameters can also be defined.

-The location of the base of the robot with respect to a general fixed frame (matrix Z).

-joint velocities (QPj) and accelerations (QDPj).

-external forces of each link on the environment (FXj, FYj, FZj)

-external moments of each link on the environment (Cxj, CYj,CZj).

-The speed and the acceleration of the base of the robot.

-the acceleration of gravity: g = [G1 G2 G3 ] T .

3-THE GEOMETRIC MODELS

The following models can be obtained under menu Geometric.

3.1

The direct geometric model: The direct geometric model, gives the position and orientation of the end effector as function of the motorized joints variables. It can be obtained by the multiplication of the transformation matrices along the path between the base of the robot and the terminal link.

A frame j will be defined with respect to frame r, by the following (4x4) transformation matrix :

r T j =         r A j r P j 0 0 0 1 (1-a)
where * r A j defines the orientation of frame j with respect to frame r.

* r P j defines the position of the origin of frame j with respect to frame i.

The frame j coordinate will be defined with respect to frame i, with i =a(j), by the following (4x4) matrix :

i T j =        
Cγ j Cθ j -Sγ j Cα j Sθ j -Cγ j Sθ j -Sγ j Cα j Cθ j Sγ j Sα j d j Cγ j +r j Sγ j Sα j Sγ j Cθ j +Cγ j Cα j Sθ j -Sγ j Sθ j +Cγ j Cα j Cθ j -Cγ j Sα j d j Sγ j -r j Cγ j Sα j Sα j Sθ j Sα j Cθ j Cα j r j Cα j +b j

0 0 0 1 (1-b)
where : * S= sin, C= cos,

In the case of open loop robots the direct geometric model is defined by the transformation matrix of the terminal link referred to the base. In the case of closed loop robots the constraint equations defining the relation between the passive joint variables, in the path between the terminal link and the base, and the motorized joints outside this path can be obtained by solving the geometric constraint equations (section 3.4), in order to obtain the location of the end effector as function of motorized joints only.

Fast geometric model

This function gives the transformation matrix between two frames in customized form, using intermediate variables in order to minimize its calculation cost.

The inverse geometric model

The inverse geometric model is the closed form solution giving all the configurations of the robot corresponding to a given location of the end effector. In our software package three methods are used :

* The first is derived from the work presented in [START_REF] Pieper | The kinematics of manipulators under computer control[END_REF][START_REF] Bennis | Modèle géométrique inverse des robots à structure découplable: Application à la résolution des équations de contrainte des boucles fermées[END_REF]. This method gives the solution of six degrees of freedom robot provided that one of the following conditions is verified:

-the robot contains three translational joints.

-the robot has three rotational joints defining a spherical joint (their axes are intersecting).

* The second method is derived from the work given in [START_REF] Paul | Robot manipulators : mathematics, programming and control[END_REF][START_REF] Khalil | On the explicit derivation of the inverse geometric models of robots[END_REF]. This method can provide the solution of most of the current industrial robots.

* the third method is derived from the general method of Raghavan and Roth [START_REF] Raghavan | Kinematic analysis of the 6R manipulator of general geometry[END_REF][START_REF] Khalil | On the general solution of the inverse kinematics of sixdegrees-of-freedom manipulators[END_REF], where the solution of one variables is given as a polynomial equation of degree 16 at most, then the other variables can be obtained.

In the case of closed loop robots, the programs (Pieper, Paul, or general method) give the solution of the joints on the direct path between the base of the robot and the end effector. To get the values of the motorized joint positions outside this path the geometric constraint equations of the loops must be resolved, as seen in the following section.

j i z α i' L L d j k x i x k α k b j j L j γ x j j z θ j r j z i
z k z k+B z i x i x k+B x k z j a(k+B)=j a(k)=i L i L j . . .

Geometric constraint equations of loops

The direct and inverse geometric models of closed loop robots can be considered as the direct and inverse geometric robot of the serial path from the base to the terminal link plus the solution of the geometric constraint equations of the closed loops of the robot.

We make use of the method developed in [START_REF] Bennis | Modèle géométrique inverse des robots à structure découplable: Application à la résolution des équations de contrainte des boucles fermées[END_REF]. The solution is ensured analytically for each closed loop with less than five passive joints.The geometric constraint equations can be obtained by solving the transformation matrix along the loop (figure 3) which is equal to the identity matrix:

B T i ... jT k+B = I 4 (2)
with I 4 is the 4x4 identity matrix.

4-THE KINEMATIC MODELING

The direct kinematic model defines the velocity of the end effector as function of the joint velocities. It can be written as:

        r V n r ω ω ω ω n = r J n . q (3)
where: . q is the vector of joint velocities, r V n , r ω ω ω ω n define the translational and angular velocities of frame n referred to frame r, r J n is the Jacobian matrix of link n, referred to frame r.

The following models can be generated from the menu kinematic

Calculation of the Jacobian matrix

SYMORO+ calculates the Jacobian matrix using the method developed by Renaud [START_REF] Renaud | Calcul de la matrice jacobienne nécessaire à la commande coordonnée d'un manipulateur[END_REF] . It gives the Jacobian matrix as the product of three matrices, two of them are of full rank and the third contains simple terms such that :

r J n =         r A i 0 3 0 3 r A i         I 3 -i Lj,n 0 3 I 3 i J n,j (4)
with 0 3 and I 3 denotes the zero and identity (3x3) matrix respectively, the symbol ^ means the 3x3 matrix of the vector product; such that ax b= â b.

The vectoriel matrix J n,j is given as :

J n,j=       σ f a f + - σ f (a f xL f,j ) … σ n a n + - σ n (a n xL n,j ) - σ f a f … - σ n a n ( 5
)
where f is the first joint on the path between the base and link n, such that f=1 in the case of serial robots.

a j is the unit vector along the axis z j L i,j is the position vector connecting the origin of frame i to that of frame j σ j is equal to 1 if j is rotational, and equal to 0 if j is translational

Calculation of the velocities of the links

The translational and angular velocities of links can be calculated using relation (3), they can be calculated more efficiently, from the number of operations point of view, by the following recursive algorithm, for j=1,...,N L ,

j ω ω ω ω i = j A i i ω ω ω ω i (6a) j ω ω ω ω j = j ω ω ω ω i + - σ j q . j j a j (6b) j V j = j A i ( i V i + i ω ω ω ω i x i P j ) + σ j q . j j a j (6c)
where i= a(j) denotes the link antecedent to j, and j a j is the unit vector [0 0 1] T . The initial values 0 ω ω ω ω 0 and 0 V 0 are the velocities of the base.

Calculation of links accelerations

Differentiating equation (3), we get the translational and rotational acceleration of link n as :

      . V n . ω ω ω ω n = . J n . q + J n .. q (7)
The calculation of link accelerations using ( 7) is time consuming, it is more efficient to calculate the link accelerations by the following recursive equations, which can be obtained by differentiating equations ( 6):

j ω ω ω ω . j = j A i i ω ω ω ω . i + - σ j ( q .. j j a j + j ω ω ω ω i x q . j j a j ) ( 8 ) j U j = j ω ω ω ω . ^ j + j ω ω ω ω ^ j j ω ω ω ω ^ j (9) j V . j = j A i [ i V . i + i U i i P j ]+σ j [q .. j j a j +2 j ω ω ω ω i x q . j j a j ] (10)
where: i= a(j) , j ω ω ω ω i and j ω ω ω ω j are calculated using equations (6).

Calculation of . J . q of the links

In some applications (task space control), we need to calculate the vector . J . q . From equation [START_REF] Raghavan | Kinematic analysis of the 6R manipulator of general geometry[END_REF] it can be seen that . J . q can be obtained by the use of the recursive equations (8,..10) and by assuming q .. j equal zero [START_REF] Khalil | An efficient algorithm for the dynamic control of robots in the cartesian space[END_REF].

Kinematic constraint equations

This function gives the relation between the velocities of the passive joints as function of the velocities of the motorized joints. It can be calculated by equating the velocities on the terminal frame of each loop using the two sides of the loop. From figure 3, we get:

      V k ω ω ω ω k = k J k . q 1 = k+B J k+B . q 2 (11)
where . q 1 and . q 2 denote the velocities of the joints from the root of the loop to the opened joint, along each side of the loop.

As function of the type of the loop, spatial or planar, and the type of the opened joint, redundant rows in equation ( 11) can be automatically eliminated [START_REF] Zghaib | Génération Symbolique Automatique des Equations de la Dynamique des Systèmes mécaniques Complexes avec Contraintes Cinématiques[END_REF]. Doing this procedure on all the loops we obtain the kinematic constraint equation as :

      Wa Wp 0 Wac Wpc Wc         . q a . q p . q c = 0 (12)
where . q a , . q p , . q c represent the velocity of active (motorized), passive, and opened joints respectively.

from the first row of equation ( 12) we obtain :

q . p = -W -1 p W a q . a = W q . a (13)
The acceleration constraint equation can also be given by differentiating equation [START_REF] Khalil | Minimum operations and minimum parameters of the dynamic model of tree structure robots[END_REF] with respect to the time as:

        W a W p 0 W ac W pc Wc         q .. a q .. p q .. c +         ψ ψ ψ ψ Φ Φ Φ Φ = 0 (14)
In fact ψ ψ ψ ψ and Φ Φ Φ Φ are calculated using a recursive method similar to that calculating . J . q as given in section 4.4.

Inverse kinematic model

The inverse kinematic model gives . q as function of the link velocities V n , ω ω ω ω n . The solution implies to calculate the inverse of the Jacobian matrix, which is reduced to the inversion of i J j . The determinant is also provided to study the singular configurations of the robot.

5-THE DYNAMIC MODELING

The following models can be obtained in the menu dynamic.

Inverse dynamic model (I.D.M)

The inverse dynamic model gives the motor torques or forces as a function of the joint positions, velocities and accelerations. The calculation of this model represent the computed control law which decouples and linearizes the equation of motion of the robots. The model generated by our program has a reduced number of operations, such that it can be used for dynamic control applications, and this is thanks to the use the base inertial parameters (see section 6) and a customized Newton Euler algorithm which is linear in the inertial parameters [START_REF] Khalil | Minimum operations and minimum parameters of the dynamic model of tree structure robots[END_REF].

The I.D.M using Newton Euler algorithm is given as follows :

The following notations will be used: F j total forces on link j, M j total moments on link j around the origin o j. f j force on link j due to motor j and its antecedent link i, m j moment on link j due to motor j and the antecedent link i.

f e,j the force of link j on the environment, m e,j the moments of link j on the environment, Γ Γ Γ Γ motor torques or forces vector, g acceleration of gravity,

The algorithm consists of two recursive calculations : a-The forward calculation, for j=1,…,N L , is given as: j F j = M j j V . j + j U j j MS j (15) j M j = j J j j ω ω ω ω . j + j ω ω ω ω j x ( j J j j ω ω ω ω j ) + j MS j x j V . j [START_REF] Khalil | The use of the generalized links to determine the minimum inertial parameters of robots[END_REF] where j ω ω ω ω j , j ω ω ω ω . j , j V . j, j U j are calculated using equations. [START_REF] Khalil | On the explicit derivation of the inverse geometric models of robots[END_REF][START_REF] Khalil | On the general solution of the inverse kinematics of sixdegrees-of-freedom manipulators[END_REF][START_REF] Renaud | Calcul de la matrice jacobienne nécessaire à la commande coordonnée d'un manipulateur[END_REF][START_REF] Khalil | An efficient algorithm for the dynamic control of robots in the cartesian space[END_REF].

To take into account the gravity forces the gravity acceleration will be subtracted from the translational acceleration of the base such that :

0 V . 0 = 0 V . b -g , 0 ω ω ω ω . 0 = 0 ω ω ω ω . b , 0 ω ω ω ω 0 = 0 ω ω ω ω b
b-By studying the equilibrium of each link, the backward recursive calculation, for j=N L ,…,1, gives:

j f j = j F j + ∑ j f k + j f e,j (17)
i f j = i A j j f j (18) j m j = j M j + ∑ j A k k m k + ∑ j P k x j f k + j m e,j (19)
projecting i f j or j m j on the joint axis and taking into account the inertia and frictions of the actuators, we get the motor torque as :

Γ j = (σ j j f j + σ j j n j ) T j a j + Ia j q .. j + F vj q . j + F sj sign (q . j ) [START_REF] An | Estimation of inertial parameters of rigid body links of manipulators[END_REF] the sum sign in equations ( 17) and ( 19) is on k, which denote all the links successors of j such that a(k)=j. For serial robot, the maximum number of operations of this algorithm is equal to 92N L -127 multiplications and 81N L -117 additions, for N L = 6 these relations give 425 multiplications and 369 additions. This number will be further reduced for real robots, where many of the geometric parameters of distances are zero and those of angles are equal to kπ/2, with k integer, and in particularly if the links are supposed symmetric. Table 2 gives the number of operations for the 6 d.o.f PUMA robot (using base inertial parameters as defined in section 6.4) .

Dynamic model of Closed loop robots

The joint positions of the equivalent tree structure of a closed loop robot can be written as :

q tr =      
respectively. Γ Γ Γ Γ tr can be obtained using Newton-Euler algorithm as given in section 5.1, while G can be obtained from the resolution of kinematic constraint equations.

Direct dynamic model (D.D.M)

Initialization : for j = 1,..., N L : j J + j = j J j , j MS + j = j MS j , M + j = M j for j =N L ,..., 2 and a(j) ≠ 0 : a(j) MS j = a(j) A j j MS j ( 27) a(j) J + a(j) = a(j) J + a(j) + a(j) A j j J j j A a(j) -[ a(j) P j a(j) MS j + ( a(j) P j a(j) MS j ) T ] + a(j) P j a(j) PT j M j (28) a(j) MS + a(j) = a(j) MS + a(j) + a(j) A j j MS j + a(j) P j M j ( 29)

M + a(j) = M + a(j) + M j (30)
Using the conditions [START_REF] Khalil | An efficient algorithm for the calculation of the filtered dynamic model of robots[END_REF] in Newton Euler algorithm, the recursive forward calculation will be reduced to:

j F j = σ j [ 0 0 M j + ] T + - σ j [ -MY j + MX j + 0] T (31)
j M j = - σ j [ XZ j + YZ j + ZZ j + ] T + σ j [ MY j + -MX j + 0 ] T (32)
The recursive backward calculation of the I.D.M will be reduced to j f j = j F j (33)

j m j =σ j [ MY j + -MX j + 0 ] T + - σ j [ XZ j + YZ j + ZZ j + ] T (34)
A j,j = σ j M j + + σ j ZZ j + + Ia j (35) where: A j,j is the (j,j) element of the matrix A

To calculate the other elements of the column j (A a(j),j ,… , A sj(0),j ), where sj(0) indicates the succeeding link of the base on the path between link 0 and link j, we continue the backward calculations which gives the following relations for k=j, a(j), a(a(j)),…,sj(0) :

a(k) f a(k) = a(k) A k k f k (36) a(k) m a(k) = a(k) A k k m k + a(k) P k x a(k) f a(k) (37) A a(k),j = a(k) a T a(k) (σ j a(k) f a(k) + - σ j a(k) m (ak) ) (38)
A i,j = 0 if link i is not on the path between link j and link 0.

Table 2, gives the number of operations for calculating the inertia matrix for the PUMA robot (using base inertial parameters).

Direct dynamic model without calculating the inertia matrix

An algorithm derived from the work of [START_REF] Featherstone | The calculation of robot dynamics using articulated body inertias[END_REF][START_REF] Brandl | A very efficient algorithm for the simulation of robots and multibody systems without inversion of the mass matrix[END_REF] is programmed in customized form to get directly the joint accelerations as function of joint positions, velocities and torques without calculating nor inverting the inertia matrix. This algorithm can be used only for open loop robots. Table 2 gives the number of operations for calculating the D.D.M for the PUMA robot (using base inertial parameters) .

Dynamic model using Lagrange equation

This function calculates the elements of the matrices A, B, and Q of the dynamic model defined in equation [START_REF] Middleton | Adaptive computed torque control for rigid link manipulators[END_REF]. Two methods has been developed to get the elements of B : the first based on differentiating the inertia matrix of the robot obtained in section 5.3; the other method uses the algorithm developed in [START_REF] Renaud | Iterative analytical computation of the dynamic of a robot manipulator[END_REF]. Table 2 gives the number of operations for calculating the coefficients of the matrices A and B for the PUMA robot (using base inertial parameters).

6-THE IDENTIFICATION MODEL

In order to use the dynamic model in simulation or control we need to know the values of the dynamic (inertial and friction) parameters. The most appropriate method to evaluate these parameters is the use of the identification techniques. Many identification models which are linear in the dynamic parameters are proposed:

the dynamic model, the filtered dynamic model, and the energy model. All these models are generated in SYMORO+ as given in the following sections.

The dynamic identification model:

The dynamic model can be used to identify the inertial parameters [START_REF] An | Estimation of inertial parameters of rigid body links of manipulators[END_REF][START_REF] Khosla | Parameter identification of robot dynamics[END_REF][START_REF] Gautier | Identification of the dynamic parameters of a closed loop robot[END_REF]. Since the dynamic model is linear in the dynamic parameters, as defined in section 2.2, it can be written as follows:

Γ Γ Γ Γ = D(q,q . , .. q ) K ( 39
)
where K is the (N p x1)vector of dynamic parameters.

D is (N L xNp) matrix. N P is the number of inertial and friction parameters.

To identify K a sufficient number of equations can be obtained by calculating relation (39) at different times. The least squares solution is generally used in the solution SYMORO+ gives the symbolic expressions of the elements of the matrix D using customized symbolic calculation. The calculation of the column j of the matrix D is obtained from the Newton Euler algorithm of the inverse dynamic model, section 5.1, with the assumption that the the j th inertial parameters, denoted as K j, is equal to one while all the other inertial parameters are equal to zero. It is to be noted that the calculation of j ω ω ω ω j , j ω ω ω ω . j , j V . j, j U j for j = 1,...,N L is calculated only once for all the columns.

The filtered dynamic identification model

The filtered dynamic model is proposed in [START_REF] Middleton | Adaptive computed torque control for rigid link manipulators[END_REF][START_REF] Hsu | Adaptive identification and control for manipulators without using joint accelerations[END_REF][START_REF] De | Parameters identification of robots manipulators via sequential hybrid estimation algorithms[END_REF] in order to get a model which is not function of the joint accelerations. We have proposed and programmed a new method to get this model in SYMORO+ [START_REF] Khalil | An efficient algorithm for the calculation of the filtered dynamic model of robots[END_REF]. The main idea of the proposed algorithm is derived from the Lagrangian equation which is given as:

Γ Γ Γ Γ = d dt ∂L ∂q . - ∂L ∂q (40) with :
L the Lagrangien of the system, equal to E -U, E the kinetic energy of the system, U the potential energy of the system.

Equation (40), can be written as :

Γ Γ Γ Γ = d D 1 (q, q . ) K dt + D 2 (q, q . ) K (41)
where

D 1 (q, q . ) K = ∂ E ∂ q . ( 42
)
and

D 2 (q, q . ) K = - ∂E ∂q + ∂U ∂q (43)
Applying a numerical filter, denoted F, of second order or higher, on the two sides of equation (41) gives :

Γ Γ Γ Γ f = D 1df (q, q . ) K + D 2f (q, q . ) K (44)
where the subscript "f" means the application of the filter F, and "df" means the application of a filter F ' which associate to the filter F a derivative action. For instance in the case of using second order filter F and F ' could be taken as: :

F(s) = a 2 (s+a) 2 (45) F '(s) = a 2 s (s+a) 2 (46)
SYMORO+ gives the symbolic expressions of the elements of the matrices D 1 and D 2 using customized symbolic calculation using the algorithm presented in [START_REF] Khalil | An efficient algorithm for the calculation of the filtered dynamic model of robots[END_REF].

The energy identification model

Using the dynamic model in the identification need to estimate or measure the joint accelerations. To overcome this difficulty a model based on the energy theorem has been proposed [START_REF] Gautier | On the identification of the inertial parameters of robots[END_REF].

To simplify the writing we assume that the friction is neglected. From the energy theorem we get:

⌡ ⌠ t 1 t 2 . q T Γ Γ Γ Γ dt = H(t 2 ) -H(t 1 ) = ∆ ∆ ∆ ∆h(q,q . ) K (47)
where : H(t i ) is the total energy (kinetic and potential) at time t i , it is linear in the inertial parameters and can be written as :

H = h(q,q . ) K (48)
h is a row matrix function of q and q .

. Since the kinetic and potential energies of link j can be written as :

E j = 1 2 [ j ω ω ω ω j T j J j j ω ω ω ω j +M j j V j T j V j +2 j MS j T ( j V j x j ω ω ω ω j )] (49)
U j = -0 g T ( 0 P j + 0 A j j MS j )

where, 0 g is the acceleration of gravity referred to frame zero. 0 P j is the position of the origin of frame j w.r.t. frame zero. 0 A j is the 3x3 transformation matrix giving the orientation of frame j w.r.t. frame 0, such that

: 0 A j = [ 0 s j 0 n j 0 a j ] (51)
with s j , n j , a j are the unit vectors along the x, y, and z axis respectively.

The coefficients of the inertial parameters in the vector h can be given as follows :

h XXj = 1 2 ω 1,j ω 1,j h XYj = ω 1,j ω 2,j h XZj = ω 1,j ω 3,j h YYj = 1 2 ω 2,j ω 2,j h YZj = ω 2,j ω 3,j h ZZj = 1 2 ω 3,j ω 3,j

h MXj = ω 3,j V 2,j -ω 2,j V 3,j -0 g T 0 s j h MYj = ω 1,j V 3,j -ω 3,j V 1,j -0 g T 0 n j h MZj = ω 2,j V 1,j -ω 1,j V 2,j -0 g T 0 a j h Mj = 1 2 j V j T j V j -0 g T 0 P j where, j ω ω ω ω j and j V j are represented by the vectors : j ω ω ω ω j = [ω 1,j ω 2,j ω 3,j ] T (53)

j V j = [V 1,j V 2,j V 3,j ] T (54)
To identify K a sufficient number of equations can be obtained by calculating relation (47

)
between different intervals of time. The least squares solution is generally used in this identification. SYMORO+ calculates the elements of the matrix h using customized symbolic calculation.

The base inertial parameters

The base inertial parameters are defined as the minimum parameters which can be used to get the dynamic model. They represent the set of parameters which can be identified using the dynamic or energy model, thus its determination is essential for the identification of the inertial parameters of robots. They constitute also the parameters to be adapted during an adaptive dynamic control strategy. The use of the base parameters in all the dynamic models presented in section (5) leads to reduce their computation complexity [START_REF] Khalil | Minimum operations and minimum parameters of the dynamic model of tree structure robots[END_REF].

These parameters can be obtained from the classical inertial parameters by eliminating those which have no effect on the dynamic model and by regrouping some others. In [28,...,33] we have presented symbolic and numerical methods to calculate these parameters for serial, tree structured, or closed loop robots. These algorithms have been programmed in SYMORO+.

The program generates a new file of inertial parameters which can be used directly in the Dynamic or Identification menus.

OPTIMIZATION OF THE MODELS

The models obtained under customized symbolic form can be optimized from the number of operations point of view using the optimizer menu. The optimized output file can be given in either of the following format: Fortran, Mathematica, C, Maple, Matlab, such that the generated model can be used directly by these systems.

CONCLUSION

Figure 1 :

 1 Figure 1 : Main page of SYMORO+

Figure 2 :

 2 Figure 2: Definition of link frames.

Figure 3 :

 3 Figure 3: Frames associated with the opened joint.

Table 1 :

 1 Functions and models of the menus of SYMORO+

q a q p[START_REF] Khosla | Parameter identification of robot dynamics[END_REF] where q a , q p represent the positions of active and passive joints respectively.The dynamic model of a closed-loop structured robot can be obtained as a function of the corresponding tree structure dynamic model using the following relation[START_REF] Kleinfinger | Dynamic modelling of closed-chain robots[END_REF][START_REF] Dombre | Modélisation et commande des robots[END_REF]:Γ Γ Γ Γ m = G T Γ Γ Γ Γ tr = = Γ Γ Γ Γ a + W T Γ Γ Γ Γp(22)where: Γ Γ Γ Γ m is the (mx1) vector of the torques of motorized joints, G =(∂q tr /∂q a ) is the Jacobian matrix of q tr with respect to q a , W is defined in equation (13). Γ Γ Γ Γ tr is the (N L x1) vector of the joint torques (or forces) of the corresponding tree structure, which is supposed as:Γ Γ Γ Γ tr =       Γ Γ Γ Γ a Γ Γ Γ Γ pwhere Γ Γ Γ Γ a and Γ Γ Γ Γ p denote the vectors of the torques corresponding to active and passive joints

This paper presents the software package of symbolic modelling of robots SYMORO+.

This package generates all the models needed in the simulation, identification, control and design of robots. The algorithms used in deriving these models have been presented, they have been programmed using MATHEMATICA system. A friendly user interface is developed in C. The paper gives also an overview of the best algorithms which can be used in modelling the robots. Current extension is concerned with the modelling of flexible robots [34] structure and walking legged robots.

ACKNOWLEDGMENT

This work has been supported by the "CRITT Productique de Pays de Loire", I would like to thank also all the members of the the robotics team of the "Laboratoire d'Automatique de Nantes" who have participated in this project, either by developing new algorithms in the modelling of robots or by testing this package. Particular thanks to C.Chevallereau, F.Bennis, Ph. Seigle, P.Restrepo, and D.Murareci who have been involved in programming some modules of SYMORO+.

The direct dynamic model is used to simulate the robot dynamics, it permits to calculate the joint accelerations as a function of the input torques or forces. Since the general form of the inverse dynamic model can be written as:

where :

A is the inertia matrix of the robot, B(q, q . )q . gives the Coriolis and centrifugal forces, Q is the gravity forces.

The direct dynamic model can be obtained as follows:

..

where:

Thus the calculation of A, and H represent the direct dynamic model.

Calculation of the vector H

The vector H can be calculated by the Newton-Euler method, given in section 5.1 by noting that, see equation [START_REF] Middleton | Adaptive computed torque control for rigid link manipulators[END_REF], it is equal to Γ Γ Γ Γ under the condition that .. q = 0. To calculate H efficiently a separate function is given to calculate it using a customized symbolic calculation.

Table 2 gives the number of operations for calculating H for the PUMA robot (using base inertial parameters).

Calculation of the inertia matrix A

The calculation of the matrix A is given by an algorithm similar to the algorithm of Newton-Euler [START_REF] Walker | Efficient dynamic computer simulation of robotics mechanism[END_REF]. By using the fact that the j th column of A is equal to Γ Γ Γ Γ if : q . = 0, g = 0, .. q = e j ( 26) with e j is the unit (N L x 1) vector, whose elements are equal to zero except the j th component which is equal to 1.

To increase the efficiency of this method, we make use of the symmetrical property of A and we use the generalized link inertial parameters. The generalized link is defined as the fictitious link composed of link j and all the succeeding links articulated on it [START_REF] Khalil | The use of the generalized links to determine the minimum inertial parameters of robots[END_REF]. The algorithm is thus composed of two steps, the calculation of the inertial parameters of the generalized links and a customized Newton-Euler algorithm which take into account the conditions of equation [START_REF] Khalil | An efficient algorithm for the calculation of the filtered dynamic model of robots[END_REF].

The inertial parameters of the generalized link j are denoted as: M + j , j J + j , j MS + j , they will be calculated by the following algorithm :


Jean-Baptiste Mallye
email: jean-baptiste.mallye@u-bordeaux.fr

Salvador Bailon

Corentin Bochaton

Monica Gala

Nathalie Serrand

Arnaud Lenoble

Blanchard Cave 2: A historical period Audubon's shearwater (Puffinus lherminieri) nesting site in Marie-Galante (Guadeloupe islands, FWI)

Keywords: Audubon's shearwater, Palaeobiogeography, Taphonomy, Human-induced impact, Lesser Antilles, Caribbean

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Oceanic islands are known to be fragile environments where indigenous biotas are strongly vulnerable to human-induced modifications [START_REF] Van Der Geer | Evolution of Island Mammals: Adaptation and Extinction of Placental Mammals on Islands[END_REF]. This remark is especially valid for Caribbean islands which are home to numerous wild animal species threatened with extinction or which have already disappeared [START_REF] Myers | Biodiversity hotspots for conservation priorities[END_REF]. This area presents the worldwide highest extinction rate in the world of its endemic non-flying mammals, greater than 95% following [START_REF] Dávalos | West Indian mammals: the old, the new, and the recently extinct[END_REF]. Indigenous Squamates and bats also have very high extinction rates depending of the island (e. g. 60% to 70% on Marie-Galante Island: - [START_REF] Bailon | New data on Pleistocene and Holocene herpetofauna of Marie-Galante (Blanchard Cave, Guadeloupe Islands, French West Indies): insular faunal turnover and human impact[END_REF][START_REF] Stoetzel | Late quaternary changes in bat palaeobiodiversity and palaeobiogeography under climatic and anthropogenic pressure: new insights from Marie-Galante, lesser Antilles[END_REF]. Concerning birds, many extinct species have been described [START_REF] Turvey | Holocene Extinctions[END_REF] and the description of the past Caribbean avian diversity is still ongoing (e.g. [START_REF] Gala | Evidence of the former existence of an endemic macaw in Guadeloupe, lesser Antilles[END_REF][START_REF] Takano | A new species of woodcock (Aves: Scolopacidae: Scolopax) from Hispaniola, West Indies[END_REF][START_REF] Steadman | A new extinct species of snipe (Aves: Scolopacidae: Gallinago) from the West Indies[END_REF]. During the last decade several studies have highlighted the preponderant role of Man in these numerous extinction events [START_REF] Steadman | Asynchronous extinction of late quaternary sloths on continents and islands[END_REF][START_REF] Bailon | New data on Pleistocene and Holocene herpetofauna of Marie-Galante (Blanchard Cave, Guadeloupe Islands, French West Indies): insular faunal turnover and human impact[END_REF][START_REF] Soto-Centeno | Fossils reject climate change as the cause of extinction of Caribbean bats[END_REF][START_REF] Stoetzel | Late quaternary changes in bat palaeobiodiversity and palaeobiogeography under climatic and anthropogenic pressure: new insights from Marie-Galante, lesser Antilles[END_REF] and the minor role of natural modifications of the environment in these processes. However, much work has still to be done to determine the exact timing of extinctions and the exact relation between extinctions and human behaviours or historical event.

Amongst extinct species, marine bird species have seen an important reduction in their numbers [START_REF] Lowrie | Seabird breeding atlas of the Lesser Antilles[END_REF]. The Audubon's shearwater (Puffinus lherminieri), a small, land-nesting bird, is currently considered one of the most threatened, particularly due to the introduction of mammal species, such as the black rat, which feed on its eggs and hatchlings. In fact, these rodents have been shown to heavily impact marine bird colonies [START_REF] Atkinson | The spread of commensal species of Rattus to oceanic islands and their effects on island Avifaunas[END_REF][START_REF] Towns | Have the harmful effects of introduced rats on islands been exaggerated?[END_REF][START_REF] Jones | Severity of the effects of invasive rats on seabirds: a global review[END_REF]. Indeed, the extent to which black rat predation can impact breeding colonies of Audubon's shearwater was recently revealed following an attempt to eradicate the rodent from the islets of Saint-Anne, Martinique [START_REF] Pascal | Conséquences sur l'avifaune nicheuse de la réserve naturelle des îlets de Sainte-Anne (Martinique) de la récente invasion du rat noir (Rattus rattus), établies à l'issue d'une tentative d'éradication[END_REF].

Today, Audubon's shearwater nesting grounds are for the most part located on small coastal islets (Birdlife International, 2014;[START_REF] Lowrie | Seabird breeding atlas of the Lesser Antilles[END_REF][START_REF] Bright | Seabird surveys on Dog Island, Anguilla, following eradication of black rats find a globally important population of red-billed tropicbirds (Phaethon aethereus)[END_REF]. However, the discovery of Audubon's shearwater bones in archaeological or paleontological contexts on Caribbean islands where this bird is no longer present, such as Mona, Anguilla, Antigua and Barbuda [START_REF] Wing | Vertebrate remains from Indian sites on Antigua, West Indies[END_REF][START_REF] Pregill | Late quaternary vertebrate faunas of the lesser Antilles: historical components of Carribbean biogeography[END_REF][START_REF] Nieves-Rivera | Bones of Puffinus Lherminieri lesson (Aves: Procellaridae) and two other vertebrates from Cueva Del Agua, Mona Island, Puerto Rico (West Indies)[END_REF], suggests a reduction in nesting areas as a consequence of European colonisation, especially due to the intentional or accidental introduction of exogenous mammals such as the rat or mongoose. This, however, does not exclude the possibility of an earlier human impact.

In fact, the human exploitation of nesting ground marine birds is described as having almost immediately followed the settling of the first European colonists in the region in the early 17th century.

For example, historical accounts mention that sterns on the islets of Les Saintes were heavily exploited by Europeans (Breton, 1647: 32) and that marine birds formed part of the subsistence strategies of indigenous populations (Du Tertre, 1667: 274-275;Breton, 1665: 164). However, the use of marine birds is likely to date to considerably earlier. For instance, the disappearance of Black-capped petrel from Martinique was advanced by [START_REF] Wetmore | A record for the black-capped petrel, Pterodroma Hasitata, in Martinique[END_REF] on the basis of the discovery of a bone of this species in pre-Columbian middens of the Ceramic age period (ca. -300 BCE to 1500 CE). A much earlier human impact on native bird populations is even possible. Indeed, excavations at a preceramic site on Dog Island (Anguilla), one of the most important nesting grounds in the Caribbean [START_REF] Hodge | Anguilla[END_REF]), led Crock (2005) to suggest that human predation on birds dates to the earliest occupation of the archipelago, some 5000 y BP.

The question is even more pertinent for the Aubudon's shearwater, one of the most frequent marine birds found in Amerindian middens [START_REF] Grouard | Subsistance, systèmes techniques et gestion territoriale en milieu insulaire antillais précolombien : exploitation des vertébrés et des crustacés aux époques Saladoïdes et Troumassoïdes de Guadeloupe[END_REF][START_REF] Wing | Native American Use of Animals in the Caribbean[END_REF], several of which amongst the oldest known in the archipelago [START_REF] Hofman | Archipelagic resource procurement and mobility in the northern lesser Antilles: the view from a 3,000-year-old tropical forest campsite on Saba[END_REF].

In addition to Amerindian middens, Aubudon's shearwater remains have also been documented from caves and rock shelters (e.g. [START_REF] Pregill | Late quaternary vertebrate faunas of the lesser Antilles: historical components of Carribbean biogeography[END_REF][START_REF] Morgan | Late Quaternary fossil vertebrates from the Cayman Islands[END_REF][START_REF] Nieves-Rivera | Bones of Puffinus Lherminieri lesson (Aves: Procellaridae) and two other vertebrates from Cueva Del Agua, Mona Island, Puerto Rico (West Indies)[END_REF].

Nevertheless, these sites often lack chronological resolution and the agents responsible for the deposits (human and/or animal predation or nesting sites) more often than not remain difficult to determine. As a result, the historical biogeography of this species proves difficult to document.

Numerous studies have however provided criteria for determining the agent responsible for the accumulation of bird bones in fossil contexts. While the identification of butchery marks remains the most reliable evidence for human exploitation [START_REF] Laroulandie | Taphonomie et archéozoologie des Oiseaux en grotte : applications aux sites paléolithiques du Bois-Ragot (Vienne), de Combe Saunière (Dordogne) et de La Vache (Ariège)[END_REF][START_REF] Laroulandie | Les traces liées à la boucherie, à la cuisson et à la consommation d'oiseaux: apport de l'expérimentation[END_REF][START_REF] Laroulandie | Anthropogenic Versus Non-anthropogenic Bird Bone Assemblages: New Criteria for their Distinction[END_REF][START_REF] Vigne | Contribution archéozoologique à l'histoire récente des oiseaux marins de l'île Lavezzi (Corse -XIVe-XXe Siècles)[END_REF], the relative proportions of certain anatomical parts can provide arguments concerning the origin of avian material in both archaeological and natural contexts [START_REF] Baales | Accumulations of bones of Lagopus in Late Pleistocene sediments. Are they caused by man or animals?[END_REF][START_REF] Bovy | Differential avian skeletal part distribution: explaining the abundance of wings[END_REF][START_REF] Bochenski | Owls, Diurnal Raptors and Humans: Signatures on Avian Bones[END_REF][START_REF] Ericson | Interpretations of archaeological bird remains: a taphonomic approach[END_REF][START_REF] Livingston | The Taphonomic interpretation of avian skeletal part frequencies[END_REF]. Besides, in a few rare cases, it has been possible to demonstrate that bird bone accumulations result from the use of caves as nesting sites, where numerous complete remains of both young and adult individuals occur in the context of a near total absence of evidence for either animal or human predation [START_REF] Laroulandie | Taphonomie et archéozoologie des Oiseaux en grotte : applications aux sites paléolithiques du Bois-Ragot (Vienne), de Combe Saunière (Dordogne) et de La Vache (Ariège)[END_REF][START_REF] Laroulandie | Alpine Chough Pyrrhocorax Graculus from Pleistocene Sites between Pyrenees and Alps: Natural Versus Cultural Assemblages[END_REF][START_REF] Mourer-Chauviré | Les oiseaux du Pléistocène moyen et supérieur de France[END_REF][START_REF] Mourer-Chauviré | Les oiseaux dans les habitats paléolithiques : gibier des hommes ou proies des rapaces?[END_REF]).

Here we discuss evidence from a site in Guadeloupe, Blanchard 2, where historic-period Audubon's shearwater remains dominate the vertebrate assemblage. Sedimentological data, a direct date on faunal material, and an analysis of mollusc, crustacean and vertebrate remains combined with a taphonomic approach indicates the site to have been a nesting ground during the island's colonial period. This taphonomic approach also allows the causes leading to the abandonment of the site as a nesting ground to be discussed, and by extension, investigate hypotheses concerning the dynamics of Audubon's shearwater nesting as a response to the region's prehistoric and historic settlement.

Site background

Blanchard 2 (15°52′55.82″N, 61°14′2.27″W, 10 m NGF) is a small crevice cave formed in the rhodolith limestone of Pliocene age [START_REF] Lenoble | Formes et remplissages du karst littoral guadeloupéen[END_REF]. The cave opens onto the Capesterre cliff, not far from Blanchard 1. The site lies a few metres above the Petite Anse coastal plain in a dry deciduous forest on the southern part of Marie-Galante (Fig. 1) [START_REF] Rousteau | Carte Écologique de La Guadeloupe[END_REF]. Discovered in 2007, vertebrate and malacological material, together with a pierced, lozenge-shaped conch shell button and an unworked conch shell identified on the surface near the entrance alongside a flint flake recovered further within the cave suggested an Amerindian occupation. Moreover, the site is located in an area rich in sites of both cultural and natural importance. Several dozen metres to the west on the same cliff face can be found the rock shelters of Cadet. Cadet 3 rock shelter contains Late Glacial to Holocene fossil deposits [START_REF] Stouvenot | L'abri sous roche Cadet 3 (Marie-Galante): un gisement à accumulations de faune et à vestiges archéologiques[END_REF], and Cadet 2 rock sheleter, an Amerindian funerary cave also contains natural deposits and documents the islands final Pleistocene vertebrates [START_REF] Courtaud | La cavité sépulcrale de Cadet 2 (Capesterre-de-Marie-Galante, Guadeloupe) et la question du cannibalisme[END_REF][START_REF] Bochaton | Fossil and subfossil herpetofauna from Cadet 2 Cave (Marie-Galante, Guadeloupe Islands, F.W.I.): evolution of an insular herpetofauna since the Late Pleistocene[END_REF]. A test pit practised under the porch of the nearby Blanchard 1 site also produced an Amerindian burial (Stouvenot, 2005;[START_REF] Grouard | Fréquentation amérindienne des cavités des Petites Antilles[END_REF], while the zone deeper within the cave contains substantial Pleistocene fossil deposits [START_REF] Lenoble | Formes et remplissages du karst littoral guadeloupéen[END_REF][START_REF] Goetz | Des fouilles dans la grotte Blanchard, près de Capesterrede-Marie-Galante[END_REF][START_REF] Bailon | New data on Pleistocene and Holocene herpetofauna of Marie-Galante (Blanchard Cave, Guadeloupe Islands, French West Indies): insular faunal turnover and human impact[END_REF][START_REF] Stoetzel | Late quaternary changes in bat palaeobiodiversity and palaeobiogeography under climatic and anthropogenic pressure: new insights from Marie-Galante, lesser Antilles[END_REF]. Amerindian settlements are also known from the vicinity of the cave, including the Saladoid and Troumassoid site of Tourlourous [START_REF] Colas | Direction des Affaires Culturelles de Guadeloupe[END_REF] located a little more than a kilometre to the northeast on the same plain, and the Troumassoid site of Petite Anse, a few hundred metres from Blanchard 2 on the littoral belt of the Anse [START_REF] Lenoble | Nettoyage du fossé de Petite Anse (Capesterre-de-Marie-Galante, Guadeloupe)[END_REF][START_REF] Casagrande | Direction des Affaires Culturelles de Guadeloupe[END_REF]. Blanchard 2 comprises a main corridor that follows the cliff line and from which several smaller, lower chambers can be accessed (Fig. 2). This configuration is typical of crevice caves that form following slope failure [START_REF] Halliday | Crevice caves[END_REF]. The cave has two entrances, the first, however, is impractical due to rock fall. The more accessible eastern entrance, although low (~1 m) is 8 m wide, allowing access to a slope leading to an 8 m long and 2 to 2.5 m wide corridor filled with rock fall. This north-south corridor runs approximately 15 m and cuts several pre-existing cavities that are too low to enter (0.3 to 0.5 m). It is likely that these smaller, interconnected chambers continue for several metres in the rock mass.

Material and methods

Following a comprehensive topographic survey of the cave's surface, a half-square metre test pit (test pit 1) was excavated towards the front of the central chamber in the northern section of the cave (Fig. 2). Excavations consisted of eleven 5-cm-thick spits that followed the geometry of the deposit.

The excavated sediment was dry-sieved using a 2.8 mm mesh in order to recover all osseous materials. Additionally, bones were collected from the surface of the cave's eastern entrance. The stratigraphic analysis combined pedological (colour, texture and structure) and sedimentological criteria (bedding, clast type and orientation) complemented by a consideration of the geometry of the stratigraphic units and the nature of the contact between layers. A second 0.5 m by 0.5 m test pit (test pit 2) was dug to no deeper than 0.2 m on the slope overlying the site in order to control for the possible introduction of faunal material into the site.

Mollusc and crustacean remains were identified at the Institut national de recherches archéologiques préventives (Inrap) of Guadeloupe by S. Serrand. Vertebrate remains were determined to taxa with reference to the Comparative Anatomy collections held by the Muséum national d'Histoire naturelle in Paris (MNHN) and the PACEA laboratory at the University of Bordeaux. A fair portion of the bird remains could not be identified due to their heavily fragmented state, the presence of young individuals presenting no clear diagnostic traits and numerous problems in identifying bones of small Passeriformes. This non-identifiable material was therefore grouped in four size classes: small passerines (class 1, e.g. Loxigilla noctis), specimens similar in size to pigeons (class 2, e.g. Zenaida aurita) or Audubon's shearwater (class 3, e.g. Puffinus lherminieri) and those equal to or larger than herons (class 4, Nyctanassa violacea). All of the vertebrate remains were counted by spit (number of identified specimen, NISP) in order to avoid overestimating the minimum number of individuals (MNI) [START_REF] O'connor | The Archaeology of Animal Bones[END_REF]. On the other hand, the MNI could be estimated for the less-fragmented mollusc and crustacean remains.

All faunal material was examined under a binocular loupe at between ×10 and ×40 magnification in order to identify mechanical, biological or chemical surface alterations. Age of death for the birds was estimated as a function of the degree of long bone fusion and bone porosity [START_REF] Lefèvre | Les modifications post-mortem chez les oiseaux : l'exemple de l'avifaune holocène de Patagonie australe: micromammifères, oiseaux et problèmes taphonomiques[END_REF][START_REF] Laroulandie | Taphonomie et archéozoologie des Oiseaux en grotte : applications aux sites paléolithiques du Bois-Ragot (Vienne), de Combe Saunière (Dordogne) et de La Vache (Ariège)[END_REF]. Finally, breakage patterns were assessed with reference to criteria published by [START_REF] Villa | Breakage patterns of human long bones[END_REF] concerning long bone fracture morphologies and shaft fragmentation.

Results

Topography

A series of surface depressions recognisable in the chamber adjacent to the test pit were mapped (Fig. 3). The largest, measuring 70 cm in diameter and some 15 cm deep, cut two older depressions.

These surface structures are accessible by a shallow gulley dug into a small, approximately 10 cm high surrounding bank formed by displaced sediment. The form and dimensions of these features are consistent with nests or dens.

Stratigraphy

Two stratigraphic units were documented in the 60 cm deep test pit 1 (Fig. 4). The upper unit (Unit 1) is 40 to 50 cm thick and contains a dark brown (7,5YR 3/2) granular organic loam with numerous small roots. This poorly-sorted sediment, rich in sands and limestone granules, presents a poorlyexpressed granular substructure with friable, millimetre-sized rounded aggregates. This unit presents a crude stratification expressed by (a) less than 1 m long lenses of sediments richer in limestone granules and (b) banded colour variations that follow the stratification plane defined by these lenses.

Nearly all the faunal material recovered from the test pit derives from this unit (Fig. 5).

The lower unit (unit 2) is at least 20 cm thick and contains a massive light brown (7,5YR 5/4) loam with numerous limestone granules and blocks of various sizes. There is no contrast between the facies in terms of mineral content; both are composed of loams that infiltrated the cavity via the fissure responsible for the formation of the corridor and limestone fragments connected to the disaggregation of the cave walls. Conversely, the two facies differ in sedimentary structure, organic content, and bone and gastropod composition, all of which are significantly more abundant in the upper unit.

The unit 1 facies is easily explained by its relation to the current topography of the soil surfacethe lens of limestone granules at the top of the unit corresponds to the bank of the nest bordering the test pit (Fig. 4). This lens, identical in both extension and facies to the buried structures, indicates that unit 1 is connected to the repeated accumulation of sediments displaced by nesting animals. This is further supported by colour differences observable in the sedimentary bedding, demonstrating the deposit to reflect the superimposition of successive digging/displacement events, as has been documented in other fossil contexts [START_REF] Zeitoun | The cave of the monk (Ban Fa Suai, Chiang Dao wildlife sanctuary, northern Thailand)[END_REF]. Unit 2 is composed of sediment trapped in the cave that preserves evidence for a biogenic structure related to animal activity.

In sum, the preserved stratigraphy of Blanchard Cave 2 represents the natural accumulation of slope deposited sediments, with the upper unit reorganised and enriched with organic debris connected to the cave's use by sheltering or burrowing animals responsible for the nests.

Radiocarbon dating

Although a sample from an Audubon's shearwater bone collected from the base of unit 1 (spit 9)

could not be dated due to insufficient collagen preservation, a second sample recovered from the middle of the same unit (spit 6) returned a non-calibrated date of 760 ± 30 BP (Lyon-9663_Sac-31250).

When calibrated using Calib 7.0 and the Marine13 isotope curve, the Audubon's shearwater having an exclusively marine based diet, this date produces an age range of between 1479 and 1638 CE (95.4% probability).

4.4. The Blanchard 2 faunal assemblage: composition and taphonomic observations from the test pit 1

The Blanchard 2 faunal material comprises 1333 remains (Table 1, Table 2), a quarter (NISP= 357) recovered from the surface of the cave. This fairly large assemblage is dominated by molluscs (NISP =609) and bird remains (NISP= 399), whose proportions vary by depth (Fig. 5), and a slightly smaller sample of crustaceans (NISP =325).

Molluscs (Table 1)

Over 600 mollusc shells representing at least 214 individuals (MNI) were recovered from the test pit 1 (Table 1). The majority is composed of land gastropods (NISP= 397, 65.5%) alongside several different marine taxa (NISP= 209, 34.5%). The first category is dominated by Amphibulima patula (NISP =207, MNI = 87) and Pleurodonte josephinae (NISP = 144, MNI = 52), which are found together throughout the deposits. Amongst the marine molluscs, only a single bivalve fragment (Pectinidae)

was identified in the surface material, with the remainder of the marine assemblage dominated by gastropods, particularly the West Indian top shell (Cittarium pica). This large sea snail is represented by 165 remains, largely found on the surface (NISP =91), corresponding to only 14 individuals.

The number of shell fragments is generally very high in all spits, particularly spits 11, 10, 7 and 5, and never falls below 60%. This is especially the case with the A. patula, P. josephinae and C. pica shells.

A large majority of the C. pica shells (90%) belong to small-sized individuals while most of other marine taxa are small-sized species (e.g. Thais, Littorina, Cenchritis). This pattern combined with marks of shell wear strongly suggests the introduction of most of the marine shells by hermit crabs, apart from the bivalve and the two polyplacophoran fragments that are unsuited to this crustacean.

Amongst the Amphibulima shell fragments, only seven bare clear traces of impact on the edge of the opening located on the columella edge, or occasionally both on both opposite sides (Fig. 6). These semi-elliptical impacts are wider than they are deep and result from a blow delivered from the exterior to the interior of the test. These types of traces are particularly clear on nearly-complete shells (4 of 207 remains), and it is very likely that additional similar traces were unnoticed due to the heavily fragmented nature of the mollusc assemblage.

Crustaceans (Table 1)

Apart from several elements attributable to the hermit crab Coenobita clypeatus (14 remains of 3 individuals), crustaceans are represented by 311 fragments from at least 49 terrestrial land crabs of the genus Gecarcinus. Although documented in every spit and particularly high in the surface collection and first spit, the number of Gecarcinus remains decreases with depth, apart from a slight increase in spit 6 (z = 30 cm). Breakage is relatively low and a fair number of almost complete crab legs (dactyls and propodi) were documented.

Vertebrates

Vertebrates comprise almost a third (NISP = 399) of the faunal material recovered from the test pit (Table 2). Unequally distributed in the deposit, the vertebrate assemblage is dominated by bird bones, which represent more than two thirds of the identifiable remains, followed by mammals (18%).

4.4.3.1. Fish and herpetofauna. Squamates are primarily represented by large-sized anoles, likely

Anolis ferreus (NISP= 38), the sole endemic species of Marie-Galante Island (Bochaton et al., 2017a(Bochaton et al., , 2017b)), as well as a single Pholidoscelis possibly also the endemic species of Marie-Galante (Bochaton et al., 2017a(Bochaton et al., , 2017b) (cf. Pholidoscelis turukaeraensis) and three bones attributable to Iguana sp. that does not present any characters allowing for a specific attribution [START_REF] Bochaton | Osteological differentiation of the Iguana Laurenti, 1768 (Squamata: Iguanidae) species: Iguana iguana (Linnaeus, 1758) and Iguana delicatissima Laurenti, 1768, with some comments on their hybrids[END_REF]. Two fish backbone fragments and the femur of an Eleutherodactylid frog (Eleutherodactylus sp.) were also recovered from test pit 1.

4.4.3.2. Birds. Bird bones comprise 60% (NISP= 238) of the vertebrate assemblage from Blanchard 2. The most well represented species is the Audubon's shearwater (P. lherminieri), totalling nearly 100 identifiable remains (Fig. 7), followed by 111 bones that can fairly confidently be assigned to size class 3. Juvenile remains are relatively numerous, comprising nearly 49% (NISP = 74) of assemblage.

Alongside the less well-represented species, we identified a claw of a burrowing owl (Athene cunicularia) in spit 8 (z = 75 cm) and five bones of a yellow-crowned night-heron (Nyctanassa violacea)

on the surface of the cave and in spit 1. The remains of juvenile passerines (size class 2) were also found in spits 1 and 6. uniquely in spit 1, those attributable to rat (Rattus sp.) were recovered both from the surface of the cave and more sporadically in spits 1 through 3. Rodents are also represented by a single agouti bone (Dasyprocta sp.) and 23 bones of a giant rice rat (Megalomys sp.). These 23 specimens could be assigned to the oryzomini undescribed species B of Steadman, which is a now-extinct large-sized rodent known to have inhabited the islands of Montserrat, Guadeloupe, Marie-Galante, Barbuda and Antigua in pre-Columbian times [START_REF] Steadman | Vertebrates from archaeological sites on Montserrat, West Indies[END_REF]. Bat bones, including the Antillean fruit-eating bat (Brachyphylla cavernarum), Jamaican fruit bat (Artibeus jamaicensis) and a free-tailed bat (Mollosidea) were found more dispersed in the deposits.

The vertebrate remains are heavily fragmented (72%), particularly the bird and mammals bones.

Fractures are most frequently observed on long bone extremities (82.5%), with post-mortem (53.8%) breakage being significantly more common than green bone fractures (18%). Green bone breakage varies by spit; virtually no broken bones were recovered from the lowermost spits (8, 10, 11), medium values (< 15%) were recorded for the middle spits (spits 2, 3, 6, 7, 9) and higher values (> 20%) for spits from the middle and top of the test pit (spits 1, 4, 5) as well as the cave surface.

Of the 90 long bone extremities with typical green bone fractures, 16 are associated with traces of predation (e.g. crushing, perforations, notches, Fig. 8, Table 3). These traces can be observed on Puffinus remains (n = 9) as well as the bones of the yellow-crowned night heron and anoles (n = 4), all of which were recovered from either the surface (n = 12) or the first three spits of the test pit 1 (n =2).

The morphology of the traces and their association with a high degree of bone breakage are consistent with a carnivore contribution in the formation of the assemblage [START_REF] Laroulandie | Taphonomie et archéozoologie des Oiseaux en grotte : applications aux sites paléolithiques du Bois-Ragot (Vienne), de Combe Saunière (Dordogne) et de La Vache (Ariège)[END_REF] Figure 8. Gnawing marks on Audubon's shearwater remains (spits 1 to 3). T: tooth marks, n: notches, c: crenulated edges.

Discussion

Site formation

The absence of archaeological material, traces of human activity (ash, charcoal) or cut-marked bones demonstrate that the Blanchard 2 deposits to result from natural accumulation processes.

Consequently, the artefacts collected from the cave surface (a button, flint flake and a spider conch shell) can likely be connected to the accumulation of slope-derived colluvium within the site. That being said, the site probably had a more complex formation history with several accumulation processes contributing to the composition of the faunal assemblage.

Use of the cave by animals

The numerous Audubon's shearwater remains, the lack of traces referable to predation on the majority of these remains, and the high proportion of juvenile remains (between 14% and 40% according to spit) argue in favour of a natural death assemblage linked to Blanchard 2 being a nesting site. This is further supported by the fact that this ground-nesting marine bird nests only in rock crevices or burrows [START_REF] Lee | Audubon's shearwater. West Indian Breeding Seabird Atlas[END_REF]. While Audubon's shearwater nests sometimes take the form of burrows dug into sandy substrates, they are most often found, at least in the Lesser Antilles, in caves and rock shelters [START_REF] Précheur | Étude de la dynamique de population du Puffin d'Audubon (Puffinus lherminieri) de la réserve naturelle des îlets de Sainte-Anne[END_REF][START_REF] Burke | Barbados[END_REF]Collier and Brown, 2008a, 2008b, 2008c). For example, at Hardy cave in the Sainte-Anne islets off the southern coast of Martinique, Audubon's shearwaters were found to nest in the deepest and most inaccessible parts of the cave, where they lay their eggs in small nests in the ground (Pinchon, 1976: 26). The limited size (ca. 70 cm) of the Blanchard 2 nests, which could accommodate a bird of this size, and the fact that they are similarly located in small niches or against cave walls reinforce the likelihood of the cave being a nesting site. In behavioral terms, the nesting habits of the Audubon's shearwater recall those of tropicbirds (genus Phaethon), another burrow-nesting seabird that also prefers natural cavities and is known to burrow into loose ground to build their nests, producing similar depressions (Fig. 9).

Furthermore, the bioturbation facies of unit 1 provides clear evidence for a biogenic sedimentary structure of a deposit connected to the repeated digging of nests. This, combined with the predominance of Audubon's shearwaters remains, especially juvenile individuals, supports the hypothesis of the cave having been used as a nesting site throughout the development of unit 1.

Figure 8. Nest dug by a tropicbird in a cave on the island of Saint Barthélemy (photo A. Lenoble).

Both hermit and land crabs are also known to frequent karstic contexts, where they take refuge during the day. While the abundance of terrestrial gastropod shell fragments is consistent with the cave being 200 m from the shore, the presence of marine gastropods (e.g. Cittarium, Nerita) is likely tied to the reuse of tests by hermit crabs that are still present in the cave's vicinity. All of these detrivore and scavenger crustaceans (hermit crabs and Gecarcinus sp.) have also been documented in Audubon's shearwater nesting sites, where they feed on the corpses of hatchlings (Pinchon,. The commensality between crabs and shearwaters implies the remains of the former to represent an attritional death assemblage. This is confirmed by the low degree of breakage observable in the crustacean remains coupled with the presence of leg fragments (dactyles and propodi). Such a commensality is also suggested by the fact that correlations in the relative abundance of crab and shearwater (Fig. 5) remains are not significantly different (Fischer's exact test p= 0.36).

Finally, gnawing marks observed on bones recovered from the surface of the cave and the summit of the fossil deposit further highlight the frequentation of the cave by a carnivore preying on Audubon's shearwater. Three carnivores have been identified in the mammalian faunal assemblages from sites on Marie-Galante; dogs (Canis familiaris), introduced during the Amerindian period [START_REF] Grouard | Dog burials associated with human burials in the West Indies during the early pre-Columbian ceramic age (500 BC-600 AD)[END_REF] alongside domestic cats (Felis catus) and the small Asian mongoose, both of which were first brought by European colonists [START_REF] Lorvelec | Impacts des mammifères allochtones sur quelques espèces autochtones de l'Îlet Fajou (Réserve Naturelle du Grand Cul-de-Sac Marin, Guadeloupe), établis à l'issue d'une tentative d'éradication[END_REF]. While the island is not home to feral dogs and cats, the mongoose is frequent on the island and its traces around the site are easily visible, making it the most likely agent responsible for the traces on the bones. This interpretation is consistent with the heavy fragmentation of the bones and the traces of gnawing on the entirety of the vertebrate assemblage. In fact, these traits are more typical of a viverrid rather than a small felid (Fig. 8 and Laroulandie, 2000: Figs. 49-52), which is in agreement with the presence of this taxon in the Blanchard 2 faunal assemblage.

Intrusive elements

The organo-mineral nature of the cave's infill and the morphology of the open crevice cave demonstrate that all or part of the sediment composing unit 1 represent slope-derived colluvium trapped in the site. As the colluvium in the test pit 2 on the slope above the site (test pit 2) contains both malacofauna and faunal material, it is safe to assume that at least part of the material recovered from the cave may derive from down-slope colluviation.

The fact that the Blanchard 2 land snail assemblage is dominated by P. josephinae and A. patula clearly illustrates this likelihood. P. josephinae are known to occur in the litter of non-degraded forests [START_REF] Robinson | The land mollusca of Dominica (Lesser Antilles), with notes on some enigmatic or rare species[END_REF], while A. patula, equally found in humid environments [START_REF] Schramm | Note sur l'habitat du Succinea patula[END_REF][START_REF] Mazé | Note sur les moeurs et les habitudes de l'Amphibulima patula[END_REF], are unknown from the east of the island [START_REF] Mazé | Découverte de l'Amphibulima patula, Bruguière, à Marie-Galante[END_REF] and have not be observed alive since the late 19th century (Lenoble et al. in press).

Moreover, numerous A. patula bear notches identical to those left by birds preying on molluscs [START_REF] Snyder | A comparative study of mollusc predation by limpkins, everglade kites, and boat-tailed grackles[END_REF]. Importantly, even though shell-feeding birds are known in the region, such as the hook-billed kite (Chondrohierax uncinatus; [START_REF] Smith | Feeding habits and bill Polymorphism in Hook-billed Kites[END_REF][START_REF] Johnson | Systematics and conservation of the hook-billed kite including the island taxa from Cuba and Grenada[END_REF] or snail kite (Rostrhamus sociabilis; [START_REF] Snyder | A comparative study of mollusc predation by limpkins, everglade kites, and boat-tailed grackles[END_REF], none of them lives in caves and the accumulations they produce are found on the ground immediately below nesting sites or in forests.

Therefore, the presence of numerous A. patula shells in the cave is more parsimoniously explained by the introduction of colluvium that reworked older shell deposits associated with markedly different climatic conditions. This interpretation is corroborated by the fact that in the nearby site of Cadet 3, A. patula shells are abundant in levels assigned to the mid-Holocene [START_REF] Hodell | Reconstruction of Caribbean climate change over the past 10, 500 years[END_REF], which is associated with regionally more humid conditions, and then become rare in more recent levels [START_REF] Stouvenot | L'abri sous roche Cadet 3 (Marie-Galante): un gisement à accumulations de faune et à vestiges archéologiques[END_REF].

Other elements of the faunal assemblage may also have accumulated in the site. For example, as the test pit 2 assemblage also contains bones, it is impossible to exclude the possibility that at least part of the faunal material recovered within the cave is in secondary position. This would include the rare agouti or passerine remains as well as the few archaeological pieces recovered from this site. This being said, this component of the Blanchard 2 assemblage has little bearing on the interpretation of the site as an Audubon's shearwater nesting ground (see above). This interpretation is supported by the bioturbation facies of unit 1 connected to the repeated burrowing of nesting birds and the abundance of hatchling remains. This is in stark contrast with the majority of other caves that yielded Audubon's shearwater remains, where human intervention is most often evoked to explain their presence [START_REF] Frank | Vertebrate Paleontology of Isla de Mona, Puerto Rico[END_REF][START_REF] Kaye | Geology of Isla de Mona, Puerto Rico and notes of the age of Mona passage[END_REF][START_REF] Wetmore | Birds remains from the West Indies[END_REF] whereas natural accumulations remain rare [START_REF] Morgan | Late Quaternary fossil vertebrates from the Cayman Islands[END_REF][START_REF] Nieves-Rivera | Bones of Puffinus Lherminieri lesson (Aves: Procellaridae) and two other vertebrates from Cueva Del Agua, Mona Island, Puerto Rico (West Indies)[END_REF].

In sum, the site of Blanchard 2 provides evidence for a faunal assemblage linked to the use of the cave as a nesting site, including a dominance of Audubon's shearwater remains, a high percentage of juvenile birds, a bioturbation facies linked to repeated burrowing, and the presence of numerous terrestrial crustaceans. These characteristics provide sound criteria for identifying fossil nesting sites.

Age of the fossil assemblage

The small Asian mongoose was introduced to Guadeloupe at the end of the 19th century and then spread from this island to Marie-Galante in the beginning of the 20th century [START_REF] Horst | The mongoose in the West Indies: the biogeography and population biology of an introduced species[END_REF].

Further, the black rat was introduced substantially earlier, appearing in the Greater Antilles immediately after Columbus arrived in the Americas [START_REF] Wing | Native American Use of Animals in the Caribbean[END_REF]. The presence of this rodent is attested to from the outset of the French colonisation of Guadeloupe (Du Tertre, 1667; Breton, 1647),

if not earlier, in connection with the Spanish presence in the archipelago during the 16th century.

Unlike the rat remains, which are dispersed in the upper third of the fossil layer, mongoose bones are limited to the cave surface, suggesting that the deposit formed during the historical period or slightly before. This is consistent with a radiocarbon date obtained on an Audubon's shearwater bone from spit 6, which, when calibrated, falls between 1470 and 1638 CE.

Consequently, Blanchard 2 likely served as an Audubon's shearwater nesting site during a period shortly before the French colonisation of Marie-Galante that began in 1648 [START_REF] Du Tertre | Histoire générale des Antilles habitées par les Français[END_REF]. On the other hand, carnivore traces (see above) on Audubon's shearwater bones recovered from the final three spits (1 to 3) of the test pit 1 demonstrate that these birds were still nesting in the cave after the early 20th century introduction of the mongoose. While it is difficult to fix a precise chronology for the cave's use, the fact that no current inhabitants recall Audubon's shearwaters nesting on the island suggests that it ceased to be a nesting site no later than the mid-twentieth century.

The nature of the bioturbated deposits does, however, not rule out the possibility that the dated bone was reworked from deeper sediment by burrowing animals. Consequently, two chronological hypotheses can be imagined for the time period over which fossil unit 1 formed. If the dated bone derives from the base of unit 1, the fossil deposit would have formed over a 400-year period beginning shortly before the arrival of the first French colonists and ending sometime in the early 20 th century.

On the other hand, if we assume the dated bone to be associated with spit 6 and a constant sedimentation rate for the underlying 20 cm, unit 1 would have begun to form between the 13th and 14 th centuries or just before Columbus arrived on the island. Notwithstanding this uncertainty, radiometric, sedimentological and biological data all favour the cave being occupied during both the 'contact period' between Columbus' arrival when the indigenous Carib population still controlled the island and the ensuing period of European colonisation (i.e, 1648 to the beginning of the 20th century).

Landscape change

It is interesting to note that the dated Audubon's shearwater bone was recovered from a depth of 40 cm, too deep to be explained uniquely by the surface burrows, which are no deeper than 15 cm. This suggests that the sediment accumulated in the cave during its use by nesting birds. As discussed

above, the open morphology of the crevice cave likely trapped colluvium, which, based on the stratigraphy, account for some 50 cm of the deposits accumulated in the cave during the historic period. However, this type of sedimentation is not observable in the preceding millennia otherwise the cave would have been quickly and completely infilled even if we assume a sedimentation rate of several centimetres per century. Moreover, if colluviation had continued into the early 20th century, the burrows documented on the cave surface would have been obscured. Taken together, this suggests that sediment contemporaneous with the use of the cave by Audubon's shearwater accumulated over a relatively short period of time during the historical period and can be connected to human alteration of the local environment.

These different elements support soil erosion to have followed the deforestation of the Guadeloupe islands after the arrival of European populations. This "erosional crisis" during the historical period has been documented on other islands of the archipelago. For example, [START_REF] Lasserre | La Guadeloupe: étude géographique[END_REF] considers it to be tied to "itinerant" Creole gardening of less fertile areas of the island coupled with slash and burn practices and charcoal production. These activities would have concerned the entirety of the limestone hills throughout the Guadeloupe Islands. This would effectively exclude the fossil deposit developing before the landscape had been significantly altered by erosion. Consequently, a large part of deposit would have formed during the historical period, making the shorter, 400-year chronology for the deposition of unit 1 the most likely.

Evolution of Audubon's shearwater nesting sites

The Audubon's shearwater is the only shearwater that currently nests in the Lesser Antilles [START_REF] Lowrie | Seabird breeding atlas of the Lesser Antilles[END_REF], present in Barbados, Dog Island, Martinique, Marie-Galante, La Désirade, Saint Martin, Saint Barthélemy and Saba. Their nesting areas are found on small deserted islets off the coasts of the larger islands [START_REF] Burke | Barbados[END_REF]Collier andBrown, 2008b, 2008c;[START_REF] Lemoine | Martinique[END_REF]Levesque et al., 2008;[START_REF] Bright | Seabird surveys on Dog Island, Anguilla, following eradication of black rats find a globally important population of red-billed tropicbirds (Phaethon aethereus)[END_REF] or, as in the case of Marie-Galante, La Désirade and Saba, on inaccessible cliffs (Levesque and Mathurin, 2008;Collier and Brown, 2008a).

The case of Blanchard 2 shows the nesting area of these birds to have been larger in the past, with breeding sites found in zones other than the most inaccessible ones. The main causes that biologists assumed to underlie this reduction is the introduction of predators, such as the mongoose, but, above all, the rat [START_REF] Pascal | Conséquences sur l'avifaune nicheuse de la réserve naturelle des îlets de Sainte-Anne (Martinique) de la récente invasion du rat noir (Rattus rattus), établies à l'issue d'une tentative d'éradication[END_REF][START_REF] Daltry | Evidence that Eradicating Black Rats Has Boosted the Recovery of Rare Reptiles and Seabirds on Antiguan Islands[END_REF][START_REF] Bright | Seabird surveys on Dog Island, Anguilla, following eradication of black rats find a globally important population of red-billed tropicbirds (Phaethon aethereus)[END_REF]. The probable presence of the rat on Marie-Galante from the very outset of European colonisation, as well as the presence of rat bones in the upper third of the fossil deposit, suggests that this intrusive rodent did not have a sufficiently important impact, at least on the scale of several centuries, to provoke the abandonment of the nesting site. Moreover, gnawing marks on bones from the upper portion of the deposit reveal that a mammalian carnivore, in this case, the small Asian mongoose, to likely have been the contributing factor to Blanchard 2 ceasing to function as an Audubon's shearwater nesting site.

Moreover, this carnivore is known regionally as the main predator of ground-breeding birds [START_REF] Pinchon | Faune des Antilles françaises Les oiseaux. 2nde édition[END_REF][START_REF] Lescure | Le peuplement en reptiles et amphibiens des Petites Antilles[END_REF], including sea birds [START_REF] Lorvelec | Impacts des mammifères allochtones sur quelques espèces autochtones de l'Îlet Fajou (Réserve Naturelle du Grand Cul-de-Sac Marin, Guadeloupe), établis à l'issue d'une tentative d'éradication[END_REF][START_REF] Lewis | Conservation implications of small Indian mongoose (Herpestes auropunctatus) predation in a hotspot within a hotspot: the Hellshire Hills, Jamaica[END_REF]. In this respect, the Blanchard 2 data are consistent with the mongoose having a decisive impact on the island's historical Audubon's shearwater populations. Accordingly, human predation during the historical period, as clear from the example of the black-capped petrel (Pterodroma hasitata) in Guadeloupe and Dominique [START_REF] Labat | Nouveau voyage aux isles de l'Amérique[END_REF][START_REF] Feilden | The deserted domicile of the Diablotin in Dominica[END_REF][START_REF] Goetz | Conservation action plan for the blackcapped petrel (Pterodroma hasitata)[END_REF], would have had only a moderate, if any impact on Audubon's shearwater populations on Marie-Galante. Moreover, the possibility that historical exploitation destabilised Audubon's shearwater colonies is contradicted by the lack of evidence for human intervention. The fact that Blanchard 2 was occupied prior to the arrival of European colonists and continued thereafter suggests only a limited historical impact, whether in the form of direct predation or the modification of the landscape for agricultural purposes. On the other hand, it is interesting to note that the initial use of the cave as a nesting site some 500 y ago followed the abandonment of the southern coast of the island by Amerindian groups.

Indeed, the southern coast of Marie-Galante was occupied during the ceramic period (ca. -300 BCE to 1500 CE, [START_REF] Delpuech | Guadeloupe amérindienne[END_REF], as made evident by the nearby sites of Tourlourous and Petit Anse [START_REF] Serrand | Direction des Affaires Culturelles of Guadeloupe[END_REF]Lenoble et al., 2011;[START_REF] Casagrande | Direction des Affaires Culturelles de Guadeloupe[END_REF]. One the one hand, the most recent dates come from Amerindian burials at Blanchard and Cadet 2, which demonstrate a continuous Amerindian presence into the 14th century (Stouvenot, 2005;[START_REF] Courtaud | La cavité sépulcrale de Cadet 2 (Capesterre-de-Marie-Galante, Guadeloupe) et la question du cannibalisme[END_REF] and on the other one, seventeenthcentury European sources describe Marie-Galante as free of indigenous settlement [START_REF] Breton | Relations de l'île de la Guadeloupe[END_REF][START_REF] Coppier | Histoire et voyage des Indes Occidentales et de plusieurs autres régions maritimes[END_REF][START_REF] Pacifique De Provins | Brève relation du voyage des îles de l'Amérique[END_REF][START_REF] Du Tertre | Histoire générale des Antilles habitées par les Français[END_REF]. This is in accordance with the choice to move villages to difficult access positions [START_REF] Breton | Relations de l'île de la Guadeloupe[END_REF][START_REF] Breton | Dictionnaire caraïbe-français[END_REF] in response to the expanded influence of Tainos populations in the Greater Antilles in the decades preceding the arrival of Europeans in the Americas [START_REF] Hofman | Attractive ideas, desirable goods: examining the late ceramic age relationships between greater and lesser Antillean societies[END_REF] or as a consequence of regular confrontations with European colonists during the contact period (e. g. [START_REF] Lafleur | Les Caraïbes des Petites Antilles[END_REF]. Regardless the reasons underlying these changes, it is clear that between the 14th century, which sees the island's final Amerindian occupation, and the arrival of European colonists in the 17th century, coastal plains of islands such as Marie-Galante and especially its southern coast no longer saw a permanent Amerindian presence. Interestingly, it is during exactly this same period that Audubon's shearwaters first occupied Blanchard 2, suggesting that these birds used the southern coast of Marie-Galante during a 400-year period following the departure of the final Amerindian groups.

This would imply that Amerindian populations had a significant impact on the distribution of Audubon's shearwater reproductive colonies, a likelihood reinforced by the sequence from the nearby Cadet 3 rock shelter. Audubon's shearwater first appears in the Cadet 3 vertebrate faunal assemblage at the beginning of the Holocene and subsequently decreases from the mid-Holocene onwards, when the first Amerindian populations arrived in the archipelago [START_REF] Hofman | Archipelagic resource procurement and mobility in the northern lesser Antilles: the view from a 3,000-year-old tropical forest campsite on Saba[END_REF]. This is also in agreement with archaeological evidence that shows Audubon's shearwater remains, while the most frequent sea bird found in pre-Columbian sites [START_REF] Grouard | Subsistance, systèmes techniques et gestion territoriale en milieu insulaire antillais précolombien : exploitation des vertébrés et des crustacés aux époques Saladoïdes et Troumassoïdes de Guadeloupe[END_REF][START_REF] Wing | Native American Use of Animals in the Caribbean[END_REF], can also form a large part if not dominate Amerindian middens [START_REF] Wing | Vertebrate remains from Indian sites on Antigua, West Indies[END_REF][START_REF] Hofman | Archipelagic resource procurement and mobility in the northern lesser Antilles: the view from a 3,000-year-old tropical forest campsite on Saba[END_REF]. Taken together, this data and the evidence from Blanchard 2 support the hypothesis of a non-negligible Amerindian impact on Audubon's shearwater breeding sites.

Conclusion

A combination of paleontological, taphonomic, sedimentological, and chronological information clearly demonstrates Blanchard 2 Cave to have been an Audubon's shearwater nesting site from sometime around the middle of the second millennia to the beginning of the 20 th century. Moreover, the use of the cave as a nesting site began after the abandonment of the area by Amerindian population, suggesting that the exploitation of marine birds by Amerindian societies significantly influenced Audubon's shearwater colonies. Additionally, traces of chewing on bones found in the upper portion of the test pit, as well as mongoose bones collected from the cave surface, indicate a recent reduction in the nesting area of Audubon's shearwater on Marie-Galante primarily due to the introduction of this carnivore. This is not to say that predation by humans, landscape changes connected to agricultural practices or the introduction of the rat did not have consequences for Audubon's shearwater colonies. Rather, the arrival of the mongoose had a major influence, which, alone or in combination with others factors, led to the abandonment of Blanchard 2 as a nesting site.

Finally, determining the exact role each factor played in the reduction of seabird nesting sites on Marie-Galante and other Caribbean Islands requires complementary data from additional sites. The biogenic sedimentary structure of the Blanchard 2 deposits and the unique faunal associations characteristic of this type of site do, however, already provide new criteria for identifying Audubon's shearwater nesting sites in the fossil record.

Figure 1 .

 1 Figure 1. Location of the sites discussed in this study. 1: Blanchard 2 cave; 2: Blanchard 1 cave; 3, Cadet 2 rock shelter; 4: Cadet 3 rock shelter; 5: Petite Anse; 6: Tourlourous.

Figure 2 .

 2 Figure 2. Cave topography. Left, view from the top; right, view from the East side of the cliff.

Figure 3 .

 3 Figure 3. Plan of the nest to the right of the test pit within the cave.

Figure 4 .

 4 Figure 4. Stratigraphy of test pit 1. The numbers to the right indicate the different spits.

Figure 5 .

 5 Figure 5. Vertical changes in the number of faunal remains by category. A -bones, B -terrestrial malacofauna, C -marine malacofauna, D -crustaceans.

Figure 6 .

 6 Figure 6. Notches and edge damage on A. patula shells.

Figure

  Figure 7. Audubon's shearwater remains from Blanchard. Comparison with a specimen from the type locality of Saint Barthélemy. 1: Humerus, 2: First Phalange of the third digit, 3: carpometacarpus, 4: Coracoid, 5: Tarsometatarsus, 6: Tibiotarsus. For each bone, comparative specimen is on the left and the fossil one is on the right.

Table 1 .

 1

List of molluscs and crustaceans including number of remains (NISP) by spit. Table 2. List of vertebrates including number of remains (NISP) by spit.

Table 3 .

 3 Fragmentation and bone surface modifications on the faunal assemblage recovered from testCompared to the first test pit, the faunal assemblage from test pit 2 is very limited, comprising two Audubon's shearwater bones, a redrumped agouti (Dasyprocta leporina), a rat femur and incisor, a bat radius and an anole vertebrae. On the other hand, malacofauna are numerous(NISP= 506, MNI

	pit 1.
	4.5. The faunal assemblage from test pit 2

= 224) and, although in different frequencies, comprise essentially the same species as test pit 1 (Table

4

). For example, gastropod species are the same as those found in test pit 1 and are all small individuals. Additionally, like test pit 1, hermit and land crabs (Gerarcinidae sp.) are well represented and a chiton fragment was also documented. The major difference between the two test pits is the proportion of different gastropods, the numbers of A. patula and P. josephinae decrease, C. crenulatum are abundant and B. cf. guadalupensis decrease significantly in the cave infill. The latter two taxa are almost certainly part of a natural death assemblage. Finally, breakage is generally high, except for the Chondropoma.

Table 4

 4

	. Comparison
	of fauna recovered
	from test pits 1 and 2.
	NISP: Number of
	Identified Specimens;
	MNI: Minimal
	Number of
	Individuals.

Acknowledgements

We would like to thank the Natural Reserve of Saint Bartholomew for donating modern specimens of P. lherminieri used for osteological comparison. We also wish to thank Alain Queffelec and Myriam Boudadi-Maligne for their help during the excavations at Blanchard 2.

This study was conducted as part of the BIVAAG Program established by the Centre National de la Recherche Scientifique (CNRS), with support from a European PO-FEDER grant 2007-2013 n°2/2.4/-33456, the Guadeloupe Regional Council, the Direction de l'Environnement et de l'Aménagement du Territoire (DEAL) of Guadeloupe, and the Direction des Affaires Culturelles (DAC) of Guadeloupe.