A Pashkevich
email: anatol.pashkevich@emn.fr

A Klimchik

S Caro

D Chablat
email: damien.chablat@irccyn.ec-nantes.fr

Stiffness modelling of parallelogram-based parallel manipulators

Keywords: stiffness modeling, parallel manipulators, parallelogram-based linkage, external loading, linear approximation

The paper presents a methodology to enhance the stiffness analysis of parallel manipulators with parallelogram-based linkage. It directly takes into account the influence of the external loading and allows computing both the non-linear "load-deflection" relation and relevant rankdeficient stiffness matrix. An equivalent bar-type pseudo-rigid model is also proposed to describe the parallelogram stiffness by means of five mutually coupled virtual springs. The contributions of this paper are highlighted with a parallelogram-type linkage used in a manipulator from the Orthoglide family.

Introduction

In the last decades, parallel manipulators have received growing attention in industrial robotics due to their inherent advantages of providing better accuracy, lower mass/inertia properties, and higher structural rigidity compared to their serial counterparts [START_REF] Ceccarelli | A stiffness analysis for CaPaMan (Cassino Parallel Manipulator)[END_REF][START_REF] Company | Modelling and preliminary design issues of a 4-axis parallel machine for heavy parts handling[END_REF][START_REF] Merlet | Parallel Robots[END_REF]. These features are induced by the specific kinematic structure, which eliminates the cantilever-type loading and decreases deflections due to external wrehches. Accordingly, they are used in innovative robotic systems, but practical utilization of the potential benefits requires development of efficient stiffness modeling techniques, which satisfy the computational speed and accuracy requirements of relevant design procedures.

Amongst numerous parallel architectures studied in robotics literature, of special interest are the parallelogram-based manipulators that employ special type of linkage constraining undesirable motions of the end-platform. However, relevant stiffness analysis is quite complicated due to the presence of internal closed kinematic chains that are usually replaced by equivalent limbs [START_REF] Majou | Parametric stiffness analysis of the Orthoglide[END_REF], Pashkevich et al., 2009a) whose parameters are evaluated rather intuitively. Thus, the problem of adequate stiffness modelling of parallelogram-type linkages, which is in the focus of this paper, is still a challenge and requires some developments.

Another important research issue is related to the influence of the external loading that may change the stiffness properties of the manipulator. In this case, in addition to the conventional "elastic stiffness" in the joints, it is necessary to take into account the "geometrical stiffness" due to the change in the manipulator configuration under the load (Kovecses and Anjeles, 2007). Moreover buckling phenomena may appear [START_REF] Timoshenko | Theory of Elasticity[END_REF]) and produce structural failures, which must be detected by relevant stiffness models.

This paper is based on our previous work on the stiffness analysis of overconstrained parallel architectures (Pashkevich et al., 2009a(Pashkevich et al., , 2009b) ) and presents new results by considering the loading influence on the manipulator configuration and, consequently, on its Jacobian and Hessian. It implements the virtual joint method (VJM) introduce by Salisbary [START_REF] Salisbury | Active stiffness control of a manipulator in Cartesian coordinates[END_REF] and Gosselin [START_REF] Gosselin | Stiffness mapping for parallel manipulators[END_REF]) that describes the manipulator element compliance with a set of localized six-dimensional springs separated by rigid links and perfect joints. The main contribution of the paper is the introduction of a non-linear stiffness model of the parallelogram-type linkage and its linear approximation by 6x6 matrix of the rank 5, for which it is derived an analytical expression.

Problem statement

Let us consider a typical parallel manipulator that is composed of several kinematic chains connecting a fixed base and a moving platform (Figure 1a). It is assumed, that at least one chain includes a parallelogram-based linkage that may introduce some redundant constraints improving the mechanism stiffness. Following the VJM concept, the manipulator chains are usually presented as a serial sequence of pseudorigid links separated by rotational or translational joints of one of the following types: (i) perfect passive joints; (ii) perfect actuated joints, and (iii) virtual flexible joints that describe compliance of the links, joints and/or actuators. To derive a parallel architecture without internal kinematic loops, the parallelogram based linkage must be replaced by an equivalent bar-type element that has similar elastic properties. Thus, let us consider the VJM model of the kinematic par-allelogram assuming its compliance is mainly due to the compliance of the longest links, which are oriented in the direction of the linkage (Figure 1b,c). In order to be precise, let us also assume that the stiffness of the original bar-elements is described by a 6 × 6 matrix K b whose elements are identified using FEA-based modeling.

In the framework of the VJM approach, the geometry of the parallelogram-based linkage can be descried by the following homogeneous transformations

T i = T z (η d/2) R y (q i1 ) T x (L) T x (θ i1 ) T y (θ i2 ) T z (θ i3 ) R x (θ i4 ) R y (θ i5 ) R z (θ i6 ) R y (q i2 ) T z (-η d/2) R y (-q i2 ) (1)
where η = (-1) i , T i is a 4 × 4 homogenous transformation matrix, T x , ..., R z are the matrices of elementary translation and rotation, L, d are the parallelogram geometrical parameters, q i j are the passive joint coordinates, θ i j are the virtual joint coordinates, i = 1, 2 defines the number of the kinematic chain, j identifies the coordinate number within the chain. It should be noted that using this notation, the closed loop equation can be expressed as

T 1 = T 2 .
For further computational convenience, the homogenous matrix equations (1) may be also rewritten in the vector form as

t = g i (q i , θ i ), (2)
where vector t = (p, ϕ) is the output frame pose, that includes its position p = (x, y, z) T and orientation ϕ = (ϕ x , ϕ y , ϕ z ), vector q i = (q i1 , q i2 ) contains the passive joint coordinates, and vector θ i = (θ i1 , ..., θ i6 ) collects all virtual joint coordinates of the corresponding chain. Using the above assumptions and definitions, let us derive the stiffness model of the parallelogram-based linkage.

Static equilibrium

Let us derive first the general static equilibrium equations of the parallelogram assuming that it is virtually divided into two symmetrical serial kinematic chains with the geometrical and elastostatic models t = g i (q i , θ i ), τ i = K i θ i , i = 1, 2 where the variables θ i correspond to the deformations in the virtual springs, K i is the spring stiffness matrix, and τ i incorporates corresponding forces and torques. Taking into account that the considered mechanism is under-constrained, it is prudent to assume that the end-points of both chains are located at a given position t and to compute partial forces F i corresponding to the static equilibrium.

To derive equations, let us express the potential energy of the mechanism as

E(θ 1 , θ 2 ) = 1 2 2 ∑ i=1 θ T i K θ i θ i (3)
In the equilibrium configuration, this energy must be minimised subject to the geometrical constraints t = g i (q i , θ i ), i = 1, 2. Hence, the Lagrange function is

L(θ 1 , θ 2 , q 1 , q 2 ) = 1 2 2 ∑ i=1 θ T i K θ i θ i + 2 ∑ i=1 λ T i (t -g i (θ i , q i )) (4)
where the multipliers λ i may be interpreted as the external forces F i applied at the end-points of the chains. Further, after computing the partial derivatives of L(...) with respect to θ i , q i , λ i and setting the derivatives to zero, equations of the static equilibrium can be presented as

J T θ i λ i = K i θ i ; J T qi λ i = 0; t = g i (q i , θ i ); i = 1, 2 (5)
where J θ i = ∂ g i (...)/∂ θ i and J qi = ∂ g i (...)/∂ q i are the kinematic Jacobians. Here, the configuration variables θ i , q i and the multipliers λ i are treated as unknowns. Since the derived system is highly nonlinear, the desired solution can be obtained only numerically. In this paper, it is proposed to use the following iterative procedure

λ ′ i q ′ i = J θ i K -1 θ i J T θ i J qi J T qi 0 t -g i + J qi q i + J θ i θ i 0 ; θ ′ i = K -1 θ i J T θ i λ ′ i (6)
where the symbol " ′ " corresponds to the next iteration. Using this iterative procedure, for any given location of the end-point t, one can compute both the partial forces F i and the total external force, which allows to obtain the desired "forcedeflection" relation F = f(t) for any initial unloaded configuration.

Stiffness matrix

To compute the stiffness matrix of the considered parallelogram-based mechanism, the obtained "force-deflection" relations must be linearized in the neighborhood of the static equilibrium. Let assume that the external forces F i , i = 1, 2 and the endpoint location t of both kinematic chains are incremented by some small values δ F i , δ t and consider simultaneously two equilibriums corresponding to the state variables (F i , θ i , q i , t) and (F i + δ F i , θ i + δ θ i , q i + δ q i , t + δ t). Under these assumptions, the original system (5) should be supplemented by the equations

(J θ i + δ J θ i ) T (λ i + δ λ i ) = K θ i (θ i + δ θ i ) ; (J qi + δ J qi ) (λ i + δ λ i ) = 0; t + δ t = g i (q i , θ i ) + J θ i δ θ i + J qi δ q i ; (7)
where δ J θ i and δ J qi are the differentials of the Jacobians due to changes in (θ i , q i ).

After relevant transformation, neglecting high-order small terms and expanding the differentials via the Hessians of the function Ψ i = g i (q i , θ i ) T λ i :

H (i) qq = ∂ 2 Ψ i /∂ q 2 i , H (i) qθ = ∂ 2 Ψ i /∂ q ∂ i θ i , H (i) θ θ = ∂ 2 Ψ i /∂ θ 2 i
, the system of equations may be rewritten as

J θ i δ λ i + H (i) qθ δ q i + H (i) θ θ δ θ i = K θ i δ θ i ; J qi δ λ i + H (i) qq δ q i + H (i) qθ δ θ i = 0; J θ i δ θ i + J qi δ q i = δ t (8)
After analytical elimination of the variable δ θ i and defining k

(i) θ = (K θ -H (i) θ θ ) -1 , one can obtain a matrix equation δ λ i δ q i = J θ i • k (i) θ • J T θ i J qi + J θ i • k (i) θ • H (i) θ q J T qi + H (i) qθ • k (i) θ • J T θ i H (i) qq + H (i) qθ • k (i) θ • H (i) θ q -1 • δ t 0 (9)
which yields the linear relation δ λ = K ci δ t defining the Cartesian stiffness matrix K ci for each kinematic chain. Taking into account the architecture of the considered mechanism, the total stiffness matrix may be found as K c = K c1 + K c2 . These expressions allow numerical computation of the Cartesian stiffness matrix for a general case, both for loaded and unloaded equilibrium configuration. However, in the case of unloaded equilibrium, the above presented equations may be essentially simplified:

K p (q) = 2             K 11 0 0 0 0 0 0 K 22 0 0 0 K 26 0 0 0 0 0 0 0 0 0 K 44 + d 2 C 2 q K 22 4 0 d 2 S 2q K 22 8 0 0 0 0 d 2 C 2 q K 11 4 0 0 K 26 0 d 2 S 2q K 22 8 0 K 66 + d 2 S 2 q K 22 4             , (10)
where C q = cos q, S q = sin q, S 2q = sin 2q, q = q 1i (Figure 1b,c), K i j are the elements of the 6 × 6 stiffness matrix of the parallelogram bars K b which are assumed here to be the only elements of the linkage that posses non-negligible stiffness. In the following example, this expression is evaluated from point of view of accuracy and applicability to stiffness analysis of the parallelogram-based manipulators.

Illustrative example

To demonstrate validity of the proposed model and to evaluate its applicability range, let us apply it to the stiffness analysis of the Orthoglide manipulator (Figure 1a). It includes three parallelogram-type linkage where the main flexibility source is concentrated in the bar elements of length 310 mm. Using the FEA-based software tools and dedicated identification technique (Pashkevich et al., 2009c), the stiffness matrix of the bar element was evaluated as

K b =          2.20 • 10 4 0 0 0 0 0 0 1.81 • 10 1 0 0 0 -2.84 • 10 3 0 0 7.86 • 10 1 0 1.25 • 10 4 0 0 0 0 3.48 • 10 4 0 0 0 0 1.25 • 10 4 0 2.66 • 10 6 0 0 -2.84 • 10 3 0 0 0 5.85 • 10 5          , (11)
where for linear/angular displacements and for the force/torque are used the following units:

[mm], [rad], [N], [Nmm]
. This bar-element was also evaluated with respect to the structural stability under compression, and the FEA-modelling produced three possible buckling configurations presented in Figure 2. It is obvious that these configurations are potentially dangerous for the compressed parallelogram. However, the parallelogram may also produce some other types of buckling because of presents of the passive joints. For the parallelogram-base linkage incorporating the above bar elements, expression (11) yields the following stiffness matrix

K p = 2         
         , (12)
corresponding to the straight configuration of the linkage (i.e. to q = 0). Being in good agreement with physical sense, this matrix is rank deficient and incorporates exactly one zero row/column corresponding to the z-translation, where the parallelogram is completely non-resistant due to specific arrangement of passive joints. Also, as follows from comparison with doubled stiffness matrix (11) that may be used as a reference, the parallelogram demonstrates essentially higher rotational stiffness that mainly depends on the translational stiffness parameters of the bar (moreover, the rotational stiffness parameter K 55 of the bar is completely eliminated by the passive joints). The most significant here is the parallelogram width d that explicitly presented in the rotational sub-block of K p .

To investigate applicability range of the linear model based on the stiffness matrix (10), it was computed a non-linear "force-deflection" relation corresponding to the parallelogram compression in the x-direction. This computation was performed using an iterative algorithm presented in Section 3. As follows from obtained results, the matrix (10) ensures rather accurate description of the parallelogram stiffness in small-deflection area. However, for large deflections, corresponding VJM-model detects a geometrical buckling that limits applicability of the matrix (10).

Similar analysis was performed using the FEA-technique, which yielded almost the same "force-deflection" plot for the small deflections but detected several types of the buckling, with the critical forces may be both lower and higher then in the VJM-modelling. Corresponding numerical values and parallelogram configurations are presented in Figure 3.

The developed VJM-model of parallelogram was verified in the frame of the stiffness modelling of the entire manipulator. Relevant results are presented in Table 1. They confirm advantages of the parallelogram-based architectures with respect to the translational stiffness and perfectly match to the values obtained from FEAmethod. However, releasing some assumptions concerning the stiffness properties of the remaining parallelogram elements (other than bars) modifies the values of

Conclusions

The paper presents new results in the area of enhanced stiffness modeling of parallel manipulators with parallelogram-based linkage. In contrast to other works, it explicitly takes into account influence of the loading and allows both computing the non-linear load-deflection relation and detecting the buckling phenomena that may lead to manipulator structural failure. There is also derived an analytical expression for relevant rank-deficient stiffness matrix and proposed equivalent bartype pseudo-rigid model that describes the parallelogram stiffness by five localized mutually coupled virtual springs. These results are validated for a case study that deals with stiffness modeling of a parallel manipulator of the Orthoglide family, for which the parallelogram-type linkage was evaluated using both proposed technique and straightforward FEA-modeling.

Fig. 1

 1 Fig. 1 Parallel manipulators with parallelogram-based linkages.

Fig. 2

 2 Fig. 2 Critical forces and buckling configurations of a bar element employed in the parallelograms of Orthoglide.

Fig. 3

 3 Fig. 3 Force-deflection relations and buckling configurations for parallelogram compression (modelling methods: VJM, FEA, and LIN -linear model with proposed stiffness matrix).

Table 1

 1 Stiffness parameters of the Orthoglide manipulator for different assumptions concerning parallelogram linkage. the translational and rotational stiffness. The latter gives a new prospective research direction that targeted at more general stiffness model of the parallelogram.

	Model of linkage		VJM-model		FEA-model
		K tran	K rot	K tran	K rot
		[mm/N]	[rad/N]	[mm/N]	[rad/N]
	2x-bar linkage	3.35 • 10 3	0.13 • 10 6	-	-
	Parallelogram linkage
	(solid axis)	3.23 • 10 3	4.33 • 10 6	3.33 • 10 3	4.10 • 10 6
	Parallelogram linkage
	(flexible axis)	3.08 • 10 3	4.06 • 10 6	3.31 • 10 3	4.07 • 10 6

Acknowledgements

The work presented in this paper was partially funded by the Region "Pays de la Loire" (project RoboComposite).


Mohamed Kamel Doraï

PALESTINIAN REFUGEE CAMPS IN LEBANON MIGRATION, MOBILITY AND THE URBANIZATION PROCESS

Introduction

United Nations High Commissioner for Refugees (UNHCR) and United Nations Relief and Works Agency for Palestine Refugees in the Near East (UNRWA), make a clear distinction between refugees in camps, and refugees outside camps, whether in urban areas or rural settlement. This categorization is linked to the implementation of their policies of assistance and eventual protection. The rapid development of refugee movements in the Middle East since the 1990's, such as Sudanese in Cairo or Iraqis in Damascus and Amman, and the permanence of the Palestinian question invite us to reconsider the modes of settlement of refugee groups. Refugee camps, that focus attention of many observers are not, according to UNHCR statistics, the main location of refugees in the world. Refugee camps gather around 25% of the whole refugee population worldwide. In the Palestinians case, the situation is quite similar to the other refugees, even is the proportion of refugees living in camps is higher -around one third of the registered refugees are still living in camps -and variable depending on the area where they are settled. In reality, the number of refugees living in camps is higher if we add the refugees living in unofficial camps or informal gathering that do not benefit from the same assistance and services from international organizations or host states.

The boundary between camps and gatherings is blurring, and some refugees live in camp-like situation.

Palestinians are the oldest refugee community in the region since World War 2. Today in Lebanon, more than 50% of the registered refugees still reside in the UNRWA camps, which symbolize the vulnerability of the Palestinian presence in Lebanon. On the one hand, Palestinians face a wide range of legal, political and social constraints in Lebanon. On the other hand, from the Cairo Agreement in 1969 to the Israeli invasion in 1982, Palestinians have enjoyed in this country a freedom of action that no other host state ever gave them. Sixty years of exile have generated new forms of local integration, especially in urban areas where refugee camps are now part of the cities that surround them. Since late 1940's, refugee camps transformed deeply from tents to highly dense built up areas. As mentioned by Jihane Sfeir (2008: 208), since the 1950's the places where Palestinians settled in the suburbs of Beirut where not only Palestinians areas, but poor and segregated neighbourhoods where marginalized migrants, such as Syrians, Kurds or Armenians also settle. Parallel to the urbanization process, refugee camp population has profoundly changed due to emigration, internal displacement and social mobility. The Palestinian experience in Lebanon challenges the dichotomy between urban refugees vs. camp dwellers, and leads us to reconsider the boundary between these two categories.

After discussing the problematic use of the categories of urban refugees and camp dwellers, the urbanization process of Mar Elias camp is developed, to analyze the role of migration and mobility in the redefinition of the boundaries between the camp and the city. This article is based on semi-structured interviews with Palestinian refugees living or working in Mar Elias refugee camps and foreign workers living in the camp conducted between 2005 and 2007. A total of 20 interviews have been made (15 with Palestinians and 5 with foreign workers) as well as regular informal discussion with key informants in Mar Elias and Shatila. Observations on daily mobility of the camp dwellers and outside visitors have also been made during regular visits to the camp and it surroundings during the same period 2 .

Urban refugees vs. camp dwellers

Blurring categories

Refugee studies have produced since the 1970's a wide range of categories to describe refugee movement or settlement such as urban refugees, camp dwellers, self settled refugees, etc. [START_REF] Black | Refugees and displaced persons: geographical perspectives and research directions[END_REF][START_REF] Fábos | Urban refugees: Introduction[END_REF][START_REF] Kunz | Exile and Resettlement: Refugee Theory[END_REF][START_REF] Rogge | A Geography of Refugees: some Illustrations from Africa[END_REF][START_REF] Zetter | More Labels, Fewer Refugees: Remaking the Refugee Label in an Era of Globalization[END_REF]. Recently, researchers have shown a growing interest on the issue of urban refugees in the world, pointing mainly the problem of protection and access to services they face in the third world big cities [START_REF] Jacobsen | Editorial Introduction. Refugees and Asylum Seekers in Urban Areas: A livelihoods Perspective[END_REF] 3 . The differences between urban refugee and camp dwellers have been studied, but the transformation of refugee camps into urban areas spaces has not been studied as such except in a few cases [START_REF] Agier | De nouvelles villes, les camps de réfugiés -éléments d'ethnologie urbaine[END_REF][START_REF] Michel | Between War and City: Towards an Urban Anthropology of Refugee Camps[END_REF][START_REF] Bulle | Une urbanité dans l'épreuve : le mur de séparation à Jérusalem et ses bordures -Eléments pour une analyse des actions situées en milieu problématique[END_REF][START_REF] Hyndman | Managing Displacement : Refugees and the Politics of Humanitarianism[END_REF][START_REF] Malkki | Purity and Exile: Violence, Memory, and National Cosmology Among Hutu Refugees in Tanzania[END_REF][START_REF] Seren | Rapport de recherche : L'urbanisation des camps de réfugiés dans la bande de Gaza et en Cisjordanie[END_REF]).

The classical distinction operated between refugee camps dwellers and urban refugees is mainly an operational one produced by international organisations. This categorization has to be differentiated from the evolution of refugee camps and from the practices developed by the refugees themselves. Refugee camps are not closed areas even when they are geographically isolated, they can be connected to a wider urban environment through mobility or transnational connections such as remittances [START_REF] Doraï | Palestinian Emigration from Lebanon to Northern Europe: Refugees, Networks, and Transnational Practices[END_REF][START_REF] Fresia | Des "réfugiés-migrants" : Les parcours d'exil des réfugiés mauritaniens au Sénégal[END_REF][START_REF] Horst | Xawilaad: The Importance of Overseas Connections in the Livelihoods of Somali Refugees in the Dadaab Refugee Camps of Kenya[END_REF].

In the Palestinian case, due to the rapid urbanization of the Middle Eastern countries most of the refugee camps are part of the capitals and main cities in their respective countries or host regions, as mentioned by Ishaq Al-Qutub :

In the case of the Palestinian Arab refugee camps -such as those existing in the occupied West Bank and Gaza Strip, Lebanon, Jordan and Syria -they are prevailing features of the urban structures of these states. […] The camp cities, both small and large, can be considered as urban conglomerations in the demographic and ecological sense. […] These cities represent a unique urban pattern, which has special features, problems, structures, and consequently requires a special classification in the study of urban societies in the Middle East. (Al-Qutub 1989: 91, 107) Refugee camps can become parts of urban areas or may become themselves urban centres due to their demographic weight and the variety of activities developed, such as socio-economic activities, political centres of decision making, and the central role they play in the Palestinian society in exile. In some specific cases, the categorization depends upon the institution in charge of the refugees. For example in Damascus, Yarmouk is considered a refugee camp by the Syrian authorities whereas UNRWA does not recognize it as such. At an operational level (international responsibility, access to services, legal context, etc.) a clear distinction exists between camp dwellers and urban refugees. But the analysis of the geographical development of refugee camps in their local context leads us to consider the refugee camps as urban areas. Camps tend to look increasingly like that of the poorer informal urban areas nearby. The temporal dimension of the Palestinian exile is also a key element to take into consideration. Sixty years of exile have led to a specific relation with their host societies, with a strong local integration linked to a rapid urbanization of the different host countries parallel to a strong segregation due to the socio-political and legal context.

Place and mobility

The categories of urban refugee and camp dweller are often considered through their residency place and not according to their short and/or long term spatial practices.

Mobility is a key practice to take into consideration because it reveals the complementarities of different urban spaces, and the different kinds of relations they have. Refugees living in camps experience different scales of mobility (daily movements, temporary and long term emigration, forced displacement, etc.) and develop a wide range of practices (economic, political, cultural and/or social activities) that extends beyond the camp's boundaries. Mobility and migrations have to be understood in their different temporalities [START_REF] Rémy | Mobilités et ancrages : vers une autre définition de la ville[END_REF]. On the long term, the refugee camp population changes, some refugees leaving the camp to settle elsewhere, newcomers come to settle in the camp for a variety of reasons that will be developed.

Different generations of refugees have experienced life in exile, each of them having a specific relation to the camp, due to specific socio-historical context. Individual trajectories also contribute to blur the distinction between urban refugees and camp dwellers. Many refugees reside successively inside and outside camps during their life to access different kind of resources (Hyndman 2000: 158-162;[START_REF] Fresia | Des "réfugiés-migrants" : Les parcours d'exil des réfugiés mauritaniens au Sénégal[END_REF].

Refugee camps themselves host temporarily or more permanently different waves and groups of refugees. New immigrant communities also settle in the camps and/or around the camps. In Damascus for example, Iraqis refugees settle in neighbourhoods composed of Palestinian refugees, internally displaced Syrians from the Golan and internal migrants coming from the countryside. The same pattern can be found in most of the Middle Eastern cities such as Beirut, Amman or Cairo (Al Ali 2004: 9-12).

Urban margins, where refugees and migrants settle, are not disconnected from the urban dynamics of the surrounding cities.

"The city is never simply the spatial organization of the mosaic of territories: territories of second settlements upset sooner or later this organisation, to produce more confused moral, composed of cultural hybrids themselves produced by successive migrant population belonging to the same community or to different ones" 4 (Joseph 1998: 93) Refugee camps develop ties with their urban environment and even if they are segregated and marginalized, they are part of the urban settings that host them. Daily mobility crosses camp boundaries. Refugees can live in a camp and work or study outside the camp, or vice versa. Urban practices -such as shopping, visiting family or friends, accessing services or assistance -often lead refugees to go in and out of the camps and to frequent other neighbourhoods.

The camp and the city: local forms of urban integration

Since the arrival of the refugees in 1948, the Palestinian presence in the Lebanon has always been problematic, and the Lebanese authorities oscillated for a long time between support for the Palestinian cause and the will to control the Palestinian Mar Elias camp is a densely built-up area, with narrow streets, in most of them cars cannot enter. If old small house of one-floor still exist, most of the buildings are now composed of two or three floors. As it is the case for most refugee camps in Lebanon, the only way to accommodate more refugees is to add new storeys and to existing buildings and take advantage of each unoccupied squared meters inside the perimeter of the camp. The prohibition of any spatial extension of the camps since their creation has led to the existence of highly dense spaces. The camp contrasts strongly with its environment by its density in an urban environment with higher buildings separated by large streets. The quality of the building also characterizes the camp. While Palestinian buildings are irregular, most of them are in bad condition as well as the camp infrastructure (sewage system, electricity, roads, etc.). The camp appears as a pocket of poverty in an urban environment in rapid evolution since the end of the civil war, even if, according to its inhabitants it is perceived as one of the best camp in Beirut, where Palestinians with a certain level of income live. Mar Elias residents often emphasize the relative quality of the infrastructure (water and electricity) as well as the quality of life due to the small size of the camp. It is often opposed in the discourse to Shatila camp which is described as overcrowded with a very deficient infrastructure. The fact that a large proportion of the residents in Shatila are not Palestinians is also pointed out and often linked to the rise of criminal activities, which are not described in Mar Elias. Two different small commercial areas have been developed along the eastern and western boundaries of the camp. The western one is situated on the main urban highway from southern suburbs of Beirut to Ras Beirut. The activities (garage, manufacturer of furniture, grocery, etc.) are not geared towards the camp population, but to customers coming from other areas of Beirut. The eastern one is mainly composed of small groceries, fruit and vegetable sellers, etc. This area is frequented both by camp dwellers and inhabitants of the neighbouring districts. Prices are generally lower in refugee camps than in other places. The same phenomenon can be found in other camps, such as Al Buss in Tyre (south Lebanon) or Jabal Hussein in Amman (Jordan).

Mar Elias can be qualified as an urban space due to four main elements. First, the temporary camp has been transformed in a dense built-up area, composed of two to three storeys buildings connected to the infrastructures such as water, electricity, telephone, etc. Second, originally conceived as a place to accommodate Palestinian refugees and offer them a temporary shelter and humanitarian assistance, the social composition of the refugee camps have deeply changed, some refugees leaving the camp other settling in along with a growing presence of non Palestinians in certain camp such as Mar Elias or Shatila. The camp became one stage in the migratory trajectory of the Palestinians. Third, the location in or outside the camp is the result of a residential choice, linked to renting prices, work opportunities like in other neighbourhoods in Beirut where migrants settle. The camp can no longer be considered only as a place of refuge. Fourth, the development of economic activities tend integrate the camp in its local environment; Mar Elias is visited by outside non-Palestinians individuals. As mentioned by [START_REF] Michel | Between War and City: Towards an Urban Anthropology of Refugee Camps[END_REF]: "Due to their very heterogeneity, camps may become the genesis of unexpected cities, new social environments, relationships and identification". A new social environment emerges due to the high level of in and out migrations in the camp and the diversity of social backgrounds of its inhabitants since the end of the civil war in Lebanon.

The key role of migration in the evolution of the refugee camps

Migration plays a crucial role in the social evolution of the Palestinian refugee camp in Lebanon. As shown by Rosemary Sayigh, Palestinians refugees have been internally displaced several times during the Lebanese wars. Three refugee camps have been totally destroyed -and not reconstructed -during the wars. 90% of the refugees have been forced to leave their home at least once during the war (Sayigh 2004: 5). State The camp is a stage in the professional career, in the family life or in the migration process. Most of the refugees who are currently living in the camp originally came from other camps, even if they reside outside the city for long periods. The main purpose of the internal Palestinian migration towards the camps in Beirut is to find employment and improve the quality of life, such as the families settling in Shatila (Sayigh 1994: 65).

The first case illustrates the plural roles played by the refugee camp as a temporary refuge at the end of the Lebanese civil war, before being able to access to a better housing situation outside the camp. "Like all urban migrants, asylum seekers are attracted to urban centres because economic resources and opportunities, including education for their children, are concentrated there, and in cities migrants can access the social networks and ethnic enclaves that supports newcomers, and which initiate the process of integration".

My parents left

Through the settlement of international migrants, the camp is playing a role in the city Lebanon a few weeks ago. We never see them; they wake up in the morning, they go to work, and they come back at night. Sometimes they buy food here."

The house rent in the refugee camps is lower then in other places, especially compared to central Beirut where accommodation is expensive, hence attracts poor, new migrants (Sudanese, Bangladeshis, Sri Lankans, etc.). The fact that security forces do not enter the camps is an advantage to undocumented migrants who feel more protected from eviction. Thus, for a variety of reasons, a refugee camp such as Mar Elias, but also Shatila, or Fakhani -a vast Palestinian urban gathering -became host places for new migrants. The presence of recently arrived migrant workers or refugees generates a variety of interactions and complementarities with migrant population settled in Beirut for decades like the Palestinians.

Palestinian refugee camps -or areas with strong Palestinian concentration -host an increasing number of migrants since the mid-1990s. Some of them are refugees or asylum seekers as is the case of Iraqis in the Shatila camp in 2003, the majority being irregular migrant workers. Various groups of immigrants live in the camps: low income immigrant workers, irregular migrants, asylum seekers, unrecognized refugees, domestic workers, etc. Some settle there on the long term with their family, othersmainly newcomers -transit through the camp before finding employment and activating community networks of solidarities which will allow them to improve their situation and move to another neighbourhood or to pursue their migratory journey towards western countries. The presence of this non-Palestinian population in refugee camps leads us to reconsider the traditional perception of refugee camps and to view them as spaces of urban relegation. The migrant communities participate to the blurring of the boundaries between the camp and the city.

Accommodation of migrant populations in precarious situation obey certain imperatives: a relatively low cost given the low level of migrants' resources, a secure space that allows them to avoid being exposed to police controls and a rather central location that facilitate access to employment and minimize the distance between employment and residence for security reasons. The Palestinian refugee camps, as well as Palestinian gatherings, offer numerous advantages for irregular migrants. A Sudanese living in Mar Elias explains why he settled there. I left south Sudan because of the insecurity which reigns there. I moved initially towards Khartoum. I could not stay there because it was impossible to find a job, and because of the discrimination we face. I have then decided to leave. Egypt tightened its migration policy and it was not easy for us to cross the border to go there. I thus took a plane ticket for Damascus, Syria does not ask for a visa for the Sudanese. Once in Damascus, I found other Sudanese already leaving there. As there is no work in Syria, one advised me to go to Lebanon, the wages are better, and the Sudanese find there rather easily work, especially in the cleaning sector. I crossed the border illegally, by paying a smuggler and I arrived in Beirut. As I do not have residency permit I prefer to live in a refugee camp where the police does not enter. Also it is cheaper and close to my work. Forms of socio-economic complementarities emerge between migrant and refugee populations originating from different waves in these areas of urban relegation. A renting market develops in the Palestinian camps and gatherings, which satisfy the host (Palestinians) and hosted (newcomers) populations. The small grocers stores which we find in camps or the Palestinian gatherings also benefit from the presence of these customers according to the interviews with Palestinian small shopkeepers settled in these areas.

Conclusion

The Palestinian refugee camps tend to evolve by becoming integrated into the economic activity and into their urban environment. Even if they are still marginalized and segregated areas they are now part of the major cities in the Middle East. Economic activities, daily mobility, the presence of new international migrants, and the strong political and cultural significance for the Palestinian refugees, are the different elements that characterize the contemporary Lebanese refugee camps as urban settlements. Mar Elias situation is not isolated in Lebanon. Al Buss camp, in Tyre region, witnesses a similar development. It is becoming increasingly difficult to discern the western boundary of the camp. Many Lebanese Shiite families driven from the Israeli-occupied southern border zone built an informal neighbourhood next to the camp. The numerous businesses that have been established along the main road on the northern and eastern sides of the camp, developed both by Palestinians and by Lebanese, serve to integrate the outer fringe of the camp into the townscape. It is the geographical situation of the camp of Al Buss at the entrance of the city and at the crossroads of main roads that has facilitated this evolution [START_REF] Doraï | Les camp de réfugiés palestiniens d'Al Buss à Tyr : Ségrégation et précarité d'une installation durable" in Géographies[END_REF].

Most of the studies on refugees establish a clear distinction between refugees in camps and urban refugees living in cities. The Palestinian case, due to its exceptional duration and the context of rapid urbanization of their host countries, invites us to partly reexamine this dichotomy. If some refugee camps appear to be isolated from their urban environment -such as Aïn al Helweh, Borj al Shamali or Rashidiyyeh -, most of the refugee camps in Lebanon (as well as in most of the Middle Eastern cities) are now part of the main Lebanese urban areas. A diachronic analysis of the different camp's evolution should be developed to retrace their specific history and the ties with their local environment. On the one hand camps appear to be marginalized and segregated areas due to the special -and often changing -regulation and mode of controls as well as the legal status of their Palestinian residents. On the other hand, refugee camps are strongly connected to their urban environment through the daily mobility of Palestinian refugees, the growing presence of other groups of refugees and migrants (such as Syrian or Asian workers and Iraqi or Sudanese asylum seekers and refugees), and the development of commercial activities that blur the boundaries of the refugee camps, making it a part of the city.

1 Researcher -CNRS (National Centre for Scientific Research) IFPO -French Institute for the Near East, Damascus; Mohamed-Kamel.Dorai@univ-poitiers.fr 2 I would like to thank J. Abu Hawash and the Palestinian Human Rights Organization for their help and fruitful discussion all along my fieldwork.

5 Nahr al Bared is a Palestinian refugee camp established in 1949 in the northern part of Lebanon, 16 km north of Tripoli, hosting more than 30,000 individuals. Between May and September 2007, fighting between Fatah al Islam and the Lebanese army has led to the displacement of the camp population and the destruction of the camp infrastructure and houses. The camp was composed of two parts: the "old camp", the official one was totally destroyed, while the "new camp", the unofficial extension was partly destroyed. There is an ongoing project to reconstruct the camp, but until now most of the refugees are still displaced living in precarious shelters in and around Beddawi neighbouring camp and elsewhere. Some families have managed to return to the camp in prefabricated houses in March 2008. 6 Interviews in Mar Elias camp have been conducted in 2006 and 2007. I would like to thank Jaber Abu Hawash for his precious help. 7 It is estimated that two thirds of the Palestinian refugees in Lebanon are from a rural origin [START_REF] Khalidi | All that remains. The Palestinian villages occupied and depopulated by Israel in 1948[END_REF] 8 See for example [START_REF] Berthomière | Décrire les migrations internationales. Les expériences de la co-présence[END_REF] or [START_REF] Deboulet | Ethiopiennes, Philippines et Soudanais : voisinages migrants et confrontation aux sociétés d'accueil à Beyrouth[END_REF]

  political and military activities. The PLO, created in 1964, developed its institutions in its different host countries. The rise of Palestinian power, which coincided with the decline of the Lebanese state's authority, was sustained by an important institutional development in the Palestinian camps. There is a clear link between military and political empowerment of the PLO on the one hand, and the growth of Palestinian social and economic institutions as well as the development of infrastructures in camps. Since 1975, the different conflicts (civil war, Israeli invasions and Syrian interventionism) have aggravated the tensions between the PLO and its allies on one side, and certain parts of the Lebanese society and militias on the other one. After the Israeli invasion of 1982, the eviction of the PLO-leadership and militia has deeply changed the relations between Palestinians and Lebanese. The present situation of the Palestinian refugees in Lebanon is the result of sixty years of contested residence in the country. From their arrival in Lebanon until the signing of the Cairo Agreement (1969), Palestinians were marginalized by the host society and deprived of basic rights. The aim of this agreement was to regulate Palestinian -Lebanese relations. It has opened new opportunities for Palestinians, giving them the possibility to work and develop their political activities as well as their military presence in the camps. The military activities of the Palestinian commandos were allowed, theoretically in coordination with Lebanese authorities. The self organisation of the camp residents was also promoted through the creation of local committees (Klaus 2000: 57-68). Palestinian Refugees in Lebanon, a marginalized population. Palestinian refugees in Lebanon are currently one of most underprivileged of the diaspora communities in the Middle East. They suffer from a wide range of legal restrictions. The Lebanese legislation is very strict regarding Palestinians since the arrival of the first refugees in 1948. This legislation has been modified according to the agreements -and the disagreement -, between the PLO and the various Lebanese governments. It limits the access of the refugees to the labour market, to the education system, to the international mobility, to the social and health system, as well as to the formal property market (Al-Natour 1997; Said 2001). The legal status of the Palestinians has important implications on the socio-spatial organization of this community in the Lebanon. The refugees tend to be confined in the informal sector or in the least profitable labour activities which do not require an official work permit. Furthermore, the departure of the PLO in 1982 deprived a number of refugees of jobs developed by the strong presence of the Palestinian political institutions in Lebanon. In the fragile Lebanese economic context since the end of the Civil war, the Palestinians are marginalized on the informal Lebanese labour market with the arrival of low-cost foreign manpower (Jureidini 2003; Chalcraft 2006). Today the situation is more and more precarious for Palestinians and due to the political confrontation between the current parliamentary majority and the opposition. Although the Palestinians are not strictly speaking part of the internal political conflict in Lebanon, the question of their permanent settlement in Lebanon (tawtin, "implantation") as well as the presence of weapons in camps challenges the political neutrality of the Palestinians, in a conflict with a strong regional dimension that tends to implicate them de facto. Since 2003, some improvements -most of them very marginal -show a possibility of partial normalization of the Palestinian-Lebanese relations. For example, relations with Lebanon improved with the partial -and controversial -abolition of the ban on working in certain job sectors. In December, 2005 posters produced by the organizations of the Palestinian civil society to claim the right to work, to own a property, for security in Lebanon beside the right of return, indicate the evolution of some parts of the Palestinian civil society in its relation to its host society. The destruction of Nahr al Bared refugee camp in Tripoli symbolizes the reversibility of the Palestinian settlement in the country 5 . The case of Nahr al Bared shows that refugee camps -as well as urban informal areas -because they are considered as temporary spaces can always be subject to destruction or unilateral state intervention. Camps in Lebanon are also under strict control of the authorities. The situation varies between the camps considered, depending on "security" reasons. The camps in the South are under strict control because of the presence of Palestinian political parties and their militias (such as Rashidiyyeh camp south of Tyre), or jihadist groups (such as Aïn al Helweh near Saïda). Mar Elias, the focus of this chapter, is not controlled by the Lebanese army. Other camps in Saïda, Tyre or Tripoli are under the strict control of the army, non Palestinians have to ask for an authorization to enter some camps. The control of the entrance of refugee camps plays an important role in the degradation of the Palestinian housing situation. These measures have been applied in a very strict way from the beginning of the 1990's until 2005. Some camps still have a single entrance controlled by the Lebanese Army. This only applies to the camps in the south and Nahr al Bared in the north. The other camps are not surrounded by the Lebanese army. Cars can be searched and it was forbidden to bring any building material inside without a previous authorization. These limitations prevent any maintenance or renovation of houses. Another consequence is the difficulty for the young couples to settle in the camps in single-family units, because they cannot build new houses. The control of the camp materializes in the space by the physical presence of the army at the entry of the camp, and the closure of all the secondary roads that connect the camp with its immediate spatial environment, leaving only a single entry for the cars. These limitations were sometimes abolished and new constructions appear in the camps, depending on the relations between Palestinian organizations and the Lebanese State. Since spring 2005, some camps, such as Al Buss in Tyre, witness an important densification of buildings, numerous families taking advantage of the abolition of the limitations to add a storey to their house, others to build new rooms. Yet, these limitations can be again imposed on the refugees by decision of the Lebanese authorities (Doraï 2006). Mar Elias, a camp in the city Mar Elias camp was been created in 1952 by the UNRWA, on a small area of 5,400 m² in the South West of the Beirut municipality. In 1958, according to UNRWA figures, the camp hosted 449 registered refugees and 616 in 2006. The first inhabitants of the camp arrived by boat following the creation of the State of Israel in 1948. Most of them had an urban background mainly coming from two Palestinian cities, Haïfa and Jaffa. The vast majority of them was Christian and they have been first accommodated in the Greek Orthodox convent of Mar Elias. In 1952, they have resettled in a camp set up in the wood close to the convent. The large movement of emigration -especially GreekOrthodox families -characterizes this camp, which population remained stable since its creation(Sfeir 2008: 238-240).Today, the camp is situated at the crossroad between Beirut Southern suburbs of Bir Hassan and Ouzaï, Ras Beirut and Cola intersection. Most of the refugee camps are located far from the city centres. Mar Elias is located in privileged area at the crossroad between the one of the main centre of Beirut (Ras Beirut) and the southern suburbs that gather poor internal migrants and internally displaced people from the south of Lebanon, as well as more recently arrived refugees such as Sudanese and Iraqis. Cola intersection is one of the most important transport hub in Beirut that connect to most neighbourhoods of Beirut and all the southern part of Lebanon. This central location facilitates circulation both for camp dwellers who can easily reach other parts of Beirut, and for people from outside the camp wishing to come whether they are from Beirut or from other regions of Lebanon, especially the Southern coastal region.

  The camp's location has facilitated the development of different sets of activities. A number of Palestinian NGOs are present in the camp, such as Beit Atfal Al-Soumoud, Palestinian Martyrs' Association, Ghassan Kanafani Cultural Foundation, Al Inaach, Palestinian Red Crescent Society, Aidoun, Palestinian Human Rights Association, etc. They developed their activities in the camp because of its central location in Beirut and the relative freedom they can enjoy as being Palestinian NGOs, even if they must have a Lebanese to head them, who can be from Palestinian background. The camp also host offices from most of the Palestinian political parties. Commercial activities developed in the camp, some dedicated to the Palestinian population of the camp and others to Lebanese customers living in the neighbourhood or workers employed in the surrounding manufacturing businesses. A typical example is the experience of the Palestinian refugee who first opened a restaurant in the camp before deciding to reside there 6 . I arrived in Mar Elias in 1992. I lived previously in Tell al Zaatar and Damour. I bought a house in the camp and opened a small restaurant. In 1994 I came to live here. I opened the first restaurant in the camp. It was a good economic opportunity. Most of the customers come from the enterprises around the camp as well as daily workers working in the construction sector. Eating here is less expensive for them than in other restaurants, that's why they come here.

  instability and spatial mobility are two major aspects of Palestinian life in Lebanon. This leads to blur the distinction between camp and urban settlement. If we consider Palestinian journeys over the long term (i.e. individual, familial and intergenerational itineraries), most of the refugees have experienced life in and outside the camp. International emigration also plays an important role in creating a dynamic house and rental market, and bringing in new migrants who move in and settle in the camps and Palestinian gatherings. Palestinian migrations Due to high level of international and internal emigration, only a few of the families that settled in Mar Elias camp in 1952 is still there. They have been gradually replaced by Palestinians coming from other refugee camps like Tell al Zaatar in Beirut after its destruction in 1976, or Rashidiyyeh south of Tyre. Other refugees settled in Mar Elias in 1991 after having been internally displaced during the civil war. The high level of both internal and international emigration can be explained by three factors. First, a large part of the Palestinian refugees who came to the Mar Elias Convent had Lebanese family ties through marriage. Their local integration has thus been eased by the kinship connection they had with their host society. Second, their urban origin from Haïfa and Jaffa has facilitated their integration in the city, contrary to most of the refugees coming from a rural background 7 in other camps such as Shatila or Burj al Barajneh in the Beirut urban area. Third and last, this point is related to the previous one, their high education levels have lead to higher international emigration. A combination of these interrelated factors -urban origin, religious affiliation, family networks -has facilitated their movement from the camp to other Beirut neighbourhoods and abroad. The Palestinian population currently living in Mar Elias camp is the result of different migratory experiences, be it internally displaced population by war and/or other forms of internal migration (economic opportunity, renting market, geographical location). The case-studies that follow show the plural role of the camp in the life-long migration process. Most of the refugees interviewed between 2005 and 2008 have lived successively in and outside refugee camps during their life. Their migratory experience is also made up of a combination of forced displacement -for example after the destruction of Tell al Zaatar -and search of a better economic and/or housing situation.

  Palestine in 1948 and settled in Tell al Zaatar. In 1976, when the camp was destroyed we left to settle in Damour. We stayed there until 1982. Then I went back to Beirut where I squatted a free apartment in the Raouché area. I opened my small workshop in Verdun, but rent price increased and I began to look for another place. At the end of the civil war in 1991, I had to leave the apartment and I decided to buy one in Mar Elias from a Palestinian who left to Denmark. I chose Mar Elias for two main reasons. First, it was a central location where my previous customers could easily come. Second, prices for accommodation and for my workshop were lower than in other parts of the city because Mar Elias is a refugee camp. Buying an apartment in a refugee camp is easy for a Palestinian, you just have to register the transaction at the popular committee and then at the UNRWA. As I have children I decided to move with my family and rent an apartment outside Beirut. I rent mine in Mar Elias to a Palestinian NGO for 200 USD per month and now I live in a largest apartment in Naameh [a village south of Beirut] for the same price. I come here everyday to work. This skilled workman has combined his residency outside the camp -while keeping a property in the camp -and his economic activity inside the camp. The end of the civil war and the prohibition of buying properties for Palestinians have generated attraction towards the camps, while it is impossible to buy a house in other parts of the city. The legal discrimination faced by Palestinians in Lebanon leads them to buy property in the refugee camps where they can access some of the basic rights denied them outside the camp like property ownership and paid work, as mentioned in this second case. I was working in the 1970's for the PLO in Rashidiyyeh camp then I moved to the southern suburbs of Beirut in the 1980's. I stayed in Haret Hreik until 1987 and then I moved to Raouché. I came to Mar Elias because I was working for a Palestinian political organization here in 1997. As I was displaced during the war, the Ministry of Displaced gave me 7,000 USD at the beginning of the 1990's. I bought a house in the camp with this money. In parallel, I opened a grocer's shop in the camp. I rent the store for 100 USD per month. I appreciate to live in Mar Elias, it's a quiet area with a good infrastructure compared to the other camps in Beirut. Due to the fragile political environment in Lebanon, some Palestinians prefer to live in a camp, where most of the inhabitants are Palestinians. Along with the relative low prices for accommodation, camps are still playing a strong symbolic role in the Palestinian society in exile, attracting refugees as it is expressed in this third case. I am originally from Tell al Zaatar. When the camp was destroyed I left to Damour and then to Saida. In 1988 I came back to Beirut to find employment. After working in different sectors I decided to settle in Mar Elias because it is a quite place with a central location. In 2000, I bought my shop for 14,000 USD and I work as a hairdresser. Most of the customers come from outside the camp because they know me and also because it is cheaper here. With four other Palestinians we bought a piece of land in the middle of the camp and built a four-storey building. Each apartment will cost us around 30,000 USD. I decided to buy something here because my mother wanted to live with Palestinians in a quiet area. We feel more secure to live in a Palestinian environment. One of his employees is a Palestinian from Iraq who came back in Lebanon in 1999 when the situation there deteriorated. The growing foreign presence in the camps Parallel to the Palestinian secondary migration towards camps, these areas has witnessed the arrival of Palestinian refugees coming from other parts of Lebanon as well as arrivals of few Lebanese citizens and Syrian, Asian and African immigrants who settled with the end of the civil war at the beginning of the 1990's. Migrants in transit, staying for longer or shorter periods in Beirut, tend to settle in the marginalized "popular" neighbourhoods of the city. Their presence is visible in the public space, through the development of businesses with bilingual signboards on the streets and by the periodic influx of migrant populations -mainly housemaids but also construction workers -on Sundays in certain neighbourhoods like Dawra (East of Beirut). This attachment of migrants to specific urban areas is stressed by Karen Jacobsen (2006: 276):

  as a host area for newcomers, mainly poor immigrants. The camp has thus hosted different waves of refugees and immigrants, like other deprived areas of the southern and eastern suburbs of Beirut metropolitan area. 8 These migratory patterns are crucial parts of the urban dynamics. The camp plays the role of temporary "host space" for waves of migrants settling in the city for different reasons. Palestinian refugees living in poor, low -income areas are well placed to play host the new immigrants. Deprived of the right to work in Lebanon, they can earn money from renting property. Some of them decide to add a new floor on the top of their house and rent it to migrants for 100 to 150 $ per month. It gives them a supplementary income. A Palestinian owner in Mar Elias camp summarizes the reason why he decided to extend his apartment and rent it. "As I am Palestinian I am not allowed to work legally in Lebanon. In the mid-1990's I decided to build a supplementary floor on the top of my house. I constructed a two-rooms apartment plus one single room separated by a small terrace. This brings me around 200 USD per month. The room is rented by two Sri-Lanki domestic workers and the apartment by a Sudanese family with two children." Thus, because many Palestinian families have emigrated abroad or changed their place of residence in Lebanon -when they have enough money many refugee families leave the camp to rent apartments in the city -some houses are free to be rented to other families. A Palestinian grocer in Fakhany explains: "My sister and her husband own an apartment in the neighbourhood. They left for Denmark in 2005, where they live now. They left their apartment for rent and, as I am living here, I manage the renting for them. Presently a group of six young Sudanese share the apartment for 200 USD. They just arrived in

A special issue of the Journal of Refugee Studies on Urban refugees (vol. 19, n°3) has been published in September 2006.

"La ville n'est jamais simplement l'organisation spatiale de la mosaïque de territoires : les territoires de deuxième implantation viennent tôt ou tard bousculer cette organisation pour fabriquer un moral bien plus confus, composés d'hybrides culturels produits par la succession des populations migrantes, appartenant à la même communauté ou à des communautés différentes".


Beatriz M Funatsu
email: beatriz.funatsu@lmd.polytechnique.fr

Vincent Dubreuil

Chantal Claud

GDR 2839 MSG-ATR

We examine the diurnal cycles of deep convection and precipitation in the Southern Amazon region, using mainly the Advanced Microwave Sounding Unit, contrasting a year of drought (2004)(2005) with a "normal" year (2005)(2006). MODIS 550nm aerosol optical thickness data and TRMM 3B42 precipitation products were also used to investigate whether a significant relationship between these two parameters can be identified. We use a simple, tractable method to identify convective areas based on humidity channels of AMSU. We found through comparisons with independent ground precipitation data that the diurnal cycle of convection in regional scale can be realistically described by this method. We observe that deep convective areas have a slow build-up early in the rainy season of the dry year, and rapid in the normal year, and in both cases, convection starts around midday. As the rainy season advances, the peak hour of maximum convective activity shifts towards mid-afternoon, and eventually spreads towards the night. During the peak of rainy season (December through February) convective activity was up to 12% larger in the "normal" year. Convective overshooting was found to be 50% less frequent in the drought year compared to the following year. A simple spatial correlation analysis of MODIS 550nm and TRMM 3B42 monthly mean data showed that by the end of dry season, aerosol loading and precipitation amounts have a positive (albeit low) significant correlation.

I. INTRODUCTION

The Amazon region is the largest continuous tropical ecosystem in the world, and plays an important role in controlling the climate in the tropics. However, its southern edge has been changing dramatically since the 70's due to the expansion of the "pioneer frontier", with large sections of the forest being replaced by agricultural fields or pastures for cattle grazing. Biomass burning is increased by the end of the dry season for clearing areas and preparing the soil for seeding. The impact of increased aerosol loading and transformation of land cover on convection and precipitation are not well known, especially so in regional and local scales. In order to identify anthropogenic-related changes it is imperative to know the climatology and natural variability of the climate system. This work thus provides a description of the diurnal cycle of convection and precipitation in Southern Amazon, particularly, for the state of Mato Grosso, in central Brazil, stressing the similarities and differences between a year of extreme drought (2005) and a "normal" year (2006).

II. DATA AND METHODS

In the tropical region, deep convection and precipitation are key elements. Measurements of these quantities in the Amazon region rely nearly entirely on satellite-derived data, due to the scarcity of ground measurements. In particular, the diurnal cycle of convection is generally described with infrared data from geostationary satellites, due to their high temporal sampling. In the present work, we choose alternatively to use data from the Advanced Microwave Sounding Unit (AMSU), onboard NOAA-15 through -17 satellites covering the period July 2004 -June 2006. Notice that the rainy season in the southern part of Amazon starts around October and ends towards April, while dry season runs from June through September. In addition to AMSU, we use as well the daily 3-hourly precipitation products from the Tropical Rainfall Measurement Mission (TRMM) 3B42 with a resolution of 0.25°x0.25°, and daily and monthly MODIS 550nm 1°x1° aerosol optical depth, for the same period. The target area is bounded by [18°S, 8°S] [63°W, 50°W], which comprised the entire state of Mato Grosso, Brazil, and include immediate neighboring regions.

Deep convective areas were identified using a criterion proposed by [START_REF] Hong | Detection of tropical deep convective clouds from AMSU-B water vapor channels measurements[END_REF], based on the three water vapor channels (183.3 ± 1, 183.3 ± 3, 183.3 ± 7 GHz) of AMSU-B. The criterion ∆T 17 ≥ 0 K, ∆T 13 ≥ 0 K, ∆T 37 ≥ 0 K (i.e., differences between water vapor channels, e.g., ∆T 17 = difference between brighteness temperatures of channels 183.3 ± 1 and 183.3 ± 7) is able to discriminate convective clouds, while convective overshooting is detected whenever ∆T 17 ≥ ∆T 13 ≥ ∆T 37 > 0 K. AMSU-B data was first re-sampled to a grid of 0.25°x0.25°; then the relative area of deep convection (DC) and convective overshooting (COV) relative to the area covered by the satellite pass for every available hour was computed. Because we use data from the three NOAA KLM series (-15 through 17) it provides a 6-daily coverage of the target region. Figure 1a shows the number of passes per hour for each month (from July 2004 through June 2006); some months do not contain information for some hours. Therefore, we opt to make a 3 hour running mean for every day to obtain the daily cycle, and then we average over the total number of measurements in the month.

III. RESULTS

a. Diurnal cycles of Deep Convection and Convective Overshooting

The diurnal cycles of DC is displayed in Figure 1b. We observe that for both years, there is a rapid build-up of DC at around midday starting from October. Deep convection peaks between 14-17 Local Time (LT) and slightly later in 2005/2006. As the rainy season advances, DC "extends" into the night hours. By the end of rainy season, DC finishes rather abruptly. The maximum area of DC in the drought year had a maximum of around 3%, while the next year had a maximum of around 3.5%, i.e., more than 15% larger than the previous year. In order to examine whether the daily cycle description is realistic, we compared the annual hourly mean DC relative area with precipitation amounts from the station of Alta Floresta (near 10°S, 55°W) for the hydrological year 2006-2007 (Figure 2). The correlation between these two parameters is 0.73 (with 99.9% significance). Because these datasets are completely independent, we conclude that AMSU provides a realistic description of the diurnal cycle of convection. The daily amounts of AMSU have a very good correlation with TRMM data as well (> 0.8 for daily values, > 0.9 for monthly values), but these datasets are not independent since TRMM use AMSU data (among other sources) for precipitation estimates.

The convective overshooting area represents on average 20 to 40% of the monthly mean DC area in agreements with results of [START_REF] Hong | Detection of tropical deep convective clouds from AMSU-B water vapor channels measurements[END_REF] who estimated the ratio (COV/DC) of 26% for the whole tropical belt for the period of March 2003-February 2004; in our study this ratio can reach 100% in a daily mean (not shown), indicating the large day-to-day variability. Convective overshooting is fairly intense in October 2004 but its relative area shrinks as the drought conditions persist (Figure 1c). The contrast of COV area between this drought-affected year and the following year is remarkable, with a 100% increase in COV fraction compared to the previous year.

b. Comparison with aerosol optical depth

In the southern edge of the Amazon forest, biomass burning season starts around July and has a peak in September.

The impact of aerosols on cloud development and precipitation is not conclusive. Some studies indicate that there is suppression of convection and precipitation with increased aerosol content (e.g., Rosenfeld 1999, Jiang et al. 2006), while others observe the opposite effect (e.g. [START_REF] Lin | Effects of biomass-burning-derived aerosols on precipitation and clouds in the Amazon Basin: a satellite-based empirical study[END_REF]. [START_REF] Koren | Smoke invigoration versus inhibition of clouds over the Amazon[END_REF] proposes that these two observations coexist and the final effect depends on the initial cloud fraction: while cloud fields with large cloud coverage will be invigorated through microphysical effects, fields with low fraction will be strongly inhibited by aerosol absorption.

Here we performed a simple correlation analysis of monthly mean aerosol optical depth (AOD) and TRMM precipitation (used here as a proxy for convection since they have high correlation) for the end of dry season, i.e., July through September, when the region experiences consistent meteorological conditions. We found that there is positive (albeit small) and significant correlation between these two quantities (Figure 3). As the rainy season starts, there is a sharp decrease in AOD due to wet deposition, and the possible relationship between AOD and DC is overruled by dynamical effects.

IV. CONCLUSIONS AND PERSPECTIVES

This work presents an effort to characterize the regional scale variability of deep convection and precipitation in the southern Amazon region. To our knowledge, it is the first time that a constellation of polar orbiting satellites has been used to this purpose. Our results show that convective activity was decreased in ~10% in a year of severe drought, but convective overshooting was reduced by 50%. These results have implications for example, to climate modeling as it is desirable that they will capture such variability, and for troposphere-stratosphere exchanges. Future work include extending this work for the entire span of AMSU data availability, as well as studying the variability of convection at local scale using geostationary data.

Figure 1 :

 1 Figure 1: (a) Number of passes per hour and per month covering at least 30% of the target area [18°S, 8°S] [63°W, 50°W], (b) monthly mean relative area of DC per hour, and (c) monthly mean realative area of convective overshooting.

Figure 2 :

 2 Figure 2: Annual hourly mean of (a) relative area of DC (%) for the period July 2006 to July 2007, and (b) accumulated precipitation (mm/day; bars) for the period August 2006 to September 2007, with superimposed 3-hour mean curve (blue).

Figure 3 :

 3 Figure 3: Monthly mean AOD (dimensionless, contours), and TRMM 3B42 accumulated precipitation (mm/day; shaded), for months/years indicated on top of each panel. Correlation r is significant at 98%.

Acknowledgements

NOAA AMSU data was provided by the French Mixed Service Unit ICARE. MODIS 550nm AOD and TRMM 3B42 data was obtained using the GES-DISC Interactive Online Visualization ANd aNalysis Infrastructure (Giovanni) as part of the NASA's Goddard Earth Sciences (GES) Data and Information Services Center (DISC). We acknowledge the mission scientists and Principal Investigators who provided the data used in this research effort.


Guillaume Moroz
email: guillaume.moroz@irccyn.ec-nantes.fr

Damien Chablat
email: damien.chablat@irccyn.ec-nantes.fr

Philippe Wenger
email: philippe.wenger@irccyn.ec-nantes.fr

Fabrice Rouillier
email: fabrice.rouiller@inria.fr

F Rouiller

Cusp points in the parameter space of RPR-2PRR parallel manipulators

Keywords: Kinematics, Singularities, Cusp, Parallel manipulator, Symbolic computation

come   L'archive ouverte pluridisciplinaire

Introduction

It is well known that the workspace of a parallel manipulator is divided into singularity-free connected regions [START_REF] Chablat | Working modes and aspects in fully parallel manipulators[END_REF]. These regions are separated by the so-called parallel singular configurations, where the manipulator loses its stiffness and gets out of control. The so-called cuspidal manipulators have the ability to change their assembly-mode without running into a singularity, which thus may increase the size of the singularity-free regions [START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF][START_REF] Husty | Non-singular assembly mode change in 3-RPR-parallel manipulators[END_REF]. The word "cuspidal" stems from the notion of cuspidal configuration, defined as one configuration where three direct kinematic solutions coalesce. A cuspidal configuration in the manipulator joint space allows non-singular assembly-mode changing motions. Thus, determining cuspidal configurations is an important issue that has attracted the attention of several researchers [START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF][START_REF] Hernandez | Defining conditions for nonsingular transitions between assembly modes[END_REF][START_REF] Zein | Singular curves in the joint space and cusp points of 3-rpr parallel manipulators[END_REF][START_REF] Bamberger | Assembly mode changing in parallel mechanisms[END_REF]. In particular, [START_REF] Zein | Singular curves in the joint space and cusp points of 3-rpr parallel manipulators[END_REF] (resp. [START_REF] Bamberger | Assembly mode changing in parallel mechanisms[END_REF]) has analyzed the cuspidal configurations of planar 3-RPR (resp. 3-PRR) manipulators 1 . More recently, [START_REF] Hernandez | Defining conditions for nonsingular transitions between assembly modes[END_REF] studied the RPR-2PRR, a simpler planar 3-DOF manipulator that lends itself to algebraic calculus [START_REF] Hernandez | Defining conditions for nonsingular transitions between assembly modes[END_REF]. In both papers, the cusp configurations were determined by looking for the triple roots of a univariate polynomial. This approach may yield spurious solutions. In this paper, the cuspidal configuration are determined directly from the Jacobian of the whole set of geometric constraints of the robot, which guaranties that only true solutions are obtained. Then, we classify the parameter space of a family of RPR-2PRR manipulators according to the number of cuspidal configurations. It is shown that these manipulators have either 0 or 16 cuspidal configurations. The proposed method is based on the notion of discriminant varieties and cylindrical algebraic decomposition, and resorts to Gröbner bases for the solutions of systems of equations.

Mechanism under study 2.1 Kinematic equations

A 1 A 2 A 3 B 1 (x, y) B 2 B 3 θ 1 θ 2 θ 3 α ρ 1 ρ 2 ρ 3 a b L 2 L 3 Fig. 1 A RPR-2PRR parallel manipulators with (a = 1, b = 2, L 2 = 2, L 3 = 2, x = 1/2,y = 1, θ = 0.2).
The actuated joint symbols were filled in gray.

A RPR-2PRR parallel manipulator is shown in Fig. 1. This manipulator was analyzed in [START_REF] Hernandez | Defining conditions for nonsingular transitions between assembly modes[END_REF]. It has

B 3 are aligned, a= (B 1 , B 2 ), b= (B 1 , B 3 ), L 2 = (A 2 , B 2 ) and L 3 = (A 3 , B 3 ).
The geometric constraints can be expressed by the following 5 equations [START_REF] Hernandez | Defining conditions for nonsingular transitions between assembly modes[END_REF]:

f 1 : ρ 2 1 =x 2 + y 2 f 2 : x =ρ2 + L 2 cos(θ 2 ) -b cos(α) f 4 : x =L 3 cos(θ 3 ) -a cos(α) (1)
f 3 : y =L 2 sin(θ 2 ) -b sin(α) f 5 : y =ρ3 + L 3 sin(θ 3 ) -a sin(α)
Without loss of generality, we fix a = 1 in the rest of the article.

An algebraic model

The singular and cuspidal equations were previously computed using the three following steps [START_REF] Hernandez | Defining conditions for nonsingular transitions between assembly modes[END_REF]:

1. Reduce the equation system to a polynomial equation depending on the articular variables ρ 1 , θ 1 , θ 2 and one pose variable α: g(ρ 1 , θ 1 , θ 2 , α) = 0 (this step is done by eliminating variables, either with resultants or with Gröbner basis ) 2. Add the constraint ∂ g ∂ α = 0 to define the parallel singularities 3. Add the constraint ∂ g ∂ α = 0 and ∂ 2 g ∂ α 2 = 0 to define the cuspidal configurations This approach has the advantage of reducing the problem of computing the cusp configurations, to the problem of analysing the triple roots of a single polynomial. However, this only gives a necessary condition for the manipulator to have cusp configurations. In particular it is possible that 3 configurations of the robot coalesce in one coordinate but not in the others.

Let us come back to the theoretical definitions, using Jacobian matrices to define directly the triple roots of the original system of equations in all the input and output variables. If P is a list of polynomials and X a list of variables, let J k (P, X) be the union of P = {p 1 , . . . , p m } and of all the k × k minors of the Jacobian matrix of the p i with respect to the X i .

For the analysis of the RPR-2PRR manipulator, we introduce:

Y := [x, y, α, ρ 2 , ρ 3 ] W := [b, L 2 , L 3 , ρ 1 , θ 2 , θ 3 ] S := { f 1 , .
. . , f 5 }. Using these notations, the parallel singularities of the manipulator are defined by {v ∈ R 11 , p(v) = 0, q(v) > 0, ∀p ∈ J 5 (S,Y ), ∀q ∈ {b, L 2 , L 3 , ρ 1 } so that the cuspidal configurations are fully characterized by :

S = {v ∈ R 11 , p(v) = 0, q(v) > 0, ∀p ∈ J 5 (J 5 (S,Y ),Y ), ∀q ∈ {b, L 2 , L 3 , ρ 1 }}

Main tools from computational algebra

The algebraic problem to be solved is basically related to the resolution of polynomial parametric systems.

More specifically, one needs to solve a system of the following form :

E = {v ∈ R n , p 1 (v) = 0, . . . , p m (v) = 0, q 1 (v) > 0, . . . q l (v) > 0}
where p 1 , . . . , p m , q 1 , . . . , q l are polynomials with rational coefficients depending on the unknowns X = [X 1 , . . . , X n ] and on the parameters

U = [U 1 , . . . ,U d ].
There are numerous possible ways of solving parametric systems in general. Here we focus on the use of Discriminant Varieties (DV, [START_REF] Lazard | Solving parametric polynomial systems[END_REF]) and Cylindrical Algebraic Decomposition (CAD, [START_REF] Collins | Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition[END_REF]) for two reasons. It provides a formal decomposition of the parameter space through an exactly known algebraic variety (no approximation). It has been already successfully used for similar mathematical classes of problems (see [START_REF] Corvez | Using computer algebra tools to classify serial manipulators[END_REF]).

To reduce the dimension of the parameter space to three so that it can be displayed, we set L 2 = L 3 . Not that the proposed method can treat the general case L 2 = L 3 without any problems. When L 2 = L 3 , the system to solve is S with the unknowns [x, y, α, ρ 2 , ρ 3 , θ 2 , θ 3 ] and the parameters [b, L 2 , ρ 1 ].

Basic black-boxes

First experiments are often performed for specific values of the parameters, especially singular and/or degenerated cases. Here, we mainly use exact computations, namely formal elimination of variables (resultants, Gröbner bases) and resolution of systems with a finite number of solutions, including univariate polynomials.

Let us describe the global solver for zero-dimensional systems. It will be used as a black box in the general algorithm we describe in the sequel.

Given any system of equations

p 1 = 0, . . . , p m = 0 of polynomials of Q[X 1 , . . . , X n ],
one first computes a Gröbner basis of the ideal < p 1 , . . . , p m > for any ordering.

At this stage, one can detect easily if the system has or has not finitely many complex solutions.

If yes, then compute a so called Rational Univariate Representation or RUR (see [START_REF] Rouillier | Solving zero-dimensional systems through the rational univariate representation[END_REF]) of < p 1 , . . . , p m >, which is, in short, an equivalent system of the form :

{ f (T ) = 0, X 1 = g 1 (T )
g(T ) , . . . , X n = g n (T ) g(T ) }, T being a new variable that is independent of X 1 , . . . , X n , equipped with a so called separating element (injective on the solutions of the system) u ∈ Q[X 1 , . . . , X n ] and such that :

V ( p 1 , . . . , p m ) u -→ V ( f ) u -1 --→ V ( p 1 , . . . , p m ) (x 1 , . . . , x n ) → β = u(x 1 , . . . , x n ) → g 1 (β ) g(β ) , . . . , g n (β ) g(β )
defines a bijection between the (real) roots of the system (denoted by V (p 1 , . . . , p m )) and the (real) roots of the univariate polynomial (denoted by V ( f )).

We then solve the univariate polynomial f , computing so called isolating intervals for its real roots, say non-overlapping intervals with rational bounds that contain a unique real root of f (see [START_REF] Rouillier | Efficient isolation of polynomial real roots[END_REF]). Finally, interval arithmetic is used for getting isolating boxes of the real roots of the system (say non overlapping products of intervals with rational bounds containing a unique real root of the system), by studying the RUR over the isolating intervals of f . In practice, we use the function RootFinding[Isolate] from Maple software, which performs exactly the computations described above.

For example, with (b = 2, L 2 = 2, L 3 = 3, ρ 1 = 2), the polynomial system S defining the cuspidal configurations has 16 real solutions. One of these solutions is (ρ 2 = 2.30, ρ 3 = 1.86, φ = -2.36, x = 1.79, y = 0.885, θ 2 = -2.88, θ 3 = -0.999). We observe the three coalescing configurations around this root by calculating the direct kinematic solutions with θ 2 = -2.892 and θ 3 = -1.007. These solutions are shown in Figure 2.

A 1 A 3 A 2 B 1 B 2 B 3 A 1 A 1 A 1 A 3 A 3 A 3 A 2 A 2 A 2 B 1 B 1 B 1 B 2 B 2 B 2 B 3 B 3 B 3
Fig. 2 A cuspidal configuration (left), and the three converging configurations (right).

Discriminant varieties

The above method allows one to study instances of the problem and may be used together with a discretization of the parameter space to get a first idea of the complexity of the general problem to be solved. But the true issue addressed in this paper is to find criteria on the parameters that allow classifying the configurations to be studied (for example to distinguish the manipulators having cuspidal configurations from the others). This leads to a more general problem since one then has to study non zero-dimensional, semi-algebraic sets.

Let p 1 , . . . , p m , q 1 , . . . , q l be polynomials with rational coefficients depending on the unknowns X 1 , . . . , X n and on the parameters U 1 , . . . ,U d . Let us consider the constructible set :

C = {v ∈ C n , p 1 (v) = 0, . . . , p m (v) = 0, q 1 (v) = 0, . . . q l (v) = 0}
If we assume that C is a finite number of points for almost all the parameter values, a discriminant variety V D of C is a variety in the parameter space C d such that, over each connected open set U satisfying U ∩V D = / 0, C defines an analytic covering. In particular, the number of points of C over any point of U is constant.

Let us now consider the following semi-algebraic set :

S = {v ∈ R 11 , p(v) = 0, q(v) > 0, ∀p(v) ∈ J 5 (J 5 (S,Y ),Y ), ∀q(v) ∈ {b, L 2 , L 3 , ρ 1 }}
If we assume that S has a finite number of solutions over at least one real point that does not belong to V D , then V D ∩ R d can be viewed as a real discriminant variety of S , with the same property : over each connected open set U ⊂ R d such that U ∩ V D = / 0, C defines an analytic covering. In particular, the number of points of R over any point of U is constant.

Discriminant varieties can be computed using basic and well known tools from computer algebra such as Gröbner bases (see [START_REF] Lazard | Solving parametric polynomial systems[END_REF]) and a full package computing such objects in a general framework is available in Maple software through the RootFinding[Parametric] package. Figure 3 represents the discriminant variety of the cuspidal configurations of the RPR-2PRR manipulator.

The complementary of a discriminant variety

At this stage, we know, by construction, that over any simply connected open set that does not intersect the discriminant variety (so-called regions), the system has a constant number of (real) roots.

The goal of this part is now to provide a description of the regions for which the number of solutions of the system at hand is constant. For that, we compute an open CAD ( [START_REF] Collins | Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition[END_REF][START_REF] Dolzmann | Efficient projection orders for CAD[END_REF]).

Let P d ⊂ Q[U 1 , . . . ,U d ] be a set of polynomials. For i = d -1 . . .0, we introduce a set of polynomials P i ⊂ Q[U 1 , . . . ,U d-i ] defined by a backward recursion:

• P d : the polynomials defining the discriminant variety

• P i : {Discriminant(p,U i ),LeadingCoefficient(p,U i ), Resultant(p, q,U i ), p, q ∈ P i+1 }
We can associate to each P i an algebraic variety of dimension at most i-1 : V i = V (∏ p∈P i p). Figure 3 and 4 represent respectively V 3 and V 2 for the manipulator at hand.

The V i are used to define recursively a finite union of simply connected open subsets of R i of dimension i: ∪ n i k=1 U i,k such that V i ∩ U i,k = / 0, and one point u i,k with rational coordinates in each U i,k .

In order to define the U i,k , we introduce the following notations. If p is a univariate polynomial with n real roots: Moreover, if p is a n-variate polynomial, and v is a n -1-uplet, then p v denotes the univariate polynomial where the first n -1 variables have been replaced by v.

Root(p, l) =    -∞ if l ≤ 0 the l th real roots of p if 1 ≤ l ≤ n +∞ if l > n
Roughly speaking, the recursive process defining the U i,k is the following:

• For i = 1, let p 1 = ∏ p∈P 1 p. Taking U 1,k =]Root(p, k); Root(p, k + 1)
[ for k from 0 to n where n is the number of real roots of p 1 , one gets a partition of R that fits the above definition. Moreover, one can chose arbitrarily one rational point u 1,k in each U 1,k . • Then, let p i = ∏ p∈P 1 p. The regions U i,k and the points u i,k are of the form:

U i,k = {(v 1 , ..., v i-1 , v i ) | v := (v 1 , ..., v i-1 ) ∈ U i-1, j , v i ∈]Root(p v i , l), Root(p v i , l + 1)[} u i,k = (β 1 , ..., β i-1 , β i ), with (β 1 , ..., β i-1 ) = u i-1, j β i ∈]Root(p u i-1, j i , l), Root(p u i-1, j i , l + 1)[
where j, l are fixed integer.

For our example, we get for p 3 a trivariate polynomial of degree 33, for p 2 a bivariate polynomial of degree 113, and for p 1 a univariate polynomial of degree 59. The zero-dimensional solver then provides the positive real roots of p 1 (Table 1), from which we easily deduce the open intervals u 1,k . We then use the zerodimensional solver to solve every p 2 (u 1,k ,U 2 ) and deduce all the tests points of the U 2,k ′ in each cells of Figure 4. Finally we use the zero-dimensional solver to solve every p 3 (u 2,k ,U 3 ) and deduce all the tests points of the U 3,k ′ , describing so the complementary of the discriminant variety.

Discussing the number of solutions of the parametric system.

At this stage, we have a full description of the complementary of the discriminant variety of the system to be solved : a recursive process for the construction of each cell U d,k and a test point (with rational coordinates) in each of these cells. By definition of the discriminant variety, we know that the system has a constant finite number of solutions over each of these cells and computing this number for each cell is the only remaining step. This can be done simply by solving all the systems S |U=u d,k , k = 1 . . . n d using the zero-dimensional solver.

For our example, the process described in 3.3 returns 344 cells of dimension 3 ( U 3,1 , ..., U 3,344 ). We solve the system S for each of the 344 associated sample points, and we get always either 0 or 16 solutions. By selecting only the cells where the manipulator has 16 cuspidal configurations, we obtain the 58 cells shown in Figure 5. Table 2 provides the different formula bounding the three dimensional cells U 3,1 , ..., U 3,344 and Table 3 represents the 58 cells of Figure 5, where the manipulator has 16 cuspidal configurations.

Table 2 Formula describing the boundaries of the cells in Table 3.

b 1 = 0, b 2 = Root(8 b 6 -11 b 4 + 6 b 2 -1, 2) L 2 1 (b) = Root b 2 + 1 3 L 6 2 -3 b 2 + 3b + 1 b 2 -3b + 1 (b -1) 2 (b + 1) 2 L 4 2 + b 3 = Root(4 b 2 + b 6 -3 b 4 -1, 2), b 4 = Root(b 8 + 3 b 6 + 3 b 4 + b 2 -1, 2) 3 b 2 + 1 (b -1) 4 (b + 1) 4 L 2 2 -(b -1) 6 (b + 1) 6 , 2 b 5 = Root(-2 b 4 + b 6 + 3 b 2 -1, 2), b 6 = 1/ √ 2, b 7 = 1, b 8 = (2) L 2 2 (b) = 1 -b 2 , L 2 3 (b) = 1/ 1 -b 2 , L 2 4 (b) = (1 -b 2 )/b, L 2 5 (b) = ∞ b 9 = Root(2 b 2 +b 6 -3 b 4 -1, 2), b 10 = Root(b 8 -b 6 -3 b 4 -3 b 2 -1, 2) L 2 6 (b) = b 2 / 1 -b 2 , L 2 7 (b) = 1/ b 2 -1, L 2 8 (b) = b 2 / b 2 -1 b 11 = Root(-4 b 4 + b 6 + 3 b 2 -1, 2) L 2 9 (b) = (b 2 -1)/b, L 2 10 (b) = b 2 -1 b 12 = Root(b 6 -6 b 4 + 11 b 2 -8, 2), b 13 = ∞ ρ 1 1 (b, L 2 ) = Root(-ρ 6 1 b 6 + 3 b 4 L 2 2 b 2 + 1 -L 2 2 ρ 4 1 -3 b 2 -7 L 2 2 b 2 + 7 L 2 2 + L 4 2 + L 4 2 b 4 -2 L 4 2 b 2 + 1 ρ 2 1 + L 2 2 b 2 + 1 -L 2 2 3 , 2) ρ 1 2 (b, L 2 ) = Root(ρ 6 1 + -3 b 4 + 3 L 2 2 b 2 -3 L 2 2 ρ 4 1 + 21 L 2 2 b 6 + 3 L 4 2 b 4 -6 L 4 2 b 2 + 3 b 8 -21 L 2 2 b 4 + 3 L 4 2 ρ 2 1 + L 2 2 b 2 -b 4 -L 2 2 3 , 2) ρ 1 2 (b, L 2 ) = b 2 , ρ 1 3 (b, L 2 ) = 1/b
Table 3 Cells of R 3 where the manipulator has cuspidal configurations.

]b 1 b 2 [ (]L 2 1 L 2 2 [, ]ρ 1 1 ρ 1 2 [), (]L 2 2 L 2 3 [, ]ρ 1 3 ρ 1 2 [), (]L 2 3 L 2 4 [, ]ρ 1 3 ρ 1 2 [), (]L 2 4 L 2 5 [, ]ρ 1 3 ρ 1 4 [) ]b 2 b 3 [ (]L 2 1 L 2 6 [, ]ρ 1 1 ρ 1 2 [), (]L 2 6 L 2 2 [, ]ρ 1 1 ρ 1 2 [), (]L 2 2 L 2 3 [, ]ρ 1 3 ρ 1 2 [), (]L 2 3 L 2 4 [, ]ρ 1 3 ρ 1 2 [), (]L 2 4 L 2 5 [, ]ρ 1 3 ρ 1 4 [) ]b 3 b 4 [ (]L 2 1 L 2 6 [, ]ρ 1 1 ρ 1 2 [), (]L 2 6 L 2 2 [, ]ρ 1 1 ρ 1 2 [), (]L 2 2 L 2 4 [, ]ρ 1 3 ρ 1 2 [), (]L 2 4 L 2 3 [, ]ρ 1 3 ρ 1 4 [), (]L 2 3 L 2 5 [, ]ρ 1 3 ρ 1 4 [) ]b 4 b 5 [ (]L 2 1 L 2 6 [, ]ρ 1 1 ρ 1 2 [), (]L 2 6 L 2 2 [, ]ρ 1 1 ρ 1 2 [), (]L 2 2 L 2 4 [, ]ρ 1 3 ρ 1 2 [), (]L 2 4 L 2 3 [, ]ρ 1 3 ρ 1 4 [), (]L 2 3 L 2 5 [, ]ρ 1 3 ρ 1 4 [) ]b 5 b 6 [ (]L 2 1 L 2 2 [, ]ρ 1 1 ρ 1 2 [), (]L 2 2 L 2 6 [, ]ρ 1 3 ρ 1 2 [), (]L 2 6 L 2 4 [, ]ρ 1 3 ρ 1 2 [), (]L 2 4 L 2 3 [, ]ρ 1 3 ρ 1 4 [), (]L 2 3 L 2 5 [, ]ρ 1 3 ρ 1 4 [) ]b 6 b 7 [ (]L 2 1 L 2 2 [, ]ρ 1 1 ρ 1 2 [), (]L 2 2 L 2 4 [, ]ρ 1 3 ρ 1 2 [), (]L 2 4 L 2 6 [, ]ρ 1 3 ρ 1 4 [), (]L 2 6 L 2 3 [, ]ρ 1 3 ρ 1 4 [), (]L 2 3 L 2 5 [, ]ρ 1 3 ρ 1 4 [) ]b 7 b 8 [ (]L 2 1 L 2 9 [, ]ρ 1 2 ρ 1 1 [), (]L 2 9 L 2 10 [, ]ρ 1 4 ρ 1 1 [), (]L 2 10 L 2 7 [, ]ρ 1 4 ρ 1 3 [), (]L 2 7 L 2 8 [, ]ρ 1 4 ρ 1 3 [), (]L 2 8 L 2 5 [, ]ρ 1 4 ρ 1 3 [) ]b 8 b 9 [ (]L 2 1 L 2 9 [, ]ρ 1 2 ρ 1 1 [), (]L 2 9 L 2 7 [, ]ρ 1 4 ρ 1 1 [), (]L 2 7 L 2 10 [, ]ρ 1 4 ρ 1 1 [), (]L 2 10 L 2 8 [, ]ρ 1 4 ρ 1 3 [), (]L 2 8 L 2 5 [, ]ρ 1 4 ρ 1 3 [) ]b 9 b 10 [ (]L 2 1 L 2 7 [, ]ρ 1 2 ρ 1 1 [), (]L 2 7 L 2 9 [, ]ρ 1 2 ρ 1 1 [), (]L 2 9 L 2 10 [, ]ρ 1 4 ρ 1 1 [), (]L 2 10 L 2 8 [, ]ρ 1 4 ρ 1 3 [), (]L 2 8 L 2 5 [, ]ρ 1 4 ρ 1 3 [) ]b 10 b 11 [ (]L 2 1 L 2 7 [, ]ρ 1 2 ρ 1 1 [), (]L 2 7 L 2 9 [, ]ρ 1 2 ρ 1 1 [), (]L 2 9 L 2 10 [, ]ρ 1 4 ρ 1 1 [), (]L 2 10 L 2 8 [, ]ρ 1 4 ρ 1 3 [), (]L 2 8 L 2 5 [, ]ρ 1 4 ρ 1 3 [) ]b 11 b 12 [ (]L 2 1 L 2 7 [, ]ρ 1 2 ρ 1 1 [), (]L 2 7 L 2 9 [, ]ρ 1 2 ρ 1 1 [), (]L 2 9 L 2 8 [, ]ρ 1 4 ρ 1 1 [), (]L 2

Conclusion

We have proposed a general method to describe rigorously the design parameters for which a manipulator has cuspidal configurations. This method can be applied directly to other mechanisms, such as the ones studied in [START_REF] Corvez | Using computer algebra tools to classify serial manipulators[END_REF][START_REF] Wenger | Classification of 3R positioning manipulators[END_REF] for example. The tools used to perform the computations were implemented in a Maple library called Siropa 2 . For 3D illustration purposes, we have detailed the main computations to be performed with manipulators satisfying L 2 = L 3 . However, the proposed method allows one directly to solve the general case (L 2 = L 3 ) by computing a discriminant variety of the system with 4 parameters b, L 2 , L 3 , ρ 1 , and by decomposing R 4 with a CAD adapted to the discriminant variety. This description generalizes and completes the analyse done in [START_REF] Hernandez | Defining conditions for nonsingular transitions between assembly modes[END_REF]. There is still some limitations though. In particular, when the system that defines the cuspidal configurations has no solution, it may mean that there exists a manipulator with no cuspidal configurations, but it may also mean that no manipulator can be assembled with these design parameters. Thus it is essential to be able to describe precisely the set of design parameter values for which a manipulator can be assembled. Fig. 5 The cells of R 3 where the manipulator admits cuspidal configurations, front view (left) and back view (right).

Fig. 3 Fig. 4 V

 34 Fig. 3 Discriminant Variety of the cuspidal configurations

Table 1 1 b b 1 b 2 b 3 b 4 b 5 b 6 b 7 b 8 b 9 b

 119 Numerical values of the positive roots of p 10 b 11 b 12 0.0 0.533 0.564 0.617 0.656 0.707 1.0 1.41 1.52 1.62 1.77 1.88

  1 actuated prismatic joint ρ 1 , and 2 passive prismatic joints ρ 2 and ρ 3 . The two revolute joints cen- tered in A 2 and A 3 are actuated while the ones centered in A 1 , B 1 , B 2 and B 3 are passive. The pose of the moving platform is described by the position coordinates (x, y) of B 1 and by the orientation α of the moving platform B 1 B 2 . The input variables (actuated joints values) are defined by ρ 1 , θ 2 and θ 3 . The points B 1 , B 2 and

The underlined letter means an actuated joint 1

http://www.irccyn.ec-nantes.fr/ moroz/siropa/doc

Acknowledgements

The research work reported here was made possible by SiRoPa ANR Project.