Latifah Nurahmi
email: latifah.nurahmi@irccyn.ec-nantes.fr

Josef Schadlbauer
email: josef.schadlbauer@uibk.ac.at

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

Manfred Husty
email: manfred.husty@uibk.ac.at

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

Caro

Kinematic Analysis of the 3-RPS Cube Parallel Manipulator

Keywords: 3-RPS-Cube, parallel manipulators, singularities, operation mode, motion type, Darboux motion

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Since the development of robot technology, the lower-mobility parallel manipulators have been extensively studied. One parallel manipulator of the 3-dof family is the 3-RPS Cube and was proposed by [START_REF] Huang | Motion Characteristics and Rotational Axis Analysis of Three DOF Parallel Robot Mechanisms[END_REF] [START_REF] Huang | Motion Characteristics and Rotational Axis Analysis of Three DOF Parallel Robot Mechanisms[END_REF]. The 3-RPS Cube parallel manipulator, shown in Fig. 1, is composed of a cube-shaped base, an equilateral triangular-shaped platform, and three identical legs. Each leg is composed of a revolute joint, an actuated prismatic joint and a spherical joint mounted in series.

By referring to the design of the 3-RPS Cube manipulator, the type synthesis of 3dof rotational manipulators with no intersecting axes was discussed in [START_REF] Chen | Type Synthesis of 3-DOF Rotational Parallel Mechanisms With No Intersecting Axes[END_REF]. The kinematic characteristics of this mechanism were studied in [3][START_REF] Huang | Identification of Principal Screws of 3-DOF Parallel Manipulators by Quadric Degeneration[END_REF][START_REF] Huang | Analysis of Instantaneous Motions of Deficient Rank 3-RPS Parallel Manipulators[END_REF], by identifying the principal screws, and the authors showed that the manipulator belongs to the general third-order screw system, which can rotate in three dimensions and the axes do not intersect.

In [6], Huang et al. showed that the mechanism is able to perform a motion along its diagonal, which is known as the Vertical Darboux Motion (VDM). Several mechanical generators of the VDM were later revealed by Lee and Hervé [START_REF] Lee | On the Vertical Darboux Motion[END_REF], in which one point in the moving platform is compelled to move in a plane.

Later in [START_REF] Huang | A 3DOF Rotational Parallel Manipulator Without Intersecting Axes[END_REF], the authors showed that the manufacturing errors have little impact on the motion properties of the 3-RPS Cube parallel manipulator. By analysing the Instantaneous Screw Axes (ISA), Chen et al. showed in [START_REF] Chen | Axodes Analysis of the Multi DOF Parallel Mechanisms and Parasitic Motion[END_REF] that this mechanism performs parasitic motions, in which the translations and the rotations are coupled.

By using an algebraic description of the manipulator and the Study kinematic mapping, a characterisation of the operation mode, the direct kinematics, the general motion, and the singular poses of the 3-RPS Cube parallel manipulator are discussed in more detail in this paper, which is based on [10][11][START_REF] Schadlbauer | A Complete Kinematic Analysis of the 3-RPS Parallel Manipulator[END_REF][START_REF] Schadlbauer | Operation Modes in Lower Mobility Parallel Manipulators[END_REF][START_REF] Schadlbauer | The 3-RPS parallel manipulator from an algebraic viewpoint[END_REF]. The derivation of the constraint equations is the first essential step to reveal the existence of only one operation mode and to solve the direct kinematics problem.

In 1897, Darboux [START_REF] Bottema | Theoretical Kinematics[END_REF] studied the 3-dof motion where the vertices of a triangle are compelled to remain in the planes of a trihedron respectively. The three planes are mutually orthogonal and this is the case of the 3-RPS Cube parallel manipulator. Darboux showed that in this 3-dof motion, the workspace of each point of the moving platform is bounded by a Steiner surface, while the vertices of the moving platform remain in the planes.

Under the condition that the prismatic lengths remain equal, the moving platform of the manipulator is able to perform the VDM. It follows from Bottema and Roth [START_REF] Bottema | Theoretical Kinematics[END_REF] that this motion is the result of a rotation about an axis and a harmonic translation along the same axis. In this motion, all points in the moving platform (except the geometric center of the moving platform) move in ellipses and the path of a line in the moving platform is a right-conoid surface.

The singularities are examined in this paper by deriving the determinant of the Jacobian matrix of the constraint equations with respect to the Study parameters. Based on the reciprocity conditions, Joshi and Tsai in [START_REF] Joshi | Jacobian Analysis of Limited-DOF Parallel Manipulators[END_REF] developed a procedure to express the Jacobian matrix J of lower-mobility parallel manipulators that comprises both actuation and constraint wrenches. In this paper, this matrix is named the extended Jacobian matrix (J E ) of the lower-mobility parallel manipulators, as explained in [START_REF] Amine | Singularity Analysis of the H4 Robot Using Grassmann-Cayley Algebra[END_REF][START_REF] Amine | Lower-Mobility Parallel Manipulators: Geometrical Analysis, Singularities and Conceptual Design[END_REF][START_REF] Amine | Singularity analysis of 3T2R parallel mechanisms using Grassmann Cayley algebra and Grassmann geometry[END_REF][START_REF] Amine | Conceptual Design of Schonflies Motion Generators Based on the Wrench Graph[END_REF][START_REF] Amine | Singularity Conditions of 3T1R Parallel Manipulators With Identical Limb Structures[END_REF]. The rows of J E are composed of n linearly independent actuation wrenches plus (6 -n) linearly independent constraint wrenches.

In a general configuration, the constraint wrench system, W c , must be reciprocal to the twist system of the moving platform of the parallel manipulator. A constraint singularity occurs when the (6-n) constraint wrench system W c degenerates. In such a configuration, at least one of the initial constrained motions will no longer be constrained. As a result, the mechanism gains one or several dof . This can lead to a change in the motion pattern of the mechanism, which then can switch to another operation mode.

By locking the actuated joints of the parallel manipulator, the moving platform must be fully constrained, i.e., the system spanned by the actuation wrench system, W a , and constraint wrench system, W c , must span a 6-system. An actuation singularity hence occurs when this overall wrench system of the manipulator degenerates, i.e., is not a 6-system any more, while the manipulator does not reach a constraint singularity. This concept will be applied in this paper to illustrate the singularities of the extended Jacobian matrix (J E ) of the 3-RPS Cube parallel manipulator. It allows us to investigate the actuation and constraint singularities that occur during the manipulator motion. This paper is organized as follows: A detailed description of the manipulator architecture is given in Section 2. The constraint equations of the manipulator are expressed in Section 3. These equations are used to identify the operation mode(s) and the solutions of the direct kinematics of the manipulator in Section 4. In Section 5, the conditions on the leg lengths for the manipulator to reach a singularity configuration are presented. Eventually, the general motion and the Vertical Darboux Motion (VDM) of the manipulator are reviewed in Sections 6 and 7.

Manipulator Architecture

The 3-RPS Cube parallel manipulator shown in Fig. 1, is composed of a cube-shaped base, an equilateral triangular-shaped platform and three identical legs. Each leg is composed of a revolute joint, an actuated prismatic joint and a spherical joint mounted in series.

The origin O of the fixed frame Σ 0 is shifted along σ 0 = [h 0 , h 0 , h 0 ] from the center of the base in order to fulfill the identity condition (when the fixed frame and the moving frame are coincident), as shown by the large and red dashed box in Fig. 1. Likewise, the origin P of the moving frame Σ 1 is shifted along σ 1 = [h 1 , h 1 , h 1 ] as described by the small and blue dashed box in Fig. 1. The revolute joint in the i-th (i = 1 . . . 3) leg is located at point A i , its axis being along vector s i , while the spherical joint is located at point B i , the i-th corner of the moving platform. The distance between the origin O of the fixed frame Σ 0 and point A i is equal to h 0 √ 2. The axes s 1 , s 2 and s 3 are orthogonal to each other. The moving platform has an equilateral triangle shape and its circumradius is equal to

d 1 = h 1 √ 6/3.

Each pair of vertices

A i and B i (i = 1, 2, 3
) is connected by a prismatic joint. The prismatic length is denoted by r i . Since the i-th prismatic length is orthogonal to the revolute axis s i , the leg A i B i moves in a plane normal to s i .

As a consequence, there are five parameters, namely r 1 , r 2 , r 3 , h 0 , and h 1 . h 0 and h 1 are design parameters, while r 1 , r 2 , and r 3 are joint variables that determine the manipulator motion.

Constraint Equations

In this section, the constraint equations are expressed whose solutions illustrate the possible poses of the moving platform (coordinate frame Σ 1 ) with respect to Σ 0 . In the following, we use projective coordinates to define the position vectors of points A i and B i . The coordinates of points A i and points B i expressed in Σ 0 and Σ 1 are respectively:

r 0 A 1 = [1, 0, -h 0 , -h 0 ] T , r 1 B 1 = [1, 0, -h 1 , -h 1 ] T , r 0 A 2 = [1, -h 0 , 0, -h 0 ] T , r 1 B 2 = [1, -h 1 , 0, -h 1 ] T , r 0 A 3 = [1, -h 0 , -h 0 , 0] T , r 1 B 3 = [1, -h 1 , -h 1 , 0] T (1)
To obtain the coordinates of points B 1 , B 2 and B 3 expressed in Σ 0 , the Study parametrization of a spatial Euclidean transformation matrix M ∈ SE(3) is used as follows:

M =    x 2 0 + x 2 1 + x 2 2 + x 2 3 0 T 3×1 M T M R    (2)
where M T and M R represent the translational and rotational parts of transformation matrix M, respectively, and are expressed as follows:

M T =       2(-x 0 y 1 + x 1 y 0 -x 2 y 3 + x 3 y 2 ) 2(-x 0 y 2 + x 1 y 3 + x 2 y 0 -x 3 y 1 ) 2(-x 0 y 3 -x 1 y 2 + x 2 y 1 + x 3 y 0 )       , M R =       x 2 0 + x 2 1 -x 2 2 -x 2 3 2(x 1 x 2 -x 0 x 3 ) 2(x 1 x 3 + x 0 x 2 ) 2(x 1 x 2 + x 0 x 3 ) x 2 0 -x 2 1 + x 2 2 -x 2 3 2(x 2 x 3 -x 0 x 1 ) 2(x 1 x 3 -x 0 x 2 ) 2(x 2 x 3 + x 0 x 1 ) x 2 0 -x 2 1 -x 2 2 + x 2 3       (3)
The parameters x 0 , x 1 , x 2 , x 3 , y 0 , y (x 0 : x 1 : x 2 : x 3 : y 0 : y 1 : y 2 : y 3 ) T = (0 : 0 : 0 : 0 : 0 : 0 : 0 : 0) T

Every projective point X will represent a spatial Euclidean displacement, if it fulfills the following equation and inequality:

x 0 y 0 + x 1 y 1 + x 2 y 2 + x 3 y 3 = 0, x 2 0 + x 2 1 + x 2 2 + x 2 3 = 0 (5)
Those two conditions will be used in the following computations to simplify the algebraic expressions. The coordinates of points B i expressed in Σ 0 are obtained by:

r 0 B i = M r 1 B i i = 0, . . . , 3 (6)
As the coordinates of all points are given in terms of Study parameters, design parameters and joint variables, the constraint equations can be obtained by examining the manipulator architecture. The leg connecting points A i and B i is orthogonal to the axis s i of the i-th revolute joint, expressed as follows:

s 1 = [0, 1, 0, 0] T s 2 = [0, 0, 1, 0] T s 3 = [0, 0, 0, 1] T (7)
Accordingly, the scalar product of vector (r 0 B ir 0 A i ) and vector s i vanishes, namely:

(r 0 B i -r 0 A i ) T s i = 0 (8)
After computing the corresponding scalar products and removing the common denom-

inators (x 2 0 + x 2 1 + x 2 2 + x 2 3
), the following three equations come out:

g 1 : -h 1 x 0 x 2 + h 1 x 0 x 3 -h 1 x 1 x 2 -h 1 x 1 x 3 -x 0 y 1 + x 1 y 0 -x 2 y 3 + x 3 y 2 = 0 g 2 : h 1 x 0 x 1 -h 1 x 0 x 3 -h 1 x 1 x 2 -h 1 x 2 x 3 -x 0 y 2 + x 1 y 3 + x 2 y 0 -x 3 y 1 = 0 g 3 : -h 1 x 0 x 1 + h 1 x 0 x 2 -h 1 x 1 x 3 -h 1 x 2 x 3 -x 0 y 3 -x 1 y 2 + x 2 y 1 + x 3 y 0 = 0 (9)
To derive the constraint equations corresponding to the leg lengths, the joint variables r i are given and we assume that the distance between points A i and B i is constant, i.e.

r i = const. It follows that point B i has the freedom to move along a circle of center A i and the distance equation can be formulated as (r 0

B i -r 0 A i ) 2 = r 2 i .
As a consequence, the following three equations are obtained:

g 4 : 2h 2 0 x 2 0 + 2h 2 0 x 2 1 + 2h 2 0 x 2 2 + 2h 2 0 x 2 3 -4h 0 h 1 x 2 0 + 4h 0 h 1 x 2 1 + 2h 2 1 x 2 0 + 2h 2 1 x 2 1 + 2h 2 1 x 2 2 + 2h 2 1 x 2 3 -8h 0 h 1 x 2 x 3 -r 2 1 x 2 0 -r 2 1 x 2 1 -r 2 1 x 2 2 -r 2 1 x 2 3 -4h 0 x 0 y 2 -4h 0 x 0 y 3 -4h 0 x 1 y 2 + 4h 0 x 1 y 3 + 4h 0 x 2 y 0 + 4h 0 x 2 y 1 + 4h 0 x 3 y 0 -4h 0 x 3 y 1 + 4h 1 x 0 y 2 + 4h 1 x 0 y 3 + 4y 2 0 -4h 1 x 1 y 2 + 4h 1 x 1 y 3 -4h 1 x 2 y 0 + 4h 1 x 2 y 1 -4h 1 x 3 y 0 + 4y 2 1 -4h 1 x 3 y 1 + 4y 2 2 + 4y 2 3 = 0 g 5 : 2h 2 0 x 2 0 + 2h 2 0 x 2 1 + 2h 2 0 x 2 2 + 2h 2 0 x 2 3 -4h 0 h 1 x 2 0 + 4h 0 h 1 x 2 2 + 2h 2 1 x 2 0 + 2h 2 1 x 2 1 + 2h 2 1 x 2 2 + 2h 2 1 x 2 3 -8h 0 h 1 x 1 x 3 -r 2 2 x 2 0 -r 2 2 x 2 1 -r 2 2 x 2 2 -r 2 2 x 2 3 -4h 0 x 0 y 1 -4h 0 x 0 y 3 + 4h 0 x 1 y 0 -4h 0 x 1 y 2 + 4h 0 x 2 y 1 -4h 0 x 2 y 3 + 4h 0 x 3 y 0 + 4h 0 x 3 y 2 + 4h 1 x 0 y 1 + 4h 1 x 0 y 3 + 4y 2 0 -4h 1 x 1 y 0 -4h 1 x 1 y 2 + 4h 1 x 2 y 1 -4h 1 x 2 y 3 -4h 1 x 3 y 0 + 4y 2 1 + 4h 1 x 3 y 2 + 4y 2 2 + 4y 2 3 = 0 g 6 : 2h 2 0 x 2 0 + 2h 2 0 x 2 1 + 2h 2 0 x 2 2 + 2h 2 0 x 2 3 -4h 0 h 1 x 2 0 + 4h 0 h 1 x 2 3 + 2h 2 1 x 2 0 + 2h 2 1 x 2 1 + 2h 2 1 x 2 2 + 2h 2 1 x 2 3 -8h 0 h 1 x 1 x 2 -r 2 3 x 2 0 -r 2 3 x 2 1 -r 2 3 x 2 2 -r 2 3 x 2 3 -4h 0 x 0 y 1 -4h 0 x 0 y 2 + 4h 0 x 1 y 0 + 4h 0 x 1 y 3 + 4h 0 x 2 y 0 -4h 0 x 2 y 3 -4h 0 x 3 y 1 + 4h 0 x 3 y 2 + 4h 1 x 0 y 1 + 4h 1 x 0 y 2 + 4y 2 0 -4h 1 x 1 y 0 + 4h 1 x 1 y 3 -4h 1 x 2 y 0 -4h 1 x 2 y 3 -4h 1 x 3 y 1 + 4y 2 1 + 4h 1 x 3 y 2 + 4y 2 2 + 4y 2 3 = 0 (10)
The Study equation in Eq. ( 5) is added since all solutions have to be within the Study quadric, i.e.:

g 7 : x 0 y 0 + x 1 y 1 + x 2 y 2 + x 3 y 3 = 0 (11)
Under the condition (x 2 0 + x 2 1 + x 2 2 + x 2 3 = 0), we can find all possible points in P 7 that satisfy those seven equations. To exclude the exceptional generator (x 0 = x 1 = x 2 = x 3 = 0), we add the following normalization equation:

g 8 : x 2 0 + x 2 1 + x 2 2 + x 2 3 -1 = 0 (12)
It assures that there is no point of the exceptional generator that appears as a solution.

However, for each projective solution point, we obtain two affine representatives. This has to be taken into account for the enumeration of the number of solutions.

Solving the System

Solving the direct kinematics means finding all possible points in P 7 that fulfill the set of equations {g 1 , ..., g 8 }. Those points are the solutions of the eight constraint equations that represent all feasible poses of the 3-RPS Cube parallel manipulator. They also depend on the design parameters (h 0 , h 1 ) and the joint variables (r 1 , r 2 , r 3 ).

The set of eight constraint equations are always written as a polynomial ideal with variables {x 0 , x 1 , x 2 , x 3 , y 0 , y 1 , y 2 , y 3 } over the coefficient ring C[h 0 , h 1 , r 1 , r 2 , r 3 ]. Although the solutions of the direct kinematics can be complex, they are still considered as solutions.

To apply the method of algebraic geometry, the ideal is now defined as:

I =< g 1 , g 2 , g 3 , g 4 , g 5 , g 6 , g 7 , g 8 > (13)
The vanishing set V(I) of the ideal I comprises all points in P 7 for which all equations vanish, namely all solutions of the direct kinematic problem. At this point, the following ideal is examined, which is independent of the joints variables r 1 , r 2 and r 3 :

J =< g 1 , g 2 , g 3 , g 7 > (14)
The primary decomposition is computed to verify if the ideal J is the intersection of several smaller ideals. The primary decomposition returns several J i in which J = i J i .

In other words, the vanishing set is given by V(J ) = i V(J i ). It expresses that the variety V(J ) is the union of some other or simpler varieties V(J i ).

The primary decomposition geometrically tells us that the intersection of those equa-

Paper JMR-14-1262, corresponding author's last name: CARO tions will split into smaller parts. Indeed, it turns out that the ideal J is decomposed into two components J i as:

J = 2 i=1 J i (15)
with the results of primary decompositions 1 as follows:

J 1 = < x 0 y 0 + x 1 y 1 + x 2 y 2 + x 3 y 3 , ... > J 2 = < x 0 , x 1 , x 2 , x 3 > (16)
An inspection of the vanishing set V(J 2 ∪ g 8 ) yields an empty result, since the set of

polynomials {x 0 , x 1 , x 2 , x 3 , x 2 0 + x 2 1 + x 2 2 + x 2
3 -1 = 0} can never vanish simultaneously over R or C. Therefore, only one component is left and as a consequence, the manipulator has only one operation mode, which is defined by J 1 . To complete the analysis, the remaining equations have to be added by writing:

K i = J i ∪ < g 4 , g 5 , g 6 , g 8 > ( 17
)
Since there is only one component, the vanishing set of I is now defined by:

V(I) = V(K 1 ) (18)
From the primary decomposition, it is shown that the ideal I cannot be split and K 1 is named I hereafter.

Solutions for arbitrary design parameters

The 3-RPS Cube manipulator generally has only one operation mode, which is described by the ideal I. The solutions of the direct kinematic problem in this operation mode will be given for arbitrary values of design parameters (h 0 , h 1 ). To find out the Hilbert dimension of the ideal I, the certain values of design parameters are chosen as h 0 = 2 m

dim(I) = 0 ( 19
)
The dim denotes the dimension over C[h 0 , h 1 , r 1 , r 2 , r 3 ] and shows that the number of solutions to the direct kinematic problems is finite in this general mode. The number of solutions and the solutions themselves were computed via an ordered Gröbner basis, which led to a univariate polynomial of degree 32. As two solutions of a system describe the same pose (position and orientation) of the moving platform, the number of solutions has to be halved into 16 solutions.

| V(I) |= 16 (20)
Therefore, there are at most 16 different solutions for given general design parameters and joint variables, i.e., there are theoretically 16 feasible poses of the moving platform for given joint variables. Notably, for arbitrarily values of design parameters and joint variables, some solutions might be complex.

Solutions for equal leg lengths

In the following subsection, it is assumed that all legs have the same length. The corresponding prismatic lengths are r 1 = r 2 = r 3 = r. Similar computations can be performed which were done in the previous subsection to enumerate the Hilbert dimension of the ideal. The Hilbert dimension is calculated and it follows that:

dim(I) = 0 ( 21
)
This shows that the solutions of the direct kinematics problem with equal leg lengths are finite. When the number of solutions is computed for the system, it has to be halved and the following result is obtained:

| V(I) |= 16 (22)
The number of solutions for equal leg lengths is the same number as the solutions for arbitrary design parameters. Due to the fact that there are fewer parameters, the Gröebner basis can be computed without specifying any value. The solutions of Study parameters in the case of equal leg lengths are x 1 = x 2 = x 3 and y 1 = y 2 = y 3 . One manipulator pose with equal prismatic lengths leads to the following solutions of Study parameters:

x 0 = 1 2 -h 1 3h 2 0 -2h 2 1 + 3r 2 -3h 0 -2h 1 h1 x 1 = 1 6 √ 3 h 1 3h 2 0 -2h 2 1 + 3r 2 -3h 0 + 2h 1 h1 y 0 = 1 12 √ 3 h 1 3h 2 0 -2h 2 1 + 3r 2 -3h 0 + 2h 1 3/2 h1 2 y 1 = -1 12h 1 -h 1 3h 2 0 -2h 2 1 + 3r 2 -3h 0 -2h 1 2h 1 -3h 0 + 3h 2 0 -2h 2 1 + 3r 2 x 1 = x 2 = x 3 y 1 = y 2 = y 3 (23)

Operation mode analysis

In the previous section, the joint variables (r 1 , r 2 , r 3 ) were fixed. In this section, they can change, i.e., the behaviour of the mechanism is studied when the prismatic joints are actuated. The joint variables (r 1 , r 2 and r 3 ) are used as unknowns and the computation of the Hilbert dimension shows that:

dim(I) = 3 ( 24
)
where dim denotes the dimension over C[h 0 , h 1 ] and shows that the manipulator has 3 dof in general motion.

The matrix M ∈ SE(3) in Eq. ( 2) represents a discrete screw motion from the pose corresponding to the identity condition, where Σ 0 and Σ 1 are coincident, to the transformed pose of Σ 1 with respect to Σ 0 . A discrete screw motion is the concatenation of a rotation about an axis and a translation along the same axis. The axis A, the translational distance s, and the rotational angle ϕ of the discrete screw motion can be computed from the matrix M. This information can also be obtained directly from the Study parameters, as they contain the information on the transformation. The Plücker coordinates L = (p 0 : p 1 : p 2 : p 3 : p 4 : p 5 ) of the corresponding discrete screw motion are expressed as:

p 0 = (-x 2 1 -x 2 2 -x 2 3 )x 1 , p 1 = (-x 2 1 -x 2 2 -x 2 3 )x 2 , p 2 = (-x 2 1 -x 2 2 -x 2 3 )x 3 , p 3 = x 0 y 0 x 1 -(-x 2 1 -x 2 2 -x 2 3 )y 1 , p 4 = x 0 y 0 x 2 -(-x 2 1 -x 2 2 -x 2 3 )y 2 , p 5 = x 0 y 0 x 3 -(-x 2 1 -x 2 2 -x 2 3 )y 3 . (25)
The unit vector of an axis A of the corresponding discrete screw motion is given by [p 0 , p 1 , p 2 ] T . The Plücker coordinates of a line should satisfy the following condition [START_REF] Pottmann | Computational Line Geometry[END_REF]:

p 0 p 3 + p 1 p 4 + p 2 p 5 = 0 (26)
The rotational angle ϕ can be enumerated directly from cos ϕ 2 = x 0 , whereas the translational distance s of the transformation can be computed from the Study parameters, as follows:

s = 2y 0 x 2 1 + x 2 2 + x 2 3 ( 27
)
The following example shows the manipulator poses by solving the direct kinematic problem. Arbitrary values are assigned to the design parameters and joint variables as The first solution of the direct kinematics is depicted in Fig. 2(a), with (x 0 : x 1 : x 2 :

follows: h 0 = 2 m, h 1 = 1 m, r 1 = 1.2 m,
x 3 : y 0 : y 1 : y 2 : y 3 ) = (-0.961 : -0.189, -0.153 : 0.128 : -0.007 : 0.304 : -0.250 : 0.089).

The discrete screw motion of the moving platform from identity into the actual pose in

Singularity Conditions of the Manipulator

The manipulator reaches a singular configuration when the determinant of the Jacobian matrix vanishes. The Jacobian matrix is the matrix of all first order partial derivatives of eight constraint equations {g 1 , g 2 , g 3 , g 4 , g 5 , g 6 , g 7 , g 8 } with respect to {x 0 , x 1 , x 2 , x 3 , y 0 , y 1 , y 2 , y 3 }.

Since the manipulator has one operation mode, the singular configurations occur within this operation mode only. In the kinematic image space, the singular poses are computed by taking the Jacobian matrix from I:

The vanishing condition det(J) = 0 of the determinant J is denoted by S. The factorization of the equation of the Jacobian determinant splits it into two components, namely S 1 : det 1 (J) = 0 and S 2 : det 2 (J) = 0.

det(J) = 0 det 1 (J) det 2 (J) = 0 (29)
It shows that the overall determinant will vanish if either S 1 or S 2 vanishes or both S 1 and S 2 vanish simultaneously. By adding the expression of the Jacobian determinant into the system I, the new ideal associated with the singular poses can be defined as:

L i = I ∪ S i i = 1, 2 (30)
The ideals now consist of a set of nine equations L i =< g 1 , g 2 , g 3 , g 4 , g 5 , g 6 , g 7 , g 8 , g 9 >.

The ninth equation is the determinant of the Jacobian matrix.

In mechanics, the singularity surface is desirable also in the joint space R, where R is the polynomial ring over variables r 1 , r 2 and r 3 . To obtain the singularity surface in R, the following projections are determined from ideals L i :

L i → R i = 1, 2 (31)
Algebraically, each projection is an elimination of Study parameters from the ideal L i and is mapped onto one equation generated by r 1 , r 2 and r 3 . It was not possible to compute the elimination in general, thus we assigned some values to the design parameters, namely h 0 = 2 m and h 1 = 1 m. The eight Study parameters x 0 , x 1 , x 2 , x 3 , y 0 , y 1 , y 2 , y 3

were eliminated to obtain a single polynomial in r 1 , r 2 and r 3 .

For the system L 1 , the elimination yields a polynomial of degree four in r 1 , r 2 and r 3 in Eq. ( 32) and its zero set of polynomial is plotted in Fig. 3. By taking a point on this surface, we are able to compute the direct kinematics of at least one singularity pose.

r 4 1 -r 2 1 r 2 2 + r 4 2 -r 2 1 r 2 3 -r 2 2 r 2 3 + r 4 3 -2r 2 1 -2r 2 2 -2r 2 3 -20 = 0 (32)
Due to the heavy elimination process of Study parameters from ideal L 2 , some arbitrary values have been assigned to the joint variables r 1 = 2 m and r 2 = 1.7 m. Then the elimination can be carried out and the result is a univariate polynomial of degree 64 in

r 3 .
Let us consider one singularity configuration of the manipulator when the moving frame Σ 1 coincides with the fixed frame Σ 0 and all joint variables have the same values.

The system L 2 is now solved by assigning the joint variables as r 1 = r 2 = r 3 . The elimination process returns a univariate polynomial of degree 24 in r 3 . The real solutions of joint variables in this condition is

r 1 = r 2 = r 3 = √ 2 m.
The coordinates of points B 1 , B 2 and B 3 can be determined by solving the direct kinematics. Accordingly, we can form the extended Jacobian matrix (J E ) of the manipulator, which is based on the screw theory. The rows of the extended Jacobian matrix J E are composed of n actuation wrenches W a and (6 -n) constraint wrenches W c . Since the Paper JMR-14-1262, corresponding author's last name: CARO By considering that the prismatic joints are actuated, each leg applies one actuation force whose axis is along the direction of the corresponding actuated joint u i , as follows:

F a1 = [ u 1 , r 0 B 1 × u 1 ] F a2 = [ u 2 , r 0 B 2 × u 2 ] F a3 = [ u 3 , r 0 B 3 × u 3 ] W a = span( F a1 , F a2 , F a3 ) (33)
Due to the manipulator architecture, each leg applies one constraint force, which is perpendicular to the actuated prismatic joint and parallel to the axis s i of the revolute joint, written as:

F c1 = [ s 1 , r 0 B 1 × s 1 ] F c2 = [ s 2 , r 0 B 2 × s 2 ] F c3 = [ s 3 , r 0 B 3 × s 3 ] W c = span( F c1 , F c2 , F c3 ) (34)
By collecting all components of the extended Jacobian matrix, we obtained:

J T E = F a1 F a2 F a3 F c1 F c2 F c3 (35)
The degeneracy of matrix J E indicates that the manipulator reaches a singularity configuration. We can observe the pose of the manipulator when r 1 = r 2 = r 3 = √ 2 m, the matrix J E in this pose is rank deficient, while neither the constraint wrench system nor the actuation wrench system degenerates, i.e. rank(J E ) = 5, rank(W a ) = 3, and rank(W c ) = 3. This means that the manipulator reaches an actuation singularity.

By examining the null space of the degenerate matrix J E , the uncontrolled motion (infinitesimal gain motion) of the moving platform can be obtained. This uncontrolled motion is characterized by a zero-pitch twist that is reciprocal to all constraint and actuation wrenches. It is denoted by s λ and is described in Eq. ( 36). This singularity posture is depicted in Fig. 4, the uncontrolled motion of the moving platform is along the purple line.

s T λ = 1 1 1 0 0 0 (36)

General Motion

The set of eight constraint equations is written as a polynomial ideal I with variables

x 0 , x 1 , x 2 , x 3 , y 0 , y 1 , y 2 , y 3 over the coefficient ring C[h 0 , h 1 , r 1 , r 2 , r 3 ].
Paper JMR-14-1262, corresponding author's last name: CARO I =< g 1 , g 2 , g 3 , g 4 , g 5 , g 6 , g 7 , g 8 >

The general motion performed by the 3-RPS Cube parallel manipulator is characterized by solving the ideal I. The equations g 1 , g 2 , g 3 , g 4 , g 5 , g 6 , g 7 from ideal I can be solved linearly for variables y 0 , y 1 , y 2 , y 3 , R 1 , R 2 , R 3 [START_REF] Schadlbauer | A Complete Kinematic Analysis of the 3-RPS Parallel Manipulator[END_REF], R i being the square of the prismatic lengths, i.e., R i = r 2 i , and

δ = x 2 0 + x 2 1 + x 2 2 + x 2 3 .
Hence, the Study parameters become:

y 0 = h 1 (x 2 1 x 2 + x 2 1 x 3 + x 1 x 2 2 + x 1 x 2 3 + x 2 2 x 3 + x 2 x 2 3 ) δ y 1 = - h 1 (x 2 0 x 2 -x 2 0 x 3 + x 0 x 2 2 + x 0 x 2 3 -x 2 2 x 3 + x 2 x 2 3 ) δ y 2 = h 1 (x 2 0 x 1 -x 2 0 x 3 -x 0 x 2 1 -x 0 x 2 3 -x 2 1 x 3 + x 1 x 2 3 ) δ y 3 = - h 1 (x 2 0 x 1 -x 2 0 x 2 + x 0 x 2 1 + x 0 x 2 2 -x 2 1 x 2 + x 1 x 2 2 ) δ (38)
The terms R i 2 are also expressed in terms of x 0 , x 1 , x 2 , x 3 . The remaining Study parameters are still linked in equation g 8 :

x 2 0 + x 2 1 + x 2 2 + x 2
3 -1 = 0, which amounts to a hypersphere equation in space (x 0 , x 1 , x 2 , x 3 ). Accordingly, the transformation matrix is obtained. However, only the translational part of the transformation matrix depends on parameters (x 0 , x 1 , x 2 , x 3 ).

M T =       2h 1 (x 0 x 2 -x 0 x 3 + x 1 x 2 + x 1 x 3 ) -2h 1 (x 0 x 1 -x 0 x 3 -x 1 x 2 -x 2 x 3 ) 2h 1 (x 0 x 1 -x 0 x 2 + x 1 x 3 + x 2 x 3 )       (39)
This parametrization provides us with an interpretation of the general motion performed by the manipulator. The moving platform of the manipulator is capable of all orientations determined by (x 0 , x 1 , x 2 , x 3 ). The translational motion is coupled to the orientations via Eq. (39).

The position of any point in the moving platform ([1, x, y, z] T ) with respect to the fixed frame Σ 0 ([1, X, Y, Z] T ) during the motion is determined by:

x 0 = 1, x 1 = x 2 = x 3 = 0 : r 0 C 0 = [1, -h 1 , -h 1 , -h 1 ] T x 1 = 1, x 0 = x 2 = x 3 = 0 : r 0 C 1 = [1, -h 1 , h 1 , h 1 ] T x 2 = 1, x 0 = x 1 = x 3 = 0 : r 0 C 2 = [1, h 1 , -h 1 , h 1 ] T x 3 = 1, x 0 = x 1 = x 2 = 0 : r 0 C 3 = [1, h 1 , h 1 , -h 1 ] T (42)
C 0 , C 1 , C 2 and C 3 are the vertices of a tetrahedron C as shown in Fig. 5. Those points correspond to the poses of the moving platform subjected to the actuation singularities.

The uncontrolled motions of the moving platform are characterized by zero-pitch twists that intersect the geometric center of the moving platform and the corresponding vertices.

If two parameters are null, for instance x 2 = x 3 = 0, the motion of point Q will be determined by:

X = -h 1 Y = -h 1 (x 2 0 -x 2 1 )/(x 2 0 + x 2 1 ) Z = -h 1 (x 2 0 -x 2 1 )/(x 2 0 + x 2 1 ) (43)
This means that point Q moves along the edge C 0 C 1 , covering the closed interval between the two vertices. If only one parameter is zero, for instance if x 0 = 0, the point Q will occupy the closed triangle C 1 C 2 C 3 . Eventually, if none of the parameters is null, then point Q will move inside the tetrahedron C.

Let us consider an arbitrary point R in the moving platform such that:

(x + h 1 )(y + h 1 )(z + h 1 ) = 0 (44)
For example, take a point at the geometric center of the triangular-shaped platform, of coordinates

r 1 R = [1, -2 3 h 1 , -2 Figure 6: Pseudo-tetrahedron D. X = -2 3 h 1 Y = -2 3 h 1 (x 2 0 + x 0 x 1 -x 2 1 ) x 2 0 + x 2 1 Z = -2 3 h 1 (x 2 0 -x 0 x 1 -x 2 1 ) x 2 0 + x 2 1 ( 46
)
This represents an ellipse e 01 that passes through the vertices D 0 and D 1 and lies in the plane X = -2 3 h 1 . Accordingly, the four vertices of the pseudo-tetrahedron D are joined by six ellipses, as shown in Fig. 6.

When only one parameter is equal to zero, for instance x 0 = 0, the trajectory of point R will follow a particular surface, called the Steiner surface F 03 . It passes through the Then the expressions of the trajectory of point R are given by:

X = -2 3 h 1 (x 2 1 -x 1 x 2 -x 1 x 3 -x 2 2 -x 2 3 ) (x 2 1 + x 2 2 + x 2 3 ) Y = 2 3 h 1 (x 2 1 + x 1 x 2 -x 2 2 + x 2 x 3 + x 2 3 ) (x 2 1 + x 2 2 + x 2 3 ) Z = 2 3 h 1 (x 2 1 + x 1 x 3 + x 2 2 + x 2 x 3 -x 2 3 ) (x 2 1 + x 2 2 + x 2 3 ) (47)
Therefore, the trajectory of an arbitrary point of the moving platform forms the shape of pseudo-tetrahedron D and contains four vertices D i (i = 0, 1, 2, 3). These vertices are joined by six ellipses and any three of the vertices are linked by a Steiner surface F j (j = 0, 1, 2, 3). Any two Steiner surfaces (F i and F j ) share one ellipse e ij in common.

Let us analyse the motion of a special point S that does not fulfill Eq. (44). For instance, the point S is at one vertex of the triangular-shaped platform, B 3 (Fig. 1). If three parameters (among four parameters x i , i = 0, 1, 2, 3) are equal to zero, the positions of the point S are determined by:

x 0 = 1, x 1 = x 2 = x 3 = 0 : r 0 E 0 = [1, -h 1 , -h 1 , 0] T x 1 = 1, x 0 = x 2 = x 3 = 0 : r 0 E 1 = [1, -h 1 , h 1 , 0] T x 2 = 1, x 0 = x 1 = x 3 = 0 : r 0 E 2 = [1, h 1 , -h 1 , 0] T x 3 = 1, x 0 = x 1 = x 2 = 0 : r 0 E 3 = [1, h 1 , h 1 , 0] T (48)
Those points are coplanar and are the vertices of a rectangle as shown in Fig. 8. If two parameters are zero, for example x 2 = x 3 = 0, the path of point S is along the edge E 0 E 1 .

Accordingly, in a general configuration the point S always moves in the plane Z = 0.

Another special point which does not fulfill Eq. ( 44) is the origin of the moving frame P . According to Eq. ( 40), the positions of point P are given by:

δ = x 2 0 + x 2 1 + x 2 2 + x 2 3 X = 1 δ 2h 1 (x 0 x 2 -x 0 x 3 + x 1 x 2 + x 1 x 3 ) Y = 1 δ 2h 1 (-x 0 x 1 + x 0 x 3 + x 1 x 2 + x 2 x 3 ) Z = 1 δ 2h 1 (x 0 x 1 -x 0 x 2 + x 1 x 3 + x 2 x 3 ) (49)
If three parameters (among four parameters x i , i = 0, 1, 2, 3) are equal to zero, the positions of the point P will be always coincident with the origin of the fixed frame O. Paper JMR-14-1262, corresponding author's last name: CARO linked to the eighth equation in x 2 0 + 3x 2 1 -1 = 0, which is simply an ellipse equation in the space x 0 and x 1 . This ellipse equation can be parametrized by x 0 = cos(u) and

x 1 = 1 3 sin(u) √ 3.
As a result, the workspace of the manipulator performing the VDM is parametrized by the parameter u. Hence, the Study parameters are expressed as:

x 0 = c(u) x 1 = 1 3 s(u) √ 3 y 0 = 2 3 s(u) 3 √ 3 y 1 = - 2 3 c(u)s(u) 2 x 2 = 1 3 s(u) √ 3 x 3 = 1 3 s(u) √ 3 y 2 = - 2 3 c(u)s(u) 2 y 3 = - 2 3 c(u)s(u) 2 (52)
where s(u) = sin(u), c(u) = cos(u).

Therefore, the possible poses of the moving platform can be expressed by the following transformation matrix:

T =                 1 0 0 0 a 4 3 c(u) 2 -1 3 -2 3 s(u)(c(u) √ 3 -s(u)) -2 3 s(u)(c(u) √ 3 -s(u)) a -2 3 s(u)(c(u) √ 3 -s(u)) 4 3 c(u) 2 -1 3 -2 3 s(u)(c(u) √ 3 -s(u)) a -2 3 s(u)(c(u) √ 3 -s(u)) -2 3 s(u)(c(u) √ 3 -s(u)) 4 3 c(u) 2 -1 3                 (53)
where a = 4 3 sin(u) 2 .

Trajectory of the moving platform performing the Vertical Darboux Motion

Let us consider the point B 1 moving in the plane X = 0 and the geometric center R of the moving platform as shown in Fig. 1. The paths followed by those two points are obtained by setting u = -π 2 . . . π 2 by using the transformation matrix T defined in Eq. ( 53).

It appears that those two paths are different as shown in Fig. 10 4 . Point R moves 4 The animation of the trajectories is shown in: that is parallel to the plane X = 0.

Let us take all segments joining point B 1 to any point of segment B 2 B 3 and plot the paths of all points on those segments. All those paths are planar ellipses, except the path followed by point R. Accordingly, the set of all paths forms a ruled surface called Right-conoid surface, which is illustrated in yellow in Fig. 11

                (55)
The instantaneous screw axis of the moving platform is obtained from the components of matrix A as explained in [START_REF] Schadlbauer | Operation Modes in Lower Mobility Parallel Manipulators[END_REF], after normalization:

ISA = 1 √ 3 1 √ 3 1 √ 3
All twists of the manipulator are collinear. As a consequence, the fixed axode generated by the ISA is a straight line of unit vector [1/ √ 3, 1/ √ 3, 1/ √ 3] T . In the moving coordinate frame, the moving axode corresponding to this motion is congruent with the fixed axode as depicted in Fig. 12. However, the moving axode does not appear clearly as it is congruent with the fixed axode. Indeed, the moving axode internally slides and rolls onto the fixed axode.

Conclusions

In this paper, an algebraic geometry method was applied to analyse the kinematics and the operation mode of the 3-RPS Cube manipulator. Primary decomposition of an ideal of eight constraint equations revealed that the manipulator has only one general operation mode. In this operation mode, the direct kinematics was solved and the number of solutions was obtained for arbitrary values of design parameters and joint variables. The singularity conditions were computed and represented in the joint space. It turns out that the manipulator reaches the singularity when the moving frame coincides with the fixed frame and all joint variables are equal. The uncontrolled motion of the moving platform in this singularity configuration was investigated and geometrically interpreted.

Figure 1 :

 1 Figure 1: The 3-RPS Cube Parallel Manipulator.

r 2 =

 2 2 m, and r 3 = 1.5 m. By considering only the real solutions, the manipulator has two solutions for those design parameters and joint variables.

Fig. 2 (

 2 Fig.2(a) is along the axis A 1 . In Plücker coordinates, it is given by (p 0 : p 1 : p 2 : p 3 : p 4 : p 5 ) = (0.014 : 0.011 : -0.009 : 0.021 : -0.020 : 0.007). The rotational angle and translational distance along the screw axis A 1 are ϕ 1 = 5.725 rad and s 1 = -0.057 m, respectively.

Figure 2 (

 2 Figure 2(b) illustrates the second solution of the direct kinematic problem, with (x 0 : x 1 : x 2 : x 3 : y 0 : y 1 : y 2 : y 3 ) = (0.962 : 0.056 : -0.021 : -0.265 : 0.001 : -0.293 : 0.232 : -0.076). The moving platform is transformed from the identity into the final pose via the axis A 2 in Fig. 2(b), with rotational angle ϕ 2 = 0.552 rad and translational distance s 2 = 0.01 m. The Plücker coordinate vector of the discrete screw motion is defined by (p 0 : p 1 : p 2 : p 3 : p 4 : p 5 ) = (-0.004 : 0.001 : 0.019 : -0.021 : 0.017 : -0.006).

Figure 2 :

 2 Figure 2: Solutions of the Direct Kinematics.

Figure 3 :

 3 Figure 3: Singularity Surface of L 1 .

Figure 4 :

 4 Figure 4: Singularity Pose at the Identity Condition.

Figure 5 :

 5 Figure 5: Tetrahedron C.

Figure 7 :

 7 Figure 7: Steiner Surface F 0 .

Figure 8 :

 8 Figure 8: Rectangle E.

Figure 9 :

 9 Figure 9: Steiner Surface G 0 .

Figure 10 :

 10 Figure 10: Trajectories of points B 1 and R.

5 .

 5 This type of ruled surfaces is generated by moving a straight line such that it intersects perpendicularly a fixed straight line, called the axis of the Right-conoid surface. The fixed straight line followed by point R is axis of the Right-conoid surface.

Figure 11 :

 11 Figure 11: Right-conoid Surface of the VDM.

7. 2

 2 Axodes of the manipulator performing the Vertical Darboux MotionHaving the parametrization of the VDM performed by the 3-RPS Cube parallel manipulator in terms of Study parameters, it is relatively easy to compute the ISA. The possible poses of the moving platform as functions of time in this special motion only allow the orientations that are given by one parameter u. The ISA are obtained from the entries of the velocity operator:A = Ṫ T -1(54)By setting u = t, matrix A becomes:

Figure 12 :

 12 Figure 12: ISA Axodes of VDM.

  1 , y 2 , y 3 , which appear in matrix M, are called Study parameters. These parameters make it possible to parametrize SE(3) with dual quaternions. The Study kinematic mapping maps each spatial Euclidean displacement of SE(3) via transformation matrix M onto a projective point X [x 0 : x 1 : x 2 : x 3 : y 0 : y 1 : y 2 : y 3 ]

	in the 6-dimensional Study quadric S ∈ P 7 [14], such that:
	SE(3) → X ∈ P 7

Paper JMR-14-1262, corresponding author's last name: CARO

h 1 , -2 3 h 1 ] T . If any of the three parameters is zero, then the corresponding positions of point R will become:x 0 = 1, x 1 = x 2 = x 3 = 0 : r 0 D 0 = [1, -2 3 h 1 , -2 3 h 1 , -2 3 h 1 ] T x 1 = 1, x 0 = x 2 = x 3 = 0 : r 0 D 1 = [1, -2 3 h 1 , 2 3 h 1 , 2 3 h 1 ] T x 2 = 1, x 0 = x 1 = x 3 = 0 : r 0 D 2 = [1, 2 3 h 1 , -2 3 h 1 , 2 3 h 1 ] T x 3 = 1, x 0 = x 1 = x 2 = 0 : r 0 D 3 = [1, 2 3 h 1 , 2 3 h 1 , -2 3 h 1 ] T(45)D 0 , D 1 , D 2 and D 3 are the vertices of a pseudo-tetrahedron D as shown in Fig.6and it was verified that these vertices amount to the singularities of the 3-RPS Cube manipulator. If two parameters are equal to zero, for instance x 2 = x 3 = 0, the point Q will move along the edge C 0 C 1 , while the path of point R will be given by:Paper JMR-14-1262, corresponding author's last name: CARO

The motion animation of point R that is bounded by the Steiner surface, is shown in: http://www.irccyn.ec-nantes.fr/ ~caro/ASME_JMR/JMR_14_1262/animation_steiner.gif

http://www.irccyn.ec-nantes.fr/ ~caro/ASME_JMR/JMR_14_1262/animation_trajectories.gif[START_REF] Huang | Analysis of Instantaneous Motions of Deficient Rank 3-RPS Parallel Manipulators[END_REF] The animation of the right-conoid surface is shown in: http://www.irccyn.ec-nantes.fr/ ~caro/ASME_JMR/JMR_14_1262/animation_rightconoid.gif Paper JMR-14-1262, corresponding author's last name: CARO

Acknowledgments

The authors would like to acknowledge the support of the Österreichischer Austauschdienst/OeAD, the French Ministry for Foreign Affairs (MAEE) and the French Ministry for Higher Education and Research (MESR) (Project PHC AMADEUS). Moreover, Prof. Manfred Husty acknowledges the support of FWF grant P 23832-N13, Algebraic Methods in Collision Detection and Path Planning.

T , which is a special point in the cube of a moving frame Σ 1 as shown in Fig. 5. Then, its positions with respect to the fixed frame Σ 0 according to Eq. ( 40) are:

The coordinates of point Q depend on (x 0 , x 1 , x 2 , x 3 ). There are four possible positions corresponding to the three parameters (among four parameters x i , i = 0, 1, 2, 3) are equal to zero. These corresponding positions of Q are:

Paper JMR-14-1262, corresponding author's last name: CARO

Vertical Darboux Motion

The condition for the manipulator to generate the VDM is that all prismatic lengths are equal, i.e., r 1 = r 2 = r 3 . By solving the direct kinematics of the manipulator with the same prismatic lengths, the Study parameters obtained to perform the VDM yield

By substituting those values into the ideal I, the set of eight constraint equations becomes:

It follows from Eq. (50) that the first three constraint equations are the same. Likewise, the next three equations are identical. Mathematically, one has to find the case of 1-dof motion, as known as cylindrical motion, with one parameter that describes the VDM.

Equation (50) can be solved linearly for the variables R i , y 0 , y 1 in terms of x 0 , x 1 , as follows:

From Eq. (51), it is apparent that the manipulator can perform the VDM if and only if all prismatic lengths are the same. The remaining Study parameters x 0 and x 1 are still


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

Stéphane Caro Kinematic

Kinematic Modeling and Twist Feasibility of Mobile Cable-Driven Parallel Robots

Keywords: Cable-Driven Parallel Robots, Mobile Bases, Kinematic Modeling, Available Twist Set

come   L'archive ouverte pluridisciplinaire

Introduction

A Cable-Driven Parallel Robot (CDPR) is a type of parallel manipulator with limbs as cables, connecting the moving-platform with a fixed base frame. The platform is moved by appropriately controlling the cable lengths or tensions. CDPRs contains numerous advantages over conventional robots, e.g, high accelerations [START_REF] Kawamura | High-speed manipulation by using parallel wire-driven robots[END_REF], large payload capabilities [START_REF] Albus | The nist robocrane[END_REF], and large workspace [START_REF] Lambert | Implementation of an aerostat positioning system with cable control[END_REF].

However, a major drawback in classical CDPRs having fixed cable layout, i.e, fixed exit points and cable configuration, is the potential collisions between the cables and the surrounding environment which can significantly reduce the robot workspace. Better performances can be achieved with an appropriate CDPR architecture. Cable robots with a possibility of undergoing a change in their geometric structure are known as Reconfigurable Cable-Driven Parallel Robots (RCDPRs). Different strategies have been proposed for maximizing the robot workspace or increasing platform stiffness in the recent work on RCDPRs [START_REF] Gagliardini | Discrete reconfiguration planning for cable-driven parallel robots[END_REF]. However, reconfigurability is typically performed manually for most existing RCDPRs.

To achieve autonomous reconfigurability of RCDPRs, a novel concept of Mobile Cable-Driven Parallel Robots (MCDPRs) was introduced in [START_REF] Rasheed | Tension distribution algorithm for planar mobile cable-driven parallel robots[END_REF]. The first MCDPR prototype has been designed and built in the context of Echord++ FASTKIT project 1 . The targeted application for such MCDPR prototype is logistics. Some papers deals with the velocity analysis of parallel manipulators [START_REF] Merlet | Efficient computation of the extremum of the articular velocities of a parallel manipulator in a translation workspace[END_REF]. However, few focus on the twist analysis of CDPRs [START_REF] Lessanibahri | Twist feasibility analysis of cabledriven parallel robots[END_REF]. This paper deals with the kinematic modeling of MCDPRs that is required to analyze the kinematic performance of the robot. The paper is organized as follows. Section 2 presents the kinematic model of MCDPRs. Section 3 deals with the determination of the Available Twist Set (ATS) for MCDPRs using the kinematic modeling of the latter. The ATS can be used to obtain the twist capacities of the moving-platform. Section 4 presents the twist capacities of the moving-platform for the MCDPRs under study. Finally, conclusions are drawn and future work is presented in Section 5.

Kinematic modeling

A MCDPR is composed of a classical CDPR with m cables and a n degree-offreedom (DoF) moving-platform mounted on p Mobile Bases (MBs). The jth mobile base is denoted as M j , j = 1, . . . , p. The ith cable mounted onto M j is named as C i j , i = 1, . . . , m j , where m j denotes the number of cables carried by M j . u i j denotes the unit vector along the cable C i j . Each jth mobile base along with its m j number of cables is denoted as jth PD (pd j ) module. Each pd j consists of a proximal (prox j ) and a distal (dist j ) module. dist j consists of m j cables between M j and the moving-platform. In this paper, cables are assumed to be straight and massless, thus can be modeled as a Universal-Prismatic-Spherical (UPS) kinematic chain. Generally MBs are four-wheeled planar robots with two-DoF translational motions and one-DoF rotational motion, thus, prox j can be modeled as a virtual RPP kinematic chain between the base frame F 0 and the frame F b j attached to M j . An illustrative example with p = 4 MBs and m = 8 cables is shown in Fig. 1a. A general kinematic architecture of a MCDPR is shown in Fig. 1b.

Kinematics of the Distal Module

A classical CDPR is referred as distal modules in MCDPR. The twist 0 t dist P of the moving-platform due to the latter is expressed as [START_REF] Gouttefarde | Interval-analysis-based determination of the wrench-feasible workspace of parallel cable-driven robots[END_REF][START_REF] Roberts | On the inverse kinematics, statics, and fault tolerance of cable-suspended robots[END_REF]:

A 0 t dist P = l, (1)
where A is the (m × n) parallel Jacobian matrix, containing the actuation wrenches due to the cables on the mobile platform. The twist 0 t P = [ω ω ω, ṗ] T is composed of the platform angular velocity vector ω ω ω = [ω x , ω y , ω z ] T and linear velocity vector ṗ = [ ṗx , ṗy , ṗz ] T , expressed in F 0 . 0 t dist P denotes the platform twist due to the distal module motion. l is a m-dimensional cable velocity vector. Here, Eq. ( 1) can be expressed as:

          A 1 A 2 . . . A j . . . A p           0 t dist P =           l1 l2 . . . l j . . . lp           , (2)
where l j = [ l1 j , l2 j , . . . , lm j j ] T . A j is expressed as:

A j =      [( 0 b 1 j -0 p) × u i j ] T u T 1 j [( 0 b 2 j -0 p) × u 2 j ] T u T 2 j . . . [( 0 b m j j -0 p) × u m j j ] T u T m j j      , (3)
where ith row of A j is associated with the actuation wrench of the ith cable mounted onto M j . 0 b i j denotes the Cartesian coordinate vector of the anchor points B i j in F 0 . 0 p denotes the platform position in F 0 .

Kinematic modeling of a MCDPR

The twist 0 t j P of the moving-platform due to pd j can be expressed in F 0 as:

0 t j P = 0 t prox j P + 0 t dist j P (4)
where 0 t prox j P ( 0 t dist j P , resp.) is the twist of the moving-platform due to the motion of the proximal (distal, resp.) module of pd j expressed in F 0 . Equation ( 4) take the form:

0 t j P = b j Ad P 0 t prox j b j + 0 R b j b j t dist j P (5)
where b j Ad P is called the adjoint matrix, which represents the transformation matrix between twists expressed in F b j and twist expressed in F P , b j Ad P = I 3 0 3 -b j rP I 3 [START_REF] Kawamura | High-speed manipulation by using parallel wire-driven robots[END_REF] where b j rP is the cross-product matrix of vector --→ 0 b j P expressed in F 0 . b j t dist j P is the moving-platform twist due to dist j expressed in F b j . The augmented rotation matrix ( 0 R b j ) is used to express b j t

dist j P in F 0 : 0 R b j = 0 R b j 0 3 0 3 0 R b j (7)
where 0 R b j is the rotation matrix between frames F b j and F 0 . As the Proximal module is being modeled as a virtual RPP limb, 0 t prox j b j from Eq. ( 4) can be expressed as:

0 t prox j b j = J b j qb j (8)
where J b j is a (6 × 3) serial Jacobian matrix of prox j and qb j is the virtual joint velocities of the latter, namely,

0 t prox j b j = k 0 0 3 0 3 k 0 × 0 p 0 R b j i 0 0 R b j j 0   θ j ρ1 j ρ2 j   (9)
where i 0 , j 0 and k 0 denotes the unit vector along x 0 , y 0 and z 0 respectively. Upon multiplication of Eq. ( 5) with A j :

A j 0 t j P = A j b j Ad P J b j qb j + A j 0 R b j b j t dist j P . (10)
As

A j 0 R b j b j t dist j P
represents the cable velocities of the dist j (see Eq. ( 2)), Eq. ( 10) can also be expressed as:

A j 0 t j P = A j b j Ad P J b j qb j + l j . ( 11
)
The twist of the moving-platform t P and the twists generated by the limbs are the same, namely, 0 t 1 P = 0 t 2 P = 0 t 3 P . . . = 0 t j P . . . = 0 t p P = t P (12)

Thus, the twist of the moving-platform in terms of all the p number of limbs can be expressed as:

     A 1 A 2 . . . A p      t P =      A 1 b1 Ad P J b1 0 0 • • • 0 0 A 2 b2 Ad P J b2 0 • • • 0 . . . . . . . . . . . . 0 0 0 • • • A p bp Ad P J bp      qb + l (13)
where qb = [ qb1 , qb2 , . . . , qbp ] T and l = [ l1 , l2 , . . . , lp ] T . Equation ( 13) can be expressed in the matrix form as:

At P = B b qb + l ( 14
)
At P = B q (15)
where B = [B b I m ] is a (m × (3p + m))-matrix while q = [ qb l] T is a (3p + m)dimensional vector containing all joint velocities. Equation (15) represents the first order kinematic model of MCDPRs.

Available Twist Set of MCDPRs

This section aims at determining the set of available twists for MCDPRs that can be generated by the platform. For a classical CDPR, the set of twist feasible poses of its moving platform are known as Available Twist Set (ATS). A CDPR posture is called twist-feasible if all the twists within a given set, can be produced at the platform, within given joint velocity limits [START_REF] Lessanibahri | Twist feasibility analysis of cabledriven parallel robots[END_REF].

According to [START_REF] Lessanibahri | Twist feasibility analysis of cabledriven parallel robots[END_REF], ATS of a CDPR corresponds to a convex polytope that can be represented as the intersection of the half-spaces bounded by its hyperplanes known as Hyperplane-Shifting Method (HSM) [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]. Although HSM can be utilized to determine the ATS of MCDPRs, the approach in [START_REF] Lessanibahri | Twist feasibility analysis of cabledriven parallel robots[END_REF] is not directly applicable due to the difference in the kinematic models of CDPR (Eq. 1) and MCDPR (Eq. 15) as matrix B = I.

The kinematic model of the MCDPRs is used to determine the ATS of the moving-platform. In case m = n, A is square, Eq. ( 15) can be expressed as:

t P = A -1 B q =⇒ t P = J q (16)
where J is a Jacobian matrix mapping the joint velocities onto the platform twist. The ATS will correspond to a single convex polytope, constructed under the mapping of Jacobian J.

In case m = n, matrix A is not square, however there exist in total C n m (n × n) square sub-matrices of matrix A, denoted by A k , k = 1, . . . ,C n m , obtained by removing mn rows from A. For each sub-matrix we can write:

tk p = A k -1 B k =⇒ tk p = J k q, k = {1, . . . ,C n m } (17)
where tk p is the twist generated by the kth sub-matrix A k out of C n m (n × n) square sub-matrices of matrix A. B k is a sub matrix of B using corresponding rows that are chosen in A k from A. HSM 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] is directly applicable to compute all the hyperplanes for C n m convex polytopes knowing the minimum and maximum joint velocity limits. Thus, the ATS of MCDPRs is the region bounded by all of the foregoing hyperplanes.

Results

This section deals with the twist feasibility analysis of two different case studies. From the ATS acquired using the kinematic model of a given MCDPR configuration, we aim to study the difference in the moving platform twist considering fixed and moving MBs. The first case study is a planar MCDPR with a point mass end-effector shown in Fig. 2a. The MBs have only one degree of freedom along i 0 . The joint velocity limits are defined as:

-0.8 m.s -1 ≤ ρ1 j ≤ 0.8m.s -1 , -2m.s -1 ≤ li j ≤ 2m.s -1 , i = {1, 2}, j = {1, 2}, (18)
x Matrix A has six 2 × 2 sub-matrices Thus, ATS is the region bounded by the hyperplanes formed by these six convex polytopes. The difference of ATS between fixed (correspond to a classical CDPR) and moving MBs can be observed in Figs. 2b and2c. To illustrate the difference, a Required Twist Set (RTS) equal to [1.15m.s -1 , 1.675m.s -1 ] T is considered, depicted by a red point in Figs. 2b and2c. For fixed MBs, it should be noted that RTS is outside the ATS. By considering moving MBs, RTS is within the ATS. The same approach is adopted to determine the ATS for a given FASTKIT configuration in Fig. 3a. The joint velocity limits are defined as:

-0.2 m.s -1 ≤ θ j , ρ1 j , ρ2 j ≤ 0.2 m.s -1 , j = {1, 2}, (19)
-2 m.s -1 ≤ li j ≤ 2 m.s -1 , j = {1, 2}, i = {1, . . . , 4}, (20)
The maximum absolute twist that the platform can achieve in each Cartesian direction by considering fixed and moving MBs are illustrated in Figs. 3b and3c.

The maximum absolute wrench of the moving platform is illustrated in red where f x , f y , f z and m x , m y , m z represent the forces and the moments that can be generated by the cables onto the moving platform. For the analysis, the cable tensions are bounded between 0 and 20 N respectively. It can be observed the twist capacity of the moving-platform is increased when MBs are moving. On the contrary, high velocity capability of the moving-platform in certain directions also results in very less wrench capability in the respective directions. Thus, this velocity is unattainable outside certain dynamic conditions.

Conclusion

This paper dealt with the kinematic modeling of Mobile Cable-Driven Parallel Robots (MCDPRs) that can be used to analyze its kinematic performance. The developed kinematic model was utilized to determine the Available Twist Set (ATS) of MCDPRs. It considers the joint velocity limits for cables and the Mobile Bases (MBs). Using ATS, the twist capacities of the moving-platform was determined. Two case studies have been used in order to illustrate the effect of the moving MBs onto the platform twist. Future work will focus the trajectory planning of of MCD-PRs and experimental validations with FASTKIT prototype.

Fig. 1 :

 1 Fig. 1: (a) MCDPR Parameterization (b) Kinematic Architecture of MCDPRs, active joints are highlighted in gray, passive joints are highlighted in white

4. 1

 1 Case study: p = 2, m = 4 and n = 2 DoF MCDPR

Fig. 2 :

 2 Fig. 2: (a) Configuration under study of p = 2, m = 4 and n = 2 MCDPR (b) ATS in green for fixed MBs (c) ATS in green for moving MBs

4. 2

 2 Case study: p = 2, m = 8 and n = 6 DoF MCDPR

Fig. 3 :

 3 Fig. 3: (a) A FASTKIT configuration (b, c) Maximum absolute twist and wrenches that FASTKIT platform can generate about each Cartesian direction

https://www.fastkit-project.eu/

Acknowledgements This research work is part of the European Project ECHORD++ "FASTKIT" dealing with the development of collaborative and mobile cable-driven parallel robots for logistics.


Thomas Stigger
email: thomas.stigger@uibk.ac.at

Abhilash Nayak
email: abhilash.nayak@ls2n.fr

Stéphane Caro
email: stephane.caro@ls2n.fr

Philippe Wenger
email: philippe.wenger@ls2n.fr

Martin Pfurner
email: martin.pfurner@uibk.ac.at

Manfred Husty
email: manfred.husty@uibk.ac.at

Algebraic Analysis of a 3-RUU Parallel Manipulator

Keywords: 3-RUU, kinematic analysis, direct kinematics, algebraic geometry

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

For theoretical and practical purposes, the kinematic analysis of a parallel manipulator (PM) is essential to understand its motion behavior. Kinematic constraints can be transformed via Study's kinematic mapping into algebraic constraint equations. Every configuration of the PM is thereby mapped to a point in a projective space, P 7 [START_REF] Husty | 21st Century Kinematics, chap. Kinematics and Algebraic geometry[END_REF][START_REF] Husty | Algebraic methods in mechanism analysis and synthesis[END_REF]. Consequently, well developed concepts of algebraic geometry [START_REF] Cox | Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra[END_REF] can be used to interpret the algebraic constraint equations to obtain necessary information about the PM.

In that vein, many PMs were investigated using algebraic geometry concepts. Resultant methods were adopted to solve the direct kinematics of Stewart-Gough platforms [START_REF] Husty | An algorithm for solving the direct kinematics of general Stewart-Gough platforms[END_REF]. A complete kinematic analysis including the characterization of operation modes, solutions to direct kinematics and determination of singular poses was performed for the 3-RPS PM [START_REF] Schadlbauer | Operation Modes in Lower-Mobility Parallel Manipulators[END_REF][START_REF] Schadlbauer | The 3-RPS parallel manipulator from an algebraic viewpoint[END_REF], the 3-RPS cube PM [START_REF] Nurahmi | Kinematic analysis of the 3-RPS Cube Parallel Manipulator[END_REF] and 3-PRS PMs with different arrangements of prismatic joints [START_REF] Nurahmi | Operation modes and singularities of 3-PRS parallel manipulators with different arrangements of P-joints[END_REF]. In the foregoing papers, the prismatic joints were considered to be actuated, which makes the analysis inherently algebraic. A more challenging kinematic analysis of an over-constrained 4-RUU PM with square base and moving platform was accomplished by decomposing it into two 2-RUU PMs [START_REF] Nurahmi | Reconfiguration analysis of a 4-RUU parallel manipulator[END_REF]. The constraint equations of a 3-RUU PM are derived in this paper and its direct kinematics problem is solved. Nevertheless, a complete characterization of the manipulator operation modes has not been obtained yet.

The paper is organized as follows: Section 2 describes the manipulator architecture. Section 3 deals with the derivation of algebraic constraint equations with two approaches and their comparison. Section 4 presents the solutions to direct kinematics for arbitrary design parameters and hints the recognition of a translational operation mode.

Manipulator Architecture

The 3-RUU PM is shown in Figure 1a. Each limb consists of a revolute joint and two universal joints mounted in series with the first revolute joint as the active joint. The moving platform and the fixed base form equilateral triangles with vertices C i and A i , respectively, i = 1, 2, 3. The unit vectors of the revolute joint axes within the i-th limb are denoted as s i j , i = 1, 2, 3; j = 1, ..., 5. s i5 and s i1 are tangent to the circumcircles (with centers P and O) of the moving platform and the base triangles, respectively. Vectors s i1 and s i2 are always parallel, so are vectors s i3 and s i4 . The origin of the fixed coordinate frame, F O is at O and the z O -axis lies along the normal to the base plane whereas the origin of the moving coordinate frame F P is at P and the z P -axis lies along the normal to the moving platform plane. x O and x P axes are directed along OA 1 and PC 1 , respectively. r 0 and r 1 are the circumradii of base and the moving platform, respectively. a 1 and a 3 are the link lengths. θ i1 is the angle of rotation of the first revolute joint about the axis represented by vector s i1 measured from the base plane whereas θ i2 is the angle of rotation of the second revolute joint about the axis represented by vector s i2 measured from the first link.

s 35 A 3 B 3 C 3 s x O z O y O O x P z P y P O P F P A 1 B 1 C 1 A 2 B 2 C 2 θ 11 θ 12 h 1 h 2 a 1 F O a 3 (a) The 3-RUU PM in a general configuration s 3 s 2 s 1 s 4 s 5 x 0 y 0 z 0 x 1 y 1 z 1 θ 1 θ 2 A B C F 1 F 0 a 1 a 3 (b) A RUU limb

Constraint Equations

The constraint equations of the 3-RUU PM are derived using a geometrical approach and the Linear Implicitization Algorithm (LIA) [START_REF] Walter | On implicitization of kinematic constraint equations[END_REF]. First, canonical constraint equations for a limb of the PM are derived by attaching fixed and moving coordinate frames to the two extreme joints of a RUU limb as shown in Fig. 1b. Each U-joint is characterized by two revolute joints with orthogonal and intersecting axes and Denavit-Hartenberg (DH) convention is used to parameterize each limb. F 0 and F 1 are the fixed and the moving coordinate frames with their corresponding z-axes along the first and the last revolute joint axes, respectively. Later on, general constraint equations are derived for the whole manipulator.

Derivation Using a Geometrical Approach

Canonical Constraints In order to derive the geometric constraints for a RUU limb, the homogeneous coordinates4 of points A, B,C (a, b, c, respectively) and vectors s j , j = 1, ..., 5, shown in Fig. 1b are expressed as follows:

0 a = [1, 0, 0, 0] T 0 b = [1, a 1 cos(θ 1 ), a 1 sin(θ 1 ), 0] T 1 c = [1, 0, 0, 0] T 0 s 1 = [0, 0, 0, 1] T 0 s 2 = [0, 0, 0, 1] T 0 s 3 = [0, cos(θ 1 + θ 2 ), sin(θ 1 + θ 2 ), 0] T 0 s 4 = [0, cos(θ 1 + θ 2 ), sin(θ 1 + θ 2 ), 0] T 1 s 5 = [0, 0, 0, 1] T (1)
where θ 1 and θ 2 are the angles of rotation of the first and the second revolute joints. Study's kinematic mapping is used to express the vectors c and s 5 in the fixed coordinate frame F 0 , using the transformation matrix 0 T 1 consisting of Study parameters x i and y i , i = 0, 1, 2, 3:

0 c = 0 T 1 1 c and 0 s 5 = 0 T 1 1 s 5 ,
where 0 T 1 = 1 ∆        ∆ 0 0 0 d 1 x 0 2 + x 1 2 -x 2 2 -x 3 2 -2 x 0 x 3 + 2 x 1 x 2 2 x 0 x 2 + 2 x 1 x 3 d 2 2 x 0 x 3 + 2 x 1 x 2 x 0 2 -x 1 2 + x 2 2 -x 3 2 -2 x 0 x 1 + 2 x 2 x 3 d 3 -2 x 0 x 2 + 2 x 1 x 3 2 x 0 x 1 + 2 x 2 x 3 x 0 2 -x 1 2 -x 2 2 + x 3 2        (2)
with

∆ = x 0 2 + x 1 2 + x 2 2 + x 3 2 = 0 and d 1 = -2 x 0 y 1 + 2 x 1 y 0 -2 x 2 y 3 + 2 x 3 y 2 , d 2 = -2 x 0 y 2 + 2 x 1 y 3 + 2 x 2 y 0 -2 x 3 y 1 , d 3 = -2 x 0 y 3 -2 x 1 y 2 + 2 x 2 y 1 + 2 x 3 y 0 .
All vectors are now expressed in the base coordinate frame F 0 and hence the geometric constraints can be derived. The following constraints are already satisfied:

1. The first and the second revolute joint axes are parallel: s 1 = s 2 2. Third and fourth revolute joint axes are parallel:

s 3 = s 4 3.
-→ AB is perpendicular to the first and the second revolute joint axes: (b -a) T s 1 = 0 4. The second revolute joint axis is perpendicular to the third revolute joint axis:

s T 2 s 3 = 0 5. Length of the link AB is a 1 : ||b -a|| 2 = a 1
The remaining geometric constraints are derived as algebraic equations 5 : The second revolute joint axis, the fifth revolute joint axis and link BC lie in the same plane. In other words, the scalar triple product of the corresponding vectors is null:

g 1 : (b -c) T (s 2 × s 5 ) = 0 (3)
Vector -→ BC is perpendicular to the third and the fourth revolute joint axes:

g 2 : (b -c) T s 4 = 0 (4)
The fourth and the fifth revolute joint axes are perpendicular:

g 3 : s T 4 s 5 = 0 (5)
Length of the link BC is a 3 :

g 4 : ||b -c|| -a 3 = 0 (6)
Furthermore, Study's quadric equation S : x 0 y 0 + x 1 y 1 + x 2 y 2 + x 3 y 3 = 0 must be taken into account. The five geometric relations g 1 , g 2 , g 3 , g 4 , S describe the RUU limbs of the PM under study. As a matter of fact, when the first revolute joint is actuated, each limb has four DoF and it should be possible to describe it by only two constraint equations. Eqs. ( 4) and ( 5) contain the passive joint variable v 2 along with the active joint variable v 1 . Eliminating v 2 from g 2 and g 3 results in an equation that amounts to g 1 . Therefore, the two constraint equations in addition to the Study quadric describing a RUU limb are g 1 and g 4 , namely Eqs. ( 3) and [START_REF] Nurahmi | Operation modes and singularities of 3-PRS parallel manipulators with different arrangements of P-joints[END_REF]. The polynomials g 1 , g 4 and S define an ideal, which is a subset of all polynomials in the Study parameters:

I 1 = g 1 , g 4 , S ⊆ k[x 0 , x 1 , x 2 , x 3 , y 0 , y 1 , y 2 , y 3 ]. (7)
Explicitly these polynomials take the form:

g 1 := (x 0 x 1 -x 2 x 3 ) (v 1 2 -1) + (-2x 0 x 2 -2x 1 x 3 ) v 1 (x 2 0 + x 2 1 + x 2 2 + x 2 3 )a 1 -2((x 2 0 + x 2 3 )(x 1 y 1 + x 2 y 2 ) + 2(x 2 1 + x 2 2 )(x 0 y 0 + x 3 y 3 ))(v 2 1 -1) = 0, (8)
g 4 := -x 0 2 + x 1 2 + x 2 2 + x 3 2 v 1 2 + 1 a 1 2 + 4 (y 1 x 0 -y 0 x 1 + y 3 x 2 -y 2 x 3 ) v 1 2 + 8 (-x 0 y 2 + x 1 y 3 + x 2 y 0 -x 3 y 1 ) v 1 + 4 (y 2 x 3 -y 3 x 2 -y 1 x 0 + y 0 x 1 )) a 1 + x 0 2 + x 1 2 + x 2 2 + x 3 2 a 3 2 -4 y 2 2 + y 3 2 + y 0 2 + y 1 2 v 1 2 + 1 = 0. ( 9
)
General Constraints g 1 and g 4 are the constraint equations of an RUU limb with specially adapted coordinate systems. To assemble the PM one has to transform these equations so that the limbs get into the positions of Fig. 1a. It is well known [9] that 5 cosine and sine of angles are substituted by tangent half-angles to render the equations algebraic; cos

(θ i ) = 1-v 2 i 1+v 2 i sin(θ i ) = 2v i 1+v 2 i where v i = tan(θ i /2), i = 1, 2
the necessary transformations are linear in the image space coordinates. Due to lack of space these transformations are only shown for the derivation of the constraint equations using the LIA in Sec.3.2 (Eq.14). One ends with six constraint equations g i1 , g i4 , i = 1, 2, 3 which form together with S = 0 and the normalization condition N :

x 2 0 + x 2 1 + x 2 2 + x 2 3 -1 = 0 an ideal I = g 11 , g 14 , g 21 , g 24 , g 31 , g 34 , S , N ⊆ k[x 0 , x 1 , x 2 , x 3 , y 0 , y 1 , y 2 , y 3 ] (10)

Derivation Using a Linear Implicitization Algorithm

Canonical Constraints The canonical pose of a RUU limb is chosen such that the rotation axes coincide with the z-axes and the common normals of these axes are in the directions of the x-axes of the coordinate systems in order to derive the canonical constraint equations using LIA. It computes implicit equations of lowest possible degree out of parametric equations by comparing coefficients with an arbitrary system of implicit equations with the same degree. An extended explanation is given in [START_REF] Walter | On implicitization of kinematic constraint equations[END_REF]. To describe the RUU kinematic chain using the usual Denavit-Hartenberg (DH) parameters, the following 4 × 4 matrices are defined: T = M i .G i , i = 1, . . . , 5, where the M i -matrices describe a rotation about the z-axis with u i as the rotation angle. The G i -matrices describe the transformation of one joint coordinate system to the next.

M i =     1 0 0 0 0 cos (u i ) -sin (u i ) 0 0 sin (u i ) cos (u i ) 0 0 0 0 1     , G i =     1 0 0 0 a i 1 0 0 0 0 cos (α i ) -sin (α i ) d i 0 sin (α i ) cos (α i )     . (11)
The parameters in G i are DH parameters encoding the distance along x-axis a i , the offset along z-axis d i and the twist angle between the axes α i . The DH parameters for the RUU limb are

α 2 = π 2 , α 4 = -π 2 , d 1 = a 2 = d 2 = d 3 = a 4 = d 4 = α 1 = α 3 = 0.
Computing the Study-Parameters based on the transformation matrix T yields the parametric representation of the limb [START_REF] Husty | Algebraic methods in mechanism analysis and synthesis[END_REF]. Applying LIA yields the following quadratic canonical constraint equations S , f 1 and f 2 :

J = f 1 , f 2 , S ⊆ k[x 0 , x 1 , x 2 , x 3 , y 0 , y 1 , y 2 , y 3 ], (12)
where

f 1 := (x 0 x 1 -x 2 x 3 ) (v 1 2 -1) -(2x 0 x 2 + 2x 1 x 3 ) v 1 a 1 + 2 v 1 2 + 1 (x 0 y 0 + x 3 y 3 ) = 0 f 2 := -x 0 2 + x 1 2 + x 2 2 + x 3 2 v 1 2 + 1 a 1 2 + 4 (y 1 x 0 -y 0 x 1 + y 3 x 2 -y 2 x 3 ) v 1 2 + 8 (-x 0 y 2 + x 1 y 3 + x 2 y 0 -x 3 y 1 ) v 1 + 4 (y 2 x 3 -y 3 x 2 -y 1 x 0 + y 0 x 1 )) a 1 + x 0 2 + x 1 2 + x 2 2 + x 3 2 a 3 2 -4 y 2 2 + y 3 2 + y 0 2 + y 1 2 v 1 2 + 1 = 0 (13)
General Constraints To obtain the constraint equations of the whole mechanism from the canonical constraint equations, coordinate transformations are applied in the base and moving platform. To facilitate the comparison of the constraint equations derived by two different approaches, the coordinate transformations should be consistent with the global frames F O and F P as shown in Fig. 1a. The necessary transformations can be done directly in the image space P 7 [START_REF] Pfurner | Analysis of spatial serial manipulators using kinematic mapping[END_REF] by the mapping

            x 0 x 1 x 2 x 3 y 0 y 1 y 2 y 3             →             2 v 0 2 + 1 x 0 -2 v 0 2 x 1 + 4 v 0 x 2 + 2 x 1 2 v 0 2 + 1 x 3 2 v 0 2 x 2 + 4 v 0 x 1 -2 x 2 ((r 0 -r 1 ) x 1 + 2 y 0 ) v 0 2 -2 x 2 (r 0 -r 1 ) v 0 + (-r 0 + r 1 ) x 1 + 2 y 0 ((r 0 -r 1 ) x 0 -2 y 1 ) v 0 2 + 4 v 0 y 2 + (r 0 -r 1 ) x 0 + 2 y 1 ((-r 0 -r 1 ) x 2 + 2 y 3 ) v 0 2 -2 (r 0 + r 1 ) x 1 v 0 + (r 0 + r 1 ) x 2 + 2 y 3 ((r 0 + r 1 ) x 3 + 2 y 2 ) v 0 2 + 4 v 0 y 1 + (r 0 + r 1 ) x 3 -2 y 2             , (14)
where

v 0 = tan(γ i ), i = 1, 2, 3, γ 1 = 0, γ 2 = 2π 3 and γ 3 = 4π 3
. The general constraint equations are obtained by transforming the f i of Eq.12 with Eq.14. The transformed equations are denoted f i1 = f i2 = 0, i = 1, 2, 3, and determine together with S = 0 and N = 0, the ideal J :

J = f 11 , f 12 , f 21 , f 22 , f 31 , f 32 , S , N ⊆ k[x 0 , x 1 , x 2 , x 3 , y 0 , y 1 , y 2 , y 3 ]
(15)

Ideal Comparison

A careful observation of the ideals spanned by the canonical constraint polynomials of both approaches reveals that g 4 = f 2 and

g 1 = f 1 (x 2 0 + x 2 1 + x 2 2 + x 2 3 ) -2(x 2 0 + x 2 2 )(v 2 1 + 1)S . Since x 2 0 + x 2 1 + x 2 2 + x 2
3 cannot be null, these ideals are the same. Thus, it follows that the ideals I and J spanned by the constraint equations of the whole manipulator are also contained in each other: I ⊆ J ⊆ I . Since I and J determine the same ideal, the variety of the constraint polynomials must be the same [START_REF] Cox | Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra[END_REF]. Therefore, the set of constraint equations derived in Section 3.2 is used for further computations as it contains only quadratic equations.

Direct Kinematics: Numerical Examples

Because of the complexity of the manipulator, it is not possible to compute the direct kinematics without using some numerical values. In the following subsections, the following arbitrary values are assigned to the design parameters of the manipulator, a 1 = 3, a 3 = 5, r 0 = 11, r 1 = 7.

Identical Actuated Joints Assuming the actuated joint angles are equal, θ i1 = π 2 , i = 1, 2, 3 for simplicity, the system of constraint equations in Eq. (15) yields the following real solutions and the corresponding manipulator poses are shown in Fig. 2.

(a) x 0 = √ 23023 154 , y 3 = - 3 2 x 0 , x 3 = - 3 √ 77 154 , y 0 = 3 2 x 3 , x 1 = x 2 = y 1 = y 2 = 0 , (b) x 0 = √ 23023 154 , y 3 = - 3 2 x 0 , x 3 = 3 √ 77 154 , y 0 = 3 2 x 3 , x 1 = x 2 = y 1 = y 2 = 0 , x O z O y O x P z P y P (a) x O z O y O x P z P y P (b) x O, x P z O, z P y O, y P (c) x O z O y O x P z P y P (d)
Fig. 2: A numerical example: solutions to direct kinematics corresponding to (16)

(c) {x 0 = 1, x 1 = x 2 = x 3 = y 0 = y 1 = y 2 = y 3 = 0} , (d) {x 0 = 1, x 1 = x 2 = x 3 = y 0 = y 1 = y 2 = 0, y 3 = -3} . (16)
Different Actuated Joints Substituting distinct arbitrary inputs, setting x 0 = 1 and computing a Groebner basis of the resulting polynomials with pure lexicographic ordering yields a univariate polynomial

x 3 • P(x 3 ) = 0, where degree(P(x 3 )) = 80.

Translational Operation Mode The univariate polynomial of the previous section shows that this manipulator exhibits two operation modes. The one corresponding to x 3 = 0 yields pure translational motions of the moving platform with the identity as the orientation, similar to the motion of the famous delta robot [START_REF] Clavel | Delta, a fast robot with parallel geometry[END_REF]. From S follows also y 0 = 0. The set of original constraint equations reduces to

[ 3y 3 -y 1 2 -y 2 2 -y 3 2 -4y 1 t 1 2 -6 (y 1 + 2)t 1 -y 1 2 -y 2 2 -y 3 2 -4y 1 -3y 3 , -2t 2 2 + 3t 2 + 2 y 2 √ 3 + -y 1 2 -y 2 2 -y 3 2 + 2y 1 + 3y 3 t 2 2 + (3y 1 -12)t 2 -y 1 2 -y 2 2 -y 3 2 + 2y 1 -3y 3 , 2t 3 2 + 3t 3 + 2 y 2 √ 3 + -y 1 2 -
This system of equations yields a quadratic univariate in one of the y i variables, which gives a parametrization of the motion as a function of the input variables v i1 = tan(θ i1 /2), i = 1, 2, 3.

Conclusion

In this paper, the constraint equations of a 3-RUU PM were derived by two different approaches: geometrical approach, where all possible constraints were listed based on the geometry of the manipulator and through LIA, which yields the constraints by specifying the parametric equations and the desired degree. Both approaches have benefits and disadvantages such that it is possible to miss a constraint by merely observing the manipulator geometry while it is hard to interpret the physical meaning of the equations derived through LIA. However, it turns out that the ideals spanned by the constraint polynomials with both approaches are the same. As a result, the simplest set of equations was chosen for further analysis. Due to the complexity of the mechanism, a primary decomposition of these ideals is not possible and therefore a final answer to possible operation modes can not be given. However, the factorization of the final univariate polynomial of the direct kinematics algorithm gives strong evidence that this manipulator has a translational and a general three DoF operation mode.

left superscript k denotes the vector expressed in coordinate frame F k , k ∈ {0, 1}

Acknowledgements

This work was supported by the Austrian Science Fund (FWF I 1750-N26) and the French National Research Agency (ANR Kapamat #ANR-14-CE34-0008-01).


Sana Baklouti
email: sana.baklouti@insa-rennes.fr

Stéphane Caro
email: stephane.caro@ls2n.fr

Eric Courteille
email: eric.courteille@insa-rennes.fr

Elasto-Dynamic Model-Based Control of Non-Redundant Cable-Driven Parallel Robots

This paper deals with a model-based feed-forward torque control strategy of non-redundant cable-driven parallel robots (CDPRs). The proposed feed-forward controller is derived from an inverse elastodynamic model of the CDPR to compensate for the dynamic and oscillatory effects due to cable elasticity. A PID feedback controller ensures stability and disturbance rejection. Simulations confirm that tracking errors can be reduced by the proposed strategy compared to conventional rigid body model-based control.

Introduction

Cable-driven parallel robots (CDPRs) contain a set of flexible cables that connect a fixed frame to an end-effector (EE) with a coiling mechanism for each cable. They have been used in many applications like pick-and-place [START_REF] Dallej | Towards vision-based control of cable-driven parallel robots[END_REF], rehabilitation [START_REF] Hernandez | Design Optimization of a Cable-Driven Parallel Robot in Upper Arm Training-Rehabilitation Processes[END_REF], painting and sandblasting of large structures [START_REF] Gagliardini | Discrete reconfiguration planning for cable-driven parallel robots[END_REF]. Thanks to their low inertia, CDPRs can reach high velocities and accelerations in large workspaces [START_REF] Lamaury | Dualspace adaptive control of redundantly actuated cable-driven parallel robots[END_REF]. Several controllers have been proposed in the literature to improve CDPR accuracy locally or on trajectory tracking [START_REF] Jamshidifar | Adaptive Vibration Control of a Flexible Cable Driven Parallel Robot[END_REF], [START_REF] Zi | Dynamic modeling and active control of a cable-suspended parallel robot[END_REF], [START_REF] Pott | IPAnema: a family of cable-driven parallel robots for industrial applications[END_REF]. In [START_REF] Cuevas | Assumed-Mode-Based Dynamic Model for Cable Robots with Non-straight Cables[END_REF], the control of CDPR in the operational space is presented, where the CDPR model is derived using Lagrange equations of motion for constrained systems, while considering non elastic but sagging cables through the Assumed Mode Method. In [START_REF] Merlet | Simulation of Discrete-Time Controlled Cable-Driven Parallel Robots on a Trajectory[END_REF], a discrete-time control strategy is proposed to estimate the position accuracy of the EE by taking into account the actuator model, the kinematic and static behavior of the CDPR. Multiple papers deal with the problem of controlling CDPRs while considering cable elongations and their effect on the dynamic behavior. A robust H ∞ control scheme for CDPR is described in [START_REF] Laroche | A Preliminary Study for H∞ Control of Parallel Cable-Driven Manipulators[END_REF] while considering the cable elongations into the dynamic model of the EE and cable tension limits.A control strategy is proposed for CDPRs with elastic cables in [START_REF] Khosravi | Dynamic modeling and control of parallel robots with elastic cables: singular perturbation approach[END_REF], [START_REF] Khosravi | Stability analysis and robust PID control of cable driven robots considering elasticity in cables[END_REF], [START_REF] Khosravi | Dynamic analysis and control of cable driven robots with elastic cables[END_REF]. It consists in adding an elongation compensation term to the control law of a CDPR with rigid cables, using singular perturbation theory. It requires the measurement of cables length and the knowledge of the EE pose real-time through exteroceptive measurements.

Feed-forward model-based controllers are used to fulfillaccuracy improvement by using a CDPR reference model. This latter predicts the mechanical behavior of the robot; and then generates an adequate reference signal to be followed by the CDPR. This type of control provides the compensation of the desirable effects without exteroceptive measurements. A model-based control scheme for CDPR used as a high rack storage is presented in [START_REF] Bruckmann | Design and realization of a high rack and retrieval machine based on wire robot technology[END_REF]. This research work takes into account the mechanical properties of cables, namely their elasticity. This strategy, integrating the mechanical behavior of cables in the reference signal, enhances the CDPR performances. However, it compensates for the EE positioning errors due to its rigid body behavior. The mechanical response of the robot is predicted when the mechanical behavior of the cables is not influenced by their interaction with the whole system, namely, the cable elongation is estimated. As a consequence, the main contribution of this paper deals with the coupling of a model-based feed-forward torque control scheme for CDPR with a PID feedback controller. The feed-forward controller is based on the elasto-dynamic model of CDPR to predict the full dynamic and oscillatory behavior of the CDPR and to generate the adequate reference signal for the control loop.

This paper is organized as follows: Section 2 presents the feed-forward modelbased control strategy proposed in this paper in addition to the existing rigid and elasto-static models. In Section 3, the proposed control strategy is implemented for a CDPR with three cables, a point-mass EE and three Degree-Of-Freedom (DOF) translational motions. Simulation results are presented to confirm the improvement of trajectory accuracy when using the proposed control strategy compared to conventional approaches. Conclusions and future work are drawn in Section 4.

Feed-forward model-based controller

The control inputs are mostly obtained by a combination of feed-forward inputs, calculated from a reference trajectory using a reference model of CDPR, and feedback control law, as in [START_REF] Lamaury | Control of a large redundantly actuated cable-suspended parallel robot[END_REF], [START_REF] Bayani | On the control of planar cable-driven parallel robot via classic controllers and tuning with intelligent algorithms[END_REF]. The used control scheme shown in Fig. [START_REF] Dallej | Towards vision-based control of cable-driven parallel robots[END_REF], is composed of a feed-forward block in which the inverse kinematic model is determined based on a CDPR reference model (Red block in Fig. [START_REF] Dallej | Towards vision-based control of cable-driven parallel robots[END_REF]). This latter is a predictive model of the dynamic behavior of the mechanism. Its input is the motor torque vectorζ rg ∈ R n , and its output is the reference winch rotation angle vectorq ref ∈ R n ;n being the number of actuators.

The relationship between q ref and the cable length vectorl ref ∈ R n is expressed as:

l ref -l 0 = R (q 0 -q ref ) , (1)
whereR ∈ R n×n is a diagonal matrix that is a function of the gear-head ratios and winch radius.l 0 ∈ R n is the cable length vector at the static equilibrium andq 0 ∈ R n is an offset vector corresponding to the cable length when q ref = 0. This offset is compensated at the rest time. The unwound cable length of the ith cable is calculated using the CDPR inverse geometric model.

x rg , x rg + - Reference model of CDPR q ref CDPR PID controller + + ¡ m ¡ rg

Trajectory generation

¡ corr e q .. Dynamic model (Eq. ( 2)) q m

Fig. 1: Feed-forward model-based PID control ζ rg is calculated using the following dynamic model of the CDPR, which depends on the desired EE pose x rg and acceleration ẍrg :

ζ rg = R τ rg , τ rg = W -1 rg ( w g + w e -M ẍrg ) , (2)
whereτ rg ∈ R n is a set of positive tensions.W rg ∈ R m×n is the CDPR wrench matrix,m being the DOF of its moving-platform. It is a function of x rg and maps the EE velocities to the cable velocity vector. M ∈ R m×m is the EE mass matrix,w g ∈ R m is the wrench vector due to gravity acceleration andw e ∈ R m denotes the external wrench.

In each drive, a feedback PID controller sets the vector of corrected motor torqueζ corr ∈ R n . This latter is added to ζ rg to get the motor torqueζ m ∈ R n , which results in the measured winch angular displacementsq m ∈ R n . The PID feedback control law is expressed as follows:

ζ m = ζ rg + K p e q + K d ėq + K i t + i ti e q (t) dt, (3)
whereK p ∈ R n×n is the proportional gain matrix,K d ∈ R n×n is the derivative gain matrix,K i ∈ R n×n is the integrator gain matrix, e q = q refq m is the error to minimize, leading to the correction torque vector:

ζ corr = K p e q + K d ėq + K i t + i ti e q (t) dt. (4)
It is noteworthy that ζ corr depends on the CDPR reference model used to calculate the vector q ref . the best of our knowledge, two CDPR models have been used in the literature for the feed-forward model-based control of CDPRs with non-sagging cables: (i) rigid model and (ii) elasto-static model. As a consequence, one contribution of this paper deals with the determination of the elasto-dynamic model of CDPRs to be used for feed-forward control.

Rigid model

The CDPR rigid model considers cables as rigid links. It is assumed that while applying the motor torque ζ rg , the cables tension vector is equal to τ rg and the reference signal q ref anticipates neither the cable elongation nor the oscillatory motions of the EE. The PID feedback controller uses the motor encoders response q m , which is related to the rolled or unrolled cable length l rg , which corresponds to the winch angular displacement q rg . It should be noted that the cable elongations and EE oscillatory motions are not detected here, and as a consequence, cannot be rejected.

Elasto-static model

The CDPR elasto-static model integrates a feed-forward cable elongation compensation [START_REF] Bruckmann | Design and realization of a high rack and retrieval machine based on wire robot technology[END_REF]. It is about solving a static equilibrium at each EE pose while considering cable elasticity. Assuming that each cable is isolated, the cable elongation vector δl es is calculated knowing the cable tension vector τ rg . The elastostatic cable tension vector τ es is equal to τ rg . The relationship between δl i es and τ i es on the ith cable takes the following form with a linear elastic cable model:

τ i es = τ i rg = ES δl i es δl i es + l i rg , ( 5
)
where E is the cable modulus of elasticity and S is its cross-section area.

When q rg is used as a reference signal in the feedback control scheme, the EE displacement δx es is obtained from cable elongation vector δl es . To compensate for the cable elongation effects, δl es is converted into δq es , which corrects the angular position q rg . Thus, the elasto-static reference angular displacement q es ref becomes:

q es ref = q rg -δq es . (6)
As the CDPR cables tension is always positive, δl es > 0 corresponding to δq es < 0. The reference signal q es ref corresponds to a fake position of the EE for the cable elongation compensation. Here, under the effect of cable elongations, the reference EE pose is estimated to achieve the desired pose. Although the elasto-static reference model takes into account the cable elongations, the non-compensation for the EE pose errors due to the mechanism dynamic and elasto-dynamic behavior is not considered.

Elasto-dynamic model

The CDPR elasto-dynamic model takes into account the oscillatory and dynamic behavior of the EE due to cable elongations. Here, the cables are no-longer isolated and are affected by the EE dynamic behavior. Cable elongations make the EE deviate from its desired pose x rg . The real EE pose is expressed as: x ed = x rg + δx ed . The EE displacement leads to some variations in both cable lengths and cable tensions. Indeed, the ith cable tension τ i ed obtained from the elasto-dynamic model differs from τ i rg :

τ i ed = τ i rg + δτ i ed = ES δl i ed δl i ed + l i rg , (7)
where δl i ed is the ith cable elongation assessed by considering cable elasticity and oscillations. The CDPR elasto-dynamic model takes the form:

M ẍed = W ed τ ed + w g + w e , (8)
where W ed is CDPR wrench matrix expressed at EE pose x ed . Once x ed and the cable tension vector τ ed are calculated, the cable elongation vector δl ed can be determined. This latter is converted into δq ed , which corrects the angular position vector q rg . The reference angular displacement q ed ref becomes:

q ed ref = q rg -δq ed . (9)
The proposed control strategy based on the elasto-dynamic model leads to a feed-forward controller for EE oscillatory motion compensation in addition to the conventional rigid body feedback while considering the measurements from motor encoders. It should be noted that this feed-forward controller will not disrupt the rigid body feedback stability.

3 Control of a spatial CDPR with a point-mass end-effector A spatial CDPR with three cables and three translational-DOF is considered in this section. This CDPR is composed of a point-mass EE, which is connected to three massless and linear cables. A configuration of the CREATOR prototype (Fig. (2a)), being developed at LS2N, is chosen such that the cables are tensed along a prescribed trajectory. The Cartesian coordinate vectors of the cable

(a) B 2 -2 -1 B 1 P 1 P 2 0 y (m)
1 = [2.0, -2.0, 3.5] T m, b 2 = [-2.0, -2.0, 3.5] T m, b 3 = [0.0 , 2.0 , 3.5] T m.
The EE mass is equal to 20 kg. The cables diameter is equal to 1 mm. Their modulus of elasticity is equal to 70 GPa.

Trajectory generation

A circular helical trajectory, shown in Fig. (2b), is used from static equilibrium to steady state to evaluate the efficiency of the feed-forward model-based controller while considering three CDPR reference models. The EE moves from point P 1 of Cartesian coordinate vector p 1 = [0.5, -1.0, 0.25] T m to point P 2 of Cartesian coordinate vector p 2 = [0.5, -1.0, 1.5] T m along a circular helix. The latter is defined by the following equations: The coefficients of the five-order polynomial t α are chosen in such a way that the EE Cartesian velocities and accelerations are null at the beginning and the end of the trajectory. R is the helix radius, p t is helix pitch. β 0 , β 1 and β 2 are constants. Here,

x(t) = R cos(t α ) + β 0 , y(t) = R sin(t α ) + β 1 , z(t) = p t t α + β 2 , (10)
a 5 = 24π, a 4 = -60π, a 3 = 40π, a 2 = a 1 = a 0 = 0, p t = 0.1 m, β 0 = 0.5 m, β 1 = -1.0 m, β 2 = 0.
25 m, R = 0.5 m and t sim = 15 s. The velocity maximum value is 0.8 m/s. The acceleration maximum value is 1.2 m/s 2 .

Controller tuning

The PID feedback controller is tuned using the Matlab3 PID tuning tool. This latter aims at finding the values of proportional, integral, and derivative gains of a PID controller in order to minimize error e q and to reduce the EE oscillations.

In the PID tuner work-flow, a plant model is defined from the simulation data, where the input is e q and the output is ζ corr . The gains obtained for the three control schemes are the following:

• Rigid model based: K p =3, K i =1.5 and K d =1.5

• Elasto-static model based: K p =0.53, K i =0.2 and K d =0.18

• Elasto-dynamic model based: K p =0.33, K i =0.16 and K d =0. [START_REF] Lamaury | Control of a large redundantly actuated cable-suspended parallel robot[END_REF] It is noteworthy that the gains decrease from the rigid model to the elasto-dynmic reference model.

End-effector position errors

The EE position error is defined as the the difference between its desired position x rg and its real one. This latter should be normally determined experimentally. As experimentations are not yet done, a good CDPR predictive model should be used to estimate the EE real pose. The CDPR elasto-dynamic model is the closest to the real CDPR with non sagging cables; so, it is used to predict the real behavior of the CDPR. The input of this model is ζ m , which leads to x m ed . The position error is defined as δp = x rgx m ed . To analyze the relevance of the proposed control strategy, the three control schemes under study were simulated through Matlab-Simulink . Figure (3a) shows the norm of the EE position error δp when the proposed feed-forward control law is applied while using successively the three CDPR models to generate the reference signal.

Figure (3b) illustrates the EE position error along the z-axis δz, which is the main one as the CDPR under study is assembled in a suspended configuration. The red (gree, blue, resp.) curve depicts the EE position error when the elastodynamic (elasto-static, rigid, resp.) model is used as a reference. The root-meansquare (RMS) of δp is equal to 8.27 mm when the reference signal is generated

Conclusions and future work

This paper proposed a model-based feed-forward control strategy for non-redundant CDPRs. An elasto-dynamic model of CDPR was proposed to anticipate the full dynamic behavior of the mechanism. Accordingly, a contribution of this paper deals with a good simulation model of the CDPR, including the vibratory effects, cable elongations and their interaction with the whole system, used as a reference control model. The comparison between the position errors obtained when using the proposed elasto-dynamic model or the classical rigid and elasto-static ones as control references shows meaningful differences. These differences reveal that the proposed control strategy guarantees a better trajectory tracking when adopting the proposed elasto-dynamic model to generate the reference control signal for a non-redundant CDPR. Experimental validations will be carried later on. Future work will also deal with the elasto-dynamic model-based control of redundant actuated CDPRs.

Fig. 2 :

 2 Fig. 2: (a) CREATOR prototype CAD diagram (b) End-effector desired path

Fig. 3 :

 3 Fig. 3: (a) Position error norm (b) Position error along z-axis of the end-effector

Fig. 4 :

 4 Fig. 4: Histogram of the RMS of δp and δz

  with

	tα = a5	t tsim	5	+ a4	t tsim	4	+ a3	t tsim	3	+ a2	t tsim	2	+ a1	t tsim	+ a0.

www.mathworks.com/help/slcontrol/ug/introduction-to-automatic-pid-tuning.html