S Briot

V Arakelian

V Glazunov

DESIGN AND ANALYSIS OF THE PROPERTIES OF THE DELTA INVERSE ROBOT

Introduction

Parallel manipulators have been studied in numerous publications [START_REF] Merlet | Parallel Robots[END_REF][START_REF] Ceccarelli | Fundamentals of Mechanics of Robotic Manipulations[END_REF][START_REF] Clavel | Device for Displacing and Positioning an Element in Space[END_REF][START_REF] Gosselin | Singularity Analysis of Closed-Loop Kinematic Chains[END_REF][START_REF] Arakelian | Static and Dynamic Analysis of the PAMINSA[END_REF][START_REF] Glazunov | Representations of Constraints Imposed by Kinematic Chains of Parallel Mechanisms[END_REF], hence it is impossible to go over them exhaustively. But the manipulator Delta is one of the best known and most popular due to its original architecture enabling the orientation of its moving platform to remain a constant. It is known that each kinematic chain of the manipulator Delta consists of an input link connected to the base by means of a rotating joint, and of a parallelogram connected to a moving platform. Obviously, the architecture of the parallelogram is more complicated than the architecture of the input link, therefore the weight of the parallelogram may be greater than that of the input link. In our opinion, the parallelogram is to be situated closer to the base, in order to decrease the amplitude of its displacements and velocities. Hence, it could decrease the power of the actuators.

This article focuses on a new manipulator analogous to the manipulator Delta in which the parallelogram or the link containing two universal joints is connected to the base. The constraints imposed by the kinematic chains are considered. This article partially uses the recently obtained results [START_REF] Glazunov | Representations of Constraints Imposed by Kinematic Chains of Parallel Mechanisms[END_REF]. The working volume is one the most important properties of manipulators. Here the working volumes of new manipulator sand of known manipulators are compared by using known algorithms [START_REF] Merlet | Parallel Robots[END_REF]. The singularities have an influence on the working volume that is why the special configurations of both manipulators are considered by using the approaches based on the Jacobian matrices and on the theory of screws [START_REF] Merlet | Parallel Robots[END_REF][START_REF] Ceccarelli | Fundamentals of Mechanics of Robotic Manipulations[END_REF][START_REF] Gosselin | Singularity Analysis of Closed-Loop Kinematic Chains[END_REF][START_REF] Arakelian | Static and Dynamic Analysis of the PAMINSA[END_REF].

S. Briot, V. Arakelian, 20 avenue des buttes de

Description and kinematics of the structure

Let us examine the manipulator of which the structure is described at Fig. 1. This manipulator is composed of three identical legs (Fig. 2) linked to both the fixed base and the moving platform. Each leg is composed of one actuated revolute joint Q i (i = 1, 2, 3) mounted onto the base and linked to an articulated parallelogram A i B i C i D i . Each parallelogram is connected to a rod T i P i through a revolute joint. Finally, rods T i P i are connected to the moving platform by means of another revolute joint P i . Such a structure corresponds to a Delta robot of which the legs are turned. Therefore, the degrees of freedom of the manipulator are 3 translations of the platform. It is obvious that some pairs of the structures can be changed without losing the kinematic characteristics of the moving platform. For example, the parallelograms can be replaced by one rod linked to two cardan joints.

For our analysis, let us consider that triangle

Q 1 Q 2 Q 3 is equilateral. Point O of the base frame x 0 y 0 z 0 is located at the centre of the circumscribed circle to Q 1 Q 2 Q 3 , x 0 is directed along OQ 1 and z 0 is normal to the plane Q 1 Q 2 Q 3 .
We also consider that triangle P 1 P 2 P 3 is equilateral and that point P of the moving platform is located at the centre of the circumscribed circle to P 1 P 2 P 3 . Point P is the controlled point with coordinate [x, y, z] T in the global frame.

Vectors x 1i correspond to the directions of lines OQ i and y 1i to the directions of the revolute joints Q i (Fig. 2) and they can be obtained with respect to the global frame after a rotation of angle  i around z 0 axis ( i = {0°, 120°, 240°}). Vector z 2i can be obtained from frame (O, x 1i , y 1i , z 1i ) after a rotation of angle  i around y 1i axis, vector x 3i from frame (O, x 2i , y 2i , z 2i ) after a rotation of angle  i around z 2i axis and vector x 4i from frame (O, x 1i , y 1i , z 1i ) after a rotation of angle  i around y 1i axis. The expression of the vector q = [ 1 ,  2 ,  3 ] of input coordinates can be obtained from the closure equation:

i i i i i i P T T Q OQ OP   
(1) From expressions (1) after transformations, it can be deduced that:

i b i i p Pi Qi L L x x    cos cos cos     (2) 0 sin    i p Pi Qi L y y  (3) i b i i p Pi Qi L L z z    sin sin cos    (4) where:
-R b and R p are the radii of the circumscribed circles to 2) and ( 4), and summing, it can be found that:

Q 1 Q 2 Q 3 and
    0 sin cos cos cos 2 2 2         b i i p Pi Qi i i p Pi Qi i L L z z L x x f     (5)
where the expressions of angles  i can be deduced from (3):

                                   p Qi Pi p Qi Pi i i i L y y L y y 1 1 2 1 sin , sin ,     . ( 6
)
Finally, the expression of angles  i can be obtained from equation ( 5). Differentiating equation ( 5) with respect to both articular and Cartesian coordinates, one can obtain the following expression [START_REF] Gosselin | Singularity Analysis of Closed-Loop Kinematic Chains[END_REF]:

0 q B v A    . (7) with            j i v f A ,            j i f  B ,   T z y x     v and   T 3 2 1         q , ( 8
)
where v j corresponds to the j-th coordinate of vector v. Therefore, the kinematic model can be found from (8):

q J v  
, where

B A J 1    .
(9) Thus, from expression (9), it is possible to determine the Type 1 singularity loci [START_REF] Gosselin | Singularity Analysis of Closed-Loop Kinematic Chains[END_REF] of the manipulator, which can be found by analyzing the degeneracy of matrix B. It can be shown that its determinant is equal to b 11 b 22 b 33 , which means that matrix B degenerates if b ii = 0, for i = 1, 2, 3.

Therefore, b ii is equal to zero if and only if:

- i = 90°; in this case, which corresponds to a change of working mode, a displacement of the input angle  i does not induce a displacement of the platform along y 1i axis; -the expression

    i i p Pi Qi i i i p Pi Qi i L z z L x x       sin cos cos cos cos sin      vanishes.
Introducing equations ( 2) and (4) into the following expression, we can find:

i i b i i b L L     sin cos cos sin 0    , (10)
which can be written in the form

  i i b L     sin 0 . (11)
Thus, a Type 1 singularity also exists if  i =  i + n (n). In such a case, a displacement of the input angle  i does not induce a displacement of the platform in the Ox 1i z 1i plane;

Starting with equation (9), Type 2 singularities can be found [9,10]. But for such an analysis, an augmented matrix should be used. Therefore, in order to investigate the Type 2 singularity it would be preferable to use another method which might more easily give simple results without complicated computation. Such an approach is used in the following section.

The analysis of constraints of Delta and Delta inverse robot via the theory of screws

The aim of this part is to find the Type 2 singularities of the Delta inverse robot (Fig. 3) using the theory of screw. One can find the wrenches reciprocal to the unit screws of the kinematic pairs. These wrenches for the Delta robot are determined in [START_REF] Glazunov | Representations of Constraints Imposed by Kinematic Chains of Parallel Mechanisms[END_REF]. The axes of the rotating kinematic pairs P i , Q i , T i are parallel. The parallelogram (Fig. 3) can be represented as a sliding kinematic pair S i whose axis is perpendicular to the line P i Q i and parallel to the infinite small movement of the output link of the parallelogram. Each kinematic chain determines two wrenches R 1i (0, 0, 0, r o 1ix , r o 1iy , r o 1iz ) and R 2i (0, 0, 0, r o 2ix , r o 2iy , r o 2iz ) (i = 1, 2, 3) of infinite pitch (moments) that are reciprocal to the unit screws of the axes of the kinematic pairs. Both of them are perpendicular to the axes of the pairs P i , Q i , T i . The system of constraint wrenches determining the degree of freedom of the platform has Plücker coordinates: All these six wrenches are moments, the rank of the matrix is equal to three.

When the driving pairs P i are fixed, we have nine wrenches reciprocal to the unit screws of the axes of the passive kinematic pairs. Three additional wrenches are of zero pitch, their axes are perpendicular to the axes S i of "sliding pairs" and to the vectors U i perpendicular to the axes of the pairs Q i and T i and to the lines Q i T i .

As the system presents nine wrenches, it corresponds with statically undetermined structure. The criterion of singularity is the rank of the matrix. If it is equal to five or less then the mechanism is in singular configuration. For example a singular configuration exists when the lines Q i T i (i = 1, 2, 3) of all connecting chains are coplanar or parallel.

Other structure of the Delta robot in which the mobile and immobile platforms changed the places (Fig. 4) exists. The corresponding kinematic chain consists of two universal joints P i and Q i and one turning kinematic pair T i . Each kinematic chain determines one wrench of infinite pitch (moment) R 1i (0, 0, 0, r o 1ix , r o 1iy , r o 1iz ) which is reciprocal to the unit screws of the axes of the kinematic. It is perpendicular to the axes of all turning kinematic pairs. Here the mentioned wrenches R 1i (0, 0, 0, r o 1ix , r o 1iy , r o 1iz ) are of infinite pitch and R 2i (r 2ix , r 2iy , r 2iz , r o 2ix , r o 2iy , r o 2iz ) are of the zero pitch (vectors) (i = 1, 2, 3). The axes of the wrenches R 2i intersect the points Q i as well as the axes of all passive kinematic pairs of the corresponding kinematic chains.

The criterion of singularity is the rank of the matrix (13). If it is equal to five or less then the mechanism is in singular configuration. Similarly to previous case a singular configuration exists when the lines Q i T i (i = 1, 2, 3) of all connecting kinematic chains are coplanar or parallel.

The following part concerns the comparison of the size of the workspaces of both Delta and Delta inverse manipulators.

Workspace analysis

In this part, we would like to analyze the size of the workspace of both Delta and Delta inverse robots taking into account geometric limitations. The procedure used to compute the workspace of the manipulators taking into account geometric limitations is very simple. Firstly, we discretize the Cartesian space into n points (the discretization step is fixed to 0.02 m). For a point Q to belong to the workspace of the desired manipulator, it should comply with the geometric limitations, namely:

-values of angle  i computed using the inverse geometric models have to be comprised between 0° and 90°; -for the Delta inverse robot, values of angle  i computed using the inverse geometric models comprised of between -90° and 90° in order to avoid interferences with the base; -the altitude of the platform has not to be superior to z = -0.3 m in order to avoid interferences with the base. Thus, the workspaces of both manipulators can be computed. The results are presented in the figure 5. For these two robots, the following parameters are used, which correspond to the parameters of the SurgiScope

Conclusion

In this article, a new manipulator analogous to the manipulator Delta, in which the parallelogram or the link containing two universal joints is connected with the base, is presented. The working volumes of the new manipulator and of the known manipulator with analogous links are compared. The working volume of the new manipulator is greater than the working volume of the known Delta manipulator; therefore the applications of the new manipulator are more numerous.

Fig. 1 .

 1 Fig. 1. Schematic representation of the Delta inverse robot.

Fig. 2 .

 2 Fig. 2. Schema of one leg of the Delta inverse robot.

Fig. 3 .

 3 Fig. 3. Representation of one kinematic chain of the Delta inverse robot.

Fig. 4 .

 4 Fig. 4. Other possible kinematic chain of the Delta inverse robot with cardan joints.

  ® developed by company ISIS: R b = 0.25 m, R p = 0.2 m, L p = 0.95 m, L b = 0.75 m. It can be seen that, taking into account only the geometric limitations of the manipulators, the workspace of the Delta inverse robot (1.81 m 3 ) is larger than the workspace of the Delta robot (1.48 m 3 ).

Fig. 5 .

 5 Fig. 5. Workspaces of the manipulators under study taking into account the geometric limitations.

DESIGN AND ANALYSIS OF THE PROPERTIES OF THE DELTA INVERSE ROBOT

SUMMARY

The article focuses on a new manipulator whose prototype is similar to the well known Delta robot. However, in the new manipulator, the link comprising a parallelogram or two universal joints is located on the base. In our simulations, the links of the new mechanism have the same parameters that in the existing Delta robot, used in the SurgiScope ® device commercialized by the company ISIS. It is shown that the working volume of the new manipulator is larger than the working volume of the existing manipulator.


Ramzi Bchir
email: bchir.ramzi@gmail.com

Aslam Sami Djellouli

Nadia Zitouna
email: zitounanadia@gmail.com

Didier Aurelle
email: didier.aurelle@univ-amu.fr

Gerard Pergent
email: pergent@univ-corse.fr

Christine Pergent-Martini

Habib Langar
email: habib.langar@gmail.com

Morphology and Genetic Studies of Cymodocea Seagrass Genus in Tunisian Coasts

Keywords: Cymodocea, DNA barcoding, Morphology, Mediterranean, Taxonomy

Specimens of Cymodocea (Viridiplantae, Magnoliophyta) collected on the Tunisian coasts showed a particular morphological and anatomical difference with the classical descriptions of Cymodocea nodosa (Ucria) Asch. the only species of this genus reported in the Mediterranean Sea. In order to precise the taxonomic identity of the new specimens we aimed in this work (i) to verify the identity of the new forms, (ii) to evaluate the genetic diversity of the population, (iii) to test the validity of the existing identification keys of the Tunisian Cymodocea populations. Four stations located in two regions of the Tunisian coasts were sampled. Leaf morphological and anatomical characters used in taxonomic identification were measured (e.g number of cross veins, shape of the apex). The genetic study was performed using three most common chloroplast markers for plant characterization (DNA barcodes rbcL, matK and trnHpsbA). The morphological study revealed the presence of three C. nodosa morphotypes, described here for the first time, while the molecular characterization did not allow the discrimination of these morphological types. In regard to these results, it would be wise to review the classical identification keys of the Cymodocea genus.

Introduction

Seagrasses of the genus Cymodocea K.D. Koenig (1805) are represented worldwide by four species: C. rotundata Asch. & Schweinf., C. serrulata (R.Br.) Asch & Magnus, C. angustata Ostenf. and C. nodosa (Ucria) Asch. This genus is classified to Kingdom Plantae, Tracheophyta Phylum, Angiospermae Superclass, Monocots Class, Alismatales Order and Family of Cymodoceaceae [START_REF] Taylor | Cymodoceaceae[END_REF]. All of these species colonize a habitat, characterized by a sandy and/or muddy substrate [START_REF] Terrados | Temperature effects on photosynthesis and depth distribution of the seagrass Cymodocea nodosa (Ucria) Ascherson in a Mediterranean coastal lagoon: the Mar Meno, SE Spain[END_REF] and located in the subtidal zone from -0.5 to -40 m depth. They can be established in shallow coastal lagoons [START_REF] Agostini | Growth and primary production of Cymodocea nodosa in a coastal lagoon[END_REF] and harbour areas [START_REF] Reyes | Distribution and reproductive phenology of the seagrass Cymodocea nodosa (Ucria) Ascherson in the Canary Islands[END_REF] They are widely distributed from tropical to subtropical zones, ranging from eastern Africa to the Indo-Pacific ocean, the Red Sea and the north, east and west coasts of Australia [START_REF] Short | Impacts of climate change on submerged and emergent wetland plants[END_REF]. It is worth noting that only C. nodosa has been recorded in the Mediterranean Sea [START_REF] Lipkin | The seagrasses of the eastern Mediterranean and the Red Sea[END_REF]. On the other hand, the Mediterranean Sea, which is considered as the main hotspot worldwide for macrophyte diversity, appears to be the most affected area by alien species with ~1,000 exotic taxa recorded [START_REF] Hoffman | Alien Benthic Algae and Seagrasses in the Mediterranean Sea and Their Connection to Global Warming. The Mediterranean Sea: Its history and present challenges[END_REF]- [8]. In fact, about 40-50 species of macrophytes have been directly or indirectly introduced. Frequently, the introduced species are able not only to establish themselves but they also tend to become invasive, disrupting native ecosystems [START_REF] Grosholz | Spread and potential impact of the recently introduced European green crab, Carincus maenas, in central California[END_REF]- [START_REF] Reusch | Variable responses of native eelgrass Zostera marina to a nonindigenous bivalve Musculista senhousia[END_REF]. This finding is particularly true when the introduced species belongs to the Magnoliophyta because most of them are habitat engineers or key species. Nowadays, the Suez Canal is the main entry point for Lessepsian migration from the Red Sea to the Mediterranean. Indeed, two species of the Cymodocea genus evolving in the Red Sea, could be considered as presumed candidates for introduction in the Mediterranean in regard to the unidirectional entry flow, the diversification of input vectors and the geographic proximity. This scenario is confirmed by the presence of Halophila stipulacea (Forsskål) in the Mediterranean, which was first reported in the eastern Mediterranean basin, and then in the western area [START_REF] Verlaque | Modifications des communautés algales méditerranéennes en présence de l'algue envahissante Caulerpa taxifolia (Vhal) C. Agardh[END_REF]- [START_REF] Missaoui | Apparition de la phanérogame Halophila stipulacea dans le golfe de Gabès (Tunisie) Proceedings of the second Mediterranean symposium on marine vegetation[END_REF]- [START_REF] Sghaier | Occurrence of the seagrass Halophila stipulacea (Hydrocharitaceae) in the southern Mediterranean Sea[END_REF]- [START_REF] Verlaque | Macrophytes. In: CIESM Atlas of exotic species in the Mediterranean[END_REF] and more recently by the presence of C. angustata, a new introduced species which was reported by Hattour & Ben Mustapha [START_REF] Hattour | Le Golfe de Gabés : Espèces des Eaux de Ballast, Patrimoniales et Introduites : Synthèse des Campagnes 2009 et 2010 et Actualisation[END_REF] at Mahres, Zarat and El Bibane lagoon. Additionally, in El Biban lagoon, Pergent et al. [START_REF] Pergent-Martini | Impact of fish farming facilities on Posidonia oceanica meadows: a review[END_REF], indicate the presence of reddish leaves of C. nodosa, which appear to be a relatively recent phenomenon probably due to the presence of anthocyanin as a defence system against high exposure to sunlight [START_REF] Novak | Leaf reddening in seagrasses[END_REF]. Within this context, it clearly seems that all those changes affecting biodiversity in the Mediterranean Sea have to be closely monitored and require effective tools in order to identify all incoming flow and to distinguish alien species from native Mediterranean species, and this purpose should be a scientific priority in Tunisia in regards to its geographic situation [START_REF] Cormaci | Alien macrophytes in the Mediterranean Sea: a review[END_REF]- [START_REF] Sghaier | Review of alien marine macrophytes in Tunisia[END_REF]. In fact, advancements in scientific techniques using barcoding-DNA tools provide promising avenues to understand the diverse array of organisms and to elucidate the confusing taxonomic troubles [START_REF] Hebert | Biological identifications through DNA barcodes[END_REF]- [START_REF] Coissac | From barcodes to genomes: extending the concept of DNA barcoding[END_REF]- [START_REF] Hadi | DNA barcoding green microalgae isolated from neotropical inland waters[END_REF]. The aims of this work were to study Cymodocea Tunisian populations, with (i) morphological and genetical characters, (ii) taxonomy identification of observed forms, (iii) testing the validity of existing identification key.

Materials and methods

Study area:

Four stations of Cymodocea meadows were studied from April 2014 to February 2015. They are situated in two regions of the Tunisian coast (Fig. 1.A), covering the eastern (Monastir) (Fig. 1.B) and southern sectors (El Bibane Lagoon) (Fig. 1.C). One of them is located in open sea (Monastir, Skanes) and the three others in costal lagoons (El Bibane Lagoon and Khniss Lagoon). Skanes: This site is situated on the north part of the rocky Monastir peninsula (Fig. 1

.B. station 1).

This locality is open marine water but situated in a restricted area enclosed by a dike. The sea bottom at the site is mainly sandy. Khniss Lagoon: Situated to the south of the city of Monastir (Fig. 1.B. station 2), the lagoon covers an extensive bay which is about 1 km wide in its south part and narrows towards the north. It is protected from the easterly waves by a Posidonia oceanica (L.) Delile reef barrier. The bottom of the lagoon is muddy and the average depth is about 0.5 m. El Bibane Lagoon: is a hyperhaline lagoon located along the southern Tunisian coast (Fig. 1.C station [START_REF] Agostini | Growth and primary production of Cymodocea nodosa in a coastal lagoon[END_REF][START_REF] Reyes | Distribution and reproductive phenology of the seagrass Cymodocea nodosa (Ucria) Ascherson in the Canary Islands[END_REF] where it covers an area about 230 km 2 . It is characterized by an increase in salinity from the open sea to the enclosed background of the lagoon, reaching more than 50 psu during the summer [START_REF] Guelorget | Contribution a` l'étude biologique de la Bahiret el Biban, lagune du Sud-Est Tunisien[END_REF]. The central part of the lagoon has a muddy bottom covered by an expanded Cymodocea seameadow.

Sampling and morphological studies:

Fifty shoots of Cymodocea were hand-collected from each station at a depth of 0.5-1 m (table . 2) [START_REF] Sghaier | Occurrence of the seagrass Halophila stipulacea (Hydrocharitaceae) in the southern Mediterranean Sea[END_REF]and the collected shoots were separated by a minimum distance of 2 m. Samples were identified with the keys of Den Hartog [START_REF] Hartog | The Seagrasses of the World[END_REF], Phillips & Meñez [START_REF] Phillips | Seagrasses[END_REF] and Kuo et al. [START_REF] Kuo | Seagrass morphology, anatomy, and ultrastructure[END_REF], using a binocular microscope and / or an optical microscope. Only mature leaves were selected for different measurements, essentially the number of cross veins, width of the apex (2 mm from the apex), absence or presence of teeth on the apex margins and the shape of the apices were surveyed and described. When the number of cross veins indicated uncertainties, a transverse section of the leaf blade was performed [START_REF] Macic | Anatomical features of Cymodocea nodosa growing in Montenegro (Adriatic Sea)[END_REF] with a razor blade and observed under the microscope.

Sampling materials for genetic study:

Twenty-five to thirty erect shoots of Cymodocea were collected from each station (table. 1) at a depth of 0.5 m. Furthermore, samples were harvested at least 5 m from each other (not from the same rhizome). The samples were cleaned with seawater to eliminate debris and any epiphytes were removed with a razor blade [START_REF] Lucas | Development of a DNA barcoding system for seagrasses: successful but not simple[END_REF], and after that, samples were preserved in silica gel for DNA extraction.

Statistical analysis:

For each parameter studied, a comparison of the calculated averages was performed by applying ANOVA using XLSTAT software. For all statistical analyses, the statistical significance was set at p < 0.05 % and when significant differences were detected, the Fisher test (LSD) was used to locate the differences. If the conditions of normal distribution and homogeneity of variances were not verified, the Kruscal-Wallis test was performed by the XLSTAT software. The Dunn test was also applied to locate where the differences are precisely located.

Genetic analyses:

DNA Extraction and PCR conditions:

Total DNA was extracted from dried leaves with the Qiagen DNeasy Plant Mini Kit according to manufacturer instructions. Three plastid loci were amplified and sequenced with the following primer pairs described in Lucas et al. [START_REF] Lucas | Development of a DNA barcoding system for seagrasses: successful but not simple[END_REF]:  matK (P608 5'-ACCCAGTCCATCTGGAAATCTTGGTTC-3'and P607

5'-CGTACAGTACTTTTGTGTTTACGAG-3');  rbcl (P609 5'-GTAAAATCAAGTCCACCRCG-3' and P610

5'-ATGTCACCACAAACAGAGACTAAAGC-3')  trnH-psbA spacer (P676 5'GTTATGCATGAACGTAATGCTC-3'and P677

5'-CGCGCATGGTGGATTCACAATCC-3').

For a 25 µl final PCR volume and for all markers, the composition was as follows: Promega PCR buffer (1X), MgCl 2 (2.5 mM), a mixture of dATP/ dTTP / dCTP / dGTP (0.25 mM), Forward and Reverse primer for each chloroplast region (0.5 µM), Flexigotaq polymerase (Promega) (0.625 U), and 2.5 µl of template DNA (20 ng). The PCR program was: 5 min at 94°C, 30 cycles with [1 min at 94°C, 1 min at 50°C, 1 min at 72°C], and a final extension of 10 min at 72°C. The PCR products were checked with 1, 5% agarose gel and sent for sequencing at Eurofinsgenomics.

Sequence analysis:

The obtained sequences were compared to the GenBank nucleotide sequence database to check the origin. Then sequences of different samples were aligned by hand with the BioEdit software [START_REF] Hall | BioEdit: a user-friendly biological sequence alignment editor and analysis program for Windows 95/98/NT[END_REF].

To evaluate the phylogeny, the Mega version 5.0 [START_REF] Tamura | MEGA5: molecular evolutionary genetics analysis using maximum likelihood, evolutionary distance, and maximum parsimony methods[END_REF] was used with two methods: Maximum Parsimony (MP), using the Tamura-Nei model, [START_REF] Tamura | Estimation of the number of nucleotide substitutions in the control region of mitochondrial DNA in humans and chimpanzees[END_REF] and Genetic Distances using the formula of Saitou & Nei [START_REF] Saitou | The neighbor-joining method: A new method for reconstructing phylogenetic trees[END_REF] with a bootstrap value of 1000 replications. Nucleotide and amino-acid composition, number of polymorphic sites (S), haplotype diversity, nucleotide diversity, Tajima's (D), Fu & Li (F) and (D) and Fu's (Fs) tests were calculated via DnaSP version 4.0 [START_REF] Librado | DnaSP v5: A software for comprehensive analysis of DNA polymorphism data[END_REF].

Results

Morphology and anatomy characteristics:

Leaf morphology and anatomy characteristics were absence or presence of teeth on the apex margins, the shape of the apex, leaf Tips and leaf scars (Fig. 2), revealed that three kinds of apex shape can be distinguished: However, it was necessary to make clear that B and D forms could be found on the same rhizome and even on the same shoot, in contrast to the other forms where only one type of apex was reported on each shoot (Fig. 3 A andC). To sum up, the morphological characteristics of the leaves studied allow us to recognize two distinct morphological groups among the collected specimens (Table .2). The first one comprised specimens having leaves with tiny teeth, serrate margins, obtuse apex (Fig. 3.A) and presenting an average number of veins between a minimum 7.2 ± 0.36 (Al Marsa/ El Bibane lagoon) and a maximum 7.76 ± 0.91 (Khniss lagoon) (table . 2). This group was exclusively found in lagoon stations. The second group mainly differed from the first one by the entire margin, an absence of teeth along the leaf instead of the serrate margin. It was also characterised by a varied form of apex and presented 8.7± 0.4 cross veins, which was a number significantly higher statistically than the first group (p < 0.05). It was collected only in open sea at Skanes. Furthermore, the examination of the other vegetative structures did not show any morphological difference in roots, rhizomes, stems, seeds, or even closed circular scars, in all collected specimens from the different stations.

Genetics:

Three of the most commonly used chloroplastic regions were used in this study in order to characterize four Tunisian populations of Cymodocea nodosa : matK, rbcL and trnH-psbA. These barcode markers were amplified and sequenced for 44 to 53 samples. All the sequences were first checked by using the alignment-based approach and Basic Local Alignment Search Tool (BLAST) algorithm. The results of BLAST searches confirm that all samples belonged to the unique species, C. nodosa despite the morphological divergence, the sequences presented an maximum of similarity (matK100 % , rbcL 100 % , trnH-psbA 96% ) and a higher query coverage (matK 97 % , rbcL 92% , trnH-psbA 98% ). The level of overlap between query and reference sequences had a certain impact on identity scores in particular and on the identification process in general. The matK, rbcL and trnH-psbA exhibited higher identities (almost 100%) and higher coverage (varying from 98% to 100%) than the available queries. It is worth noting that this work is the first time that the intergenic chloroplast spacer, trnH-psbA has been used for C. nodosa.

Polymorphism and genetic diversities of the nucleotide sequences:

The analysed sequences resulted in a matrix of 822, 1199 and 184 positions for matK, rbcL and trnH-psbA respectively. Of which, 45, 853 and 15 sites were excluded from the analysis because of alignment mismatches. The remaining aligned positions yielded a very low number of polymorphic sites (1, 1 and 2) representing 1, 0 and 1 parsimony informative characters. Despite the high number of gaps or indels (Table . 3), no long indels or inversions of nucleotides were reported through the multiple alignments, for the three used barcodes and no sequence variation were encountered below populations level. Two segregating mutations were reported for both matK and trnH-psbA. The very low amount of variation was confirmed by the nucleotidal diversity almost equal to 0 for the three regions, and similarly for the haplotypic index. When comparing data partitions of the base compositions, presented in (table. 4), the three chloroplast frames exhibited higher amounts of AT against GC. For the two barcodes matK and rbcL, the first, second and third positions of the codons were also calculated, as these positions was crucial for the traduction process. Guaninecytosine was enhanced in the first codon position and underrepresented in the third codon position for rbcL. The opposite could be seen for GC in matK, being low in the first position and strongly enhanced in the third position. The picture drawn for the transition (ns)/transversion (nv) can be seen in (table. 5). The ratio of transitions to transversions should be correlated to the time of divergence of two taxa, i.e. the longer the time period the lower the ns/nv coefficient. In this study, the transversional percent is higher than the transitional (0% in the case of rbcL). Three indices were calculated for studying the evolution of these molecular markers: Tajima, Fu and Li and Fs (table . 3). The indices were not significant as the negative values of Tajima and Fu and Li could converge (p > 0.10), either for a recent negative selection or for demographic population expansion. The negative values of Fs (p > 0.10) tend to apply to the demographic history attacking the C. nodosa in the Mediterranean of Tunisia coast.

Phylogenetic representations

The phylogenetic analysis was based on three methods: maximum parsimony, maximum likelihood and genetic distances. In front of the very low amount of polymorphism and the absence of parsimonious sites in this survey, only the genetic distances method was allowed. For that purpose, the Nei and Kimura formula was used to draw Neighbor-Joining dendrograms showing that the three barcodes did not distinguish between the four Tunisian populations of C. nodosa. Our NJ analysis showed that all used samples clustered into one clade with a bootstrap value of 100%. The pairwise distances indicated that the nucleotide divergence varied from 0.0% to 0.6% and the overall mean distance was 0.1 %, which is very low.

Polymorphism of the encoded regions (rbcL and matK) and degenerescence of the genetic code

To examine the patterns of synonymous codon usage we conducted a RSCU analysis and estimated the values for both matK and rbcL amino-acidic sequence (Table . 6). Relative synonymous codon usage (RSCU) is defined as the ratio of the observed frequency of codons to the expected frequency, given that all the synonymous codons for the same amino acids were used equally. The most abundantly used codons in C. nodosa cp DNA were AAA and ACU for matK (17.6) and rbcL [START_REF] Missaoui | Apparition de la phanérogame Halophila stipulacea dans le golfe de Gabès (Tunisie) Proceedings of the second Mediterranean symposium on marine vegetation[END_REF], respectively. For matK, UUU, AUU, AUA, AAU and AGU were widely distributed whereas for rbcL, CCU, AAA and GAA were the more frequent codons. All these codons were A or U-ended codons; none of the preferred codons were GC-ended. This corresponds with the low amounts of GC at the third position, reported previously in table.4. From RSCU analysis, we observed that Tunisian seagrass exhibits comparatively higher codon usage bias towards A/U-ended codons. Furthermore, analysis of over-and under-represented codons showed that codons with an RSCU>1.6 are infrequently observed in chloroplast genomes of Cymodoceae. In fact, The RSCU values of the majority of preferred and non-preferred codons fell between 0.6 and 1.6.

Discussion:

The anatomical and morphological study of Cymodocea specimens collected along the Tunisian coasts revealed the presence of two distinct morphological groups of individuals. The first one, present on the eastern coasts (Monastir Bay) as well as in the southern sectors (Lagoon of El Bibane), has toothed margins and obtuse apex. It conforms perfectly to the usual descriptions of C. nodosa and agrees with the existing identification keys [START_REF] Hartog | The Seagrasses of the World[END_REF]- [START_REF] Phillips | Seagrasses[END_REF]- [START_REF] Kuo | Seagrass morphology, anatomy, and ultrastructure[END_REF]. The second one, found only on the eastern coasts at Skanes (Monastir), with entire margins and rounded to retuse apex appears closer to the descriptions of C. rotundata from the same authors (table. 7). They have closed circular scars on shoots, absence of teeth and the form of the apex. Nevertheless, there are a few shared characters with C. nodosa such as the number of veins (9 veins) and similar morphology of roots and rhizomes (table. 7). The classical shape of C. nodosa leaf apex (obtuse apex with serrated margin) are regularly observed in open sea, as well as in lagoon, along Tunisian coastlines [START_REF] Turki | Distribution of toxic dinoflagellates along the leaves of seagrass Posidonia oceanica and Cymodocea nodosa from the Gulf of Tunis[END_REF]- [START_REF] Mabrouk | A comparison of abundance and diversity of epiphytic microalgal assemblages on the leaves of the seagrasses Posidonia oceanica (L.) and Cymodocea nodosa (Ucria) Asch in Eastern Tunisia[END_REF] but Skanes site is the only site exhibiting non-classical shape. This first location could correspond to an introduction of an alien species due to its limited spatial distribution (new arrival) and the arrival, in the Mediterranean Sea, of another seagrass species (H. stipulacea) following the opening of Suez Canal [START_REF] Boudouresque | An overview of species introduction and invasion processes in marine and coastal lagoon habitats[END_REF]. However, these Lessepsian species are initially observed in the South-Eastern part of the basin, and intermediate sites, especially for a benthic species, are reported. Indeed, among the three other Cymodecea species, only C. rotundata reported from Red Sea, Indian Ocean and Pacific Ocean could be a good candidate. The genetic studies did not show any difference between individuals of the new forms and the typical forms of C. nodosa and thus did not show any connection of the new forms to C. rotundata. Because of this, the question may be posed as follows: do specimens of the second group really belong to the C. nodosa species but cannot be clearly recognized according to the existing identification keys? Or are the used DNA barcodes really appropriate to distinguish C. nodosa from other species of the Cymodocea genus? The use of additional independent and more variable markers will be necessary here to clearly test species limits ( [START_REF] Pante | Species are hypotheses: avoid connectivity assessments based on pillars of sand[END_REF] such as the nuclear Phytochrome B (phyB) [START_REF] Daru | A Search for a Single DNA Barcode for Seagrasses of the World[END_REF]. some kind of criteria that are only available during a single stage of the life history and do not include descriptions corresponding to each stage of the complete life cycle of the leaves of the seagrass [START_REF] Hebert | Biological identifications through DNA barcodes[END_REF]. These commonly encountered problems are generally related to the difficulty in identifying closely related species, phenotypic plasticity and genetic variability. In fact, the identification of introduced species with the same type of seed, general morphology or shared identification criteria among taxa could lead novice ecologists to errors. In addition to traditional identification problems, the three DNA barcodes used in this study did not differentiate the new forms of Cymodocea from those conventionally described in the literature. However, this result showed the agreement with other studies to indicate that was no correlation between molecular and morphological analysis in species delimitation for seagrasses [START_REF] Kato | Phylogenetic analyses of Zostera species based on rbcL and matK nucleotide sequences: implications for the origin and diversification of seagrasses in Japanese waters[END_REF]- [START_REF] Talbot | Genetic characterization of Zostera asiatica on the Pacific Coast of North America[END_REF]- [START_REF] Tanaka | Phylogenetic relationships in the genera Zostera and Heterozostera (Zosteraceae) based on matK sequence data[END_REF]. The discrepancy between genetic and morphology based on taxonomy study might be explained by phenotypic plasticity, morphological convergence or interspecific hybridization and could be considered as an ''imperfect taxonomy'' [START_REF] Funk | Species-level paraphyly and polyphyly: frequency, causes, and consequences, with insights from animal mitochondrial DNA[END_REF]. In this study, intragenic spacer trnH-psbA was for the first time used for the genus Cymodocea but was also not conclusive; it did not provide any more information than those obtained from rbcL and matK in agreement with others studies [START_REF] Kusumasari | Usefulness of chloroplast genes (rbcl and matk) of dna barcoding for seagrass identification (a study of species collected from malang and lamongan water[END_REF]. Another phenomenon that could be encountered is that of the genetic bottleneck that leads to a significant decrease in genetic diversity because of genetic drift which has been reported for C. nodosa even using the microsatellite markers [START_REF] Masucci | Genetic flow directionality and geographical segregation in a Cymodocea[END_REF].

The existence of very low intraspecific variation for C. nodosa was highlighted in the present study with three chloroplast markers and agreed with other studies on might also be related to the asexual reproduction, and rarely sexual, strategy of C. nodosa, which generates a poor potential for gene dispersal due to irregular and infrequent flowering [START_REF] Procaccini | Contribution of genetics and genomics to seagrass biology and conservation[END_REF]. Moreover, C. nodosa seed dispersal is limited because the seeds remain fixed at the base of the female plants when basicarpal germination is achieved. This generates a domination of vast areas by single clones of C. nodosa [START_REF] Alberto | Genetic differentiation and secondary contact zone in the seagrass Cymodocea nodosa across the Mediterranean-Atlantic transition region[END_REF]. In order to avoid confusion, and, based on our results, which remain unreliable genetically, it can be hypothetically advanced that the new morphological forms of C. nodosa observed along Tunisian coasts could probably be morphotypes, which is the most common response to abiotic stress on seagrass. This phenomenon is known to facilitate their resistance to global change in different ecological niches [START_REF] Bradshaw | Evolutionary response to rapid climate change[END_REF] - [START_REF] Davis | Evolutionary responses to a changing climate[END_REF]- [START_REF] Peralta | Morphometric variations as acclimation mechanisms in Zostera noltii beds[END_REF]- [START_REF] Peralta | Direct effects of current velocity on the growth, morphometry and architecture of seagrasses: a case study on Zostera noltii[END_REF]- [START_REF] Puijalon | Morphological variation of two taxonomically distant plant species along a natural flow velocity gradient[END_REF] Probably, the origins of morphotype for a species, is intraspecific polymorphism, speciation in progress, incomplete derivation sorting or hybridization through introgression [START_REF] Coyer | Eelgrass meadows in the California Channel Islands and adjacent coast reveal a mosaic of two species, evidence for introgression and variable clonality[END_REF]- [START_REF] Coyer | Phylogeny and temporal divergence of the seagrass family Zosteraceae using one nuclear and three chloroplast loci[END_REF] reported for the Cymodoceaceae family [START_REF] Ito | Hybridisation in a tropical seagrass genus, Halodule (Cymodoceaceae),inferred from plastid and nuclear DNA phylogenies[END_REF].

Conclusion

To conclude, this research accentuates the need for more thorough taxonomic studies and / or an update of identification keys based on newly recorded criteria involving this important marine angiosperm. Molecular marker systems which can differentiate lower taxa levels than the current genus and species level need to be applied, such as RAD sequencing, Phytochrome B (phyB) , to challenge and delineate the boundary between species but also allow the detection of the origin of this phenotypic plasticity .

Figure 1 :

 1 Figure 1: Sampling sites of Cymodocea specimens. A: location of the studied regions along the tunisian coasts; B: sites of Monastir region; 1= Skanes ; 2= Khniss Lagoon; C: El Biban Lagoon ; 3=Jdayria; 4= Al Marsa.

Figure 2 :

 2 Figure 2: General morphology of classical Cymodocea nodosa.:A. Circular closed scars. B. leaf blade teeth. C. cross-section of leaf blade D.. fruit

Figure 2 :

 2 Figure 2: General morphology of classical Cymodocea nodosa.:A. Circular closed scars. B. leaf blade teeth. C. crosssection of leaf blade D. fruit

Table 1 .

 1 Location of sampling area.

	Location	Station	Abbreviation	Geographical coordinates
		Skanes	Pop 1 : BR	35°46'48.14"N 10°47'7.86"E
	Monastir	Lagoon of Khniss	Pop 2 : PR	35°44'45.01"N 10°49'49.09"E
	El	Jdayria	Pop 3 : BJ	33°16'38.76"N 11°17'31.33"E
	bibane	Al marsa	Pop 4 : LM	33°12'32.44"N 11°12'44.65"E

Table 2 .

 2 Morphological characteristics of the Tunisian specimens of Cymodocea genus (-significant difference for p < 0.05, = not significant).

	Character/location		Skanes	Lagoon of Khniss	Jdayria	Al Marsa
			rounded to retuse apex with
			entire margin	obtuse apex	obtuse apex	obtuse apex
	Apex forms		obtuse apex with entire margin	with serrate	with serrate	with serrate
				margin	margin	margin
			Curved and spatulate apex with
			entire margin
			obtuse apex with serrate margin
			(0%)
	Proportions	 	rounded to retuse apex with entire margin (0%) obtuse apex with entire	obtuse apex with serrate margin	obtuse apex with serrate margin	obtuse apex with serrate margin
			margin(11%)	100%	100%	100%
		 curved and spatulate apex with
			entire margin. (72%)
	Absence/presence teeth		( -)	( + )	( + )	( + )
			8.7± 0.4 a	7.76 ± 0.91 b	7.44 ±0.68 b	7.2 ± 0.36 b
	Leaf veins		Min = 7	Min = 7	Min = 7	Min = 7
			Max = 9	Max = 9	Max = 9	Max = 9
	Leaf width		2.27± 0.23 a	1.91±0.2 b	1.84±0.6 b	2.23±0.22 a

Table 3 .

 3 The genetic diversity of Cymodocea nodosa using matK and rbcL (coding barcodes) and trnH-psbA (non-coding barcode).

		matK	rbcL	trnH-psbA
	Number of samples	53	44	53
	Consensus length	822 bp	1199 bp	184 bp
	Number of indels	45	853	15
	Number of polymorphic sites	1	1	2
	Number of constant sites	766	345	168
	Number of segregating sites	1	0	1
	Nucleotid diversity	0.00084	0.00013 ± 0	0.00066 ± 0.0000001
	Haplotype diversity	0.038 ± 0.00129	0.045 ± 0.00184	0.110 ± 0.00336
			UUA, CUG, CCA,
			GCG, CAG, AAU,
	Less abundant Amino-Acid and %	GUU (0.9/ 256)	AAG, GAC, UGU,
			UGC, AGU and AGA (1/172)	Non-coding frame
	Most abundant Amino-acid and %	AAA (17.7 /256)	ACU (12 /172)
	Tajima's D test	1.096	-1.115	-1.313
	Fu and Li F* and D* tests	-1.895 and -1.858	-1.856 and -1.803	-1.195 and -0.9
	Fu's(Fu) Test p > 0.10	-1.685	-1.530	-2.394

Table 4 .

 4 The

			T	A	%AT	C	G	GC-1 GC-2 GC-3	%GC
		POP 1	30.8	39.1	35	14.5	15.6	12	15.5	18.1	15
		POP 2	30.9	38.9	34.9	14.6	15.6	11.8	15.4	18.1	15.1
	matK	POP 3	30.9	38.7	34.8	14.7	15.6	12.2 15.3	18.2	15.2
		POP 4	30.6	39.1	34.8	14.7	15.6	11.9	15.4	18.2	15.2
		Average 30.8	38.9	34.9	14.6	15.6	11.9	15.3	18.1	15.1
		1	28.6	27.9	28.2	20.6	22.8	28.3	22.8	14	21.7
		2	28.6	28	28.3	20.7	22.8	28.3	22.8	14	21.7
	rbcL	3	28.6	28	28.3	20.6	22.8	28.3	22.8	14	21.7
		4	28.6	28	28.3	20.7	22.8	28.3	22.8	14	21.7
		Average 28.6	28	28.3	20.6	22.8	28.3	22.8	14	21.7
		1	30.4	44.1	37.3	13	12.5	-	-	-	12.7
	trnH-psbA	2 3 4	30.4 30.4 30.4	44.2 44.2 44.1	37.3 37.3 37.3	13 13 13	12.4 12.4 12.5	---	---	---	12.7 12.7 12.7
		Average 30.4	44.2	37.3	13	12.5	-	-	-	12.7

T, A, C and G patterns in the three chloroplastic frames. Legend. POP : MON BR ; POP2 : BJ ; POP3 :MON PR ; POP4 : BIBEN LM .GC1-2-3 (GC' amount in the first, second and third position of the codon) were only calculated for the coding regions)

Table 5 .

 5 Maximum

	From\To	A	T	C	G

Likelihood estimate of transition/transversion for the three amplified barcodes bias using Kimura (1980) 2parameter model (Transitions are bold-written).

Table 6 .

 6 The amino-acidic composition for the two proteins encoded by matK and rbcL.

			matK		rbcL
	Amino-Acid	Codon	Count	RSCU	Count (AA)	RSCU
	Phenylalanin	UUU(F)	13.7	1.41	4	1
		UUC(F)	5.8	0.59	4	1
		UUA(L)	0	0	1	0.46
		UUG(L)	5.8	2.55	6	2.77
	Leucine	CUU(L)	4	1.74	2	0.93
		CUC(L)	1.9	0.85	0	0
		CUA(L)	1	0.42	3	1.37
		CUG(L)	1	0.44	1	0.46
		AUU(I)	13.7	1.08	4	1.7
	Isoleucine	AUC(I)	8.7	0.69	2	0.86
		AUA(I)	15.6	1.23	1	0.44
	Methionine	AUG(M)	7.8	1	0	0
		GUU(V)	0.9	0.63	6	1.85
	Valine	GUC(V)	2.2	1.46	0	0
		GUA(V)	2.8	1.89	5	1.54
		GUG(V)	0	0.03	2	0.62
		UCU(S)	2.9	1.27	2	2
		UCC(S)	3.9	1.72	3	3
		UCA(S)	0	0	0	0
	Serine	UCG(S)	3.9	1.71	0	0
		AGU(S)	2.9	1.29	1	1
		AGC(S)	0	0.01	0	0
		CCU(P)	2	0.85	8	2.48
	Proline	CCC(P)	1.2	0.54	2	0.61
		CCA(P)	6	2.59	1	0.29
		CCG(P)	0	0.02	2	0.62

Table 7 .

 7 Comparison of morphological traits of Cymodocea genus.

		Cymodocea nodosa	new forms reported	Cymodocea rotundata
	Rhizomes	Closed circular scar on the erect stem.	Closed circular scar on the erect stem.	Closed circular scar on the erect stem
		1 strongly branched root at	1 strongly branched root at each	1-3 irregularly branched roots
	Roots	each node.	node.	at each node.
		Blade with 7-9 longitudinal	Blade with 7-9 longitudinal	Blade with 9-15 longitudinal
		veins, apex obtuse serrate	veins obcordate or obtuse or	veins; apex rounded to
	Leaf	margin with very small hi-or	curved and spatulate apex with	emarginate, serrulate.
		trifurcate teeth..	entire margin .
		Leaf sheath linear to slightly	Leaf sheath linear to slightly	Slightly obconical,, ends with
	leaf	obconical,ends with two tips	obconical,ends with two tips at	two tips at the upper side
	sheath	at the upper side	the upper side

It clearly appears that the current identification keys are insufficient to distinguish C. nodosa from other species of the genus that may be invading the Mediterranean. This may be because they are based only on

Acknowledgments

This work is a part of the thesis prepared by Ramzi BCHIR for attaining his PhD degree at the Science Faculty of Tunis Tunisia. We thank the molecular biology service of the IMBE and especially Anne Haguenauer for her help. We are indebted to anonymous reviewers for their comments and constructive criticism of the manuscript.

Conflicts of Interest

The authors declare no conflicts of interest.


Vigen Arakelian
email: vigen.arakelyan@insa-rennes.fr

Sébastien Briot
email: sebastien.briot@ens.insa-rennes.fr

G C G M -Ea3913

Dynamic Balancing of the SCARA Robot

This paper deals with the complete shaking force and shaking moment balancing of the four degrees of freedom SCARA robot. Dynamic reaction forces on the frame of the manipulator are eliminated by traditional approach making the total mass center of the moving links stationary. Reaction moments on the frame of the manipulator are eliminated by optimal control of the end-effector, which rotates with prescribed acceleration. A numerical simulation carried out on the software ADAMS illustrates that such a balanced SCARA robot transmits no inertia loads to surrounding, i.e. the sum of all ground bearing forces and their moments are eliminated.

Introduction

A primary objective in balancing a linkage is to remove or reduce the variable dynamic load it transmits to its frame and surrounding structures. Different approaches and solutions have been developed and documented [START_REF] Lowen | Balancing of linkages -an Update[END_REF], [START_REF] Arakelian | A historical review of the evolution of the theory on balancing of mechanisms[END_REF], [START_REF] Arakelian | Shaking force and shaking moment balancing of mechanisms: a historical review with new examples[END_REF], but, despite its long history, mechanism balancing theory continues to develop and new approaches and solutions are constantly being reported. A new field for their application is the design of mechanical systems for fast manipulation, which are very efficient for advanced robotic applications. The cancellation of the inertia force (shaking force) transmitted to the robot frame can be achieved by traditional approach, which consists of fixation of the common center of mass of the moving links of the manipulator. The studies devoted to the inertia couple (shaking moment) balancing of high-speed manipulators may be arranged in the following groups.

(i) Balancing by counter-rotations [START_REF] Berkof | Complete fore and moment balancing of inline four-bar linkages[END_REF], [START_REF] Dresig | Vollständiger und harmonischer Ausgleich ebener Mechanismen[END_REF], [START_REF] Arakelian | Complete shaking force and shaking moment balancing of linkages[END_REF], [START_REF] Herder | A counter-rotary counterweight for light-weight dynamic balancing[END_REF]. The aim of this approach is to connect the counterweights with planetary gear trains, which generate the counter-rotations (Fig. 1). The disadvantage of such a balancing is the need for the connection of gears to the oscillating links. The oscillations of the links of the mechanism will create noise unless expensive anti-backlash gears are used.

(ii) Balancing by adding four-bar linkages [START_REF] Gosselin | Synthesis and design of reactionless tree-degree-of-freedom parallel mechanisms[END_REF], [START_REF] Ricard | On the design of reactionless parallel manipulators[END_REF]. This approach is achieved by building the mechanical system out of modules -four-bar linkages (Fig. 2). The four-bar linkages with prescribed geometric parameters allow the optimum redistribution of the moving mass to achieve the shaking moment balancing. The originality of such a solution is the balancing of the shaking moment without the counter-rotations. However, the balancing by this method is only reached by a considerably complicated design. (iii) Balancing by optimum trajectory planning [START_REF] Papadopoulos | Proceedings of the IEEE International Conference on Robotics and Automation[END_REF], [START_REF] Fattah | On the design of reactionless 3-DOF planar parallel mechanisms[END_REF]. In this case, the trajectory is planned in such a way that it brings to the minimal reactions on the base of the robot. Such an approach was applied to a redundant planar manipulator with 3 degrees of freedom and a planar 3-RRR parallel manipulator.

This paper deals with a new solution for shaking moment balancing of manipulators, which is based on the optimum control of the end-effector's rotation about its axis. We illustrate the reactionless feature of such an approach through the four degrees of freedom SCARA robot.

SCARA robot: shaking force and shaking moment

The SCARA (Selective Compliance Assembly Robot Arm) is a high-performance robot with relatively simple structure, which is composed of three revolute joints O 1 , O 2 and O 3 (Fig. 3) making it possible to move and orient the end-effector in the horizontal plane (x, y and φ) and one prismatic joint d 4 allowing its movement in the vertical direction (z). The robot was developed in the laboratory of Professor Makino at Japon's Yamanashi University [START_REF] Makino | SCARA robot and its family[END_REF] and it is successfully applied in production industries. Today there are about 240 types of SCARA robots developed by 20 Companies.

Various studies was devoted to this architecture. The singularity analysis and workspace optimization was discussed in the paper (Beiner, 1992). [START_REF] Roth | Control and mechanics of simple manipulator systems[END_REF] studied this robot to achieve the optimal design for the links to minimize the pick-and-place motion. The dynamics was studied by [START_REF] Ibrahim | Dynamics characteristics of a SCARA robot subject to NC2 velocity trajectories with different payloads[END_REF], as well as by [START_REF] Padhy | On the dynamics of SCARA robot[END_REF]. [START_REF] Chen | Local POE model for root kinematic calibration[END_REF] investigated the kinematic calibration of the 4-DOF Scara robot. The vibration control for this robot was suggested in the study of [START_REF] Kang | Robust vibration control for SCARAtype robot manipulators[END_REF]. It is well known that one of the fields of SCARA-type robots application is the fast pick-andplace manipulation. With increase of the manipulation speeds it becomes evident that the fast moving elements of mechanical systems bring about undesirable effects, such as vibration and noise. In this paper, for the first time, the dynamic balancing of the SCARA robot is studied.

In order to estimate the variable dynamic loads transmitted to the frame of a SCARA robot we carried out the simulations on the software ADAMS. For this purpose, a standard cycle of pickand-place motion (25×300 mm in 0.3 s) for fifth-order polynomial trajectory with the following initial and final locations x i = 0.15 mm, x f = -0.15 mm and y i = 0.2 mm, y f = 0.2 mm and φ i = 83 deg., φ f = 157 deg. orientation was selected. The geometric and inertia parameters of the robot are the following: l O1O2 = 0.125 mm, l O2O3 = 0.225 mm, l O1S1 = 0.0625 mm, l O2S2 = 0.1125 mm, m 1 = 1.2 kg, m 2 = 2 kg, m 3 = 0.5 kg, I 1 = 0.014 kg.m², I 2 = 0.036 kg.m², I 3 = 0.0126 kg.m², where l OiOj corresponds to the length of segment O i O j , l OiSi to the distance of the centre of masses S i from the joint centre O i , m i is the mass of link i and I i its axial moment of inertia.

Fig. 4 shows the shaking force and shaking moment variations of unbalanced robot (full line).

Shaking force and shaking moment balancing

In order to achieve the dynamic balancing of the SCARA robot, we first have to ensure that it is force-balanced, i.e. statically balanced. With regard to the vertical motion along the z axis, we can cancel the inertia forces by counterweights executing similar but opposite movements to the endeffector motion. Fig. 5 shows one of the design concepts carried out by pulley and cables. It should be noted that similar systems can be built by double slider-crank mechanisms, rhombic pantographs, and screw or cam systems.

Then, it is necessary to balance the shaking forces in the horizontal plane by making the total mass center of the moving links stationary (Fig. 4 -dotted lines). For this purpose, the following conditions may be achieved:

0 3 3 = S O l , ( 1
)
2 2 2 3 2 3 S O O O l m l m = , ( 2
) 2 1 3 2 1 1 1 ) ( O O S O l m m l m + = , (3)
taking into account that the centers of masses of links 1 and 2 are not located between the centers of rotation but outside of them. However the force balancing leads to the important increase of the moving masses of the robot, and as a result, its inertia couple (for the simulations, the following parameters for the balanced manipulator are used: l O1S1 = -0.117 mm, l O2S2 = -0.047 mm, m 1 = 6.2 kg, m 2 = 4.8 kg, I 1 = 0.175 kg.m², I 2 = 0.112 kg.m².). In Fig. 4 the shaking moment about the z axis is shown (dotted line). We can note that it is increased by a factor 3. Now that the inertia force balancing is achieved, we have to consider the cancellation of the shaking moment.

The three angles of rotation of the SCARA robot (Fig. 3) are defined by the following relationships:

⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ - + + - + + - - + ± - = - x l y x l l y x l l y x l y l O O O O O O O O O O O O O O 2 1 2 2 2 3 2 2 2 1 2 2 2 2 3 2 2 2 1 2 2 2 2 1 2 1 1 1 2 ) ( ) ( tan 2 θ , ( 4
) 1 3 2 1 2 1 1 2 cos cos θ θ θ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - = - O O O O l l x , ( 5
) 2 1 3 θ θ φ θ - - = , (6)
where, x and y are the coordinates of the end-effector. The sign " ± " in Eq. ( 4) shows that for the same end-effector position and orientation, there are two possible configurations (working modes) of the robot.

We can see from these equations that the angles θ 1 and θ 2 don't depend on the angle θ 3 , i.e. the trajectory generation in the horizontal plan is defined only by the angles θ 1 and θ 2 . It will be used for the shaking moment balancing.

After force balancing, the shaking moment is defined by the following relation: For examined robot with parameters given above and taking into account that after shaking force balancing, the axial inertia moment of the end-effector is equal to 0.0126 kg.m², we determine the law of rotation of the end-effector (Fig. 6).

It should be noted that the moment of inertia of the end-effector is relatively small compared to the moments of inertia of robot links. Thus, to ensure the balancing of the shaking moment it is necessary that it turns several times during the motion of the robot in the horizontal plane.

Finally, we would like to note that the suggested approach can be also applied to others position-orientation decoupled manipulators. Fig. 7 shows a new architecture of PAMINSA manipulator [START_REF] Arakelian | Parallel robot comprising means for setting in motion a mobile element split in two separate subassemblies[END_REF], [START_REF] Briot | Analysis and optimization of a new family of parallel manipulators with decoupled motions[END_REF] with 6 degrees of freedom, in which the generation of translations in the horizontal plane is achieved by the use of the rotating actuators M 1 , M 3 and the vertical translations by the linear actuator M v . The orientation of the end-effector is carried out by the serial wrist mounted on the platform.

The shaking moment on the frame of this manipulator can be eliminated by optimal control of the serial wrist with prescribed rotation velocities. Our future work will consist in the detailed study of such a balancing with numerical simulations carried out using ADAMS software. Although the shaking force and shaking moment are eliminated, in practice there are small deviations from the ideal zero reaction. Sources of such deviations include manufacturing errors, neglected friction and clearances in the joints and drive actuators, as well as manipulator unbalancing due to a payload. However, in all these cases the proposed balancing method will eliminate the most significant base reactions due to robot accelerating links.

Conclusion

A new field for shaking force and shaking moment balancing is the design of fast parallel manipulators, which are very efficient for advanced robotic applications. In this paper, the shaking force and shaking moment balancing method is developed for the SCARA-type robots with 4 degrees of freedom. The shaking forces of the manipulator in the horizontal plan are eliminated by optimum redistribution of masses making the total mass center of the moving links stationary. With regard to the vertical inertia forces, the opposite motion of the end-effector and the counterweights is applied. The shaking moments on the frame of the manipulator are eliminated by optimal control of the end-effector, which rotates with prescribed velocity. Numerical simulations carried out on the software ADAMS showed that after such a balancing, the manipulator transmits no inertia loads to its surroundings, i.e. the sum of all ground bearing forces and their moments are eliminated.

The advantage of such a balancing is its simplicity. The complete balancing of the shaking moment is achieved without important design modifications. We believe that the balancing of manipulators by means of optimum control of the end-effector is a very promising approach to dynamic performance improvement.

  a) Gears by external teeth b) Gears by internal teeth c) Balancing an articulated dyad

Figure 1 .

 1 Figure 1. Balancing by counter-rotation (courtesy of C.M. Gosselin).

Figure 2 .

 2 Figure 2. Balancing by adding four-bar linkages (courtesy of C.M. Gosselin).

  (a) 3D view.(b) planar view.

Figure 3 .

 3 Figure 3. Schematics of the SCARA robot developed at Yamanashi University.

  (a) Force along x-axis. (b) Moment around x-axis. (c) Force along y-axis. (d) Moment around y-axis. (e) Force along z-axis. (f) Moment around z-axis.

Figure 4 .

 4 Figure 4. Variations of the shaking force and shaking moment unbalanced (full line) and statically balanced (dotted line) robot.

Figure 5 .

 5 Figure 5. Balancing of the vertical inertia forces by using a cable and pulley arrangement.

  angular displacement θ 3 (t) can be determined by simple integration of the obtained values of 3 θ & & .

Figure 6 .

 6 Figure 6. Law of rotation of the end-effector ensuring the shaking moment balancing of the robot.

Figure 7 .

 7 Figure 7. PAMINSA with 6 DOF, in which the Cartesian displacements are carried out by the parallel structure and the orientation by the serial wrist mounted on the platform.

& & & & & & & & & & & &

In equation ( 7) we have a free parameter 3

, which defines the orientation of the end-effector between its initial and final positions but has not any influence to the prescribed trajectory.

In order to have a shaking moment of the robot equal to zero for the given trajectory, the acceleration 3 θ & & for the period t i ≤ t ≤ t f , should vary by the following law:

( )


Sébastien Briot
email: sebastien.briot.1@ens.etsmtl.ca

Ilian A Bonev
email: ilian.bonev@etsmtl.ca

A PAIR OF MEASURES OF ROTATIONAL ERROR FOR AXISYMMETRIC ROBOT END-EFFECTORS

Keywords: accuracy analysis, rotational errors, Euler angles, distance metrics

This paper deals with the problem of representing the rotational error of spatial robots with three orientational degrees of freedom (DOF). Typically, the errors on each of three Euler angles defining the orientation of an endeffector are analysed separately. However, this is wrong since an accuracy measure should depend only on the "distance" between the nominal pose and the actual one, and not on the choice of reference frame in which these are represented. Several bi-invariant metrics for rotational error exist but are single-parameter and, by definition, disregard the shape of the robot endeffector. Yet, robot end-effectors are typically axisymmetric. Therefore, we propose a two-parameter measure of rotational errors that is better suited for such robot end-effectors.

Introduction

Knowing the exact positioning accuracy is an important challenge when designing a robot. Such an analysis is useful for optimization procedures occurring in the design stage of a new robot. Many tools have been developed in order to promptly estimate the accuracy of a robot. The most common performance indices used to indirectly optimize the accuracy of a robot are the dexterity index [START_REF] Gosselin | The optimum design of robotic manipulators using dexterity indices[END_REF], the condition numbers, and the global conditioning index [START_REF] Gosselin | A global performance index for the kinematic optimization of robotic manipulators[END_REF].

Another way to estimate the accuracy of a robot is to use the first order approximation of the direct kinematic model that maps the input error vector δq to the output error vector δp through the linear relation δp = J δq, where J is the Jacobian matrix of the robot. However, such an approach will only give a rough estimate of the maximal errors occurring in the workspace. This estimate is relevant for infinitesimal errors and far from singularities, but on real robots, errors are not always small. Furthermore, all of the above tools essentially take into account only errors in the inputs. Yet, errors are due to various factors such as manufacturing tolerances, backlash, compliance, sensor errors and control er-rors. Besides, once a prototype is built, it is important to be able to simply measure and quantify these errors, which is obviously not done using the Jacobian matrix of the robot.

Clearly, from the industrial point of view, the best accuracy measure would be the maximum translational and maximum rotational errors over a given portion of the workspace [START_REF] Briot | Accuracy analysis of 3-DOF planar parallel robots[END_REF]Merlet, 2006a) or at a given nominal configuration. However, while representing the accuracy for planar robots is simple to realize [START_REF] Briot | Accuracy analysis of 3-DOF planar parallel robots[END_REF], this problem is much more complicated when the number of orientational DOFs increases.

Several papers deal with the accuracy of robots with several orientational DOFs [START_REF] Jelenkovic | Error analysis of a Stewart platform based manipulators[END_REF][START_REF] Kim | The kinematic error bound of the Stewart platform[END_REF]Merlet and Daney, 2007;[START_REF] Ropponen | Accuracy analysis of a modified Stewart platform manipulator[END_REF][START_REF] Wang | Error model and accuracy analysis of a six-DOF Stewart platform[END_REF][START_REF] Zhao | Error analysis of a serialparallel type machine tool[END_REF]. In all of these works, the authors analyse the rotational accuracy of robots by considering the errors on three Euler angles. However, such an analysis is wrong because these errors depend on the choice of reference frame in which the nominal pose and the actual one are expressed. Yet, obviously, a rotational error should depend only on the "distance" between the nominal orientation and the actual one.

For the design of mechanisms, defining metrics for measuring the distance between one pose (position and orientation) and another is of utmost importance and is still an area of ongoing research (Larochelle et al., 2007). For representing the accuracy of a robot in industry, however, it is meaningless to combine translational and rotational errors in a single measure. Fortunately, for the specific case of body orientations, there exists a family of bi-invariant metrics [START_REF] Park | Distance metrics on the rigid-body motions with applications to mechanical design[END_REF]Gupta, 1997). Certainly, the most intuitive one is the rotation angle about a unique axis that brings one reference frame to coincide with another.

Single-parameter invariant measures of rotational error are relevant only for asymmetric robot end-effectors. However, robot end-effectors usually have axial symmetry (in most industrial robots, which are sold without an end-effector, the flange on which an end-effector is mounted is axisymmetric). That is why, in this paper, we propose a pair of rotational error measures that is better suited for such end-effectors. These measures are not invariant because their raison d'être is to depend on an end-effector reference frame that takes into account the axial symmetry of the end-effector. However, our measures depend only on the orientation of the actual reference frame with respect to the nominal one.

The remainder of this paper is organized as follows. Section 2 deals with the definition of the proposed measure for orientational accuracy. In Section 3, the new error measure is illustrated on a well-known spherical robot called the Agile Eye. Conclusions are given in the last section.

2.

Representation of Rotational Errors

To describe the orientation of a body, three consecutive rotations about some of the three axes of the base reference frame are generally used. The angles of these rotations are referred to as Euler angles.

As previously mentioned, the offsets between the corresponding Euler angles associated with two reference frames varies significantly with the choice of base reference frame that is used to define these Euler angels. Therefore, such an offset cannot be used for measuring rotational errors. As a matter of fact, the same is true when considering the individual x, y and z translational errors, though these errors are bounded by the distance between the origins of the first two reference frames. This distance is, in fact, used in industry to represent translational errors, which is the only accuracy measure currently used (ISO 9283 standard).

Thus, it would be great if a similar invariant metric for the "distance" between two orientations existed. Such metrics exist and the best known one is the rotation angle about a unique axis that brings one reference frame to coincide with another. This angle is not only physically meaningful but also quite easy to compute from the trace of the rotation matrix that represents the orientation of one reference frame with respect to the other.

The only problem with such single-parameter invariant metrics is that robot end-effectors are typically axisymmetric. In some cases, when axisymmetric tools are used, the rotation about the tool axis is even irrelevant. Therefore, if industry ever becomes interested in rotational errors, it will more likely look for a two-parameter metric-one measure that reflects the deviation of the axis of symmetry and, for example, another that reflects the rotation about this axis of symmetry.

Consider two different orientations of an axisymmetric robot endeffector: a nominal one denoted by the index 0, and another one denoted by the index 1. Let d0 and d1 be two unit vectors parallel to the axis of symmetry of the robot end-effector in each of the two orientations (Fig. 1). The first meaningful parameter characterizing the change of orientations is clearly the angle α between vectors d0 and d1. Its expression is trivial:

( ) 1 0 1 cos T α - = d d . ( 1
)
Now, consider that we render the axes of symmetry of both endeffectors of Fig. 1 coincident by rotating end-effector 1 about an axis v0 normal to d0 and d1. The smallest angle to which end-effector 1 needs to be rotated about the d0-axis in order to coincide with end-effector 0 will be denoted by β and will be our second error measure. Of course, while the choice for the first measure is pretty obvious and unquestionable, there might be other alternatives for the second one. However, we believe that our choice is the most logical and intuitive one.

It can be shown that angles α and β are actually the tilt and torsion angles (to a sign difference) of the Tilt-and-Torsion (T&T) three-angle orientation parameterization introduced in [START_REF] Bonev | Geometric Analysis of Parallel Mechanisms[END_REF]. Therefore, angle β can be found using the following equation:

3,2 3,1 2,3 1,3 2,1 1,1 atan2( / sin , / sin ) atan2( / sin , / sin ) , if 0 atan2( , ) , if 0 r r r r r r α α α α α β α ⎧ - + ≠ ⎪ = ⎨ = ⎪ ⎩ (2)
where ri,j (i,j = 1,2,3) are the elements of the rotation matrix that represents the orientation of the reference frame associated with the real pose with respect to the reference frame associated with the nominal pose.

Note, that β does not depend on the choice of the x and y axes.

Obviously, 0 ≤α < 90° in the context of accuracy analysis. In this range, angles α and β are the direct equivalent of the translational and rotational errors, respectively, associated with planar 3-DOF motion.

Example: the Agile Eye

In this section, for simplicity, we address the problem of computing the rotational accuracy of a parallel robot in the presence of active-joint errors only. Specifically, we will analyse the rotational accuracy of the well known spherical parallel robot called the Agile Eye (Fig. 2a). Its kinematics, singularities (Fig. 2b), workspace, and working and assembly modes have been studied in detail in several papers [START_REF] Bonev | Singularity loci of spherical parallel mechanisms[END_REF][START_REF] Bonev | Analytical determination of the workspace of symmetrical parallel mechanisms[END_REF]Bonev et al., 2006). Therefore, we will recall only one important fact before focusing on the accuracy analysis.

The Agile Eye is a very particular parallel robot with its Type 1 (serial) and Type 2 (parallel) singularities coinciding and degenerating to six curves as shown in Fig. 2b using a T&T angles coordinate system. Each curve corresponds to finite self motions of both the platform and the legs.

Accuracy Analysis

In order to compute the rotational errors of the Agile Eye, we use T&T angles (φ, θ, σ) and the following discretization method. For a fixed torsion angle σ, we discretize the orientation space (φ, θ). At a given nominal orientation (φ 0 , θ 0 , σ 0 ) of the mobile platform, we compute the active-joint variables (q10, q20, q30) for one given working mode using the inverse kinematics. Clearly, we should stick to the same working mode throughout the whole workspace. Then, assuming that active-joint variable qi can vary from its nominal value qi0 by an error ε (in our study, ε = ±0.01°), we discretize the active-joint interval from qi0 -ε to qi0 + ε and for each set of active-joint variables solve the direct kinematics. For the Agile Eye, the direct kinematics can be found analytically. As presented in (Bonev et al., 2006), there are four non-trivial solutions which define the assembly modes of the robot. Therefore, the problem is to know which of the obtained solutions is the desired assembly mode corresponding to the nominal configuration. Fortunately, this problem has already been solved in (Bonev et al., 2006). Of course, the direct kinematics of parallel robots can rarely be solved analytically. Therefore, in such cases, a numerical method should be used, such as the Newton-Raphson algorithm. This method is much more time-consuming, but, when sufficiently far from singularities, it converges to the appropriate assembly mode of the robot.

Thus, for a given nominal configuration, it is possible to compute all possible rotational errors α and β, and to retain their maxima. These maximal errors are presented in Fig. 3 using T&T polar plots for several constant torsions. We can think of these polar plots as the equivalent of constant-orientation polar plots for planar robots.

Analysing Fig. 3, it appears that our rotational errors grow considerably near singularities (represented by crosses and dotted lines in Fig. 3), which is perfectly normal. An unexpected result is, however, the fact that, when computing the rotational errors, most of the time, the maximal output errors occur when the active-joint variables are inside their error intervals, whereas it has been demonstrated in [START_REF] Briot | Accuracy analysis of 3-DOF planar parallel robots[END_REF] that for planar parallel robots, the maximal output errors can only occur when at least two of the active-joint variables are at the extremes of their error intervals. This observation suggests that it should be impossible to find a manageable analytic expression for the output (rotational) errors of even a simple spherical parallel robot such as the Agile Eye. A simple discretization method such as ours will be sufficient but more time-consuming and less accurate. However, more efficient approaches could be used, such as interval analysis (Merlet 2006a).

It should also be noted that the curves are periodic, but not symmetric, which is due to the fact that the output errors depend on the robot's working mode.

Conclusions

In this paper, we addressed the problem of representing rotational errors of spatial robots. A review of previous works has shown that, most frequently, authors analyse the errors on each Euler angle defining the orientation of a robot end-effector. However, this is generally wrong since an accuracy measure should depend only on the "distance" between the nominal pose and the actual one, and not on the choice of reference frame in which these are expressed. In contrast, several bi-invariant metrics for rotational error exist. However, they are single-parameter and, by definition, disregard the shape of the robot end-effector. Yet, most robot endeffectors are axisymmetric. Therefore, the choice of end-effector reference frame is not arbitrary and users are likely to be interested in more than one rotational error measure. To fill this gap, we propose a pair of measures of rotational errors that is better suited for such robot end-effectors.

In order to present the efficiency of our approach, we applied it to compute the output errors of a well-known spherical parallel robot called the Agile Eye. It was shown that its accuracy depends on the working mode and is poor near singularities.

Figure 1 .

 1 Figure 1. Measuring the rotation errors of a robot end-effector.

Figure 2 .

 2 Figure 2. (a) A CAD model of the Agile Eye at its reference configuration (zero active-joint variables and φ = 0°, θ = 0° and σ = 60° T&T angles) and (b) its singularity loci.

Figure 3 .

 3 Figure 3. Rotational errors of the Agile Eye for σ ∈ [0°, 120°].

Acknowledgements

We would like to thank the anonymous reviewers for their valuable remarks that helped us improve considerably this manuscript.