Lorenzo Gagliardini
email: lorenzo.gagliardini@irt-jules-verne.fr

Stéphane Caro
email: stephane.caro@irccyn.ec-nantes.fr

Marc Gouttefarde
email: marc.gouttefarde@lirmm.fr

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

Alexis Girin
email: alexis.girin@irt-jules-verne.fr

Optimal Design of Cable-Driven Parallel Robots for Large Industrial Structures

This paper presents the preliminary studies dedicated to the design of cable-driven parallel robots (CDPRs) for industrial purposes. The goal is to transport the proper tools around a jacket, an offshore structure supporting a wind turbine, in order to perform painting and sandblasting tasks. In this paper, a simplified case study consisting of a structure composed of four tubes is investigated. A fully constrained CDPR and a suspended CDPR are studied. The design problems of the CDPRs at hand are formulated as optimization problems. They aim at determining the locations of the base anchor points of the cables that minimize the size of the CDPR, while satisfying a set of constraints. Those constraints guarantee that the moving platform can support the external wrenches and that there is no interference between the cables and between the cables and the environment, all along the path to be followed by the moving platform.

I. INTRODUCTION

Nowadays part of the worldwide energy production comes from renewable sources: offshore wind turbines represent a leading technology in the renewable energy sector [START_REF] Rock | Offshore wind energy[END_REF]. They can be installed in the sea through different supports, like jackets (offshore fixed structures, as illustrated in Fig. 1). Due to their shape and size, the fabrication of jackets is complicated and requires non standard procedures. The study of efficient technological processes aims at producing these structures in series and at limiting the risks for operators. In this context, the IRT Jules Verne is cooperating with STX Europe to investigate and develop a Cable Driven Parallel Robot (CDPR) in order to perform painting and sandblasting operations on jackets.

CDPRs can represent an appropriate technological solution for the considered tasks. Indeed, CDPR are able to cover wide spaces and their payload to weight ratio is usually very high. They are relatively cheap and can be designed in such a way to be reconfigurable and therefore adapted to different tasks.

Although no CDPR has already been designed for the application considered in this paper, research works have already been performed in similar sectors, like the naval, aeronautic and civil engineering ones. Albus et al. developed, at the beginning of the 90s, the NIST RoboCrane [START_REF] Albus | The NIST spider, a robot crane[END_REF], a CDPR able to perform several industrial operations, like painting of military aeroplanes and displacement of heavy payloads. In 2001 Holland and Cannon filed the patent US6826452 B1 [START_REF] Holland | Cable array robot for material handling[END_REF]. They proposed a robotic system composed of four cranes to be employed as a CDPR for cargo transportation. Currently, several research institutes and companies are involved in the framework of the European Project CableBOT that aims at developing modular and reconfigurable CDPRs able to perform different operations during the fabrication of largescale structures [4]. CDPRs can be employed also for other large workspace applications, e.g. the broadcasting of sport events [5], [6] and rescue operations [START_REF] Merlet | Kinematics of the wire-driven parallel robot marionet using linear actuators[END_REF].

This paper presents a preliminary design study of a CDPR transporting the tools necessary to perform painting and sandblasting operations on a jacket. The designs of a fully constrained CDPR and a suspended CDPR are investigated. Due to the complexity of a real jacket, the analysis is applied on a simplified structure made up of four tubes, as illustrated in Fig. 2. This context represents a new challenge since it introduces several constraints into the design problem. Due to the cluttered industrial environment, interferences between the cables and the environment need to be taken into account. The cable tensions and the positioning precision of the CDPR are constrained as well. In this paper, the design problem is formulated as an optimization problem, aiming at minimizing the size of the CDPR.

The paper is organized as follows: Section II describes the industrial context and the problem formulation. Section III introduces the CDPR models. Section IV presents the design of a fully constrained CDPR for the given case study. Section V describes the design of a suspended CDPR. Section VI concludes the paper and presents our future work.

II. PROBLEM DESCRIPTION

A jacket is an offshore structure composed of several tubes as shown in Fig. 1. One jacket has already been realized by STX Europe. It has a base of 30 m × 30 m, an height of 60 m and a weight of 1000 T. The upper part of the jacket, which remains outside the water, is 20 m high. This part should be painted and sandblasted, to be protected from oxidation. STX Europe planned to automatize part of the industrial process in order to produce several units per year.

The technical solution proposed in this paper consists in painting and sandblasting the structure by means of proper tools mounted on a CDPR platform: the Centre of Mass (CoM) of the CDPR follows a path along the jacket tubes approaching the tools to the jacket itself. When the CDPR is in the correct pose, the tools can perform the required tasks.

Due to the complexity of the problem, a simplified case study is introduced. Painting and sandblasting operations have to be performed on a four-bar tubular rigid structure. The structure is centered in an absolute reference frame, F b , of origin O b and axes x b , y b , z b , as shown in Fig. 2. It has an height, h s , of 10 m and a width, w s , of 5 m. The diameter of the tube, φ s , is equal to 50 cm.

The design problem can be summarized as follows. The CDPR cable exit point positions have to be determined. The CDPR should be able to follow a proper path to let the embarked painting and sandblasting tools perform the required operations along one side of the four-tube structure. The other side can be painted by a similar CDPR. Both the paths are illustrated in Fig. 2. They are 80 cm far from the structure. The CDPR should avoid cable interferences and collisions between the cables and the environment (here, the four-bar tubular structure). The cable tensions must respect given limits. The CDPR must be fully constrained during the operations, or at least in static equilibrium (in the case of a suspended design). The CDPR should be as compact as possible.

III. GEOMETRIC AND STATIC ROBOT MODELS

A CDPR is composed of a mobile platform, a support structure (the base) and m cables. Two reference frames are considered: the fixed base frame, F b , presented in Section II, and a platform frame, F p , of origin O p and axes x p , y p , z p . As illustrated in Fig. 3, the length of the i-th cable, between points A i and B i , is the norm of the vector l i expressed in F b as:

l i = a b i -t -Rb p i i = 1, . . . , m (1)
where R denotes the orientation matrix of the platform:

R = R z (φ)R x (θ)R y (ψ) = =   cφcψ -sφsθsψ -sφcθ cφsψ + sφsθcψ sφcψ + cφsθsψ cφcθ sφsψ -cφsθcψ -cθsψ sθ cθcψ   (2)
The unit vector of the i-th cable, d i , expressed in F b , is equal to:

d i = l i l i 2 i = 1, . . . , m (3)
The CDPR static model is represented by the following equilibrium equation [START_REF] Roberts | On the inverse kinematics, statics, and fault tolerance of cable-suspended robots[END_REF] Wτ + w e = 0 (4)

τ = [τ 1 , ..., τ m ]
T denotes the cable tension vector. Due to the non-rigid nature of the cables, tensions must be non-negative. Moreover, they have to remain smaller than the maximum tension value τ max . W denotes the wrench matrix composed of the wrenches w i exerted by the cables on the platform at point O p , namely,

W = d 1 d 2 . . . d m Rb p 1 × d 1 Rb p 2 × d 2 . . . Rb p m × d m (5)
w e represents the external wrench acting on the platform:

w e = f , m T = f x , f y , f z , m x , m y , m z T ( 6
)
Its components are supposed to be bounded.

f min ≤ f x , f y , f z ≤ f max (7) m min ≤ m x , m y , m z ≤ m max (8)
When the number m of cables is greater than 6, the system of equations ( 4) is underdetermined and the solution can be expressed as:

τ = τ n + τ 0 = W † w e + λn τ n ≥ 0 (9)
W † denotes the Moore-Penrose generalized inverse of W, λ ∈ R and n is a vector in the null space of W [START_REF] Roberts | On the inverse kinematics, statics, and fault tolerance of cable-suspended robots[END_REF].

IV. FULLY CONSTRAINED CDPR DESIGN The first design problem aims at minimizing the size of a fully constrained CDPR intended to paint a side of the tubular structure. To perform this operation, the CDPR CoM should follow the Path I shown in Fig. 2. The other side can be painted by a symmetrically placed CDPR following Path II. Each path is defined by a discretized curve P, composed of 600 equidistant points P . The CDPR static equilibrium is an important issue to take into consideration since the cables cannot push the platform. Therefore, the CDPR equilibrium can be satisfied only through positive cable tensions. The required robot accuracy is also considered, as well as the collisions between cables and between cables and the tubular structure. These constraints must be verified for all points P ∈ P.

The optimization is performed on a CDPR with eight cables, m = 8, which corresponds to the minimum even number of cables for a fully constrained CDPR [START_REF] Roberts | On the inverse kinematics, statics, and fault tolerance of cable-suspended robots[END_REF]. This choice should maintain the design of the CDPR as simple and cheap as possible. Steel cables are considered. They are characterized by the following properties: diameter, φ c , of 0.4 cm, Young Modulus equal to 100 GPa, elastic coefficient k of 1, 256 × 10 6 N/m and tension limit, τ max , equal to 11 650 N. In order to simplify the problem, the layout of points B i , i = 1, . . . , 8 is predefined. Points B i lie at the corners of a parallelepiped, whose width, w p , length, l p , and height, h p , are equal to 40 cm, 40 cm and 20 cm, respectively.

A. Design Variables

The decision variables of the optimization problem are represented by the position vectors of points A i . To reduce the number of variables, the points A i have been located at the vertices of parallelepiped whose edges are parallel to the axes x b , y b and z b . Hence, four variables, u x1 , u x2 , u y , u z , are sufficient to define the coordinates of points A i with respect to F b , as illustrated in Fig. 4,

a b 1 = [u x1 , u y , u z ] T , a b 2 = [u x1 , u y , -u z ] T ( 10
)
a b 3 = [u x2 , u y , u z ] T , a b 4 = [u x2 , u y , -u z ] T ( 11
)
a b 5 = [u x2 , -u y , u z ] T , a b 6 = [u x2 , -u y , -u z ] T ( 12
)
a b 7 = [u x1 , -u y , u z ] T , a b 8 = [u x1 , -u y , -u z ] T (13)
The boundaries of the design variables are defined as follows: The vector gathering the design variables is denoted x,

0.8 m ≤ u x1 ≤ 3.5 m, 0 m ≤ u x2 ≤ 0.8 m (14) 2.5 m ≤ u y ≤ 5 m, 5 m ≤ u z ≤ 7.5 m (15)
x = [u x1 , u x2 , u y , u z ] (16)
whereas q denotes the design parameter vector,

q = [m, τ max , φ c , l p , w p , h p , h s , w s , φ s ] (17)

B. Objective Function

The objective function, V (x), of the optimisation problem at hand is the volume of the parallelepiped whose vertices are the points A i , i = 1, . . . , 8:

V (x) = 4 (u x1 -u x2 ) u y u z (18)
This objective function has been selected in order to minimize the space occupied by the CDPR inside the workshop.

C. Constraints

The path to be followed by the CoM of the CDPR platform should be included in the Wrench Feasible Workspace (WFW) of the robot for the latter to be in static equilibrium and controllable [START_REF] Bosscher | rench-feasible workspace generation for cable-driven robots[END_REF], [START_REF] Gouttefarde | Interval-analysis-based determination of the wrench-feasible workspace of parallel cabledriven robots[END_REF]. The WFW is the set of platform poses that are wrench feasible. A pose is wrench feasible if the CDPR is able to balance any external wrench within a given set, with non-negative cable tensions lying in a given admissible interval [START_REF] Guay | Measuring how well a structure supports varying external wrenches[END_REF]. Hence, for the pose of the moving platform to be inside the WFW, the following condition should be fulfilled,

∀w e ∈ [w] r , ∃τ ∈ [τ ] | Wτ + w e = 0 (19)
where [w] r denotes the required wrench set, representing the set of wrenches that can be applied on the CDPR platform. Its boundaries have already been defined in Eq. ( 7) and Eq. ( 8). Their values are given in Table I.

[τ ] represents the set of admissible cable tensions, whose components are bounded as follows:

0 ≤ τ i ≤ τ max , ∀i = 1, . . . , 8 (20)
[w] r and [τ ] are defined in different spaces, connected through the static equilibrium equation by the wrench matrix.

The wrenches that can be generated by [τ ], through Eq. ( 4), is a zonotope, called the available wrench set, [w] a = -W[τ ]. In order to verify the wrench feasibility of the poses belonging to Path I, the technique described in [START_REF] Bouchard | On the ability of a cable-driven robot to generate a prescribed set of wrenches[END_REF] and [START_REF] Gouttefarde | Characterization of parallel manipulator available wrench set facets[END_REF] has been used. The algorithm verifies that all the vertices of

[w] r are included in [w] a , [w] r ⊆ [w] a .
According to [START_REF] Bouchard | On the ability of a cable-driven robot to generate a prescribed set of wrenches[END_REF] and [START_REF] Gouttefarde | Characterization of parallel manipulator available wrench set facets[END_REF], this condition can be translated into verifying the satisfaction of a finite set of linear inequalities:

Cw ≤ d, ∀w ∈ [w] r (21)
Moreover, cable-cable and cable-structure potential collisions are taken into account. Being given that two cables i and j are represented as straight line segments, a possible cable interference can be verified by computing the distance d cc i,j between the cables i and j. This distance d cc i,j is calculated by means of Lumelsky's approach [START_REF] Lumelsky | On fast computation of distance between line segments[END_REF] and has to be larger than the diameter, φ c , of the cables, i.e.,

d cc i,j ≥ φ c ∀i, j = 1, . . . , m, i = j (22)
For m = 8, the number of tests to be performed for each point P is equal to C 2 8 = 8! 2!6! = 28. Given a cable i and one of the four tubes k of the structure, cable-structure potential collisions are tested by means of the distance d cs i,k between the i-th cable and the k-th tube of the structure. The i-th cable and the k-th tube do not collide when d cs i,k is greater than the sum of the cable and tube radii.

d cs i,k ≥ (φ c + φ s ) 2 ∀i = 1, . . . , m, ∀k = 1, . . . , n c ( 23
)
where n c = 4 is the number of tubes of the structure considered in this paper. Consequently, the number of constraints to be considered is equal to m n c = 32. Due to operational requirements, the positioning errors δt x , δt y and δt z of the CoM of the moving platform along the x b , y b and x b axes should be smaller than 1 cm:

-1 cm ≤ δt x , δt y , δt z ≤ 1 cm (24)
The orientation errors δr x , δr y and δr z of the moving platform about x b , y b and z b axes should be smaller than 0.1 rad: -0.1 rad ≤ δr x , δr y , δr z ≤ 0.1 rad (25)

The elastostatic model of the CDPR is defined as follows:

w ext = Kδp = K δr T δt T T ( 26
)
where K is the stiffness matrix of the CDPR and the vector δp denotes the moving platform pose displacement generated by an external wrench w ext exerted on the moving-platform. Hence, vector δp can be computed as: From [START_REF] Bezhadipour | Stiffness of cable-based parallel manipulators with application to stability analysis[END_REF], matrix K can be expressed as

δp = K -1 w e (27)
K = K k + K t , with K k = 8 i=1 k i - τ i ρ i d i d T i d i d T i bb i T bb i T d i d T i bb i d i d T i bb i T + 8 i=1 τ i ρ i I 3x3 bb i T bb i bb i bb i T (28)
K t = - 8 i=1 τ i 0 3x3 0 3x3 0 3x3 di bb i ( 29
)
where k i denotes the i-th cable stiffness. k i is set equal to k. I 3x3 denotes the 3 × 3 identity matrix. bb i is the cross product matrix of vector b

b i = b b i,x , b b i,y , b b i,z
T defined as:

bb i =   0 -b b i,z b b i,y b b i,z 0 -b b i,x -b b i,y b b i,x 0   (30)
di denotes the cross product matrix of d i . Eq. ( 27) should be verified with respect to the bounds defined in Eq. ( 24) and Eq. ( 25) for all the vertices of [w] r .

D. Design Problem Formulation

The design problem of the CDPR can be formulated as follows:

mininimize V (x) = 4 (u x1 -u x2 ) u y u z over x = [u x1 , u x2 , u y , u z ] subject to ∀P ∈ P                  Cw ≤ d, ∀w ∈ [w] r d cc i,j ≥ φ c ∀i, j = 1, . . . , 8, i = j d cs i,k ≥ (φ c + φ s ) 2 ∀i = 1, . . . , 8, ∀k = 1, . . . , 4 -1 cm ≤ δt x , δt y , δt z ≤ 1 cm -0.1 rad ≤ δr x , δr y , δr z ≤ 0.1 rad (31)

E. Optimum Fully Constrained CDPR Design

The previous optimization problem has been solved by using the GlobalSearch Algorithm developed by Zsolt et al. [START_REF] Ugray | Scatter search and local nlp solvers: A multistart framework for global optimization[END_REF]. The optimum design is illustrated in Fig. 5. Table II provides the design variables and objective function values for the optimum fully constrained CDPR. This design is the most compact one, as visible by the short distances between the points A i along x b . This small distance prevents the cables from interfering and represents an interesting technical solution in terms of cable setting on the base.

V. SUSPENDED CDPR DESIGN

The second design optimization problem aims at finding the positions of the cable exit and connection points that minimize the size of a suspended CDPR intended to perform the task detailed in Section IV. The chosen number of cables is equal to six, i.e., m = 6. To simplify the problem, we assume that no external wrench is applied on the moving platform: only gravity is acting on the platform weighting 60 kg. The wrench produced by the platform weight is defined by the vector w g .

A. Design Variables

Unlike in Section IV, the length l p , the width w p and the height h p of the box-shaped moving platform are part of the variables of the design problem at hand. The configuration of points B i has a higher influence, in terms of static equilibrium, for suspended CDPR than for fully constrained ones. The Cartesian coordinates of points B i , i = 1, . . . , 6 are expressed in the moving platform frame as follows: (37) where The bounds on the design variables u x1 , u x2 , u y , u z are defined in Eq. ( 14) and Eq. ( 15). The bounds on variables l p , w p and h p are the following:

u x12 = (u x1 + u x2 ) /2.
0.3 m ≤ l p , w p , h p ≤ 0.6 m (38)
The design variables are collected in the vector x:

x

= [u x1 , u x2 , u y , u z , l p , w p , h p ] (39)
The design parameter vector q is defined as:

q = [m, τ max , φ c , h s , w s , φ s ] (40)

B. Design Problem Formulation

The objective function of the optimization problem is defined by Eq. (18). The constraints provided in Sec. IV-C are still valid. Hence, the design problem of the suspended CDPR can be formulated as follows:

mininimize V (x) = 4 (u x1 -u x2 ) u y u z over x = [u x1 , u x2 , u y , u z , l p , w p , h p ] subject to ∀P ∈ P                  Cw g ≤ d d cc i,j ≥ φ c ∀i, j = 1, . . . , 6, i = j d cs i,k ≥ (φ c + φ s ) 2 ∀i = 1, . . . , 6, ∀k = 1, . . . , 4 -1 cm ≤ δt x , δt y , δt z ≤ 1 cm -0.1 rad ≤ δr x , δr y , δr z ≤ 0.1 rad (41)

C. Optimum Suspended CDPR Design

The optimum design of the suspended CDPR, solution of problem (41), is shown in Fig. 7. The design variables and the objective function values associated with this design are given in Table III. Similarly to the optimum design of the fully constrained CDPR, the distance between the connection points, along x b , are reduced as much as possible, while respecting the static equilibrium constraint. The robot is in static equilibrium thanks to the gravity forces acting on the moving platform. To balance any external wrenches belonging to [x] r it would be necessary to increase the distance between points A i along the axis x b , possibly generating some collisions between the cables and the structure.

VI. CONCLUSIONS

This paper introduced a preliminary design study of CD-PRs intended to displace painting and sandblasting tools around jackets. This preliminary study considers a simplified four-tube structure in place of a jacket.

The design optimization problem has been formulated for a fully constrained CDPR considering the Cartesian coordinates of points A i as design variables. A suspended CDPR has been studied as well. The minimization of the size of the CDPRs was the objective of both the optimization problems. The size of the optimum fully constrained CDPR is smaller than the size of the optimum suspended CDPR. In the fully constrained CDPR, the connection points are very close to each other along the axis x b . The optimum fully constrained CDPR can assure the static equilibrium of the robot in presence of relevant external wrenches, along the x b and y b axes. However, the fully constrained CDPR requires a higher number of cables than the suspended CDPR. Furthermore, the suspended design frees the space below the platform from cables. Overall, if the presence of cable connected to the ground does not generate any issue, the fully constrained solution should be preferred.

The results presented in this paper could be applied to the design of a CDPR painting a single jacket side. The painting and sandblasting of a full jacket may require some reconfigurabilities. The study of reconfigurable CDPRs able to work around a jacket is part of our future work.

Fig. 1 .

 1 Fig. 1. Offshore wind turbine jacket, courtesy of STX Europe.

Fig. 2 .

 2 Fig. 2. Description of the task environment.

Fig. 3 .

 3 Fig. 3. Cable driven robot geometry. Example of a suspended CDPR. The same geometric description can be applied to a fully constrained CDPR.

Fig. 4 .

 4 Fig. 4. Layout of points A i for the fully constrained CDPR.TABLE I REQUIRED WRENCH SET DEFINITION

Fig. 5 .

 5 Fig. 5. Optimum design of the fully constrained CDPR. Solution of the optimization problem (31).

Fig. 6 .

 6 Fig. 6. Layouts of points A i and B i for the suspended CDPR.

Fig. 7 .

 7 Fig. 7. Optimum design of the suspended CDPR. Solution of the optimization problem (41).

TABLE III DESIGN

 III VARIABLE AND OBJECTIVE FUNCTION VALUES OF THE m 0.3 m 0.6 m 0.9 m 0.7 m 4.363 m 4.701 m 16.408 m 3

				OPTIMUM SUSPENDED CDPR
	hp	lp	wp	u x1	u x2	uy	uz	V
	0.3

ACKNOWLEDGMENT

This research work has been supported by the IRT Jules Verne, Bouguenais, France (Project CAROCA, Evaluation des CApacités de la RObotique à CÂbles dans un contexte industriel).

This work is supported by the IRT Jules Verne, Bouguenais, France (Project CAROCA, Evaluation des CApacités de


Lorenzo Gagliardini
email: lorenzo.gagliardini@irt-jules-verne.fr

Stéphane Caro
email: stephane.caro@irccyn.ec-nantes.fr

Marc Gouttefarde
email: marc.gouttefarde@lirmm.fr

Alexis Girin
email: alexis.girin@irt-jules-verne.fr

A Reconfiguration Strategy for Reconfigurable Cable-Driven Parallel Robots

This paper deals with Reconfigurable Cable-Driven Parallel Robots (RCDPRs). A RCDPR is able to change the locations of its cable exit points, the latter being defined as the connection points between the cables and the robot base frame. Given a RCDPR, a set of possible reconfigurations, a desired platform path and a description of the robot environment, the reconfiguration strategy proposed in this paper selects the optimal configurations to be associated to each point of the desired path. The selection of the optimal configurations can be performed with respect to several criteria such as the number of configuration changes, the number of cable reconfigurations and the robot stiffness. In this paper, the optimization is performed using a Dijkstra's based algorithm.

I. INTRODUCTION

Cable-Driven Parallel Robots (CDPR) are a particular class of parallel robots whose legs consist of cables (Fig. 1 shows an example of CDPR). CDPRs have been widely investigated in the last two decades. They are usually characterised by a large workspace size, their payload capability is quite high and their inertia is relatively low [START_REF] Gosselin | Cable-driven parallel mechanisms: state of the art and perspectives[END_REF]. Therefore, CDPRs can be employed in several applications, which require a combination of the previous characteristics: A successful example is represented by the shooting of sport events [2], [3]. Other applications have been proposed in the construction and logistic sectors, where heavy loads have to be moved [START_REF] Albus | The NIST spider, a robot crane[END_REF], [START_REF] Pott | Large-scale assembly of solar power platns with parallel cable robots[END_REF], [START_REF] Holland | Cable array robot for material handling[END_REF]. CDPRs can also be used for rehabilitation purposes [START_REF] Rosati | Performance of cable suspended robots for upper limb rehabilitation[END_REF] and as haptic devices [START_REF] Gallina | 3-dof wire driven planar haptic interface[END_REF]. CDPR studies deal also with other applications, such as rescue operations [START_REF] Merlet | Kinematics of the wire-driven parallel robot marionet using linear actuators[END_REF], artistic painting [START_REF] Gosselin | Global planning of dynamically feasible trajectories for three-dof spatial cable-suspended parallel robots[END_REF] and 3D scanning [START_REF] Dechênes | A cable-driven parallel mechanism for capturing object appearance from multiple viewpoints[END_REF].

A drawback of CDPRs is related to the potential collisions between the cables and the surrounding environment. Furthermore, the non-rigid nature of the cables influences the static equilibrium of the robot. In fact, the cables cannot push on the platform and the equilibrium depends notably on the cable layout.

When a task requires high performances, and when the working environment is cluttered, Reconfigurable Cable-Driven Robots (RCDPR) can provide better performances than standard CDPRs. By reconfigurable we mean that the RCDPR is able to change the locations of its cable exit points, defined as the connection points between the cables and the robot base frame. Thereby, by changing the cable layout, it should be possible to avoid collisions between the cables and the environment. At the same time, some robot characteristics may be improved: increasing stiffness, augmenting payload capability or decreasing the cable tensions.

The need of performing manufacturing operations over wide products is the main motivation of this work. The reconfiguration strategy proposed in this paper can notably be used to avoid unacceptable cable-environment collisions which usually take place when a non-configurable CDPR is considered.

The first studies on RCDPR have been performed in the frame of the NIST RoboCrane project [START_REF] Albus | The NIST spider, a robot crane[END_REF]. More recent investigations have been performed by Rosati et al. [START_REF] Rosati | On the design of adaptive cable-driven systems[END_REF], as well as by Zhuo et al. [START_REF] Zhou | Analysis framework for cooperating mobile cable robots[END_REF], Izard et al. [START_REF] Izard | A reconfigurable robot for cable-driven parallel robotic research and industrial scenario proofing[END_REF] and Nguyen et al. [START_REF] Nguyen | On the analysis of large-dimension reconfigurable suspended cable-driven parallel robots[END_REF], [START_REF] Nguyen | Study of reconfigurable suspended cable-driven parallel robots for airplane maintenance[END_REF]. The design of RCDPRs is one of the main goal of the European project CableBot [START_REF][END_REF].

Rosati et al. [START_REF] Rosati | On the design of adaptive cable-driven systems[END_REF] proposed a design and a reconfiguration strategy based on an analytical optimal cable layout characterization. This strategy can be applied efficiently to planar robots. However, it can hardly be extended to three-dimensional case studies, because the required cable layout analytical description is unknown for spatial CDPRs. Zhuo et al. [START_REF] Zhou | Analysis framework for cooperating mobile cable robots[END_REF] proposed to increase the Degrees of Freedom (DoF) of the robot using mobile connection points on the base. Nguyen et al. [START_REF] Nguyen | On the analysis of large-dimension reconfigurable suspended cable-driven parallel robots[END_REF] developed a reconfiguration strategy consisting of two sub-optimization problems. The first sub-problem aims at finding the bounds on the reconfiguration parameters with respect to a set of non-linear constraints. The computed bounds can be used in the second sub-problem, which is a box-constrained optimization problem. Nguyen et al. proposed several criteria, related to the cable tensions and the robot energy consumption.

For a RCDPR, several cable configurations may be appropriate to perform a given task. A cable configuration is defined as the positions of the exit points, the positions of the cable connection points on the platform and the layout (arrangement) of the cables between these two sets of points. In the sequel, we suppose that the RCDPR under study can assume a discrete set of cable configurations. Given a desired platform path and the task working conditions, the procedure is able to define the best set of configurations to assign to each point of the desired path. The selection of the optimal configurations is performed using a graph based approach. This approach associates to each point of the path a configuration by minimizing one or several criteria, such as the overall number of cable reconfigurations or the platform positioning error.

This paper is organized as follows. Section II presents the geometric, static and elasto-static models used in this work. Section III defines the main steps of a reconfiguration. Section IV proposes a reconfiguration strategy and details the corresponding algorithm, focusing on Dijkstra's algorithm [START_REF] Dikstra | A note on two problems in connection with graphs[END_REF], [START_REF] Siek | The Boost Graph Library: User Guide and Refrence Manual[END_REF], [START_REF] Caro | Algorithm for the actuation mode selection of the parallel manipulator navaro[END_REF] and on the graph construction procedures. Section V provides an illustrative example and Section VI concludes this paper.

II. GEOMETRIC AND ELASTO-STATIC ROBOT MODELS

A CDPR is composed of a mobile platform connected to a fixed base by m cables. The pose of the platform, p, is described with respect to a fixed absolute frame, F b , of origin O b and axes x b , y b and z b . The pose is composed by the vector of the Cartesian coordinates of the platform Centre of Mass (CoM), t, and the vector Φ = [φ, θ, ψ]

T containing the roll, pitch and yaw angles, φ, θ and ψ, around the axes z b , y b and x b , respectively. The cable connection points on the base, the so-called exit points, A i , i = 1, . . . , m, are defined, with respect to F b , through the Cartesian coordinate vectors a b i , i = 1, . . . , m. The platform-cable connection points, B i , i = 1, . . . , m, are defined, with respect to a local reference frame F p , through the Cartesian coordinate vectors b p i , i = 1, . . . , m. This local reference frame, of origin O p and axes x p , y p and z p , is attached to the mobile platform.

A. GEOMETRIC MODEL

The geometric model of the robot consists of the chain closure equations for the cable vectors, l b i , i = 1, . . . , m. Those vectors can be defined, with respect to F b , as: where R denotes the orientation matrix of the platform:

l b i = a b i -t -Rb p i , i = 1, . . . , m (1)
R = R z (φ) R y (θ) R x (ψ) = =   cφcθ cφsθsψ -sφcψ cφsθcψ + sφsψ sφcθ sφsθsψ + cφcψ sφsθcψ -cφsψ -sθ cθsψ cθcψ   (2)
The corresponding unit vectors are defined, with respect to F b , as:

d b i = l b i ||l b i || 2 , i = 1, . . . , m (3)
The geometric model of a generic CDPR is shown in Fig. 2.

B. STATIC MODEL

The static equilibrium of the CDPR mobile platform is described by the following equation:

Wτ + w e = 0 (4)
where τ = [τ 1 , . . . , τ m ] is the vector collecting all the cable tensions; W is the wrench matrix, whose columns are composed of the unit wrenches ŵi exerted by the cables on the platform:

W = d b 1 d b 2 . . . d b m Rb p 1 × d b 1 Rb p 2 × d b 2 . . . Rb p m × d b m ( 5
)
w e is the external wrench acting on the platform:

w e = f T , m T T = [f x , f y , f z , m x , m y , m z ] T (6)
The components of the force vector, f , as well as the components of the moment vector, m, are bounded as follows:

f min ≤ f x , f y , f z ≤ f max (7) m min ≤ m x , m y , m z ≤ m max (8)
In the following section, we will consider only fully constrained CDPRs. Hence, the inverse static problem is under-constrained and can be defined as:

τ = τ n + τ 0 = W † w e + λn τ ≥ 0 (9)
where W † denotes the Moore-Penrose generalized inverse of W, λ ∈ R and n is a vector in the null space of W [START_REF] Roberts | On the inverse kinematics, statics, and fault tolerance of cable-suspended robots[END_REF].

A possible solution to (9) can be found by minimizing the 2-norm of the cable tension vector, ||τ || 2 .

C. ELASTO-STATIC MODEL

In order to quantify the stiffness of the system, an elastostatic model is used:

δw e = Kδp ( 10
)
where the matrix K defines the relation between an infinitesimal change in the external wrench, δw e , and the corresponding infinitesimal change in the pose of the CDPR platform, δp = δt T , δΦ T , T . The reader is referred to [START_REF] Gagliardini | A reconfigurable cable-driven parallel robot for sandblasting and painting of large structures[END_REF] for a complete description of the stiffness matrix K.

III. THE TECHNICAL RECONFIGURATION PROCEDURE

This section focuses on the technical description of the reconfiguration while section IV describes the procedure to be used in order to select the optimal configurations and their reconfiguration order. The core of the proposed reconfiguration algorithm, based on a Dijkstra's algorithm, is detailed in Section IV-B. In this paper, we mean by reconfiguration that the RCDPR is able to change the locations of its cable exit points. For some RCDPRs the choice is limited to a given number of reconfigurations: this is the typical case of a RCDPR whose cables can be connected to a grid of exit points, e.g. as illustrated in [START_REF] Gagliardini | A reconfigurable cable-driven parallel robot for sandblasting and painting of large structures[END_REF]. Other RCDPRs have at disposal an infinite number of configurations: this is the typical case of a RCDPR whose exit points are sliding on rails or along linear guides, e.g. as proposed in [START_REF] Nguyen | On the analysis of large-dimension reconfigurable suspended cable-driven parallel robots[END_REF]. The RCDPRs analysed in this paper are restricted to the first category.

When the working environment is not cluttered, and the ground is free, it is possible to perform the reconfiguration on the ground. The CDPR platform is moved from the current pose to a home position on the ground. Then, when no external wrench is applied on the platform and the cables are slack, the cables can be disconnected easily from the platform. Once the exit points are reconfigured, the cables can be attached again to the platform, and the platform can be moved to a desired working pose.

The previous strategy cannot always be applied. For instance, a cluttered environment may not permit to put the CDPR platform in a home position on the ground. In the strategy proposed hereafter the platform of the CDPR should not or only slightly change its pose during the reconfiguration: a. The robot is located at the desired reconfiguration pose.

The current pose of the robot is denoted hereafter as C a .

The cables balance the weight of the platform and the external wrenches applied on the platform, according to the static equilibrium equation W a τ a + w e = 0. b. A new set of cables are connected to the proper exit point locations, according to the new desired configuration, defined hereafter as C a . The other extremities of these cables are connected to the corresponding attachment points on the platform. These cables are initially slack and they are coiled in such a way to be "just tensed", their lengths being adjusted to the platform pose. c. In order to be detached from the platform, the cables belonging to C a , but not to C a , should be slack. These cables are gradually uncoiled in such a way that the external wrenches are transferred to the cables belonging to C a . d. When the cables belonging exclusively to configuration C a are slack, they can be detached. The RCDPR is now ready to proceed with the task execution. A simple example of this procedure is given in Fig. 3.

IV. THE RECONFIGURATION ALGORITHM

In general, given a specific RCDPR, it is possible to select a combination of configurations to complete a given task. The selection can be performed with respect to one or more criteria, according to the algorithm presented hereafter and illustrated in Fig. 4.

1) The user defines the geometry of the RCDPR and provides the list of all the n C possible reconfigurations.

2) The user defines the task characteristics, including: the desired path of the platform CoM, P, discretized into n p points according to the user specifications, P = {P i , i = 1, . . . , n p }; the bounds on the external wrenches applied to the platform, f min , f max , m min , m max . 3) The algorithm computes a feasibility map. For each point P i ⊂ P, all the n C configurations should be tested, in order to verify their feasibility. For a given configuration C j , the pose associated to the i-th point of the desired path, P i,j , will be feasible, P i,j = 1, if it satisfies a list of n φ constraints, φ i,j,k , k = 1, . . . , n φ , specified by the user:

P i,j = 1 ⇐⇒ n φ k=1 φ i,j,k = 0 (11)
where:

φ i,j,k = 0 ⇐⇒ the constraint is satisfied φ i,j,k = 1 ⇐⇒ the constraint is not satisfied

The following constraints have been taken into account in this paper:

• The moving-platform pose is wrench feasible. The cables should be able to balance any possible external wrench w e , according to the ranges specified in [START_REF] Rosati | Performance of cable suspended robots for upper limb rehabilitation[END_REF] and [START_REF] Gallina | 3-dof wire driven planar haptic interface[END_REF]. This set of required external wrenches consists of an hyperectangle [w e ] r . Hence, the configuration is wrench feasible if the required external wrenches are included inside the zonotope of the available wrenches, [w e ] a , as follows:

∀ w e ∈ [w] r , ∃ τ ∈ [τ ] s.t. w e ∈ [w] a Wτ + w e = 0 (12
) This condition can be translated into a set of inequalities, according to [START_REF] Gouttefarde | Characterization of parallel manipulator available wrench set facets[END_REF]:

Cw e ≤ d, ∀ w e ∈ [w] r (13)
• No cable interference occurs. This condition is verified by measuring the distance between each pair of cables, e.g. with the procedure explained in [START_REF] Lumelsky | On fast computation of distance between line segments[END_REF]. The interference between the i-th cable and the j-th cable does not occur when the distance d cc i,j

is greater than the diameter of the cables, φ c , i.e.,

d cc i,j ≥ φ c ∀ i, j = 1, . . . , m, i = j ( 14
)
• No cable-environment collision occurs. The verification is performed using the same principle as the cable interference, being given that the obstacles are discretized into several facets. An example of a simple feasibility map is provided in Fig. 5a. The feasibility map illustrated in Fig. 5a. represents the feasible poses associated to each configuration. In this example, two configurations are taken into account, C 1 and C 2 . The poses, defined as P i,j , i = 1, . . . , 7, j = 1, 2, are equal to 1 when they are feasible and equal to 0 when they are not. We can notice that pose P 3,2 is feasible while P 4,2 is infeasible, i.e., the configuration C 2 changes from a feasible to an infeasible condition. On the contrary, between the poses P 4,2 and P 5,2 , configuration C 2 is subjected to a transition from an infeasible to a feasible condition. In both cases we will talk about Feasibility Transitions (FT). The FTs in the feasibility map of Fig. 5a are represented by three green arrows. When a FT occurs between the poses P i-1,j and P i,j , in such a way that P i-1,j is infeasible and P i,j is feasible, we will refer to the pose P i,j as a Positive Feasibility Transition (PFT). Thus:

P i,j is a PFT if P i,j = 1 P i-1,j = 0 ( 15
)
On the contrary, when a FT occurs between the points P i,j and P i+1,j , in such a way that P i,j is feasible and P i+1,j is infeasible, we will refer to the point P i,j as a Negative Feasibility Transition (NFT). Thus:

P i,j is a NFT if P i,j = 1 P i+1,j = 0 (16)

A. GRAPH NODES

In order to find the best reconfiguration strategy, using Dijkstra's algorithm, it is necessary to build a proper graph. A graph is a set of nodes connected to each others through arcs. The nodes of the graph represent the reconfigurations that the RCDPR can perform. Hereafter, the nodes will be denoted as N i,j,k , where the indices i, j and k are defined as follows:

• The index i represents the point P i to which the node is associated. • The index j represents the starting configuration C j .

• The index k represents the final configuration C k to which the RCDPR is reconfigured, starting from the configuration C j . The nodes connected by arcs will be defined hereafter as adjacent nodes. Two nodes N i,j,k and N i ,j ,k , with i > i and k = j , are adjacent when all the poses between P i,k and P i ,k are feasible:

P i ,k = 1, ∀ i s.t. i ≤ i ≤ i .

B. GRAPH BUILDING

The graph to be searched with the Dijkstra's algorithm is generated by analyzing the feasibility map. The procedure, illustrated in Fig. 6, consists of the following steps:

1) The first node, N 0 , is an empty virtual node. Its position along the desired path is 0 and it is not associated to any configuration. Its adjacent nodes, N 1,0,j , are the ones associated to the first point, P 1 , of the desired path. A node N 1,0,j is created if and only if the pose P 1,j is feasible.

2) The intermediate nodes are obtained by analyzing each point of the desired path. For a given configuration, C j , the algorithm verifies the presence of a FT at pose P i,j . When a NFT is encountered at P i,j , the RCDPR can (and should) be reconfigured in such a way to reach any configuration C k that is feasible at P i,k and P i+1,k . Thus, a node N i,j,k is created if P i,j is a NFT, P i,k = 1 and P i+1,k = 1. When P i,j is a PFT, provided that P i+1,j = 1, the RCDPR can be reconfigured in such a way to reach the configuration C j starting from any configuration C k that is feasible at P i,k . Thus, a node N i,k,j is created if P i,j is a PFT, P i,k = 1 and P i+1,j = 1. It is worth noting that, in all the cases, a node is created when the initial and the final configurations are both feasible.

3) The final node, N f , is an empty virtual node. Its position is at the end of the desired path and it is not associated to any configuration. Its adjacent nodes, N n P ,j,f , are associated to the last point, P n P , of the desired path. They are created if and only if the poses P n P ,j are feasible. The arcs are then created between the couples of adjacent nodes, the latter being defined in Section IV-A. Once the graph is built, the Dijkstra's algorithm searches the graph looking for the minimum cost path connecting nodes N 0 and N f . The cost associated to each arc of the graph is defined according to the procedure described in Section IV-C.

C. THE ARC COST FUNCTION

Let the q-th arc connect the nodes N i,j,k and N i ,j ,k . Its cost function Φ q is defined according to n λ criteria, λ t,q , t = 1, . . . , n λ , and is formulated as follows:

Φ q = n λ t=1 λ t,q µ t , µ t ∈ [0, 1] , n λ t=1 µ t = 1 (17)
The weighting factors µ t , t = 1, . . . , n λ , are defined according to the priority assigned to each criterion. The criteria λ t,q are bounded between 0 and 1. Hereafter, three criteria are considered:

λ 1,q = 1 (18)
λ 2,q = n c,q m (19)
λ 3,q = ||δt q || 2 ||δt c || 2 (20)
λ 1,q is a constant cost function equal to 1 used in order to count the number of reconfigurations performed during the task execution. λ 2,q is proportional to the number of cables, n c,q , to be reconfigured when the RCDPR passes from configuration C j to configuration C k ; λ 3,q is proportional to the mean of the norm of the positioning error:

||δt q || 2 = i i =i δt i ,k 2 i -i (21)
where δt i ,k 2 is the norm of the cable tension vector measured at the pose P i with respect to the configuration C k . In [START_REF] Caro | Algorithm for the actuation mode selection of the parallel manipulator navaro[END_REF], ||δt q || 2 is normalized with respect to the norm of the critical precision threshold vector δt c 2 .

The proposed cost functions aim at improving the efficiency of the RCDPR. By minimizing the arc cost associated to λ 1,q , the algorithm aims at reducing the total number of reconfigurations n r required to complete the task. The minimization of the arc cost associated to λ 2,q corresponds to the minimization of the total number n c of cable detaching-attaching operations required to complete the task, and consequently to the minimization of the reconfiguration set-up time. By minimizing the arc cost associated to λ 3,q , the average of the RCDPR stiffness is increased and the positioning error is reduced.

V. CASE STUDIES

A. First Case Study

A first case study is considered in order to illustrate the algorithm presented in Section. IV. A closed path P, discretized in 164 points, P i , i = 1, . . . , 164, has to be followed by the platform CoM. The path, lying in the plane (O, x b , y b ), is illustrated in Fig. 7. The orientation of the platform is constant, with zero roll, pitch and yaw angles.

An obstacle is included in the environment. It consists of a square of side length equal to 1 m. The obstacle is centered at the origin of the reference frame F b as shown in Fig. 7.

The RCDPR under study is composed of four cables. The cable exit points, A i , i = 1, . . . , 4 can be reconfigured, according to the following parametrization:

a b 1 = [u 1 + u 3 , u 2 ] T (22) a b 2 = [-u 1 + u 3 , u 2 ] T (23) a b 3 = [-u 1 + u 3 , -u 2 ] T (24) a b 4 = [u 1 + u 3 , -u 2 ] T (25)
The parameters u 1 and u 2 represent half of the distance between the exit points along x b and y b , respectively. u 3 is the offset of the cables connection point positions along x b . The robot is symmetric with respect to x b , as shown in The properties of the cables have been selected as follows: the Young Modulus, E, is equal to 100 GPa; their diameter φ c is equal to 4 mm; their stiffness coefficient, k i , is equal to 252 KN/m; the maximum allowed tension in the cables, τ max , is equal to 34 950 N. A safety coefficient η = 5 has been taken into account such that:

0 < τ i ≤ τ max η , ∀i = 1, . . . , 4 (30)
The platform can be subjected to an external wrench, w e . We assume that the components of w e are bounded as follows:

-500 N ≤ f x , f y ≤ 500 N (31) -7.5 N ≤ m z ≤ 7.5 N ( 32
)
where f x and f y stand for the external force components along x b and y b axes, respectively. m z is the external moment about the z b axis. The first analysis has been performed by optimizing the total number of reconfigurations, n r , according to the weighting factors µ 1 = 1, µ 2 = µ 3 = 0. The solution proves that four reconfigurations are sufficient to complete the proposed task. The optimal result is shown in Fig. 8. The feasibility map has been built by analyzing the constraints applied on each point of the trajectory namely, wrench feasibility, cable interferences and obstacle collisions.

For instance, configuration C 4 is the unique feasible configuration at point P 1 of the desired path. The NFT at point P 16 requires a change of configuration, from C 4 to C 3 . When a new NFT appears at point P 28 , the RCDPR adopts the configuration C 1 . A PFT appears at point P 97 . The RCDPR returns to the configuration C 3 and, later on, to the configuration C 4 , following the desired path till its end.

The optimal configurations required to complete the task are shown in Fig. 9.

A second optimization has been tested, aiming at minimizing the positioning error of the platform due to the cable elasticity, according to the weighting factors µ 1 = µ 2 = 0, µ 3 = 1. The result is similar to the one obtained by minimizing n r . In this specific case, the first reconfiguration is performed at P 18 . The RCDPR passes from the configuration C 4 to the configuration C 2 as shown in Fig. 10.

Both the optimal solutions analyzed in this case study are represented in the graph shown in Fig. 11. The red arrows indicate the path obtained by minimizing n r . The blue arrows describe the path generated by minimizing the platform positioning error. The blue optimal path leads to a reduction of the positioning error, with respect to the red optimal path, of almost 18%: the total cost of the red path, in terms of positioning error, is equal to 1.7663 while the cost associated to the blue path is equal to 1.4545.

B. Second Case Study

A second case study is now considered. The requirements in terms of desired path and external wrenches are similar to the previous case, as well as the desired orientation of the platform. The platform geometry and the cable properties are also identical. However, the number of obstacles has been increased to three, as illustrated in Fig. 12.

The robot parametrization is defined by (22-25) and (26-29). The RCDPR under study can take nine configurations. Their parameters are given in Tab. II.

The computation has been performed by assigning the same weighting factor to all the criteria composing the cost

VI. CONCLUSIONS AND FUTURE WORK

The research work presented in this paper deals with the selection of an optimal reconfiguration strategy for RCDPRs. The proposed strategy is based on the Dijkstra's algorithm and aims notably at minimizing the number of reconfigurations, as well as the reconfiguration set-up time. The proposed algorithm permits the optimization of other characteristics of the RCDPR by means of a cost function depending on multiple criteria.

Two planar case studies have been used in order to validate the proposed reconfiguration strategy. The results show that the algorithm is able to compute the optimal solutions with respect to a set of configurations proposed by the user. The results can be extended to three-dimensional problems.

The algorithm computation time is quite satisfactory when the RCDPR has a limited number of possible configurations. Each of the previous simulations required a graph resolution time lower than 0.01 s. However, the main limitation is related to the time needed to construct the feasibility map, especially when the algorithm has to deal with threedimensional RCDPRs which have a large number of possible configurations. Hence, further studies should be dedicated to the improvement of Step. 3) of the algorithm presented in Section IV.

Fig. 1 .

 1 Fig. 1. Example of a RCDPR design. The robot shown in the picture is a concept created in the framework of the CAROCA project whose intent is the painting of tubular structures.

Fig. 2 .

 2 Fig. 2. CDPR geometric description.

Fig. 3 .

 3 Fig. 3. Example of reconfiguration procedure for a 4-cable suspended planar robot. The continuous lines represent the cable associated to configuration Ca; the dashed lines represent the cables associated exclusively to configuration C a . In this example only 2 cables are reconfigured.

Fig. 4 .

 4 Fig. 4. Scheme of the reconfiguration algorithm.

Fig. 5 .

 5 Fig. 5. Example of a feasibility map and the related graph.

•

  The components of the positioning error due to the RCDPR cable elasticity, δt = [δt x , δt y , δt z ], are lower than then corresponding components of a threshold vector δt c = [δt x,c , δt y,c , δt z,c ] defined by the user. 4) The algorithm builds a graph from the feasibility map. 5) The Dijkstra's algorithm solves the optimization problem according to the criteria specified by the user. The details of Step 4. and Step 5. are given in the following subsections.

Fig. 6 .

 6 Fig. 6. Scheme of the algorithm used to build the graph for the Dijkstra's algorithm.

Fig. 7 .

 7 Four configurations have been considered in this case study. Their parameters are represented in Tab. I. The dimensions of the platform have been set to 0.2 m for the width, w p , and 0.3 m for the height, h p . The Cartesian coordinates of points B i , i = 1, . . . , 4, expressed in the moving-platform frame F p are the following: b p 1 = [w p /2, h p /2] T (26) b p 2 = [-w p /2, h p /2] T (27) b p 3 = [-w p /2, -h p /2] T (28) b p 4 = [w p /2, -h p /2] T (29)

Fig. 7 .

 7 Fig. 7. First case study and description of the exit point parametrization for the RCDPR.

Fig. 8 .

 8 Fig. 8. First case study feasibility map and optimal solution with respect the minimization of nr.

Fig. 9 .

 9 Fig. 9. First case study optimal RCDPR configurations, selected by the algorithm in order to complete the task with respect to the minimum number of configurations nr.

Fig. 10 .

 10 Fig. 10. First case study optimal solution with respect the minimization of the positioning error δt.

Fig. 11 .

 11 Fig. 11. Graph of the two optimization problems of the first case study.

Fig. 12 .

 12 Fig. 12. Second case study.

TABLE I POSSIBLE

 I CONFIGURATIONS OF THE FIRST CASE STUDY.

	Configuration	u 1	u 2	u 3
	C 1	2.25 6.25	2.50
	C 2	4.00 7.75 -1.00
	C 3	4.00 7.00 -0.25
	C 4	2.25 6.25 -2.50

TABLE II POSSIBLE

 II CONFIGURATIONS OF THE SECOND CASE STUDY.

	Configuration	u 1	u 2	u 3
	C 1	1.25 4.75	3.50
	C 2	2.75 6.25	2.00
	C 3	3.25 7.50	2.75
	C 4	3.25 7.75	3.25
	C 5	2.50 5.50	0
	C 6	3.25 7.75 -3.25
	C 7	3.25 7.50 -2.75
	C 8	2.75 6.25 -2.00
	C 9	1.25 4.75 -3.50

ACKNOWLEDGMENT

This study is part of the CAROCA project managed by IRT Jules Verne (French Institute in Research and Technology in Advanced Manufacturing Technologies for Composite, Metallic and Hybrid Structures).

This work is supported by the IRT Jules Verne, Bouguenais, France (Project CAROCA,


Lorenzo Gagliardini

Stéphane Caro

Marc Gouttefarde

Dimensioning of Cable-Driven Parallel Robot Actuators, Gearboxes And Winches According To The Twist Feasible Workspace

Cable Driven Parallel Robots (CDPRs) are a particular class of parallel robots whose legs consist of cables. CDPRs are composed of several components, e.g. winches, pulleys and actuators. The design of a CDPR requires the dimensioning of all these components, according to the task to be performed. The dimensioning of the actuators, the gearboxes and the winches are strictly related to the performances of the CDPR in terms of the platform static and kinematic equilibrium.

This paper introduces a new tool, the so called Twist Feasible Workspace (TFW), built in order to analyze the workspace of the platform twists. A pose is said to be twist feasible if the platform of the CDPR can assume a given range of linear and rotational velocities while satisfying the cable speed limits imposed by the actuators and the transmission systems. The size of the TFW is used as an optimization criterion for the dimensioning of the actuators and the winches.

I. INTRODUCTION

During the last decades, the necessity to perform industrial operations over products of large dimensions promoted the investigation of a new category of robots, the Cable-Driven Parallel Robots (CDPRs), whose legs consist of cables. One side of the cables is connected to a mobile platform while the other side is connected to a base frame. The cable connection points on the base are defined hereafter as exit points. Fig. 1 shows an example of CDPR design, realised in the framework of the IRT Jules Verne's CAROCA project.

Several CDPRs have been designed for a wide range of applications, including the displacement of heavy charges, [START_REF] Pott | Large-scale assembly of solar power plants with parallel cable robots[END_REF], [START_REF] Holland | Cable array robot for material handling[END_REF] and the painting of large products [START_REF] Albus | The NIST spider, a robot crane[END_REF], [START_REF] Gagliardini | Optimal design of cable-driven parallel robots for large industrial structures[END_REF], [START_REF]A reconfigurable cable-driven parallel robot for sandblasting and painting of large structures[END_REF]. Other possible applications are: the shooting of sport events [START_REF] Skycam | [END_REF], the rehabilitation of injured people [START_REF] Rosati | Performance of cable suspended robots for upper limb rehabilitation[END_REF] and search and rescue operations [START_REF] Merlet | A portable, modular parallel wire crane for rescue operations[END_REF]. All the previous applications take advantage of the qualities of CDPRs, such as their wide workspace, their reduced inertia and their high payload to weight ratio.

However, CDPRs are characterized by several drawbacks: cable interferences, cable collisions with the surrounding environment and the influence of the non-rigid nature of the cables over the platform positioning precision. Moreover, since the cables cannot push the platform, its static This work is supported by the IRT Jules Verne, Bouguenais, France (Project CAROCA, Evaluation des CApacités de la RObotique à CÂbles dans un contexte industriel)

1 L. Gagliardini is with IRT Jules Verne, Chemin du Chaffault, 44340, Bouguenais, France, lorenzo.gagliardini@irt-jules-verne.fr 2 S. Caro is with CNRS-IRCCyN, 1, rue de la Noë, 44321, Nantes Cedex 03, France, Stephane.Caro@irccyn.ec-nantes.fr 3 M. Gouttefarde is with the Laboratoire d'Informatique, de Robotique et de Micro-électronique de Montpellier (LIRMM-CNRS-UM), 161, rue Ada, 34392, Montpellier Cedex 05, France, marc.gouttefarde@lirmm.fr equilibrium is strictly dependant to the cable layout and the cable tension limits.

Considering the cables sufficiently resistant to bare the platform payload, the maximum tension transmitted on the cables depends mostly on the characteristics of the actuators, such as their nominal torque. The actuators influence also the twist that the CDPR platform can attain, being the nominal speed of the actuators proportional to the coiling/uncoiling speed of the cables.

Both the static and the kinematic aspects should be taken into account during the desgin of CDPRs. The static equilibrium of the platform can be analysed using the Wrench Feasible Workspace (WFW) [START_REF] Bouchard | On the ability of a cable-driven robot to generate a prescribed set of wrenches[END_REF]. By definition, the WFW is the set of wrench feasible platform poses. A pose is wrench feasible when the cables can bare a set of external wrenches and the cable tensions stay in between the given cable tension limits. In this paper, the concept of wrench feasibility will be extended to the platform kinematic analysis, introducing the so called Twist Feasible Workspace (TFW). The TFW is the set of twist feasible platform poses. A pose is twist feasible when the actuators, according to their nominal speed, are able to move the platform within a given range of linear and rotational velocities.

A procedure aiming at dimensioning the actuators and the winches of the CDPR has been developed. The design problem has been formulated as an optimization problem maximizing the size of the intersection between the TFW and the WFW.

This paper is organized as follows. Section II introduces the geometric, kinematic and static models used in this paper. Section III introduces the concept of TFW. The TFW is used as part of the design process described in Section IV. Section V provides an illustrative example and Section VI concludes this paper.

II. GEOMETRIC, STATIC AND KINEMATIC ROBOT MODELS

A CDPR consists of a mobile platform connected to a fixed base by m cables. The pose of the platform, p, is composed of the Cartesian coordinates of the platform Centre of Mass (CoM), t, and the platform roll, pitch and yaw angles, φ, θ and ψ. The platform orientation angles are collected in the vector Ω = [φ, θ, ψ]

T . The pose is described

A. Geometric model

The geometric model of a generic CDPR is shown in Fig. 2. The chain closure equations of the cable vectors, defined with respect to F b , compose the geometric model of the CDPR:

l b i = a b i -t -Rb p i , i = 1, . . . , m (1)
where R denotes the orientation matrix of the platform:

R = R z (φ) R y (θ) R x (ψ) = =   cφcθ cφsθsψ -sφcψ cφsθcψ + sφsψ sφcθ sφsθsψ + cφcψ sφsθcψ -cφsψ -sθ cθsψ cθcψ   (2)
The cable unit vectors can be derived directly by the geometric model:

d b i = l b i ||l b i || 2 , i = 1, . . . , m (3)
where

||l b i || 2 is the Euclidean norm of l b i . B. Static model
The static equilibrium of the CDPR platform is described by the following equation:

Wτ + w e = 0 ( 4
)
where W is the so called wrench matrix, whose columns are the unit wrenches ŵi exerted by the cables on the platform:

W = d b 1 d b 2 . . . d b m Rb p 1 × d b 1 Rb p 2 × d b 2 . . . Rb p m × d b m (5)
The cable tensions τ i , i = 1, . . . , m are collected in the vector τ = [τ 1 , . . . , τ m ] T . The external wrench acting on the platform, w e , is defined as follows: The components of the external force vector, f , as well as the components of the external moment vector, m, are bounded:

w e = f T , m T T = [f x , f y , f z , m x , m y , m z ] T (6)
f min ≤ f x , f y , f z ≤ f max (7) m min ≤ m x , m y , m z ≤ m max (8)

C. Kinematic model

The kinematic model of the CDPR describes the twist of the platform in terms of the cable coiling/uncoiling velocities:

J ˙l -ṗ = J ˙l - ṫ ω = 0 ( 9
)
where ṫ is the vector of the platform linear velocity, ω is the vector of the platform angular velocity and ˙l is the vector of the cable velocities, ˙l = [ l 1 , . . . , l m ]. J is the Jacobian matrix of the CDPR. The columns of the Jacobian matrix represent the twist pi associated to each cable.

Considering the duality of the geometric-static model, the static equilibrium of the platform can be rewritten as:

τ = J T w e (10)
Comparing ( 4) and [START_REF] Ziegler | Lectures on Polytopes[END_REF] we observe that:

J T w e = -W † w e (11)
The Jacobian matrix can be computed as:

J = -W † T (12)
Consequently, the kinematic model in ( 9) can be rewritten as:

W † T ˙l + ṗ = 0 (13)
where W † is the Moore-Penrose pseudo-inverse. Since the system is not square, an infinite number of solutions can be associated to Eq. 13. The pseudo-inverse provides the solution minimizing the cable lengths, preventing the cables to become slack.

A. Wrench Feasible Workspace

The platform of the CDPR is modelled as a rigid body attached to m cables. Each cable transmits on the platform a wrench, proportional to the unit wrench ŵi . The cable wrenches, w i = ŵi τ i , i = 1, . . . , m, balance the external wrench acting on the platform. We assume the external wrench acting on the platform is bounded according to [START_REF] Rosati | Performance of cable suspended robots for upper limb rehabilitation[END_REF] and [START_REF] Merlet | A portable, modular parallel wire crane for rescue operations[END_REF]. These boundaries are provided by the CDPR designer according to the task requirements. The set of external wrenches that the platform has to bare, the so called Required Wrench Set (RWS), [w e ] r , is a polytope.

Cable tensions, τ i i = 1, . . . , m, are affected as well by some limitations. Due to their non-rigid nature, cable tensions are always non-negative. A lower tension boundary, τ min > 0, can be defined by the user in order to prevent the cables to be slack. The cable tensions present also an upper bound, τ max , associated to the physical characteristics of the cables and the nominal torque of the motors. Thus, the set of cable tensions which can be transmitted by the cables is defined hereafter as the Available Cable Tension Set (ACTS):

[τ ] a = {τ | τ min ≤ τ i ≤ τ max , i = 1, . . . , m} (14)
Due to the previous constraints, cables can transfer on the platform only a bounded set of wrenches, defined as the Available Wrench Set (AWS), [w e ] a . The AWS can be computed from the ACTS recurring to the equation of the static equilibrium [START_REF] Gagliardini | Optimal design of cable-driven parallel robots for large industrial structures[END_REF]. Considering that [w e ] a is the image of [τ ] a under the linear map represented by the wrench matrix W, [w e ] a is affinely isomorphic to a particular class of polytopes called zonotopes [START_REF] Bouchard | On the ability of a cable-driven robot to generate a prescribed set of wrenches[END_REF], [START_REF] Ziegler | Lectures on Polytopes[END_REF].

According to the previous considerations, the static equilibrium of the platform can be assured only if the pose assumed by the platform is wrench feasible. Nominally, a pose is said to be wrench feasible when the CDPR can balance any external wrench w e included in [w e ] r according to cable tension limits expressed by [τ ] a :

∀w e ∈ [w e ] r , ∃τ ∈ [τ ] a s.t. w e ∈ [w e ] a Wτ + w e = 0 (15)
From a geometrical point of view, a pose is wrench feasible if [w e ] r is fully included in [w e ] a (as shown in Fig. 3):

[w e ] r ⊆ [w e ] a (16)
The feasibility can be verified according to the procedure proposed by Gouttefarde et al. [START_REF] Gouttefarde | Characterization of parallel manipulator available wrench set facets[END_REF], by measuring the distance between all the vertices of [w e ] r and the facets of

[w e ] a : Cw e ≤ d, ∀w e ∈ [w e ] r (17)
The columns of matrix C represent the facets of [w e ] a and vector d the distance of these facets from the origin of the wrench space. Alternative methods for the computation of the WFW can be found in the literature, such as the one provided by Guay et al. [START_REF] Guay | Measuring how well a structure supports varying external wrenches[END_REF] and Cruz et al. [START_REF] Ruiz | Arachnis : Analysis of robots actuated by cables with handy and neat interface software[END_REF].

B. Twist Feasible Workspace

The definition of WFW is based on the linearity of the static equilibrium equation, being W a linear map between the cable tension space and the space of the external wrench acting on the platform:

w e = m i=1 ŵi τ i ( 18
)
The kinematic problem described in (9) presents a similar structure. The geometric Jacobian matrix, J, is a linear map between the space of the cable velocities and the space of the platform twist. As a matter of fact, (9) can be rewritten as a linear combination of the cable twists, ṗi , i = 1, . . . , m (being ṗi , i = 1, . . . , m the column vectors of J):

ṗ = m i=1 ṗi li (19)
Due to the similarities of the static and kinematic problems, the considerations performed in Sec. III-A can be extended to the kinematics of the CDPR platform.

During the task execution, the platform is supposed to move within a given range of speeds:

ṫmin ≤ ṫx , ṫy , ṫz ≤ ṫmax (20) ω min ≤ φ, θ, ψ ≤ ω max (21)
where ṫx , ṫy and ṫz represent the components of the platform linear velocity, ṫ, along x b , y b and z b , respectively. φ, θ and ψ represent the components of the platform rotational velocity, ω, around x b , y b and z b , respectively. ṫmin and ṫmax are the lower and the upper bounds of the linear velocity components. Analogously, ωmin and ωmax are the lower and the upper bounds of the rotational velocity components. The set of twists, [ ṗ] r , which is limited by the previous boundaries, is defined as the Required Twist Set (RTS).

Similarly to the WFW, matrix J depends on the platform position and orientation in space. Thus, the TFW is function of the platform pose. The platform twists depends also on the power of the actuators. The nominal speed of the motors limits the cable coiling/uncoiling speed. The maximum coiling speed is defined as lmax ; for most of the actuators, considering lmin = -lmax , the cable speeds are bounded as follows:

lmin ≤ li ≤ lmax , ∀i = 1, . . . , m

The set of possible cable coiling/uncoiling speeds is defined hereafter as Available Cable Velocity Set (ACVS), [ ˙l] a :

[ ˙l] a = { ˙l | lmin ≤ li ≤ lmax , ∀i = 1, . . . , m} (23)
According to the previous limitations, the platform can assume only a given set of twists, defined hereafter as the Available Twist Set (ATS), [ ṗ] a . The ATS, which represents a polytope, can be computed from the ACVS using the Jacobian matrix as a linear map from the cable velocity space to the platform twist space.

In order to satisfy the kinematic equilibrium, the CDPR can assume only the poses contained inside the TFW. Nominally, a pose is said to be twist feasible when the CDPR can assume any twist ṗ included in [ ṗ] r according to the boundaries of the cable velocities, [ ˙l] a :

∀ ṗ ∈ [ ṗ] r , ∃ ˙l ∈ [ ˙l] a s.t. ṗ ∈ [ ṗ] a J ˙l -ṗ = 0 (24)
Alternatively, the twist feasibility condition can be described as:

[ ṗ] r ⊆ [ ṗ] a (25)
The TFW can be investigated, once again, using to the hyperplane shifting method provided by Gouttefarde et al. [START_REF] Gouttefarde | Characterization of parallel manipulator available wrench set facets[END_REF]:

C * ṗ ≤ d * , ∀ ṗ ∈ [ ṗ] r (26)

IV. CABLE-DRIVEN PARALLEL ROBOT DESIGN BASED ON THE TASK SPACE ANALYSIS

A. Description of the actuation and coiling system Designing a CDPRs is a complex task which requires the dimensioning of several components and parameters. Most of the research works involving the design of CDPRs are focused on the dimensioning of the platform and the exit point locations. This paper will focus on the dimensioning of:

• The actuators, which in this paper are represented by electric motors, whose nominal torque and speed are defined as τ M and ωM , respectively. • The winches, whose diameter is equal to φ w .

• The gearboxes, connecting the winches to the motors, whose transmission ratio is equal to ρ R . A scheme of the actuation and transmission system is shown in Fig. 4. Motors limit both the tension transmitted to the cables and the rotational speed of the winches. The gearbox modifies the torque and the speed of the motor according to the following expressions: The winches are connected to the gearbox. They can transfer on the cable a tension equal to:

τ R = ρ R τ M (27) ωR = µ R ωM = 1 ρ R ωM (28)
τ max = ρ W τ R = ρ W ρ R τ M = 2 φ w ρ R τ M ( 29
)
where the winch transmission ratio, ρ W , is equal to the inverse of the winch radius, ρ W = 2 φw . Given the winch rotational speed, ωR = ωW , the maximum cable speed can be computed as:

lmax = ωW φ w 2 (30)

B. Optimization Problem

The dimensioning of the motors, the winches and the gearboxes have a great influence on the WFW and the TFW. In order to increase the size of both the workspaces, the design procedure has been formalized as an optimization problem. The user defines the zone of the Cartesian space that the CDPR should attain. This desired zone is discretized into a set of n P points, P k , k = 1, . . . , n P ∈ P.

The user provides the dimensions of the CDPR platform, as well as the Cartesian coordinates of the exit points of the cables. These informations will be considered as part of the design constant parameters, collected in vector q. The user describes the ACTS and the ACVS, specifying as well the boundaries of the RWS and the RTS.

The objective function of the optimization problem, V , represents the percentage p f of points P ∈ P which proofs to be wrench and twist feasible at the same time:

V (x) = p f = n P k=1 F k n P ( 31
)
where F k = 1 if the the pose associated to the k-th point, P k , is both wrench and twist feasible and F k = 0 otherwise. The design variables are collected in the design variable vector,

x: x = [τ M , ωM , φ w , ρ R ] (32)
A set of constraints is verified for all the points P k ∈ P where the pose assumed by the CDPR is wrench and twist feasible. In this paper, the following constraints are taken into account:

• Cable interferences. This condition is verified by measuring the distance between each pair of cables, using the procedure proposed by Lumelsky in [START_REF] Lumelsky | On fast computation of distance between line segments[END_REF]. The interference between the i-th cable and the j-th cable does not occur when the distance d cc i,j is greater than the diameter of the cables, φ c , i.e., • Platform positioning error. This constraint is verified by analysing the platform positioning error due to elasticity of the CDPR cables. All the element of the platform positioning error, δt = [δt x , δt y , δt z ], should be lower than a threshold value, δt c . The platform positioning error can be computed from the CDPR elasto-static model proposed in [START_REF] Arsenault | Optimization of the prestress stable wrench closure workspace of planar parallel three-degree-of-freedom cable-driven mechanisms with four cables[END_REF]. The design problem is formulated as follows:

d cc i,j ≥ φ c ∀i, j = 1, . . . , m, i = j (33)
maximize: V (x) = p f over: x = [τ M , ω M , φ w , ρ R ]
subject to:

∀P k ∈ P : F k = 1 d cc i,j ≥ φ c ∀i, j = 1, . . . , m, i = j -δt c ≤ δt x , δt y , δt z ≤ δt c V. CASE STUDY

A. Case Study Description

The case study presented in this paper is based on the analysis of a spatial CDPR. The robot consists of m = 8 steel cables. Their diameter, φ c , is equal to 2 mm and their linear elastic coefficient, k c , is equal to 2.52 × 10 6 N m . The cables exit points, A i , i = 1, . . . , m, are located at the corner of the base frame, as illustrated in Fig. 5, according to the following Cartesian coordinates:

a b 1 = 1 2 -s l , -s w , -s h a b 2 = 1 2 s l , -s w , -s h (34) a b 3 = 1 2 s l , s w , -s h a b 4 = 1 2 -s l , s w , -s h (35) a b 5 = 1 2 -s l , -s w , s h a b 6 = 1 2 s l , -s w , s h (36)
a b 7 = 1 2 s l , s w , s h a b 8 = 1 2 -s l , s w , s h (37)
where the base frame length, s l , width, s w , and height, s h , are equal to 1 m, 0.6 m, 0.5 m, respectively. where w p = w p + u 0 and l p = l p + u 0 . u 0 is an offset applied to the cable-platform connection points in order to avoid cable collisions and singularities. In the specific case, this offset is equal to 1 cm. The platform is a parallelepiped, as shown in Fig. 6. Its width, w p , and its length, l p , are equal to 10 cm. Its height, h p , is equal to 5 cm. The component of the external forces acting on the platform, f x , f y and f z , have been constrained between ±20 N. The components of the external momentum, m x , m y and m z , are bounded between ±0.1 Nm:

-20 N ≤ f x , f y , f z ≤ 20 N (42) -0.1 Nm ≤m x , m y , m z ≤ 0.1 Nm (43)
Analogously, the twist components of the platform, ṫx , ṫy and ṫz , have been constrained between ±0.1 m s . The platform orientation is assumed to be constant, being the frame F p aligned to the base frame F b . The range of the design variables, τ M , ω M , φ w , ρ R , have been summarized in Tab. I.

B. Solution

The case study proposed in Sec. V-A has been solved in Matlab using the genetic algorithm toolbox and the GlobalSearch algorithm developed by Zsolt et al. [START_REF] Ugray | Scatter search and local nlp solvers: A multistart framework for global optimization[END_REF].

According to the optimal solution, 57.70% of points P k ∈ P are included in both the WFW and the TFW. The optimal dimensioning of the design variables contemplates a motor nominal torque τ M = 1.44 Nm, a motor nominal speed ωM = 195.6 rpm, a winch diameter φ w = 46.5 mm and a gearbox transmission ratio ρ R = 3.1. Fig. 7 represents the intersection of the WFW and the TFW associated to the optimal solution. Under these conditions, the TFW covers the 97.82% of the desired points P k ∈ P, as illustrated in Fig. 8; on the contrary, the WFW covers only the 57.73% of points P k ∈ P.

The correlation between the design variables, x, and the optimization function, V , has been investigated by means of the Matlab function corrcoef. A set of values has been associated to each design variable:

The different combinations of design variables have been tested computing the percentage of feasible points belonging to the WFW and the TFW. As illustrated in Fig. 9, the size of the WFW increases with respect to the nominal torque of the motor, τ M , and the gearbox transmission ratio, ρ R . The diameter of the winches, φ w , is inversely proportional to the size of the WFW. Analogously, an increasing of the nominal motor speed, ωM , induces an improvement of the TFW size, as well as the increasing of the diameter of the winches. The size of the TFW can be augmented decreasing the the gearbox transmission ratio. The performed analysis highlights that, regarding the dimensions of both the WFW and the TFW, φ w and ρ R have an higher influence with respect to the other design variables. Consequently, dimensioning the gearbox transmission ratio and the winch diameter, the designer should take into account the sensibility of these design parameters and the trade-off between the increasing of the WFW size and the decreasing of the TFW size. For example, the largest WFW can be obtained pushing the design variables to their upper limits: τ M = 1.5 Nm, ω M = 200 rpm, φ w = 50 mm and ρ R = 5. Under these assumptions, the WFW covers the 65.98% of P k ∈ P, as shown in Fig. 10. However, due to the high gearbox transmission ratio, ρ R = 5, all the points included in the WFW are not twist feasible.

VI. CONCLUSIONS AND FUTURE WORK

The research work presented in this paper dealt with the definition of a novel workspace for CDPRs, the Twist Feasible Workspace (TFW). The TFW can be used in order to evaluate the twists that the platform can achieve. Furthermore, the TFW can be successfully integrated in the dimensioning problem of the actuators, the gearboxes and the winches of a CDPR, supposing the locations of the cable exit points and the platform dimensions have been fixed a priori. The influence of the design parameters over the dimensions of the WFW and the TFW have been analysed and commented.

Despite the fact that the computation of the TFW can be performed using methods developed originally to analyse the WFW, part of the research effort should be dedicated to the design of a novel tool for the specific computation of the TFW. Furthermore, more optimization criteria should be analysed in order to dimension the CDPR components while paying attention to the component costs.

Fig. 1 .

 1 Fig. 1. Example of a CDPR design. The robot shown in the picture is a concept created in the framework of the CAROCA project, whose intent is the painting of tubular structures.

  with respect to an absolute reference frame, F b , of origin O b and axes x b , y b and z b . The Cartesian coordinate vectors a b i , i = 1, . . . , m describe the positions of the exit points, A i , i = 1, . . . , m, with respect to frame F b . The Cartesian coordinate vectors b p i , i = 1, . . . , m represent the positions of the platform-cable connection points, B i , i = 1, . . . , m, with respect to a local reference frame, F p , of origin O p and axes x p , y p and z p . This local reference frame is attached to the mobile platform.

Fig. 2 .

 2 Fig. 2. Geometric description of an example of fully constrained CDPR.

Fig. 3 .

 3 Fig. 3. Example of the mapping of [τ ]a into the wrench space, for a planar CDPR with 3 actuators.

Fig. 4 .

 4 Fig. 4. Scheme of the actuation and transmission system.

Fig. 5 .

 5 Fig. 5. Description of the case study and the desired workspace.

Fig. 6 .

 6 Fig. 6. Platform of the CDPR.

  consists of a box, illustrated in Fig. 5. The box has been discretized in n P = 1881 points, whose coordinates along x b , y b and z b are defined according to the sets of values [x], [y] and [z], respectively: [x] = {-0.45 m : 0.05 m : 0.45 m} (46) [y] = {-0.25 m : 0.05 m : 0.25 m} (47) [z] = {-0.2 m : 0.05 m : 0.2 m} (48)

  [τ M ] = {-0.5 Nm : 0.1 Nm : 1.5 Nm} (49) [ω M ] = {50 rpm : 10 rpm : 200 rpm} (50) [φ w ] = {1 cm : 1 cm : 5 cm} (51) [ρ R ] = {0.5 : 0.1 : 1, 2 : 1 : 5}

Fig. 7 .

 7 Fig. 7. Intersection of the WFW and the TFW for the optimal solution (τ M = 1.44 Nm, ωM = 195.6 rpm, φw = 46.5 mm, ρ R = 3.1).

Fig. 8 .

 8 Fig. 8. TFW of the optimal solution (τ M = 1.44 Nm, ωM = 195.6 rpm, φw = 46.5 mm, ρ R = 3.1).

Fig. 9 .

 9 Fig. 9. Analysis of the correlation between the design variables and the optimization criterion.

Fig. 10 .

 10 Fig. 10. Larger WFW, covering the 65,98% of the desired workspace, computed with respect to the design variables τ M = 1.5 Nm, ωM = 200 rpm, φw = 50 mm, ρ R = 5.

TABLE I DESIGN

 I VARIABLE BOUNDARIES.

	Design Variable	τ M	ω M	φw	ρ R
	min	0.5 Nm	50 rpm	1 cm	0.5
	M ax	1.5 Nm 200 rpm 5 cm	5

ACKNOWLEDGMENT

This study is part of the CAROCA project, managed by IRT Jules Verne (French Institute in Research and Technology in Advanced Manufacturing Technologies for Composite, Metallic and Hybrid Structures).