Martial SAUCEAU* Karl Kamar

Martial Sauceau
email: martial.sauceau@mines-albi.fr

Élisabeth Rodier

Jacques Fages

Elisabeth Rodier

Biopolymer foam production using a (SC CO2

est destinée au dépôt

INTRODUCTION

Polymers are widely used in several areas. However, due to their slow degradation and the predicted exhaustion of the world petroleum reserves, significant environmental problems have arisen. Therefore, it is necessary to replace them with bioplastics that degrade in a short time when exposed to a biologically active environment [START_REF] Bucci | PHB packaging for the storage of food products[END_REF]. Biopolymers like PHAs (polyhydroxyalkanoates) are marketed as the natural substitutes for common polymers, as they are 100% biodegradable polymers. PHAs are polyesters of various HAs which are synthesised by numerous microorganisms as energy reserve materials in the presence of excess carbon source. Poly(3-hydroxybutyrate) (PHB) and its copolymers with hydroxyvalerate (PHB-HV) are the most widely found members of this biopolymer group and were also the first to be discovered, and most widely studied PHA [START_REF] Khanna | Recent advances in microbial poly-hydroxyalkanoates[END_REF]. They possess properties similar to various synthetic thermoplastics like polypropylene and hence can be used alternatively. Specifically, PHB exhibits properties such as melting point, a degree of cristallinity and glass transition temperature, similar to polypropylene (PP). Although, PHB is stiffer and more brittle than PP, the copolymerization of PHB with 3-hydroxyvalerate (PHB-HV) produces copolymers which are less stiff and tougher. That is to say that there is a wide range of applications for these copolymers [START_REF] Gunaratne | Multiple melting behaviour of poly(3-hydroxybutyrate-cohydroxyvalerate) using step scan DSC[END_REF]. The properties of this copolymer depend on the HV content, which determines the polymer crystallinity [START_REF] Peng | Isothermal crystallization of poly(3hydroxybutyrate-co-hydroxyvalerate)[END_REF].

Extrusion is a process converting a raw material into a product of uniform shape and density by forcing it through a die under controlled conditions [START_REF] Rauwendaal | Polymer Extrusion[END_REF]. It has extensively been applied in the plastic and rubber industries, where it is the most important manufacturing process. A particular application concerns the generation of polymeric foams. Polymeric foams are expanded materials with large applications in the packaging, insulating, pharmaceutical and car industries because of their high strength/weight ratio or their controlled release properties.

Conventional foams are produced using either chemical or physical blowing agents. Various chemical blowing agents, which are generally low molecular weight organic compounds, are mixed with a polymer matrix and decompose when heated beyond a threshold temperature. This results in the release of a gas, and thus the nucleation of bubbles. This implies however the presence of residues in the porous material and the need for an additional stage to eliminate them.

Injection of scCO 2 in extrusion process modifies the rheological properties of the polymer in the barrel of the extruder and scCO 2 acts as a blowing agent during the relaxation when flowing through the die [START_REF] Sauceau | Improvement of extrusion processes using supercritical carbon dioxide[END_REF]. The pressure drop induces a thermodynamic instability in the polymer matrix, generating a large number of bubbles. The growth of cells continues until the foam is rigidified (when T<T g ). Moreover, its relatively high solubilisation in the polymer results in extensive expansion at the die. The reduction of viscosity decreases the mechanical constraints and the operating temperature within the extruder. Thus, coupling extrusion and scCO 2 would allow the use of fragile or thermolabile molecules, like pharmaceutical molecules. The absence of residues in the final material is also an advantage for a pharmaceutical application.

Our lab has developed a scCO 2 -assisted extrusion process that leads to the manufacturing of microcellular polymeric foams and already elaborated microcellular foams using a biocompatible amorphous polymer [START_REF] Sauceau | Effect of supercritical carbon dioxide on polystyrene extrusion[END_REF][START_REF] Nikitine | Residence time distribution of a pharmaceutical grade polymer/supercritical CO2 melt in a single screw extrusion process[END_REF][START_REF] Nikitine | Controlling the structure of a porous polymer by coupling supercritical CO2 and single screw extrusion process[END_REF]. In this work, this process has been applied to PHB-HV. Foam production of semi-crystalline polymer is less frequent in the literature. Crystallinity hinders the solubility and diffusion of CO 2 into the polymer and leads consequently to less uniform porous structure [START_REF] Doroudiani | Processing and characterization of microcellular foamed high-density polyethylene/isotactic polypropylene blends[END_REF]. Moreover, it has been shown that a large volume expansion ratio could be achieved by freezing the extrudate surface of the polymer melt at a reasonably low temperature [START_REF] Park | Low density microcellular foam processing in extrusion using CO2[END_REF]. Thus, in this work, in order to control and improve the porous structure of the PHB-HV, the influence of melt and die temperatures have been studied.

MATERIALS AND METHODS

PHB-HV (M w =600 kDa), with a HV content of 13 % and plasticized with 10 % of a copolyester was purchased from Biomer (Germany). Melting temperature was measured at 159°C by DSC (ATG DSC 111, Setaram). The solid density € ρ P , determined by helium pycnometry (Micromeretics, AccuPYC 1330) is about 1216 kg.m -3 . A rheological study at atmospheric pressure has been performed in oscillatory mode (MARS, Thermo Scientific). The polymer viscosity decreases when temperature and shear rate increase, which is a characteristic behaviour of a pseudoplastic fluid (Figure 1). This characterization step helped in choosing the operating conditions to process PHB-HV by extrusion. These conditions have to ensure that the polymer flows well enough through the barrel without being thermally degraded.

Figure 2 shows the experimental set up, which has previously been detailed elsewhere [START_REF] Nikitine | Residence time distribution of a pharmaceutical grade polymer/supercritical CO2 melt in a single screw extrusion process[END_REF][START_REF] Nikitine | Controlling the structure of a porous polymer by coupling supercritical CO2 and single screw extrusion process[END_REF]. The single-screw extruder has a 30 mm-screw diameter and a length to diameter ratio (L/D) of 35 (Rheoscam, SCAMEX). A great L/D ratio generally indicates a good capacity of mixing and melting but important energy consumption. The screw is divided into three parts. The first one has a length to diameter ratio of 20 and the two others have a length to diameter ratio of 7.5. Between each part, a restriction ring has been fitted out in order to obtain a dynamic gastight which prevents scCO 2 from backflowing. The first conical part allows the transport of solid polymers and then, their melting and plasticizing. Then, the screw has a cylindrical geometry from the first gastight ring to the die. This die has a diameter of 1 mm and a length of 11.5 mm. The temperature inside the barrel is regulated at five locations: T a and T b before the CO 2 injection, T c and T d after the injection and T e in the die.

Figure 1: Evolution of viscosity with pulsation

There are three pressure and two temperature sensors: P 1 after the CO 2 injector, P 2 and T 1 before the second gastight ring and P 3 and T 2 by the die. This allows measuring the temperature and the pressure of the polymer inside the extruder. Errors associated to pressure and temperature sensors were about 0.2 MPa and 3.3°C respectively. CO 2 (N45, Air liquide) is pumped from a cylinder by a syringe pump (260D, ISCO) and then introduced at constant volumetric flow rate. The pressure in the CO 2 pump is kept slightly higher than the pressure P 1 . The CO 2 injector is positioned at a length to diameter ratio of 20 from the feed hopper. It corresponds to the beginning of the metering zone, that is to say the part where the channel depth is constant and equal to 1.5 mm. The pressure, the temperature and the volumetric CO 2 flow rate are measured within the syringe pump. CO 2 density, obtained on NIST website by Span and Wagner equation of state [START_REF] Span R | A New Equation of State for Carbon Dioxide Covering the Fluid Region from the Triple-Point Temperature to 1100 K at Pressures up to 800 MPa[END_REF], is used to calculate mass flow rate and thus the CO 2 mass fraction € w CO 2 . For each experiment, only the temperature of the metering zone T d and of the die T e were changed. The three other temperatures T a , T b and T c were kept constant at 160°C. CO 2 mass fraction € w CO 2 was also kept constant at 1 %, which is much less than solubility [START_REF] Cravo | Solubility of carbon dioxide in a natural biodegradable polymer: determination of diffusion coefficientsJ[END_REF]. Three series of experiments were carried out. T d was fixed at 140°C, 135°C and 130°C respectively and T e varied from 140 down to 110°C. At lower values of T e , the extruder stopped due to too high a pressure P 3 according to the established alarm value.

Once steady state conditions are reached, extrudates were collected and water-cooled at ambient temperature in order to freeze the extrudate structure. Several samples were collected during the experiment in order to check the homogeneity of the extrudates. To calculate the apparent porosity € ρ app , samples were weighed and their volumes were evaluated by measuring their diameter and length with a vernier (Facom). To obtain this apparent density with a good enough precision, the mean of 6 measurements was carried out. Porosity, defined as the ratio of the pore volume to total volume is calculated by equation 1:

€ ε = 1- ρ app ρ P (1)
€ ρ P is the PHB-HV density and € ρ app the apparent density of the extrudates. The theoretical maximum porosity € ε max is obtained if all the dissolved CO 2 would become gaseous inside the extrudate at ambient conditions and would thus create porosity. It could be calculated by the following equation:

€ ε max = w CO 2 ρ P w CO 2 ρ P + ρ CO 2 (atm) (2)
€ w CO 2 is the CO 2 mass fraction and € ρ CO2 (atm) is the CO 2 density at ambient conditions.

To complete the characterization of the porosity structure, samples were examined by scanning electron microscopy (ESEM, FEG, Philips).

RESULTS

The results of porosity are presented in Figure 3. It is noticeable that, for all experiments, the obtained porosity is lower than the theoretical maximal porosity € ε max , which is estimated at about 90 %. The higher porosity, obtained at the lowest T d and T e , 130 and 110°C respectively, is about 70 %.

The porosity increases with decreasing temperature T d . The evolution of porosity with the temperature T e depends on the value of T d . At T d equal to 140°C, the porosity is constant, whereas at T d lower than 140°C, the porosity decreases with increasing die temperature. It was previously observed for polystyrene that, at a reasonably low temperature of polymer melt, there exists an optimal die temperature for which large volume expansion ratio are achieved by freezing the extrudate surface [START_REF] Park | Low density microcellular foam processing in extrusion using CO2[END_REF]. This effect is explained because more gas is retained in the foam at lower temperature and used for cell nucleation and growth. However, when the nozzle temperature was further decreased, the volume expansion ratio decreased because of the increased stiffness of the frozen skin layer. Our experiments might be explained in the same way. Thus T e would be still too high to obtain higher porosity. As porosity increases, it seems thus that growth phenomena occur at lower temperature. This evolution is opposite to previous results in which coalescence and growth phenomena occurred when temperature increased and led to larger porosity [START_REF] Sauceau | Effect of supercritical carbon dioxide on polystyrene extrusion[END_REF][START_REF] Park | Low density microcellular foam processing in extrusion using CO2[END_REF]. Indeed, it was believed that the polymer melt should be cooled substantially in order to increased melt strength and thus prevent cell coalescence.

Figure 2

 2 Figure 2: Experimental device

Figure 1 :

 1 Figure 1: Porosity evolution

Figure 2

 2 Figure 2 presents pictures at two different values of T d . It could be observed that the pores are large (more than 200 µm) and that they become fewer and larger when T d decreases.As porosity increases, it seems thus that growth phenomena occur at lower temperature. This evolution is opposite to previous results in which coalescence and growth phenomena occurred when temperature increased and led to larger porosity[START_REF] Sauceau | Effect of supercritical carbon dioxide on polystyrene extrusion[END_REF][START_REF] Park | Low density microcellular foam processing in extrusion using CO2[END_REF]. Indeed, it was believed that the polymer melt should be cooled substantially in order to increased melt strength and thus prevent cell coalescence.

Figure 2 :

 2 Figure 2: SEM pictures (a) Td=130°C (b) Td=140°C


Tahir Rasheed
email: tahir.rasheed@ls2n.fr

Philip Long
email: p.long@northeastern.edu

David Marquez-Gamez
email: david.marquez-gamez@irt-jules-verne.fr

Stéphane Caro
email: stephane.caro@ls2n.fr

Available Wrench Set for Planar Mobile Cable-Driven Parallel Robots

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Available Wrench Set for Planar Mobile Cable-Driven Parallel Robots

Tahir Rasheed 1 , Philip Long 2 , David Marquez-Gamez 3 and Stéphane Caro 4 , Member, IEEE Abstract-Cable-Driven Parallel Robots (CDPRs) have several advantages over conventional parallel manipulators most notably a large workspace. CDPRs whose workspace can be further increased by modification of the geometric architecture are known as Reconfigurable Cable Driven Parallel Robots(RCDPRs). A novel concept of RCDPRs, known as Mobile CDPR (MCDPR) that consists of a CDPR carried by multiple mobile bases, is studied in this paper. The system is capable of autonomously navigating to a desired location then deploying to a standard CDPR. In this paper, we analyze the Static equilibrium (SE) of the mobile bases when the system is fully deployed. In contrast to classical CDPRs we show that the workspace of the MCDPR depends, not only on the tension limits, but on the SE constraints as well. We demonstrate how to construct the Available Wrench Set (AWS) for a planar MCDPR wih a point-mass end-effector using both the convex hull and Hyperplane shifting methods. The obtained results are validated in simulation and on an experimental platform consisting of two mobile bases and a CDPR with four cables.

I. INTRODUCTION

A Cable-Driven Parallel Robot (CDPR) is a type of parallel manipulator whose rigid links are replaced by cables. The platform motion is generated by an appropriate control of the cable lengths. Such robots hold numerous advantages over conventional robots e.g. high accelerations, large payload to weight ratio and large workspace [START_REF] Pott | IPAnema: a family of cable-driven parallel robots for industrial applications[END_REF]. However, one of the biggest challenges in classical CDPRs which have a fixed cable layout, i.e. fixed exit points and cable configuration, is the potential collisions between the cables and the surrounding environment that can significantly reduce the workspace. By appropriately modifying the robot architecture, better performance can be achieved [START_REF] Gagliardini | A reconfiguration strategy for reconfigurable cable-driven parallel robots[END_REF]. CD-PRs whose geometric structure can be changed are known as Reconfigurable Cable-Driven Parallel Robots (RCDPRs). Different strategies, for instance maximizing workspace or increasing platform stiffness, have been proposed to optimize cable layout in recent work on RCDPRs [3]- [START_REF] Rosati | On the design of adaptive cable-driven systems[END_REF]. However, reconfigurability is typically performed manually, a costly and time consuming task.

Recently a novel concept of Mobile Cable-Driven Parallel Robots (MCDPRs) has been introduced in [START_REF] Rasheed | Tension Distribution Algorithm for Planar Mobile Cable-Driven Parallel Robots[END_REF] to achieve an autonomous reconfiguration of RCDPRs. A MCDPR is composed of a classical CDPR with q cables and a n degreeof-freedom (DoF) moving-platform mounted on p mobile bases (MBs). The MCDPR prototype that has been designed and built in the context of Echord++ 'FASTKIT' project is shown in Fig. 1. FASTKIT addresses an industrial need for flexible pick-and-place operations while being easy to install, keeping existing infrastructures and covering large areas. The prototype is composed of eight cables (q = 8), a six degree-of-freedom moving-platform (n = 6) and two MBs (p = 2). The overall objective is to design and implement a system capable of interacting with a high level task planner for logistic operations. Thus the system must be capable of autonomously navigating to the task location, deploying the system such that the task is within the reachable workspace and executing a pick-and-place task. In spite of the numerous advantages of the mobile deployable system, the structural stability must be considered. In [START_REF] Rasheed | Tension Distribution Algorithm for Planar Mobile Cable-Driven Parallel Robots[END_REF], a real time continuous tension distribution scheme that takes into account the dynamic equilibrium of the moving-platform and the static equilibrium (SE) of the MBs has been proposed. In this paper, we focus on the workspace analysis of MCDPRs.

The classical techniques used to analyze the workspace of CDPRs are wrench-closure workspace (WCW) [START_REF] Gouttefarde | Analysis of the wrenchclosure workspace of planar parallel cable-driven mechanisms[END_REF], [START_REF] Lau | Wrench-Closure Workspace Generation for Cable Driven Parallel Manipulators using a Hybrid Analytical-Numerical Approach[END_REF] and wrench-feasible workspace (WFW) [START_REF] Gouttefarde | Interval-analysisbased determination of the wrench-feasible workspace of parallel cable-driven robots[END_REF]. In this paper, WFW is chosen as it is more relevant from a practical viewpoint. WFW is defined as the set of platform poses for which the required set of wrenches can be balanced with wrenches generated by the cables, while maintaining the cable tension within the defined limits [START_REF] Gouttefarde | Interval-analysisbased determination of the wrench-feasible workspace of parallel cable-driven robots[END_REF]. For a given pose, the set of wrenches a mechanism can generate is defined as available wrench set (AWS), denoted as A . For classical CDPRs, AWS depends on the robot geometric architecture and the tension limits. The set of wrenches required to complete a task, referred to as the required wrench set (RWS), denoted as R, will be generated if it is fully included by A : R ⊆ A .

(1)

For MCDPRs, the classical definition of AWS for CDPRs must additionally consider the Static Equilibrium (SE) constraints associated with the MBs. The two main approaches used to represent the AWS for CDPRs are the Convex hull method and the Hyperplane shifting method [START_REF] Grünbaum | Convex Polytopes[END_REF]. Once the AWS is defined, WFW can be traced using the Capacity Margin index [START_REF] Guay | Measuring how well a structure supports varying external wrenches[END_REF], [START_REF] Ruiz | Arachnis: Analysis of robots actuated by cables with handy and neat interface software[END_REF].

This paper deals with the determination of the AWS required to trace the workspace for planar MCDPRs with point-mass end-effector. Figure 2 illustrates a Planar MCDPR with p = 2 MBs, q = 4 number of cables and n = 2 DoF point mass end-effector. In this paper, wheels are assumed to form a simple contact support with the ground and friction is sufficient to prevent the MBs from sliding. This paper is organized as follows. Section II presents the parameterization of a MCDPR. Section III deals with the SE conditions of the MBs using free body diagrams. Section IV is about the nature of AWS for MCDPRs by considering the SE of the platform as well as the SE of the MBs. Section V discusses how to trace the workspace using the capacity margin index. Section VI shows some experimental validations of the concept. Finally, conclusions are drawn and future work is presented in Section VII.

II. PARAMETERIZATION OF A MCDPR

Let us denotes the jth Mobile Base (MB) as M j , j = 1, . . . , p. The ith cable mounted onto M j is named as C i j , i = 1, . . . , q j , where q j denotes the number of cables carried by M j . The total number of cables of the MCDPR is equal to

q = p ∑ j=1 q j . (2)
Let u i j be the unit vector of C i j pointing from the pointmass effector to the cable exit point A i j . Let t i j be the cable tension vector along u i j , expressed as

t i j = t i j u i j , (3)
where t i j is the tension in the ith cable mounted on M j . The force f i j applied by the ith cable onto M j is expressed as

f i j = -t i j u i j . (4)

III. STATIC EQUILIBRIUM OF A PLANAR MCDPR

For a planar MCDPR with a point mass end-effector, the SE equation of the latter can be expressed as

f e = - p ∑ j=1 q j ∑ i=1 t i j u i j . (5)
Equation ( 5) can be expressed in the matrix form as:

Wt + f e = 0, (6)
where W is a (2 × q) wrench matrix mapping the cable tension vector t ∈ R q onto the wrench applied by the cables

A 12 A 22 A 21 A 11 C C r 1 C l 2 C r 2 B 12 u 21 11 22 u u u G 1 G 2 O f cr j f cl j C lj C rj jt h m o b il e b a se f 2j 1j f G j w g j j A 1j A 2j P e f x [m] y [m] 12 22 1 1 ∑ 2
Fig. 2: Planar MCDPR composed of two MBs (p = 2), four cables (q = 4) with two cables per mobile base (q 1 = q 2 = 2) and a two degree-of-freedom (n = 2) point-mass end-effector on the end-effector. f e = [ f x e f y e ] T denotes the external wrench applied to the end effector. t and W can be expressed as:

t = [t 1 t 2 . . . t j . . . t p ] T , (7)
W = [W 1 W 2 . . . W j . . . W p ], (8)
where

t j = [t 1 j t 2 j . . . t q j j ] T , (9)
W j = [u 1 j u 2 j . . . u q j j ]. (10)
As the MBs should be in equilibrium during the motion of the end-effector, we need to formulate the SE conditions for each mobile base, also referred to as the tipping constraints. Figure 2 illustrates the free body diagram of M j with q j = 2. The SE equations of M j carrying up to q j cables is expressed as:

w g j + q j ∑ i=1 f i j + f cl j + f cr j = 0, (11)
m O j = 0. ( 12
)
w g j denotes the weight vector of M j . m O j denotes the moment of M j about point O.

f cl j = [ f x cl j f y cl j ] T and f cr j = [ f x cr j
f y cr j ] T denote the contact forces between the ground and the left and right wheels contact points C l j and C r j , respectively. Note that the superscripts x and y in the previous vectors denote their x and y components. m O j can be expressed as:

m O j = g T j E T w g j + q j ∑ i=1 a T i j E T f i j + c T l j E T f cl j + c T r j E T f cr j , (13)
with

E = 0 -1 1 0 ( 14
)
where a i j = [a x i j a y i j ] T denotes the Cartesian coordinate vectors of point A i j , c l j = [c x l j c y l j ] T and c r j = [c x r j c y r j ] T denote the Cartesian coordinate vectors of contact points C l j and C r j , respectively. g j = [g x j g y j ] T is the Cartesian coordinate vector of the center of gravity G j . Let m Cr j be the moment generated at the right contact point C r j at the instant when M j loses contact with the ground at point C l j such that f y c l j = 0, expressed as:

m Cr j = (g j -c r j ) T E T w g j + q j ∑ i=1 (c r j -a i j ) T E T t i j (15)
Let Σ j be the set composed of M j , its front and rear wheels, the cables attached to it and the point-mass end-effector, as encircled in red in Fig. 2. From the free body diagram of Σ i , moment m Cr j can also expressed as:

m Cr j = -(p -c r j ) T E T f + (g j -c r j ) T E T w g j + p ∑ o=1,o = j q o ∑ i=1 (p -c r j ) T E T t io , (16)
where o = 1, . . . , p and o = j. p denotes the Cartesian coordinate vector of the point-mass end-effector P. f = [ f x f y ] T denotes the force applied by the cables onto the point-mass end-effector, namely,

f = -f e . (17)
Similarly, the moment m Cl j generated at the left contact point C l j on Σ j takes the form:

m Cl j = -(p -c l j ) T E T f + (g j -c l j ) T E T w g j + p ∑ o=1,o = j q o ∑ i=1 (p -c l j ) T E T t io . (18)
For M j to be stable, the moments generated by the external forces at point C r j (C l j , resp.) should be counterclockwise (clockwise, resp.), namely,

m Cr j ≥ 0, j = 1, . . . , p (19)
m Cl j ≤ 0, j = 1, . . . , p (20)

IV. AVAILABLE WRENCH SET FOR MCDPRS

In this section the nature of the AWS for MCDPRs is analyzed. The cable tension t i j associated with the ith cable mounted on M j is bounded between a minimum tension t i j and a maximum tension t i j . It should be noted that the AWS of a classical CDPR depends uniquely on its platform pose and cable tension limits and forms a zonotope [START_REF] Bouchard | On the ability of a cable-driven robot to generate a prescribed set of wrenches[END_REF]. In contrast, the tipping constraints of the MBs must be considered in the definition of the AWS of a MCDPR. The AWS A 1 for a planar CDPR with a point mass end-effector can be expressed as:

A 1 = f ∈ R 2 | f = p ∑ j=1 q j ∑ i=1 t i j u i j , t i j ≤ t i j ≤ t i j , i = 1, . . . , q j , j = 1, . . . , p . (21) 0 1 2 3 1 2 3 4 5
11

t [N] 2 1 22 t [N] x f [N] y f [N] W 11 u 22 u 21 u x [m] y [m] t [ N ] T T 1 1 2 2 P 0 0
Fig. 3: MCDPR with q = 3 cables and a n = 2 DoF pointmass end-effector, the black polytopes illustrate the TS and AWS of the CDPR at hand, whereas green polytopes illustrate the TS and AWS of the MCDPR at hand

The AWS A 2 for a planar MCDPR with a point-mass endeffector is derived from A 1 by adding the tipping constraints defined in ( 19) and (20). Thus A 2 can be expressed as:

A 2 = f ∈ R 2 | f = p ∑ j=1 q j ∑ i=1 t i j u i j , t i j ≤ t i j ≤ t i j ,
m Cr j ≥ 0, m Cl j ≤ 0, i = 1, . . . , q j , j = 1, . . . , p .

Figure 3 shows a comparison between the Tension space (TS) T 1 and the AWS A 1 of a CDPR with three cables and an point-mass platform with the TS T 2 and AWS A 2 of a MCDPR with three cables, a point-mass platform and two mobile bases. The tension space between t 11 and t 21 is reduced by the linear tipping constraint on M 1 . As M 2 is carrying a single cable C 22 , only the maximum limit t 22 is modified. The difference in the polytopes A 1 and A 2 is due to the additional constraints associated with the equilibrium of the MCDPR MBs expressed by ( 19) and (20). By considering only the classical cable tension limit (t i j and t i j ) constraints, the shape of the AWS is a zonotope. When the tipping constraints are included, the AWS is no longer a zonotope, but a convex polytope. The two methods used to represent convex polytopes are V-representation, known as the convex hull approach, and H-representation, known as the hyperplane shifting method [START_REF] Grünbaum | Convex Polytopes[END_REF]. V-representation is preferred for visualization while H-representation is used to find the relation between A and R. The convex-hull approach is used to find the vertices that form the boundary of the polytope, whereas hyperplane shifting method is a geometric method used to obtain the facets of the polytope.

y [m] t [N] t [N] v 11 v 21 v 31 v 41 v 51 v 12 v 22 v 32 v 42 P x f [N] y f [N] 2 1
Fig. 4: Comparison of TS and AWS between CDPR (in black) and MCDPR (in green) (a) MCDPR configuration with q 1 = q 2 = 2 (b) TS formed by t i1 (c) TS formed by t i2 (d) A 1 is AWS of the CDPR at hand, A 2 is AWS of the MCDPR at hand

A. Convex Hull Method

AWS is defined using the set of vertices forming the extreme points of the polytope. For the jth mobile base M j , a q j dimensional TS is formed by the attached cables. The shape of this TS depends on the mapping of the tipping constraints on the q j -dimensional TS formed by the cable tension limits t i j and t i j . Figures 4(b) and 4(c) illustrate the TS associated with each MB of the MCDPR configuration shown in Fig. 4(a). The feasible TS is formed by the cable tension limits as well as the tipping constraints of the MBs. The new vertices of the TS for MCDPRs do not correspond with the minimum/maximum cable tensions as in the classical case [START_REF] Bouchard | On the ability of a cable-driven robot to generate a prescribed set of wrenches[END_REF].

Let v j denote the number of vertices and v k j be the coordinates of kth vertex for the TS associated to the jth mobile base M j , k = 1, . . . , v j . Let V j represent the set of the vertices of the TS associated to M j :

V j = {v k j }, k = {1, . . . , v j }. ( 23
)
Let V j be a (q j × v j ) matrix containing the coordinates of the TS vertices associated with M j , expressed as:

V j = [v 1 j v 2 j . . . v v j j ]. (24)
v is the total number of vertices formed by all the q cables and is obtained by the product of the number of vertices for each MB, namely,

v = p ∏ j=1 v j . ( 25
)
Let V represent the set of all vertices in the TS which is obtained by the Cartesian product between V j , j = 1, . . . , p. Accordingly, V is (q × v)-matrix, which denotes the coordinates of all the vertices in V expressed as: where g = 1, . . . , v. v g is a q-dimensional vector representing the coordinates of the gth vertex of the MCDPR Tension Space. The image of AWS is constructed from V under the mapping of the wrench matrix W. A numerical procedure such as quickhull [START_REF] Barber | The quickhull algorithm for convex hulls[END_REF] is used to compute the convex hull forming the boundary of AWS. Figure 4(d) illustrates the AWS obtained by Convex Hull Method. A 1 is the AWS obtained by considering only the cable tension limits and is a zonotope. A 2 is the AWS obtained by considering both the cable tension limits and the tipping constraints of the two mobile bases.

V = [v 1 v 2 . . . v g . . . v v ], (26)

B. Hyperplane Shifting Method

Hyperplane Shifting Method (HFM) is a geometric approach, which defines a convex polytope as the intersection of the half-spaces bounded by its hyperplanes [START_REF] Gouttefarde | Characterization of parallel manipulator available wrench set facets[END_REF]. The classical HFM used to obtain the AWS of CDPRs is described in [START_REF] Bouchard | On the ability of a cable-driven robot to generate a prescribed set of wrenches[END_REF], [START_REF] Gouttefarde | Characterization of parallel manipulator available wrench set facets[END_REF]. Nevertheless, this approach is not sufficient to fully characterize the AWS of MCDPRs because it requires a hypercube TS (T 1 ). For instance, for the MCDPR shown in Fig. 3, it can be observed that the TS (T 2 ) is no longer a hypercube due to the additional constraints associated with the SE of the mobile bases.

As a consequence, this section presents an improved version of the HFM described in [START_REF] Rasheed | Tension Distribution Algorithm for Planar Mobile Cable-Driven Parallel Robots[END_REF][START_REF] Pott | IPAnema: a family of cable-driven parallel robots for industrial applications[END_REF] that takes into account the tipping constraints of the MCDPR mobile bases. As a result, Fig. 5 depicts the AWS of the MCDPR configuration shown in Fig. 4(a), obtained by the improved HFM. This AWS is bounded by hyperplanes H + s , H - s , s = 1, ..., q, obtained from the cable tension limits associated to the four cables attached to the point mass end-effector, and by hyperplanes H r M 1 and H l M 2 , corresponding to the tipping constraints of M 1 about point C r1 and the tipping constraint of M 2 about point C l2 , respectively.

1) Determination of H l M j and H r M j , j=1,...,p: Let r r j (r l j , resp.) be the unit vector pointing from C r j (C l j , resp.) to P, expressed as:

r r j = p -c r j p -c r j 2 , r l j = p -c l j p -c l j 2 . ( 27
)
Upon dividing (19) ((20), resp.) by pc r j 2 ( pc l j 2 , resp.), the tipping constraints can be expressed in the wrench space as:

-r T r j E T f + (g j -c r j ) T p -c r j 2 E T w g j + p ∑ o=1,o = j q o ∑ i=1 r T r j E T t io ≥ 0, (28)
-r T l j E T f + (g j -c l j ) T p -c l j 2 E T w g j + p ∑ o=1,o = j q o ∑ i=1 r T l j E T t io ≤ 0.
(29) Equations ( 28) and (29) take the form:

e T r j f ≤ d r j , e T l j f ≤ d l j . (30)
Equation ( 30) corresponds to the hyperplanes [START_REF] Gouttefarde | Characterization of parallel manipulator available wrench set facets[END_REF] for the tipping constraints of M j in the wrench space. e r j and e l j are the unit vectors normal to H r M j and H l M j , expressed as: e r j = Er r j , e l j = -Er l j .

d r j (d l j , resp.) denotes the shifted distance of H r j (H l j , resp.) from the origin of the wrench space along e r j (e l j , resp.). The shift d r j depends on the weight of M j and the combination of the cable tension t io for which

p ∑ o=1,o = j q o ∑ i=1 r T
r j E T t io is a maximum. While the shift d rl depends on the weight of M j and the combination of the cable tension t io for which

p ∑ o=1,o = j q o ∑ i=1 r T l j E T t io is a minimum, namely, d r j = (g j -c r j ) T p -c r j 2 E T w g j + p ∑ o=1,o = j max q o ∑ i=1 r T r j E T v i ko u io , k = 1, ..., v o , (32)
d l j = - (g j -c l j ) T p -c l j 2 E T w g j - p ∑ o=1,o = j min q o ∑ i=1 r T l j E T v i ko u io , k = 1, ..., v o , (33)
where v i ko corresponds to the ith coordinate of v ko . Figure 6 shows the geometric representation of the tipping hyperplanes for the MCDPR under study. From Figs. 5 and6, it can be observed that the orientation of H r M j (H l M j , resp.) is directly obtained from r r j (r l j , resp.)

2) Determination of H + s and H - s , s=1,...,q: For classical CDPRs with given cable tension limits, ∆t i j = t i jt i j is a constant, AWS is a zonotope formed by the set of vectors α i j ∆t i j u i j , where 0 ≤ α i j ≤ 1 [START_REF] Bouchard | On the ability of a cable-driven robot to generate a prescribed set of wrenches[END_REF], [START_REF] Gouttefarde | Characterization of parallel manipulator available wrench set facets[END_REF]. The shape of the zonotope depends on the directions of the cable unit vectors u i j as well as the difference between the minimum and maximum cable tension limits ∆t i j . It is noteworthy that ∆t i j is no longer a constant for MCDPRs. The property of a zonotope having parallel facets still holds as the orientation of the hyperplanes is given by the cable unit vectors u i j . However, the position of the hyperplanes is modified, forming a convex polytope with parallel facets rather than a zonotope.

H + s and H - s are obtained using the classical HFM as described in [START_REF] Bouchard | On the ability of a cable-driven robot to generate a prescribed set of wrenches[END_REF], [START_REF] Gouttefarde | Characterization of parallel manipulator available wrench set facets[END_REF] based on the TS of MCDPRs. For a planar MCDPR with a point mass end-effector, each cable unit vector u s = u i j will form a pair of parallel hyperplanes {H + s , H - s } at the origin. Each pair is associated with a unit vector e s orthogonal to its facets, expressed as:

e s = E T u s . ( 34
)
The shift of the initial hyperplanes is determined by the projection of the MCDPR tension space vertices on e s . Let l s be a q-dimensional vector containing the projections of the cable unit vectors in W on e s , expressed as:

l s = W T e s . (35)
The projection of u s will be zero as it is orthogonal to e s . The distances h + s and h - s are given by the maximum and minimum combinations of l s with the coordinates of the TS vertices V, expressed as:

h + s = max q ∑ s=1 v s g l s , g = 1, . . . , v , (36) h
- s = min q ∑ s=1 v s g l s , g = 1, . . . , v , (37)
where v s g and l s denote the sth coordinate of vector v g and l s , respectively. To completely characterize the hyperplanes, a point p + s (p - s , resp.) on H + s (H - s , resp.) must be obtained, given as:

p + s = h + s e s + p ∑ j=1 q j ∑ i=1 t i j u i j ; p - s = h - s e s + p ∑ j=1 q j ∑ i=1 t i j u i j , (38)
The respective pair of hyperplanes is expressed as:

H + s : e T s f ≤ d + s ; H - s : -e T s f ≤ d - s . (40)
The above procedure is repeated to determine the q pairs of hyperplanes associated to the q cables of the MCDPR.

V. WORKSPACE ANALYSIS Wrench-feasible workspace (WFW) is defined as the set of poses that are wrench-feasible [START_REF] Bosscher | Wrenchfeasible workspace generation for cable-driven robots[END_REF]. A well known index used to compute the wrench feasible set of poses is called Capacity Margin [START_REF] Guay | Measuring how well a structure supports varying external wrenches[END_REF], [START_REF] Ruiz | Arachnis: Analysis of robots actuated by cables with handy and neat interface software[END_REF]. It is a measure of the robustness of the equilibrium of the robot, expressed as:

s = min ( min s j,l ), (41)
where s j,l is the signed distance from jth vertex of the required wrench set R to the lth face of the available wrench set A . s j,l is positive when the constraint is satisfied, and negative otherwise. The index remains negative as long as at least one of the vertices of R is outside of A . The index is positive if all the the vertices of R are inscribed by A .

VI. EXPERIMENTS AND RESULTS

The proposed approach was tested on MCDPR prototype shown in Fig. 7(a), made up of two TurtleBot mobile bases and a 0.5 kg point mass end-effector. The WFW of the MCDPR under study is illustrated in Fig. 7(b) for R equal to the weight of the end-effector. The green region corresponds to the modified WFW where both cable tension limits and mobile base tipping constraints are satisfied. In blue area, at least one of the two mobile bases is tipping. In red area, both the cable tension limits and the mobile base tipping constraints are not satisfied to keep the end-effector in equilibrium and to avoid mobile base tipping. It can be observed that for MCDPRs, the ability of the cables to apply wrenches on the platform may be reduced due to the mobile base tipping constraints. The simulation and the experimental validation of the proposed approach can be seen in video 1 . In the latter, two different trajectories are tested and compared. Based on the proposed workspace analysis, it can be observed that if the end-effector is within the defined WFW calculated offline, both the mobile bases are in equilibrium. On the contrary, for the end-effector outside of the WFW, at least one mobile base is not in equilibrium anymore.

VII. CONCLUSION

In this paper, the Available Wrench Set required to trace the Wrench Feasible Workspace of a Mobile Cable-Driven Parallel Robot (MCDPR) has been determined. The proposed workspace considers the cable tension limits and the static equilibrium of the mobile bases. Two different approaches, the convex hull and the hyperplane shifting method, are used to illustrate how the additional constraints can be considered. The additional constraints modified the shape of the AWS, forming new facets and reducing the capability of the cables to apply wrenches on the platform. Future work will focus on extending this approach to spatial MCDPRs consisting of more than two mobile bases and taking into account wheel slipping constraints. Furthermore, the evolution of the MCDPR workspace during their deployment will be studied.

Fig. 1 :

 1 Fig. 1: Fastkit prototype, Undeployed configuration (left) Deployed configuration (right)

1 +Fig. 5 :

 15 Fig.5: AWS formed by the intersection of all the hyperplanes (red hyperplanes corresponds to the tipping constraints, blue hyperplanes corresponds to the cables tension limits)

Fig. 6 :

 6 Fig. 6: Geometric representation of e r1 and e l2 from the MCDPR configuration

Fig. 7 :

 7 Fig. 7: MCDPR prototype made up of two TurtleBot mobile bases and a 0.5 kg point mass end-effector

supported by École Centrale Nantes and Echord++ FASTKIT project.


Divya Shah

Jiuchun Gao

Anatol Pashkevich

Stéphane Caro

Benoît Courtemanche

Computer-Aided Design and Optimization of a Redundant Robotic System for Automated Fiber Placement Process

Keywords: Robotic fibre placement, Computer-Aided Design, Optimal motion planning, Collision detection, Robot programming

This paper proposes a comprehensive methodology for the computer-aided design and optimization of a robotic workcell for the automated fiber placement. The robotic cell, comprising of a 6-axis industrial manipulator and an actuated positioner, is kinematically redundant with respect to the considered task. An efficient optimization technique for managing these kinematic redundancies is proposed. The robot/positioner motion planning and robotic cell layout design in a CAD environment are presented. To confirm validity of the developed methodology, experimental results are presented that deal with automation of thermoplastic fiber placement process.

INTRODUCTION

Recently, use of fiber reinforced composite materials has drastically increased, mainly in aerospace and automotive industries [START_REF] Marsh | Automating aerospace composites production with fibre placement[END_REF]. Compared to their conventional counterparts, composite materials offer better stiffness-to-weight ratio, strength-to-weight ratio, flexibility of shaping and corrosion resistance [START_REF] Gay | Composite materials: design and applications[END_REF]. To produce components from composite materials, a new technological process AFP (Automated Fiber Placement) is increasingly implemented. In this process the workpiece molds or liners are mounted on the shaft of a rotating positioner to improve the accessibility of certain locations. An industrial manipulator equipped with a specialized technological tool is used to heat the fiber tows and place them in-situ using a pressure roller [START_REF] Peters | Handbook of composites[END_REF].

A typical AFP workcell, comprising of a 6-axis manipulator and 1-or 2-axis positioner for rotating the workpiece, has at least 7-8 degrees of freedom (DOF) whereas the fiber placement task requires only 6 DOF. This kinematic redundancy provides the user with some flexibility in programming the robot motions, and at the same time creates a problem of redundancy management in an optimal way in order to maximize the productivity. At present, a general trend in literature for redundancy resolution is based on converting this continuous path problem into a discrete one [START_REF] Li | A survey on path planning algorithms in robotic fibre placement[END_REF]. Further enhancements were successfully achieved for the applications of laser cutting in [START_REF] Dolgui | Manipulator motion planning for high-speed robotic laser cutting[END_REF] following a graph-based search. For fiber placement application, manipulator motion planning was developed in [START_REF] Martinec | Calculation of the robot trajectory for the optimum directional orientation of fibre placement in the manufacture of composite profile frames[END_REF] assuming constant tool velocity. An alternative approach was proposed in [START_REF] Debout | Tool path smoothing of a redundant machine: Application to Automated Fiber Placement[END_REF], focusing on tool path smoothing in Cartesian space. In [START_REF] Gan | Singularity-Free Workspace Aimed Optimal Design of a 2T2R Parallel Mechanism for Automated Fiber Placement[END_REF], the authors propose a singularity-free optimal design for a 2T2R parallel robot with a 2-axis positioner, thus eliminating the kinematic redundancy for the AFP process.

Another challenge in implementation of AFP is to design the optimal layout of the robotic system and to automate the robot programming process. Existing works in this area mainly deal with welding applications [START_REF] Li | Seeder Rack Welding Robot Workstation Design Based on DELMIA[END_REF]. For the fiber placement, some results were presented in [START_REF] Krombholz | ADVANCED AUTOMATED FIBRE PLACEMENT[END_REF].

However, the complete design and optimization procedure for AFP is very complicated and time-consuming. For this reason, this work concentrates on CAD-based design of workstations for AFP technology taking into account kinematic redundancy of the robotic system.

COMPUTER-AIDED DESIGN OF ROBOTIC CELL

The developed methodology for the design and optimization a robotic system for AFP is presented in Figure -1. The inputs in this process are the robot, positioner, tool and the workpiece models. The first step is the robotic cell assembly, wherein the layout of workcell is determined, and the manufacturing task is defined considering the desired placement track. The next step involves generation of optimal motions for the system taking into consideration the joint limits and kinematic singularities and checking for the admissibility of the relevant locations of the technological tool. At the final steps, the robotic cell simulations and motion programming is performed. If some problems are detected during simulations, the design process can be reconfigured and repeated with necessary modifications in the optimal motion generation or the cell assembly steps. The overall outcomes of the procedure include the cell layout along with robot programs for optimal trajectories, ready to be implemented. Each of these aspects is discussed in detail in the subsequent sections of this paper.

ROBOTIC CELL ASSEMBLY & TASK DEFINITION

The computer-aided design of the robotic cell includes selection of the workcell components and their relative positioning. To simplify this process, the robotic CAD system includes a library of commercially available robots and auxiliary devices. The positioning of the components is adjusted in a manner that allows the robot to reach the entire surface of the workpiece. In addition, certain CAD systems also include specialized tools allowing the user to create a 3D model of the workpiece and generate the fiber placement tracks. This workpiece model is also an input for robotic cell design process.

3D Modeling of Robotic Cell

The robotic system considered in this work consists of 6-axis serial anthropomorphic robot, specialized placement tool, 1-axis rotating positioner and a 1-axis linear track. Thus the system has 8 DOF, with a degree of redundancy equal to 2. Depending upon the geometry of the workpiece in consideration, the robot base location on the linear track can be fixed or changed online; thus providing the additional flexibility.

Using the standard step/iges files available from the manufacturer's catalogs, each of the components was designed in the CATIA V5 software from Dassault Systems as shown in Figure -2. All the active joints were defined so as to be able to simulate the system using joint coordinates.

Kinematic Modeling of Robot & Positioner

For developing the kinematic model of the system, a frame is attached to each component of the robotic cell as follows:  : Global frame with Z-axis in the vertical direction.  : Robot base frame with origin at the base of the robot and orientation same as the global frame.  : Positioner frame with origin at the center of rotating flange and Z-axis as the rotating axis.



: Workpiece frame with Z-axis along the direction of the workpiece liner.  : Tool frame located at Tool Center Point (TCP).  : Linear track frame with origin at the centre of the moving rail and orientation same as global frame.

The geometric model for the robot r g is obtained using the standard Denavit-Hartenberg parameters and presented in the following way:

    j j j j r r q g     6 1 1 T q , ( 1
)
where, T is the 4x4 homogenous transformation matrix, r q are the joint coordinates and j is the joint index. Similarly, the geometric model for the positioner p g is presented as a 4x4 homogenous rotation matrix

Z Rot around the Z-axis;     p Z p p q q g Rot  ; (2)
and for the linear track l g as a 4x4 homogenous translational matrix

X Trans along the X-axis;     l X l l q q g Trans  . ( 3
)
Manufacturing Task Definition In the developed system, there are some specialized tools to define fiber placement tracks for the desired workpiece. These tracks are then discretized into a set of points, called task locations, expressed in the workpiece frame. A Cartesian coordinate frame is attached to each of the task location such that the Z-axis is always normal to the workpiece surface and the X-axis follows the travelling direction of the track. The following equation represents the n task locations, indexed as i;

OPTIMAL TRAJECTORY GENERATION

For the considered AFP process, where the tool speed variations are allowed, a discrete optimization based methodology was proposed in [START_REF] Gao | Optimization of the robot and positioner motion in a redundant fiber placement workcell[END_REF] to generate timeoptimal trajectories considering one degree of redundancy, where the robot base location on the linear track is fixed. They are summarized below.

Task Graph Generation

Consider the time interval

where all notations are defined in previous section.

It is clear that this system is nonlinear and redundant. Here, it is reasonable to treat the positioner coordinate as the redundant variable, which allows using robot and positioner models independently. This redundant variable is sampled with a step of p q  as follows:

    max min , p p k p q q q  , (6)
where k=0,1,…,m and

  p p p q q q m    / min max
. Then, by computing the positioner direct kinematics and robot inverse kinematics as given by eqn. [START_REF] Debout | Tool path smoothing of a redundant machine: Application to Automated Fiber Placement[END_REF]; μ being the robot configuration index vector:

            μ q , 1 i k p p r i k r t q g g t   ; (7)
a set of all possible robot and positioner configurations for every sampling step of the positioner coordinate and for every task location of the given placement track, can be obtained:

            i k p i k r i k TL t q t , , q L  . ( 8
)
These 'location cells', expressed in the joint space, form the 'task graph' as shown in Figure -3. In case of two degrees of redundancy, similar sets of location cells can be computed for every sampling step of the linear axis coordinate l q , thus generating a two- dimensional task graph.

Admissibility & Collision Check

The next step is to verify the admissibility of these location cells on the task graph, i.e., to check for the practical feasibility of the configuration represented by the cell against the constraints. The cells violating any of these constraints are flagged "inadmissible".

Firstly, the task graph is checked for the actuator position, velocity and acceleration limits and the kinematic singularities of the robot, as given by the following equation:

      max min max min max min j i j j j i j j j i j j q t q q q t q q q t q q

               , ( 9
)
where j represents all actuator joints (robot and positioner).

The task graph, thus reduced, is then simulated on the CAD model to detect for collisions within the workcell. Various interferences among the workcell components are defined in the model and when the collisions occur, the colliding components are highlighted, as shown in Figure -4. A set of numerical values are obtained from the CAD model for each location cell, where zero represents all the collision-free candidates, thus obliging the following collision constraint:

        T t t q t p , 0 ; 0 ,    r q cols . ( 10

Optimal Trajectory Planning

To find the optimal trajectory, the edges on the task graph are weighted according to the minimum travelling time required between consequent locations, restricted by the maximum actuator velocities and accelerations. The objective function (robot processing time) can now be presented as a summation of these edge weights, expressed as dist :

      1 , , 1 1 1 ,       i k TL i k TL n i i i dist T L L , (11)
The optimization algorithm is based on the dynamic programming principle to compute the "shortest" path, with its length denoted as

i k d , :     i k TL i k TL i k k i k dist d d , ' 1 , , ' ' 1 , , min L L     , (12)
These distances are computed sequentially from the second location until the last. The indices k record the optimal solution, expressed in the joint space.

ROBOT CELL SIMULATIONS & PROGRAMMING

Debugging and analysis of the optimal robot motions is done by simulating them in the CAD environment. If the results are not acceptable, the procedure will be easily reiterated using task graphs with finer sampling step or even adjustments in the layout of the cell. The robot program for implementation is then generated. A brief comparison among the standard robot motion commands is shown in Table-1, the choice of which is critical to the performance of the system. This is because the travelling time is recomputed by the robot controller based on the motion commands. Hence, with execution of a set of polynomials and the possibility of adjusting the travelling time between nodes, Spline (SPL) commands are favorable.

EXPERIMENTAL RESULTS

To demonstrate the efficiency of the methods proposed above, the industrial case of fabricating a cylindrical pressure vessel with spherical domes (as shown in Figure -3) using thermoplastic composites was studied in collaboration with Centre Technique des Industries Mécaniques (CETIM). The laser assisted robotic fiber placement platform comprises of the following:  6-axis serial manipulator KUKA KR210 R3100 ultra.  1-axis linear track KUKA KL 2000.  1-axis rotating positioner AFPT 550 series.  Specialized placement tool from AFPT.

Starting from the individual models, the cell components were assembled together as shown in the Figure -2, such that the rotating axis falls within maximum area of robot workspace. Here, the length of the workpiece in consideration is much shorter than the horizontal reach of the robot. Thus, the robot base location on the linear axis was fixed. The best position was estimated by computing the solutions for different coordinates on the linear axis and comparing the total travelling times of the process, as shown in Figure -5. This provides the cell layout for the considered case. The desired placement track is represented by the white curve on the workpiece liner in Figure -3. To fabricate the complete pressure vessel, similar placement tracks are repeated with offsets on the rotating axis, so as to cover the entire surface area. This track was discretized into 200 task locations represented in the Cartesian space referenced to the workpiece frame. The positioner coordinate was sampled for every 1° and the solutions were generated for each to form the task graph. Following the admissibility check and trajectory planning, an optimal trajectory was generated for the process with the total travelling time 3.1sec T  . The following Figures-6 and -7 represent the positioner and robot joint coordinate and velocity curves respectively, plotted against the travelling time. The trajectory was simulated in the CAD model and the corresponding robot program, ready to be implemented on the industrial platform, was generated using the SPL commands. Compared to the initial value of sec 1 T 4  , this method reduces the total travelling time of the process 75%. This method can be adopted in fabrication of components with various shapes and sizes and complex placement track profiles. Smooth trajectories can be obtained by finer discretization of the placement track and finer sampling of the external axes. The use of the linear axis online allows for fabrication of long workpieces.

CONCLUSIONS

This paper contributes towards the complete methodology for computer-aided design and optimization of a redundant robotic system for the automated fiber placement problem. It integrates the computational tools for optimal motion planning with the CAD tools for design, simulation and verification. The theoretical results obtained in this work promise significant reduction in the processing time, thus improving the overall efficiency. Future work will involve the implementation of these results on the industrial platform at CETIM.

Figure- 1 .

 1 Figure-1. An outline of computer-aided design and optimization methodology for AFP process.

Figure- 2 .a

 2 Figure-2. CAD model of the AFP robotic workcell

  of the task locations defined in eqn. (4). Let us also, define the functions   robot and the positioner motions respectively. The optimization problem can thus be formulated as minimizing the total travelling time T . This problem is subject to the following equality constraint representing the closed loop form of the system:

Figure- 3 .

 3 Figure-3. Task-graph representation.

)Figure- 4 .

 4 Figure-4. Collision detection in CAD model.

Figure- 5 .

 5 Figure-5. Optimal robot location selection.

Figure- 6 .

 6 Figure-6. Robot and Positioner Joint Coordinates.

Figure- 7 .

 7 Figure-7. Robot and Positioner Joint Velocities.

Table -

 -

	1. Comparison of robot motion commands
		PTP	LIN/CIR	SPL
	Interpolation Joint-space Cartesian-space	Both
	Movement Start-Stop	Start-Stop	Continuous
	Time Setting	No	No	Yes


Abhilash Nayak
email: abhilash.nayak@ls2n.fr

Haiyang Li
email: haiyang.li@umail.ucc.ie

Guangbo Hao
email: g.hao@ucc.ie

Stéphane Caro
email: stephane.caro@ls2n.fr

St Éphane Caro

A Reconfigurable Compliant Four-Bar Mechanism with Multiple Operation Modes

published or not. The documents may come

INTRODUCTION

Although mechanisms are often composed of rigid bodies connected by joints, compliant mechanisms include flexible el-ements whose elastic deformation is utilized in order to transmit a force and/or motion. There are different ways to design compliant mechanisms, such as the kinematic based approaches, the building blocks approaches, and the structural optimizationbased approaches [START_REF] Gallego | Synthesis methods in compliant mechanisms: An overview[END_REF][START_REF] Olsen | Utilizing a classification scheme to facilitate rigid-body replacement for compliant mechanism design[END_REF][START_REF] Howell | Compliant mechanisms[END_REF][START_REF] Hao | Conceptual designs of multidegree of freedom compliant parallel manipulators composed of wire-beam based compliant mechanisms[END_REF]. In the kinematic based approach, the joints of a chosen rigid-body mechanism are replaced by appropriate compliant joints followed by pseudo-rigid body modeling [START_REF] Olsen | Utilizing a classification scheme to facilitate rigid-body replacement for compliant mechanism design[END_REF][START_REF] Howell | Compliant mechanisms[END_REF][START_REF] Hao | Conceptual designs of multidegree of freedom compliant parallel manipulators composed of wire-beam based compliant mechanisms[END_REF]. This method is advantageous due to the extensive choice of existing rigid-body mechanisms and their modeling tools. Parallel or closed-loop rigid-body architectures gain an upper hand here as their intrinsic properties favour the characteristics of compliant mechanisms like compactness, symmetry to reduce parasitic motions, low stiffness along the desired degrees of freedom (DOF) and high stiffness in other directions. Moreover, compliant mechanisms usually work around a given position for small range of motions and hence they can be designed by considering existing parallel manipulators in parallel singular configurations. Parallel singularity can be an actuation singularity, constraint singularity or a compound singularity as explained in [START_REF] Amine | Classification of 3T1R parallel manipulators based on their wrench graph[END_REF][START_REF] Maraje | Operation modes comparison of a reconfigurable 3-PRS parallel manipulator based on kinematic performance[END_REF][START_REF] Nurahmi | Dimensionally homogeneous extended jacobian and condition number[END_REF]. Rubbert et al. used an actuation singularity to type-synthesize a compliant medical device [START_REF] Rubbert | Using singularities of parallel manipulators for enhancing the rigid-body replacement design method of compliant mechanisms[END_REF][START_REF] Rubbert | Design of a compensation mechanism for an active cardiac stabilizer based on an assembly of planar compliant mechanisms[END_REF]. Another interesting kind of parallel singularity for a parallel manipulator that does not depend on the choice of ac-

l A D B C y 0 x 0 y 1 x 1 Ʃ 1 O 1 (a,b) O 0 (0,0) x Ʃ l Ʃ 0 FIGURE 1: AN EQUILATERAL FOUR BAR LINKAGE.
tuation is a constraint singularity [START_REF] Zlatanov | Constraint Singularities as C-Space Singularities[END_REF]. It divides the workspace of a parallel manipulator into different operation modes resulting in a reconfigurable mechanism. Algebraic geometry tools have proved to be efficient in performing global analysis of parallel manipulators and recognizing their operation modes leading to mobility-reconfiguration [START_REF] Husty | Algebraic methods in mechanism analysis and synthesis[END_REF][START_REF] Nurahmi | Reconfiguration analysis of a 4-RUU parallel manipulator[END_REF][START_REF] He | Design and Analysis of a New 7R Single-Loop Mechanism with 4R, 6R and 7R Operation Modes[END_REF]. Though there are abundant reconfigurable rigid-body mechanisms in the literature, the study of reconfigurable compliant mechanisms is limited. Hao studied the mobility and structure reconfiguration of compliant mechanisms [START_REF] Hao | Design and static testing of a compact distributed-compliance gripper based on flexure motion[END_REF] while Hao and Li introduced a position-spacebased structure reconfiguration (PSR) approach to the reconfiguration of compliant mechanisms and to minimize parasitic motions [START_REF] Hao | Positionspace-based compliant mechanism reconfiguration approach and its application in the reduction of parasitic motion[END_REF][START_REF] Li | Compliant mechanism reconfiguration based on position space concept for reducing parasitic motion[END_REF]. In this paper, one of the simplest yet ubiquitous parallel mechanisms, a planar equilateral four-bar linkage is considered at a constraint singularity configuration to synthesize a reconfigurable compliant four-bar mechanism. From our best understanding, this is the first piece of work that considers a constraint singularity to design a reconfigurable compliant mechanism with multiple operation modes, also called motion modes. This paper is organized as follows : Kinematic analysis of a rigid four-bar mechanism is performed to determine the constraint singularities and different operation modes. Rigid-body replacement design approach is followed to further synthesize a reconfigurable compliant four-bar mechanism and the motion type associated to each operation mode is verified through non-linear Finite Element Analysis (FEA).

KINEMATIC ANALYSIS AND OPERATION MODES OF A FOUR BAR LINKAGE

A planar equilateral four-bar linkage with equal link lengths, l is depicted in Fig. 1. Link AD is fixed, AB and CD are the cranks and BC is the coupler. Origin of the fixed frame (Σ 0 ), O 0 coincides with the center of link AD while that of the moving frame (Σ 1 ) O 1 with the center of BC. The coordinate axes are oriented in such a way that the position vectors of the intersection points between the revolute joint axes and the x 0 y 0 plane can be homogeneously written as follows:

r 0 A = [1, -l 2 , 0] T r 0 D = [1, l 2 , 0] T (1)
r 1 B = [1, -l 2 , 0] T r 1 C = [1, l 2 , 0] T (2)
The displacement of the coupler with respect to the fixed frame can be rendered by (a, b, φ ), where a and b represent the positional displacement of the coupler (nothing but the coordinates of point O 1 in Σ 0 ) and φ is the angular displacement about z 0 -axis (angle between x 0 and x 1 ). Thus, the corresponding set of displacements can be mapped onto a three-dimensional projective space, P 3 with homogeneous coordinates x i (i = 1, 2, 3, 4) [START_REF] Bottema | Theoretical Kinematics[END_REF]. This mapping (also known as Blashke mapping in the literature) is defined by the following matrix M :

M =       1 0 0 2x 1 x 3 + 2x 2 x 4 x 2 3 + x 2 4 -x 2 3 + x 2 4 x 2 3 + x 2 4 -2x 3 x 4 x 2 3 + x 2 4 -2x 1 x 4 + 2x 2 x 3 x 2 3 + x 2 4 2x 3 x 4 x 2 3 + x 2 4 -x 2 3 + x 2 4 x 2 3 + x 2 4       (3)
The planar kinematic mapping can also be derived as a special case of Study's kinematic mapping by equating some of the Study parameters to zero [START_REF] Husty | Algebraic methods in mechanism analysis and synthesis[END_REF]. To avoid the rotational part of M to be undefined, the following equation is defined:

H := x 2 3 + x 2 4 = 1 (4)
Without loss of generality, x i can be expressed in terms of (a, b, φ ), as follows [START_REF] Bottema | Theoretical Kinematics[END_REF] :

x 1 : x 2 : x 3 : x 4 = (au -bv) : (av + bu) : 2u : 2v with u = sin( φ 2 ), v = cos( φ 2 ) (5)

Constraint Equations

Points B and C are constrained to move along circles of centers A and D, respectively and with radius l each. The position vectors of points B and C are expressed algebraically in frame Σ 0 as follows :

r 0 B = M r 1 B ; r 0 C = M r 1 C (6) 4x 1 2 + 4x 2 2 -l 2 x 4 2 = 0 4lx 1 x 3 = 0 Q 1 Q 2 x 3 /x 4 x 1 /x 4 x 2 /x 4 L 1 L 2 C FIGURE 2: CONSTRAINT MANIFOLDS OF THE FOUR BAR LINKAGE IN IMAGE SPACE.
Therefore, the algebraic constraint equations take the form :

(r 0 B -r 0 A ) T (r 0 B -r 0 A ) = l 2 =⇒ g 1 := 4(x 2 1 + x 2 2 ) + 4lx 1 x 3 -l 2 x 2 4 = 0 (7) (r 0 C -r 0 D ) T (r 0 C -r 0 D ) = l 2 =⇒ g 2 := 4(x 2 1 + x 2 2 ) -4lx 1 x 3 -l 2 x 2 4 = 0 (8)
Since g1 ± g2 = 0 gives the same variety, the final simplified constraint equations are :

H 1 := g 1 -g 2 := 4lx 1 x 3 = 0 (9)
H 2 := g 1 + g 2 := 4(x 2 1 + x 2 2 ) -l 2 x 2 4 = 0 (10)
Equation ( 9) degenerates into two planes x 1 = x 3 = 0 into the image space and Eqn. ( 10) amounts to a cylinder with a circular cross-section in the image space. Assuming x 4 = 0, these constraint manifolds can be represented in the affine space, A 3 , as shown in Fig. 2.

Operation Modes

The affine variety of the polynomials H 1 and H 2 amounts to all the possible displacements attainable by the coupler. This variety is nothing but the intersection of these constraint surfaces in the image space [START_REF] Husty | Algebraic methods in mechanism analysis and synthesis[END_REF]. The intersections can be seen as two lines and a circle in Fig. 2. In fact, these curves can be algebraically represented by decomposing the constraint equations ( 9) and [START_REF] Zlatanov | Constraint Singularities as C-Space Singularities[END_REF]. A primary decomposition of the ideal I = H 1 , H 2 onto the field K(x 1 , x 2 , x 3 , x 4 ) results in the following sub-ideals:

I 1 = x 1 , 2x 2 -lx 4 (11) I 2 = x 1 , 2x 2 + lx 4 ( 12
)
I 3 = x 3 , 4(x 2 1 + x 2 2 ) -l 2 x 2 4 ( 13
)
It shows that this four-bar linkage has three operation modes.

The Hilbert dimension of the ideals I i including the polynomial H from Eqn. ( 4) is calculated to be one, indicating that the DOF of the four-bar mechanism is one in each of these three operation modes. I 1 and I 2 correspond to x 1 = 0 implying u = b a from Eqn. [START_REF] Amine | Classification of 3T1R parallel manipulators based on their wrench graph[END_REF]. Furthermore, for I 1 , eliminating u from 2x 2lx 4 = 0 gives

a 2 + b 2 -al = 0 (14)
which is the equation of a circle of center point B of Cartesian coordinates ( l 2 , 0) and radius l 2 as shown in Fig. 3.

l/2 D B C ( a , b ) (0,0) (0,0) (l/2,0) A D FIGURE 3: OPERATION MODE 1 : a 2 + b 2 -al = 0
Similarly, I 2 yields

a 2 + b 2 + al = 0 (15)
which is the equation of a circle of center point C of Cartesian coordinates (-l 2 , 0) and radius l 2 as shown in Fig. 4. The third ideal I 3 corresponds to x 3 = 0 and hence u = 0 implying φ = 0. The second equation of the same ideal results in

a 2 + b 2 -l 2 = 0 (16) B C l/2 A ( a , b ) (0,0) D (-l/2,0) (0,0) (0,0) (0,0) ) FIGURE 4: OPERATION MODE 2 : a 2 + b 2 + al = 0.
being the equation of a circle of center (0, 0) and radius l as shown in Fig. 5. As a result, I 1 and I 2 represent rotational modes while I 3 represents a translational mode.

l A D B C (a,b) (0,0) B FIGURE 5: OPERATION MODE 3 : a 2 + b 2 -l 2 = 0.
Ultimately, in Fig. 2, the intersection lines L 1 and L 2 of the constraint manifolds portray the rotational motion modes while the circle C portrays the translational motion mode.

Constraint Singularities

These operation modes are separated by two similar constraint singularities shown in Fig. 6.

They can be algebraically represented by 5), these singularities occur when b = 0, φ = 0 and a = ±l. These two configurations correspond to the two points Q 1 and Q 2 in the image space shown in Fig. 2. At a constraint singularity, any mechanism gains one or more degrees of freedom. Therefore, in case of the four-bar linkage with equal link lengths, the DOF at a constraint singularity is equal 2. In this configuration, points A, B, C and D are collinear and the corresponding motion type is a translational motion along the

x 1 = x 3 = 4x 2 2 - l 2 x 2 4 = 0. From Eqn. (
D B C A (0,0) (l,0) L ABCD (a) a = l, b = 0, φ = 0 A C D B (0,0) (-l,0) L ABCD (b) a = -l, b = 0, φ = 0 FIGURE 6: CONSTRAINT SINGULARITIES OF THE FOUR BAR MECHANISM.
normal to the line L ABCD passing through the four points A, B, C and D combined with a rotation about an axis directed along z 0 and passing through L ABCD . Eventually, it is noteworthy that two actuators are required in order to control the end-effector in those constraint singularities in order to manage the operation mode changing.

DESIGN AND ANALYSIS OF A COMPLIANT FOUR-BAR MECHANISM

In this section, two compliant four-bar mechanisms, compliant four-bar mechanism-1 and compliant four-bar mechanism-2, are proposed based on the operation modes and constraint singularities of the four-bar rigid-body mechanism shown in Fig. 6b. Moreover, the desired motion characteristics of the compliant four-bar mechanism-2 are verified by nonlinear FEA simulations.

Design of a compliant four-bar mechanism

Based on the constraint singularity configuration of the four-bar rigid-body mechanism represented in Fig. 6, a compliant four-bar mechanism can be designed through kinematically replacing the rigid rotational joints with compliant rotational joints [START_REF] Hao | Positionspace-based compliant mechanism reconfiguration approach and its application in the reduction of parasitic motion[END_REF]. Each of the compliant rotational joints can be any type compliant rotational joint such as cross-spring rotational joint, notch rotational joint and cartwheel rotational joint [START_REF] Howell | Compliant mechanisms[END_REF]. As shown in Fig. 7, a compliant four-bar mechanism, termed as the compliant four-bar mechanism-1, has been designed by replacing the four rigid rotational joints with three cross-spring rotational joints (RJ-0, RJ-1 and RJ-3) and one leaf-type isoscelestrapezoidal rotational joint that provides remote rotation centre (RJ-2).

For small motion ranges, the compliant four-bar mechanism-1 has the same operation modes as the fourbar rigid-body mechanism shown in Fig. 6, via controlling the rotations of the Bar-1 and Bar-3. Moreover, both the compliant four-bar mechanism-1 and the four-bar rigid-body mechanism are plane motion mechanisms. Additionally, the three cross-spring rotational joints in the compliant four-bar mechanism-1 can be replaced by other types of rotational joints, which can form different compliant four-bar mechanisms. In this paper, cross-spring rotational joints are employed due to their large motion ranges while small rotation centre shifts. However, the isosceles-trapezoidal rotational joint in the compliant four-bar mechanism-1 performs larger rotation centre shifts compared with the cross-spring rotational joint. Therefore, the compliant four-bar mechanism-1 can be improved by replacing the leaf-type isosceles-trapezoidal rotational joint with a cross-spring rotational joint. Such an improved design can be seen in Fig. 8, which is termed as the compliant four-bar mechanism-2. Note that, in Fig. 8, the RJ-0 and RJ-2, are traditional cross-spring rotational joints, while both the RJ-1 and the RJ-3 are double cross-spring joints introduced in this paper. Each of the rotational joints, RJ-1 and RJ-3, consists of two traditional cross-spring rotational joints in series. We specify that the Bar-0 is fixed to the ground and the Bar-2 is the output motion stage, also named coupler. The main body including rigid bars and compliant joints of the proposed compliant four-bar mechanism-2 can be fabricated monolithically using a CNC milling machine. It can also be 3D printed, and a 3Dprinted prototype is shown in Fig. 8. The bars of the prototype have many small through holes, which can reduce material consumption and improve dynamic performance. Additionally, two cross-shaped parts are added to the actuated bars, which are used to actuate the mechanism by hands. The operation modes of the compliant four-bar mechanism-2 as output stage are analyzed in the following sections.

OPERATION MODES OF THE COMPLIANT FOUR-BAR MECHANISM-2

Like the four-bar rigid-body mechanism shown in Fig. 6b, the output motion stage (Bar 2) of the compliant four-bar mechanism-2 has multiple operation modes under two rotational actuations (controlled by input displacements α and β ), as shown in Fig. 8. However, the compliant four-bar mechanism-2 has more operation modes than the rigid counterpart. In order to simplify the analysis, let α and β be non-negative. A coordinate system is defined in Fig. 8, which is located on Bar 2. Based on this assumption, operation modes of the compliant four-bar mechanism-2 are listed below : These operation modes are also highlighted through the printed prototype in Fig. 9. The primary motions of output motion stage (Bar-2) are the rotation in the XY plane and the translations along the X-and Y-axes; while the rotations in the XZ and YZ planes and translational motion along the Z-axis are the parasitic motions that are not the interest of this paper. Moreover, the rotation angle in the XY-plane and the Y-axis translational motion can be estimated analytically using Eqs. ( 17) and [START_REF] Zhao | A novel compliant linear-motion mechanism based on parasitic motion compensation[END_REF]. However, the X-axis translational motion cannot be accurately estimated in such a simple way, because it is heavily affected by the shift of the rotation centres of the two cross-spring rotational joints [START_REF] Zhao | A novel compliant linear-motion mechanism based on parasitic motion compensation[END_REF]. The X-axis translational motion will be analytically studied in our future work, but will be captured by non-linear FEA.

θ Z = α -β (17)
D Y = 1 2 (L B + L R )(sin α + sin β ) (18)
where θ Z is the rotation in the XY plane and D Y is the translational displacement in the Y-axis. L B and L R are the geometrical dimensions of the reconfigurable mechanism at hand, as defined in Fig. 8.

SIMULATIONS OF THE OPERATION MODES

In order to verify the operation modes of the 4R compliant mechanism-2, nonlinear FEA software is employed to simulate the motions of the compliant four-bar mechanism-2. For the FEA simulations, let L B be 100 mm, L R and L H be 50 mm, the beam thickness be 1 mm, the beam width be 23 mm, the Poissons ratio be 0.33, and the Youngs modulus be 6.9 GPa. Commercial software, COMSOL MULTIPHYSICS, is selected for the nonlinear FEA simulations, using the 10-node tetrahedral element and finer meshing technology (minimum element size 0.2 mm, curvature factor 0.4, and resolution of narrow regions 0.7). Note that the translational displacements of the Bar-2 along the X and Y axes are measured at the centre point of the top surface of the Bar-2 (termed as the interest point), as shown in Fig. Overall, for all the operation modes of the compliant four-bar mechanism-2, the obtained analytical kinematic models are accurate enough to predict the rotation angle in the XY-plane and the translation displacement along the Y-axis, under specific input actuations. Additionally, the parasitic motions are much smaller than the primary motions, which ensures that the tiny effect of the parasitic motions on the primary motions can be ignored in an acceptable way. Therefore, it has been proved that the compliant four-bar mechanism-2 can be operated in the different operation modes with high accuracy.

A PROSPECTIVE APPLICATION AS A COMPLIANT GRIPPER

The reconfigurable compliant four-bar mechanism-1 shown in Fig. 7 is used to design a reconfigurable gripper as shown in Fig. 14. It can exhibit four grasping modes based on the actuation of the linear actuator 1 (±α) or 2 (±β ) as displayed in Fig. 15. The first three grasping modes are angular, where the jaws of the gripper rotate about an instantaneous centre of rotation which is different for each grasping mode. The gripper displays an angular grasping mode when α = 0, β = 0 as shown in Fig. 15a, α = 0, β = 0 as shown in Fig. 15b or when α < 0, β < 0 as shown in the right Fig. 15c. The parallel grasping mode in which the jaws are parallel to one another is achieved when α > 0, β < 0 as shown in the left Fig. 15c. Thus, the reconfigurable compliant gripper at hand unveils an ability to grasp a plethora of shapes unlike other compliant grippers in literature that exhibit only one of these modes of grasping [START_REF] Hao | Design and static testing of a compact distributed-compliance gripper based on flexure motion[END_REF][START_REF] Hao | Conceptual design and modelling of a self-adaptive compliant parallel gripper for highprecision manipulation[END_REF]. Potential applications include micromanipulation and grasping lightweight and vulnerable materials like glass, resins, porous composites, etc. in difficult and dangerous environments. In addition, it can be used for medical applications to grasp and manipulate living tissues during surgical operations or as a gripper mounted on a parallel manipulator dedicated to fast and accurate pick-and-place operations. Figure 16 shows the prototype of the reconfigurable compliant gripper.

CONCLUSIONS AND FUTURE WORK

A novel idea of designing mobility-reconfigurable compliant mechanisms inspired by the constraint singularities of rigid body mechanisms was presented in this paper. A rhombus planar rigid four-bar mechanism was analyzed to identify its three operation modes and two constraint singularities separating those modes. The rigid joints were replaced by compliant joints to obtain two designs of a reconfigurable compliant four-bar mechanism. The second design was found to be more accurate and A novel reconfigurable compliant gripper less parasitic than the first one, which is verified by its nonlinear FEA simulations in different motion modes. Moreover, the compliant four-bar mechanism was shown to have four operation modes based on the particular actuation strategy unlike its rigid counterpart. A preliminary design of a compliant gripper has been designed based on the reconfigurable compliant fourbar mechanism introduced and studied in this paper. In the future, we will focus on the analytical kinetostatic modelling of the reconfigurable compliant mechanism at hand while exploring appropriate applications. We also intend to design mobility-reconfigurable compliant mechanisms based on the constraint singularities of spatial rigid body mechanisms.

Fig. 7 4R 2 FIGURE 7 : 1 Fig. 7 4RFIGURE 8 :

 727178 Fig. 7 4R compliant mechanisms: (a) 4R compliant mechanism-1, and (b) 4R compliant mechanism-2

1 .

 1 Operation mode I : Rotation in the XY-plane about the Axis-L, when α > 0 and β = 0, as shown in Fig. 9a, 2. Operation mode II : Rotation in the XY-plane about the Axis-R when α = 0 and β > 0, as shown in Fig. 9b, 3. Operation mode III : Rotation in the XY-plane about other axes except the Axis-L and Axis-R, when α = β > 0, as shown in Fig. 9c, and 4. Operation mode IV : Pure translations in the XY-plane along Operation mode I : Rotation in the XY-plane about the Axis-L Operation mode II : Rotation in the XY-plane about the Axis-R Operation mode III : Rotation in the XY-plane about other axes except the Axis-L and Axis-R Operation mode IV : Pure translations in the XY-plane along the X and Y axes

FIGURE 9 :

 9 FIGURE 9: OPERATION MODES OF THE COMPLIANT FOUR-BAR MECHANISM-2

  8. Results of the simulations are plotted in Figs. 10 to 13, and the following conclusions are drawn : 1. The maximum difference between the FEA results and the analytical results in terms of the Y-axis translation of the interest point (the centre of the top surface of the Bar-2) is tiny, which is less than 0.5% as shown in Figs. 10a, 11a, 12a and 13a. 2. The FEA results of the rotation in the XY-plane match the analytical results very well. The difference is less than 0.8 × 10 -3 rad (0.5% of the maximum rotation angle), which is shown in Figs. 10b, 11b and 12b.

FIGURE 10 :FIGURE 11 :

 1011 FIGURE 10: FEA RESULTS FOR OPERATION MODE I

FIGURE 12 :FIGURE 13 :

 1213 FIGURE 12: FEA RESULTS FOR OPERATION MODE III

  FIGURE 14: A novel reconfigurable compliant gripper

  (a) Angular grasping mode 1 : α = 0, β = 0 (b) Angular grasping mode 2 : α = 0, β = 0 (c) Left : parallel grasping mode (α > 0, β < 0); Right : angular grasping mode 3 (α < 0, β < 0)

FIGURE 15 :

 15 FIGURE 15: FOUR GRASPING MODES OF THE COMPLI-ANT GRIPPER

FIGURE 16 :

 16 FIGURE 16: Prototype of the reconfigurable compliant gripper

(a) Translations along the X and Y axes (b) Rotation about the Axis-L (c) parasitic motions (rotations about the X-and Y-axes and translation along the Z-axis)

ACKNOWLEDGMENT

The authors would like to express their gratitude for the Ulysses 2016 grant between Ireland and France. Mr. Tim Powder and Mr. Mike O'Shea in University College Cork are appreciated