Abhilash Nayak
email: abhilash.nayak@ls2n.fr

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

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

Philippe Wenger
email: philippe.wenger@ls2n.fr

Stéphane Caro
email: stephane.caro@ls2n.fr

Operation Mode Analysis of 3-RPS Parallel Manipulators based on their Design Parameters

Keywords: Study's kinematic mapping, 3-RPS, parallel manipulator, operation modes

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

Introduction

The 3-RPS parallel manipulator is a three degree-of-freedom (DOF) spatial mechanism, initially proposed by Hunt [START_REF] Hunt | Structural kinematics of in-parallel-actuated robot[END_REF]. This manipulator allows one pure vertical translation and two rotations about axes parallel to the horizontal plane, but since those axes do not remain fixed when the manipulator moves, the two rotations generate two parasitic horizontal translations. The mechanism is composed of three identical limbs connecting its base to its moving platform. Each limb consists of a revolute joint, a prismatic joint and a spherical joint mounted in series.

Several arrangements of the joints are possible, e.g. the R-joint axes in the base frame can be tangential to a circle, parallel or intersect at a common point.

Several research works have dealt with the kinematic analysis of the 3-RPS parallel manipulator. Huang and Fang described the constraints of the manipulator [START_REF] Huang | Motion characteristics and rotational axis analysis of three dof parallel robot mechanisms[END_REF] using screw theory. The number of solutions to the inverse kinematics was first published by Nanua et al. [START_REF] Nanua | Direct kinematic solution of a stewart platform[END_REF] and Tsai [START_REF] Tsai | Robot Analysis[END_REF], Self-motions [START_REF] Schadlbauer | Self-motions of 3-RPS manipulators[END_REF] were investigated by Schadlbauer et al. in which a spatial 3-RPS Manipulator was considered with R-joints tangential to the base circum-circle. Workspace and joint space analysis [START_REF] Chablat | Workspace and joint space analysis of the 3-RPS parallel robot[END_REF] using quaternions was done by Chablat et al., and more special configurations of the 3-RPS manipulator [START_REF] Nurahmi | Kinematic analysis of the 3-RPS cube parallel manipulator[END_REF] like the 3-RPS cube manipulator as well as the synthesis of design parameters with respect to specific operation modes [START_REF] Nurahmi | Design of 3-RPS parallel manipulators based on operation modes[END_REF] were both investigated by Nurahmi et al. Moreover, a complete algebraic analysis of the 3-RPS parallel manipulator was published, using Study's kinematic mapping in [START_REF] Schadlbauer | The 3-RPS parallel manipulator from an algebraic viewpoint[END_REF] and in [START_REF] Schadlbauer | A complete analysis of the 3-RPS parallel manipulator[END_REF]. Gallardo et al. analyzed the kinematics of the 3-RPS parallel manipulator by using screw theory [START_REF] Gallardo | Kinematics of 3-RPS parallel manipulators by means of screw theory[END_REF].

The motion capabilities of the 3-RPS parallel manipulator were exploited in telescope applications studied by Carretero et al. [START_REF] Carretero | Kinematic analysis of a three-dof parallel mechanism for telescope applications[END_REF] and in machine tool heads, investigated by Hernandes et al. [START_REF] Hernández | Transitions in the velocity pattern of lower mobility parallel manipulators[END_REF].

The application for medical purposes like human machine interactions were investigated in [START_REF] Verde | Kinematics analysis, workspace, design and control of 3-RPS and triglide medical parallel robots[END_REF], including the control of the manipulator with PID controllers.

The subject of this paper is about the determination of some conditions on the design parameters of 3-RPS manipulators with coplanar revolute joint axes for those manipulators to have two operation modes.

The paper is organized as follows: Section 2 presents the architecture and parameterization of 3-RPS manipulators with coplanar revolute joint axes. Section 3 expresses their constraint equations as a function of Study parameters and independently of joint variables. Section 4 deals with the operation mode analysis of the manipulators under study and gives some conditions on their design parameters to lead to two operation modes. The operation modes of three 3-RPS parallel manipulators with coplanar revolute joints defined based on the foregoing conditions on design parameters are analyzed in Section 5 as illustrative examples. Some discussions and conclusions are given in Sec. 6.

Manipulator Architectures

The investigated spatial parallel manipulator shown in Fig. 1 consists of a moving platform connected to a fixed base with three limbs. Each limb is composed of a revolute joint, a prismatic joint and a spherical joint mounted in series1 . The three prismatic joints are actuated. Figure 2 represents a RPS limb. The base of the 3-RPS manipulator is specified by 3 base-points A 1 , A 2 and A 3 in the fixed frame F 0 . The fixed frame is defined such that A 1 is the origin of the coordinate frame, A 2 is along the x-axis and A 3 is an arbitrary point in the XY -plane. B 1 , B 2 and B 3 are the vertices of the triangular moving-platform, B 1 is the origin of the moving-platform frame F 1 , B 2 is along the x-axis of F 1 and B 3 lies in the xy-plane.

X Y Z x y z x n 3 n 2 n 1 A 2 A 1 A 3 B 2 B 3 B 1 F 1 F 0 y B f 1 f 3 f 2
n i = [0 n 1i n 2i n 3i ] T P i A i : [1 a 1i a 2i a 3i ] T B i : [1 b 1i b 2i b 3i ] T Figure 2: A RPS limb
The ith revolute joint axis of direction n i is perpendicular to the direction of the ith prismatic joint, namely,

n i • ---→ A i B i = 0, i = 1, 2, 3 (1)

Kinematic Modeling

To derive the constraint equations of the 3-RPS parallel manipulators with coplanar revolute joint axes, the homogeneous coordinates of point A i and vector n i are firstly expressed in frame F 0 while that of the point B i are expressed in frame F 1 :

0 a 1 =(1, 0, 0, 0), 0 a 2 = (1, a 12 , 0, 0), 0 a 3 = (1, a 13 , a 23 , 0), ( 2
)
1 b 1 =(1, 0, 0, 0), 1 b 2 = (1, b 12 , 0, 0), 1 b 3 = (1, b 13 , b 23 , 0), ( 3
)
0 n 1 =(0, n 11 , n 21 , n 31 ), 0 n 2 = (0, n 12 , n 22 , n 32 ), 0 n 3 = (0, n 13 , n 23 , n 33 ), (4)
with the first entry of each vector being the homogenizing coordinate2 . Since points B i are given in the moving frame, a transformation is applied to obtain it in F 0 frame. Study's kinematic mapping can be used for this purpose. It is a mapping from the special Euclidean motion group SE(3) to a seven-dimensional projective space P 7 [START_REF] Husty | 21st Century Kinematics[END_REF]. The transformation to obtain 0 b i from 1 b i is given in Eq. ( 5)

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            (5) 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 ,
where x j , y j , j = 0, 1, 2, 3 are the so called Study parameters of the transformation 0 T 1 . A point P = (x 0 :

x 1 : x 2 : x 3 : y 0 : y 1 : y 2 : y 3 ) ∈ P 7 represents an Euclidean transformation, if and only if P lies in a 6-dimensional quadric, S 2 6 ∈ P 7 called as the Study quadric:

S 2 6 : x 0 y 0 + x 1 y 1 + x 2 y 2 + x 3 y 3 = 0 (6)
The geometric constraints of the parallel manipulator can be derived as follows. As the prismatic joints are actuated, the distance between points A i and B i is equal to the prismatic joint length r i . Therefore, the following first three distance constraints arise:

g i : ( 0 a i -0 b i ) T ( 0 a i -0 b i ) -r 2 i = 0, i = 1, 2, 3. (7)
The next three geometric constraints of the manipulator is derived from the perpendicularity between the revolute joint and the prismatic joint direction within each limb, namely, from Eq. (1):

g i+3 : 0 n T i ( 0 a i -0 b i ) = 0, i = 1, 2, 3. (8)
As a result, the six constraint equations are expressed as follows after some mathematical simplifications:

g 1 := x 0 2 + x 1 2 + x 2 2 + x 3 2 r 1 2 -4 y 0 2 -4 y 1 2 -4 y 2 2 -4 y 3 2 = 0 (9a) g 2 := x 0 2 + x 1 2 + x 2 2 + x 3 2 (r 2 2 -b 2 12 -a 12 2 ) + 2 x 0 2 + 2 x 1 2 -2 x 2 2 -2 x 3 2 a 12 b 12 + (-4 x 0 y 1 + 4 x 1 y 0 -4 x 2 y 3 + 4 x 3 y 2 )a 12 + (4 x 0 y 1 -4 x 1 y 0 -4 x 2 y 3 + 4 x 3 y 2 )b 12 -4 y 0 2 -4 y 1 2 -4 y 2 2 -4 y 3 2 = 0 (9b) g 3 := x 0 2 + x 1 2 + x 2 2 + x 3 2 (r 2 3 -a 13 2 -a 2 23 -b 2 13 -b 2 23 ) + 2 x 0 2 + 2 x 1 2 -2 x 2 2 -2 x 3 2 a 13 b 13 + (-4 x 0 x 3 + 4 x 1 x 2 ) a 13 b 23 + (-4 x 0 y 1 + 4 x 1 y 0 -4 x 2 y 3 + 4 x 3 y 2 ) a 13 + (4 x 0 x 3 + 4 x 1 x 2 ) a 23 b 13 + 2 x 0 2 -2 x 1 2 + 2 x 2 2 -2 x 3 2 a 23 b 23 + (-4 x 0 y 2 + 4 x 1 y 3 + 4 x 2 y 0 -4 x 3 y 1 ) a 23 + (4 x 0 y 1 -4 x 1 y 0 -4 x 2 y 3 + 4 x 3 y 2 ) b 13 + (4 x 0 y 2 + 4 x 1 y 3 -4 x 2 y 0 -4 x 3 y 1 ) b 23 -4 y 0 2 -4 y 1 2 -4 y 2 2 -4 y 3 2 = 0 (9c) g 4 := (2 x 0 y 1 -2 x 1 y 0 + 2 x 2 y 3 -2 x 3 y 2 ) n 11 + (2 x 0 y 2 -2 x 1 y 3 -2 x 2 y 0 + 2 x 3 y 1 ) n 21 = 0 (9d) g 5 := (2 x 0 y 1 -2 x 1 y 0 + 2 x 2 y 3 -2 x 3 y 2 ) n 12 + (2 x 0 y 2 -2 x 1 y 3 -2 x 2 y 0 + 2 x 3 y 1 ) n 22 + a 12 n 12 x 0 2 + x 1 2 + x 2 2 + x 3 2 + -x 0 2 -x 1 2 + x 2 2 + x 3 2 b 12 n 12 + (-2 x 0 x 3 -2 x 1 x 2 ) b 12 n 22 = 0 (9e) g 6 := (2 x 0 y 1 -2 x 1 y 0 + 2 x 2 y 3 -2 x 3 y 2 ) n 13 + (2 x 0 y 2 -2 x 1 y 3 -2 x 2 y 0 + 2 x 3 y 1 ) n 23 -(a 13 n 13 + a 23 n 23 ) x 0 2 + x 1 2 + x 2 2 + x 3 2 + -x 0 2 -x 1 2 + x 2 2 + x 3 2 b 13 n 13 + -x 0 2 + x 1 2 -x 2 2 + x 3 2 b 23 n 23 + (-2 x 0 x 3 -2 x 1 x 2 ) b 13 n 23 + (2 x 0 x 3 -2 x 1 x 2 ) b 23 n 13 = 0 (9f)
It should be noted that those six equations are a function of fifteen design parameters a 12 , a 13 , a 23 , b 12 , b 13 , b 23 , n ij (i, j ∈ {1, 2, 3}), three actuated prismatic joint variables r 1 , r 2 , r 3 and the Study parameters.

Operation modes

This section aims to find the conditions on the fifteen design parameters such that the 3-RPS manipulator with coplanar revolute joints can exhibit more than one operation mode. Since the R-joint axes are assumed coplanar, n 31 = n 32 = n 33 = 0.

From the standpoint of algebraic geometry, it is known that the existence of more than one operation mode requires the factorization of a polynomial belonging to the ideal of constraint polynomials (preferably the ones independent of actuated joint variables) [START_REF] Schadlbauer | The 3-RPS parallel manipulator from an algebraic viewpoint[END_REF]. In this context, an ideal I is considered such that it is a subset of the field of Study parameters:

I = g 4 , g 5 , g 6 | I ⊆ K[x 0 , x 1 , x 2 , x 3 , y 0 , y 1 , y 2 , y 3 ] (10)
From the definition of an ideal, if a polynomial g ∈ I and h ∈ K, K being the field over which the ideal I is defined, then hg ∈ I [START_REF] Cox | Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra[END_REF]. From Eqs. ( 9) and ( 10), a polynomial g is defined such that

g = h 1 g 4 + h 2 g 5 + h 3 g 6 ∈ I, where h i = 0 ∈ K[x 0 , x 1 , x 2 , x 3 , y 0 , y 1 , y 2 , y 3 ], i = 1, 2, 3 (11)
For simplicity, h i is only allowed to be a function of design parameters. This assumption forces the polynomial g to be quadratic. To this end, the problem boils down to find the coefficients h i such that g can be factorized.

In search of linear factors, two general linear equations are introduced in Eq. ( 12) and are multiplied to obtain a general quadratic polynomial s 12 in the kinematic image space, P 7 .

s 1 : m 1 x 0 + m 2 x 1 + m 3 x 2 + m 4 x 3 + m 5 y 0 + m 6 y 1 + m 7 y 2 + m 8 y 3 = 0 ( 12
)
s 2 : n 1 x 0 + n 2 x 1 + n 3 x 2 + n 4 x 3 + n 5 y 0 + n 6 y 1 + n 7 y 2 + n 8 y 3 = 0 s 12 = s 1 • s 2
where m k and n k , k = 1, ..., 8 are constants. Equating the respective coefficients of g and s 12 leads to a system of 36 linear equations in 31 unknowns. Solving for all the parameters b 12 , b 13 , b 23 , a 12 , a 13 , a 23 ,

h 1 , h 2 , h 3 , m k , n k , k = 1, ..., 8
, yields 36 solutions. It is noteworthy that some equations are dependent and the system is underdetermined. Nonetheless, the solve function in Maple parametrizes the solutions in terms of one or more unknowns. Investigating the solution set reveals that there are some trivial solutions (complex ones and the ones with h i = 0, a 12 = 0, a 13 = 0, a 23 = 0, b 12 = 0, b 13 = 0 or b 23 = 0) and some are special cases of the general one (n ij = 0, i = 1, 2, j = 1, 2, 3). The focus is on the two general solutions:

Solution 1 m 1 = n 13 ± √ n 13 2 + n 23 2 m 4 n 23 , m 2 = 0, m 3 = 0, m 4 = m 4 , m 5 = 0, m 6 = 0, m 7 = 0, m 8 = 0, (13a) n 1 = 2 n 13 ± √ n 13 2 + n 23 2 b 23 h 3 m 4 , n 2 = 0, n 3 = 0, n 4 = 2 b 23 h 3 n 23 m 4 , n 5 = 0, n 6 = 0, n 7 = 0, n 8 = 0, (13b)
m 1 = 0, m 2 = n 13 ± √ n 13 2 + n 23 2 m 3 n 23 , m 3 = m 3 , m 4 = 0, m 5 = 0, m 6 = 0, m 7 = 0, m 8 = 0, ( 14a
)
n 1 = 0, n 2 = 2b 23 h 3 n 23 2 n 13 ± √ n 13 2 + n 23 2 m 3 , n 3 = -2 b 23 h 3 n 23 m 3 , n 4 = 0, n 5 = 0, n 6 = 0, n 7 = 0, n 8 = 0, (14b)
Upon substitution of Solution 1 into Eq. ( 11) or Eq. ( 12) that describe the general quadric, the following conic comes out:

s 12 = g = 2 b 23 h 3 2 n 13 x 0 x 3 -n 23 x 0 2 + n 23 x 3 2 ( 15
)
The conic is degenerate and can be factorized as follows:

s 12 = g = 2 b 23 h 3 n 23 x 3 n 13 2 + n 23 2 + n 13 x 3 -n 23 x 0 x 3 n 13 2 + n 23 2 -n 13 x 3 + n 23 x 0 , n 23 = 0 (16) = -2b 23 h 3 n 23 x 0 -x 3 n2 + 1 + n x 0 + x 3 n2 + 1 -n , n = n 13 n 23 , n 23 = 0. ( 17
)
bringing to light two operation modes characterized by: Operation mode 1 :

x 0 -x 3 n2 + 1 + n = 0
Operation mode 2 :

x 0 + x 3 n2 + 1 -n = 0, n = n 13 n 23 , n 23 = 0. ( 18
)
In the same vein, upon substituting Solution 2 into Eq. [START_REF] Gallardo | Kinematics of 3-RPS parallel manipulators by means of screw theory[END_REF] or Eq. ( 12) results in the general quadric

s 12 = g = -2 2 n 13 x 1 x 2 -n 23 x 1 2 + n 23 x 2 2 b 23 h 3 , (19)
which splits into two polynomials characterizing the following two operation modes:

Operation mode 1 :

x 1 -x 2 n2 + 1 + n = 0
Operation mode 2 :

x 1 + x 2 n2 + 1 -n = 0, n = n 13 n 23 , n 23 = 0 (
then, the 3-RPS parallel manipulator with coplanar revolute joint axes has two operation modes characterized by Eqs. [START_REF] Amine | Classification of 3T1R parallel manipulators based on their wrench graph[END_REF] or (20). To derive these characteristic polynomials starting from the plane constraints, the scalar coefficients of their linear combination must follow the ratio The condition in Eq. ( 21) can be geometrically interpreted as the similarity (also called as homothety) between the moving platform triangle and the triangle enclosed by the three R-joint axes. This claim is proven as follows. Figure 3 and passing through a point A i is n 1i x 0 + n 2i y 0 -n T i a i , i = 1, 2, 3. Therefore, |P| = 0 implies that lines L i are concurrent, namely, planes P 1 , P 2 and P 3 intersect at line M . Thus, a 3-RPS parallel manipulator with coplanar revolute joint axes will have two operation modes if the following geometric conditions are satisfied i Moving platform triangle is homothetic to the triangle enclosed by the revolute joint axes.

h 1 : h 2 : h 3 =
A 2 A 1 A 3 B 3 B 1 n 3 n 2 n 1 B 2 b 2 b 1 b 3 β 1 γ 1 γ 3 γ 2 β 2 β 3
A 1 A 3 A 2 P 3 P 1 P 2 n 3 n 2 n 1 L 1 L 2 L 3 M

cos(β

i ) = cos(γ i ) =⇒ b 2 j + b 2 k -b 2 i 2b j b k = n j n k ||n j ||||n k || i, j, k = (123) (24)
ii The three planes normal to the three revolute joint axes, respectively, have a common line of intersection.

It can be shown that the above conditions are also necessary for the existence of two operation modes as explained thereafter.

Equations [START_REF] Amine | Classification of 3T1R parallel manipulators based on their wrench graph[END_REF] and (20) are the polynomials characterizing the two operation modes. The transition between them is when both polynomials vanish at the same time, i.e., when x 0 = x 3 = 0 for Solution 1 and when x 1 = x 2 = 0 for Solution 2. Transition pose is known to be a constraint singularity and in case of Solution 1, it corresponds to configurations in which the moving platform is upside down as explained in [START_REF] Schadlbauer | The 3-RPS parallel manipulator from an algebraic viewpoint[END_REF][START_REF] Schadlbauer | A complete analysis of the 3-RPS parallel manipulator[END_REF]. For Solution 2, the transition pose corresponds to configurations in which the moving platform is in an upright position parallel to the fixed base. The existence of two operation modes definitely implies a constraint singularity that separates those operation modes but the reciprocity is not necessarily true as explained in subsection 5.2. This fact is exploited to prove the necessary conditions for a 3-RPS PM to have two operation modes.

According to screw theory, it is well known that a parallel manipulator reaches a constraint singularity when its constraint wrench system is rank deficient [START_REF] Zlatanov | Constraint singularities of parallel mechanisms[END_REF][START_REF] Amine | Classification of 3T1R parallel manipulators based on their wrench graph[END_REF]. At this instant, the PM gains at least one dof. The constraint wrench system of a 3-RPS PM shown in Fig. 1 is spanned by three forces f 1 , f 2 and f 3 that are parallel to the revolute joint axes n 1 , n 2 and n 3 and pass through points B 1 , B 2 and B 3 , respectively. A constraint singularity implies that variety spanned by these three lines has a rank lower than 3. This can happen only when these lines reduce to a planar pencil of lines, i.e., when they are coplanar and concurrent.

The Plücker coordinates of the force lines can be written as follows:

f 1 = [ 0 n 1 , 0 b 1 × 0 n 1 ] (26a) f 2 = [ 0 n 2 , 0 b 2 × 0 n 2 ] ( 26b
)
f 3 = [ 0 n 3 , 0 b 3 × 0 n 3 ] (26c)

Coplanarity condition

Any two lines intersect when the reciprocal product of their Plücker coordinates vanishes. Therefore, the coplanarity condition can be formulated as the mutual vanishing of the reciprocal product between the force lines yielding the following three equations:

E 1 := 2 b 12 (x 0 x 2 -x 1 x 3 ) (n 11 n 22 -n 12 n 21 ) x 0 2 + x 1 2 + x 2 2 + x 3 2 = 0 (27a) E 2 := 2 (n 11 n 23 -n 13 n 21 ) (b 13 x 0 x 2 -b 13 x 1 x 3 -b 23 x 0 x 1 -b 23 x 2 x 3 ) x 0 2 + x 1 2 + x 2 2 + x 3 2 = 0 (27b) E 3 : -2 (n 12 n 23 -n 13 n 22 ) (b 12 x 0 x 2 -b 12 x 1 x 3 -b 13 x 0 x 2 + b 13 x 1 x 3 + b 23 x 0 x 1 + b 23 x 2 x 3 ) x 0 2 + x 1 2 + x 2 2 + x 3 2 = 0 (27c)
Solving the previous system of equations in Eq. ( 27) for Study parameters x 0 , x 1 , x 2 , x 3 gives two solutions:

x 0 = x 3 = 0, (28)
x 1 = x 2 = 0. ( 29
)
Calculating 0 b i , i = 1, 2, 3 with solutions (28) or (29) shows that the z-coordinates of the resulting points are the same proving that they indeed lie in a plane parallel to the XY -plane.

Concurrency condition

Case 1:

x 0 = x 3 = 0 The z-coordinate of points B i is expressed -2(x 1 y 2 -x 2 y 1 ) x 2 1 + x 2 2
. Without loss of generality, these lines can now be projected to the XY -plane to simplify the concurrency condition. Their projections have the equations:

L i := -n 2i X + n 1i Y -(-n 2i 0 b ix + n 1i 0 b iy ) = 0, i = 1, 2, 3, (30
which is the relation derived in Eq. ( 22).

Case 2:

x 1 = x 2 = 0
In this case, we obtain the following symmetric relations between the design parameters: As a conclusion, the following theorem can be stated:

b 12 = (n
Theorem 1 A 3-RPS parallel manipulator with coplanar revolute joint axes will have two operation modes if and only if the following geometric conditions are satisfied i Moving platform triangle is homothetic to the triangle enclosed by the revolute joint axes.

ii The three planes normal to the three revolute joint axes, respectively, have a common line of intersection.

Since the relationship between the number of operation modes and architecture is established, the design

In case the revolute joint axes are no longer coplanar, equating a general quadric in P 7 defined in Eq. ( 12)

with the linear combination of the plane constraint polynomials g 4 , g 5 and g 6 shown in Eqs. (9d)-(9f), does not yield any solution. The problem of finding the influence of design parameters on the operation modes of a general 3-RPS parallel manipulator is left for future work.

Examples

In this section, some example 3-RPS manipulators are considered to verify the proposed conditions. The well-known 3-RPS parallel manipulator introduced by Hunt [START_REF] Hunt | Structural kinematics of in-parallel-actuated robot[END_REF] has been the spotlight of numerous research topics and applications [START_REF] Huang | Motion characteristics and rotational axis analysis of three dof parallel robot mechanisms[END_REF][START_REF] Schadlbauer | Self-motions of 3-RPS manipulators[END_REF][START_REF] Chablat | Workspace and joint space analysis of the 3-RPS parallel robot[END_REF][START_REF] Nurahmi | Kinematic analysis of the 3-RPS cube parallel manipulator[END_REF][START_REF] Schadlbauer | The 3-RPS parallel manipulator from an algebraic viewpoint[END_REF][START_REF] Schadlbauer | A complete analysis of the 3-RPS parallel manipulator[END_REF][START_REF] Gallardo | Kinematics of 3-RPS parallel manipulators by means of screw theory[END_REF][START_REF] Verde | Kinematics analysis, workspace, design and control of 3-RPS and triglide medical parallel robots[END_REF]. Its architecture is shown in Fig. 5. The moving platform and the base are equilateral triangles with circum-radius b and a, respectively. The R-joint axes are coplanar and tangential to the base circum-circle. The P-joint in each leg is normal to its corresponding R-joint axis. The design parameters are listed below:

Example 1: 3-RPS parallel manipulator with

n 23 = 0 A 2 A 1 A 3 B 3 B 1 X Y Z x y z a b n 3 n 2 n 1 B 2
a 12 = √ 3a, a 13 = √ 3 2 a, a 23 = 3 2 a, b 12 = √ 3b, b 13 = √ 3 2 b, b 23 = 3 2 b, n 11 = 1 2 , n 21 = - √ 3 2 , n 31 = 0, n 12 = 1 2 , n 22 = √ 3 2 , n 32 = 0, n 13 = -1, n 23 = 0, n 33 = 0. (36)
√ 3 4 : ± 3 4
, which is consistent with the design parameters listed in Eq. ( 36). It is also straightforward to see that the design parameters satisfy Eq. ( 22). Thus, according to Theorem 1, the manipulator must exhibit two operation modes. To determine the characteristic equations of the operation modes, the plane constraint equations corresponding to Eq. ( 8) are considered:

g 4 := - √ 3x 0 y 2 + √ 3x 1 y 3 + √ 3x 2 y 0 - √ 3x 3 y 1 + x 0 y 1 -x 1 y 0 + x 2 y 3 -x 3 y 2 = 0 ( 37
)
g 5 := √ 3x 0 y 2 - √ 3x 1 y 3 - √ 3x 2 y 0 + √ 3x 3 y 1 + x 0 y 1 -x 1 y 0 + x 2 y 3 -x 3 y 2 + √ 3 2 (x 0 2 + x 1 2 + x 2 2 + x 3 2 )a - √ 3 2 (2 √ 3x 0 x 3 + 2 √ 3x 1 x 2 + x 0 2 + x 1 2 -x 2 2 -x 3
2 )b = 0 (38)

g 6 := -2 x 0 y 1 + 2 x 1 y 0 -2 x 2 y 3 + 2 x 3 y 2 - √ 3 2 (x 0 2 + x 1 2 + x 2 2 + x 3 2 )a - √ 3 2 (2 √ 3x 0 x 3 -2 √ 3x 1 x 2 -x 0 2 -x 1 2 + x 2 2 + x 3 2 )b = 0. (39)
Equation ( 23) can be used to find the constants h 1 , h 2 in terms of h 3 to be multiplied to the constraint polynomials g 4 , g 5 and g 6 , respectively to obtain a factorable polynomial g. For this manipulator, the design parameters yield h 1 = h 3 and h 2 = h 3 . Thus from Eqs. ( 11) and (37)

g = h 1 g 4 + h 2 g 5 + h 3 g 6 = h 3 (g 4 + g 5 + g 6 ) = -6h 3 bx 0 x 3 (40)
showing that the manipulator at hand can have two operation modes characterized by x 0 = 0 and x 3 = 0 as already presented in [START_REF] Schadlbauer | The 3-RPS parallel manipulator from an algebraic viewpoint[END_REF][START_REF] Schadlbauer | A complete analysis of the 3-RPS parallel manipulator[END_REF]. In fact, substituting n 23 = 0 in Eq. ( 15) results in the factorable polynomial

x 0 x 3 .

Example 2:

3-RPS parallel manipulator with n 13 = 0

A 2 A 1 A 3 B 3 B 1 X Y Z x y z a b n 3 n 2 n 1 B 2 Y Figure 6: A 3-RPS parallel manipulator with n 13 = 0 n 3 n 2 n 1 X Y A 2 A 1 A 3 L 2 L 3 L 1 (a) a12 = √ 3a, a13 = √ 3 2 a, a23 = 3 2 a
One operation mode

n 3 n 2 n 1 X Y A 2 A 1 A 3 L 2 L 3 L 1 2 X (b) a12 = - √ 3a, a13 = √ 3 2 a, a23 = 3 2 a
Two operation modes

n 3 n 2 n 1 X Y A 2 A 1 A 3 L 2 L 3 L 1 Ã 3 (c) a12 = √ 3a, a13 = - √ 3 2 a, a23 = 3 2 a
Two operation modes

A 2 A 1 A 3 n 3 n 2 n 1 L 2 L 3 X Y L 1 Ã 3 (d) a12 = √ 3a, a13 = √ 3 2 a, a23 = - 3 2 a
Two operation modes modes, the constraint singularity between the operation modes can be escaped by slightly modifying the design parameters such that one of the conditions in Theorem 1 is not fulfilled. . Fig. 8 shows the architecture of the manipulator at hand, where it can be pointed out that condition i. and ii. of in Eq. ( 16)

n 3 n 2 n 1 A 2 A 1 A 3 P 2 P 3 P 1 B 2 B 3 B 1
gives in Eq. ( 16) gives

g := (x 0 -x 3 (2 + √ 5))(x 0 -x 3 (2 - √ 5
g := (x 1 -x 2 (2 + √ 5))(x 1 -x 2 (2 - √ 5)) = 0 (44)
The two polynomials x 0 -x 3 (2 + √ 5) and

x 0 -x 3 (2 - √ 5) or x 1 -x 2 (2 + √ 5) and x 1 -x 2 (2 - √ 5) represent
the two operation modes of the mechanism 5 .

Assuming the prismatic joints are actuated, the direct kinematics of the manipulator can be solved by substituting arbitrary values to joint parameters r 1 = 2, r 2 = 2 and r 3 = 3. The constraint equations g 1 = 0 to g 6 = 0 and S 2 6 = 0 can be written from Eqs. ( 6) and ( 9). Adding a normalization equation

x 2 0 + x 2 1 + x 2 2 +
x 2 3 = 1 (so that ∆ = 0 in Eq. ( 5)) yields a set of eight equations to be solved for eight Study parameters. Finding the Groebner basis of the ideal of constraint polynomials with a pure lexicographic monomial ordering x 0 < lex x 1 < lex x 2 < lex x 3 < lex y 0 < lex y 1 < lex y 2 < lex y 3 results in a 16 degree univariate polynomial in variable y 3 . As anticipated, the polynomial can be factorized into two polynomials of degree 8 each corresponding to the two operation modes. It shows that a 3-RPS parallel manipulator can have at most 8 solutions to its direct kinematics in each operation mode. When the joint parameters are fixed, the direct kinematics of a 3-RPS parallel manipulator amounts to locating three points on three fixed circles with centers A i and radii r i . To this end, a corollary follows as a consequence of Theorem 1:

Corollary 2 For the 3-points on 3-circles problem, if the geometry satisfies the following conditions i normals to the planes containing the circles and passing through their centers are coplanar, ii planes containing the circles have a common line of intersection and iii the triangle formed by the three points is homothetic to the triangle enlosed by three normals to the planes passing through the centers of the circles, then the 16 degree univariate characteristic polynomial factorizes into two 8 degree polynomials.

For the above-mentioned example, eight real solutions to its direct kinematics problem are found. The solutions form four pairs of manipulator postures, one being the mirror image of another about the XYplane. Four of these solutions are displayed in Fig. 9.

The first two solutions satisfy

x 0 x 3 = 2 - √ 5 or x 1 x 2 = 2 + √ 5
and hence belong to the operation mode corresponding to x 0 -x 3 (2 -√ 5) = 0 or x 1 -x 2 (2 + √ 5) = 0, respectively while the last two satisfy

x 0 x 3 = 2 + √ 5 or x 1 x 2 = 2 - √ 5
and hence belong to the operation mode characterized by

x 0 -x 3 (2 + √ 5) = 0 or x 1 -x 2 (2 - √ 5) = 0.

Conclusions and future work

The influence of design parameters on the number of operation modes of a 3-RPS parallel manipulator with coplanar revolute joints was studied in this paper. The constraint equations of a general 3-RPS 5 To be able to factorize the polynomial g, the field of rational numbers must be extended to include n 2 23 + n 2 13 i.e. parallel manipulator were derived. The linear combination of plane constraint polynomials were equated to a general quadric in P 7 and the coefficients were solved to obtain two solutions with some relations between design parameters. These relations were substituted back into the general quadric and it factorized into two polynomials characterizing two operation modes. The conditions on the design parameters for the existence of two operation modes in 3-RPS manipulator with coplanar revolute joints was summarized as a theorem with proof. The first condition is the homothety between the moving platform triangle and the triangle enclosed by revolute joint axes, while the second condition is when three planes on which the spherical joints are confined to move have a common line of intersection. Two special cases were considered: one that has two operation modes and the other one with one operation mode. For the latter manipulator, it was shown that one can modify the design parameters to be able to have two operation modes. Finally, a numerical example was considered following the proposed theorem. Its characteristic 16 degree univariate polynomial is derived to show that it splits into two polynomials of degree 8 each, representing two operation modes.

The direct kinematic solutions lying in each operation mode were shown. Future work will include the influence of architecture on the operation modes of a 3-RPS parallel manipulator with non-coplanar revolute joints. Additionally, the proposed methodology will be extended to different parallel manipulators.

Figure 1 :

 1 Figure 1: Architecture of a 3-RPS parallel manipulator with coplanar revolute joints

  20) As a result, when the design parameters b 12 , b 13 and b 23 follow the ratio b 12 : b 13 : b 23 = (n

Figure 3 :

 3 Figure 3: Similarity condition between the moving platform triangle and the triangle enclosed by the R-joint axes.

Figure 4 :

 4 Figure 4: The planes normal to R-joint axes must have a common line of intersection.

  shows the moving platform triangle and the triangle enclosed by the revolute joints. The sides of the moving platform triangle are b 1 = (b 13 -b 12 ) 2 + b 2 23 , b 2 = |b 13 -b 12 | and b 3 = |b 12 |. Knowing the sides, the cosine of the angles β 1 , β 2 and β 3 can be determined using the cosine rule. Similarly, the cosine of the angles γ 1 , γ 2 and γ 3 between the R-joint axes n 1 , n 2 and n 3 , can be determined. Equating the cosine of respective angles results in three equations 3 .

Solving the equations for b 12 n 11 n 21 0 n 12 n 22 -n 12 a 12 n

 1212 , b 13 and b 23 yields the conditions in Eq. (21) proving that the considered triangles are similar (or homothetic).Besides, Eq. (22) has a geometrical meaning too. It can be written as the determinant of a matrix, P:|P| = 13 n 23 -n 13 a 13 -n 23 a 23 = 0 (25)|P| is the Grassmannian of three lines L 1 , L 2 and L 3 which are the projections of planes P 1 , P 2 and P 3 onto the XY -plane as shown in Fig.4. The equation of a line L i , i = 1, 2, 3, orthogonal to R-joint axis n i

) where 0 -x 2 1 )+ 2 (+ n 23 2 (

 0122 b iy and 0 b iy are the x-and y-coordinates of point B i , respectively. Therefore, the condition for concurrency of the three lines defined by equations L i = 0 is -n 21 n 11 n 21 0 b 1x -n 11 0 b 1y -n 22 n 12 n 22 0 b 2x -n 12 0 b 2y -n 23 n 13 n 23 0 b 3x -n 13 0 b 3y = (b 12 n 11 n 22 n 23 -b 12 n 13 n 21 n 22 -b 13 n 11 n 22 n 23 + b 13 n 12 n 21 n 23 -b 23 n 11 n 13 n 22 + b 23 n 12 n 13 n 21 )(x 2 2 b 12 n 11 n 12 n 23 -b 12 n 12 n 13 n 21 -b 13 n 11 n 13 n 22 + b 13 n 12 n 13 n 21 + b 23 n 11 n 22 n 23 -b 23 n 12 n 21 n 23 )x 1 x 2 to zero leads to the following relations between the design parameters: b 12 = -(n 11 n 22 -n 12 n 21 ) b 23 n 13 2 -n 13 n 22 + n 12 n 23 ) (-n 13 n 21 + n 11 n 23 ) , b 13 = -(n 12 n 13 + n 22 n 23 ) b 23 n 12 n 23 -n 13 n 22 , b 23 = b 23 (32) which are exactly those defined by Eq. (13d). Furthermore, upon substitution of the values of b 12 and b 13 and x 0 = x 3 = 0 in constraint equations g 1 = g 2 = g 3 = 0 defined by Eqs. (9d)-(9f) and eliminating b 23 , results in the following equation: (n 11 n 23 -n 13 n 21 ) a 12 n 12 + (n 12 n 21 -n 11 n 22 ) (+a 13 n 13 + a 23 n 23 ) = 0,

  75

Figure 5 :

 5 Figure 5: A 3-RPS parallel manipulator with n 23 = 0

  Calculating the right hand side of Eq. (21) gives b 12 : b 13 : b 23 =

Figure 7 :

 7 Figure 7: Slight modification of design parameters can influence the number of operation modes

5. 3 .

 3 Example 3: Arbitrary design parameters Finally, a numerical example is studied with the following arbitrary design parameters: a 13 = 2, a 23 = 2, b 23 = 3, n 11 = -3, n 21 = 5, n 31 = 0, n 12 = -3, n 22 = 2, n 32 = 0, n 13 = 2, n 23 = 1, n 33 = 0 (42)

Figure 8 : A 3 -

 83 Figure 8: A 3-RPS parallel manipulator with two operation modes characterized by x 0 -x 3 (2 + √ 5) = 0 and x 0 -x 3 (2 -√ 5) = 0

Theorem 1

 1 are satisified. Substituting the design parameters with b 12 = -135 91 and b 13 = -12 7

2 A 1 A 3 B 3 B 1 B 2 ( 2 (

 2133122 a) x0 = 0.0593, x1 = -0.2925, x2 = -0.9206, x3 = -0.2515, y0 = 0.0001, y1 = -0.8954, y2 = 0.3571, y3 = -0.2658 OR x0 = .9206, x1 = .2515, x2 = 0.0593, x3 = -.2925, y0 = -.3571, y1 = .2658, y2 = 0.0001, y3 = -.8954. b) x0 = 0.0264, x1 = 0.0940, x2 = -0.9889, x3 = -0.1119, y0 = -0.1607, y1 = -0.9782, y2 = -0.1060, y3 = 0.0775 OR x0 = -.9889, x1 = -.1119, x2 = -0.0264, x3 = -0.0940, y0 = -.1060, y1 = 0.0775, y2 = .1607, y3 = .9782.

2 (

 2 c) x0 = 0.2482, x1 = -0.6214, x2 = -0.7407, x3 = 0.0586, y0 = -0.2019, y1 = -0.7792, y2 = 0.5742, y3 = -0.1490 OR x0 = .7407, x1 = -0.0586, x2 = .2482, x3 = -.6214, y0 = -.5742, y1 = .1490, y2 = -.2019, y3 = -.7792.

2 (

 2 d) x0 = -0.6304, x1 = 0.0743, x2 = -0.7582, x3 = -0.1488, y0 = -0.3408, y1 = 0.5290, y2 = 0.1872, y3 = 0.7541 OR x0 = .7582, x1 = .1488, x2 = -.6304, x3 = 0.0743, y0 = -.1872, y1 = -.7541, y2 = -.3408, y3 = .5290.

Figure 9 :

 9 Figure 9: Solutions to direct kinematics of a 3-RPS manipulator with arbitrary design parameters

  a 12 = (n 11 n 22 -n 12 n 21 ) (a 13 n 13 + a 23 n 23 ) (n 11 n 23 -n 13 n 21 ) n 12 , a 13 = a 13 , a 23 = a 23 (n 11 n 22 -n 12 n 21 ) (n 12 n 23 -n 13 n 22 ) (n 11 n 23 -n 13 n 21 ) , b 13 = -b 23 (n 12 n 13 + n 22 n 23 ) n 12 n 23 -n 13 n 22 , b 23 = b 23 (13d) h 1 = h 3 (n 12 n 23 -n 13 n 22 ) n 11 n 22 -n 12 n 21 , h 2 = -h 3 (n 11 n 23 -n 13 n 21 ) n 11 n 22 -n 12 n 21 , h 3 = h 3

	Solution 2
				(13c)
	b 12 = -	b 23 n 13	2 + n 23	2 (13e)

  a 12 = (n 11 n 22 -n 12 n 21 ) (a 13 n 13 + a 23 n 23 ) (n 11 n 23 -n 13 n 21 ) n 12 , a 13 = a 13 , a 23 = a 23 (n 11 n 22 -n 12 n 21 ) (n 12 n 23 -n 13 n 22 ) (n 11 n 23 -n 13 n 21 ) , b 13 = b 23 (n 12 n 13 + n 22 n 23 ) n 12 n 23 -n 13 n 22 , b 23 = b 23 (14d) h 1 = h 3 (n 12 n 23 -n 13 n 22 ) n 11 n 22 -n 12 n 21 , h 2 = -h 3 (n 11 n 23 -n 13 n 21 ) n 11 n 22 -n 12 n 21 , h 3 = h 3

				(14c)
	b 12 =	b 23 n 13	2 + n 23	2

  11 n 22 -n 12 n 21 ) n 13 2 + n 23 12 , a 13 and a 23 satisfy the relation (n 11 n 23 -n 13 n 21 ) a 12 n 12 + (n 12 n 21 -n 11 n 22 ) (+a 13 n 13 + a 23 n 23 ) = 0,

2 : (n 12 n 13 + n 22 n 23 ) (-n 13 n 21 + n 11 n 23 ) : ± (-n 13 n 22 + n 12 n 23 ) (-n 13 n 21 + n 11 n 23 ) ,

(21)

and parameters a

  n 12 n 23 -n 13 n 22 : n 11 n 23 -n 13 n 21 : n 11 n 22 -n 12 n 21 .

	(23)

A revolute, prismatic and a spherical joint are denoted by R, P and S respectively.

A vector expressed in F 0 is denoted as 0 {•} whereas a vector expressed in F 1 is indicated as 1 {•}.

The notation[START_REF] Schadlbauer | The 3-RPS parallel manipulator from an algebraic viewpoint[END_REF][START_REF] Nurahmi | Kinematic analysis of the 3-RPS cube parallel manipulator[END_REF][START_REF] Hunt | Structural kinematics of in-parallel-actuated robot[END_REF] represents a cyclic permutation. It means that i, j, k are initially assigned to values 1,2,3, respectively, 2,3,1, subsequently and 3,1,2, finally.

parameters can be chosen in such a way that a constraint singularity is avoided.

When a = b, the manipulator at hand can have a constraint singularity when the moving platform is parallel to the fixed base. However, it has only one operation mode. This is because, for a general 3-RPS PM, the variety of its constraint singularity condition is a surface whereas in this case it is reduced to a line parallel to the Z-axis and passing through the intersection of the R-joint axes. It is an interesting case of a constraint singularity that does not bifurcate the workspace into multiple operation modes.

Another special case of the 3-RPS parallel manipulator is when the R-joint axes are intersecting as shown in Fig. 6. The base and the platform are still equilateral triangles with circum-radius a and b, respectively. Also, the P-joints are orthogonal to their corresponding R-joint axes in each leg. The design parameters for this manipulator are listed below:

The ratio between b 12 , b 13 and b 23 calculated using Eq. ( 21) gives b 12 :

which is consistent with the design parameters listed in Eq. ( 41). Thus, the condition i. of Theorem 1 is satisfied. It can also be verified by the fact that the R-joint axes intersect in a point, which is homothetic with the moving platform equilateral triangle. On the other hand, the left hand side of Eq. ( 22) gives

proving that this manipulator can have only one operation mode 4 . As a matter of fact, inspecting Fig. 6 reveals that the planes normal to n i and passing through A i do not have a common line of intersection and hence this manipulator cannot exhibit more than one operation mode since condition ii. of Theorem 1 is not satisfied. Figure 7a shows the projections of planes P i as lines L i onto the XY -plane.

Nonetheless, the design parameters can be altered so that the condition ii. is satisfied. Changing n ij , i = 1, 2, j = 1, 2, 3 might alter condition i. of Theorem 1, hence a 12 , a 13 or a 23 can be changed so that condition i.

is kept intact. From Eq. ( 22), writing a 12 as a function of other design parameters and substituting the values form Eq. (41) yields a 12 = -√ 3a. The design with a 12 = -√ 3a is shown in Fig. 7b and it exhibits two operation modes. Ã2 represents the initial position of point A 2 . In fact, a 13 or a 23 can also be changed similarly to obtain the designs shown in Figs. 7c and7d, respectively. In these figures, Ã3 represents the initial position of point A 3 . Calculating h i , i = 1, 2, 3 from Eq. ( 23) and substituting in the general quadric of Eq.( 11) gives the characteristic polynomial of each operation mode as x 0 -x 3 and x 0 + x 3 .

Consequently, it provides an interesting example of how architecture of a manipulator influences its number of operation modes. The procedure explained can be used to design 3-RPS parallel manipulators to have the necessary number of operation modes. Moreover, for a 3-RPS parallel manipulator with two operation


Guanglei Wu

Shaoping Bai

Stéphane Caro
email: stephane.caro@ls2n.fr

A Transmission Quality Index for a Class of Four-limb Parallel Schönflies Motion Generators

Keywords: Schönflies motion, pick-and-place, transmission index, pressure angle, singularity

This paper presents a uniform evaluation method for both transmission quality and singularity analysis for a class of parallel Schönflies-motion generators with four RRΠRR limbs. It turns out that the determinant of the forward Jacobian matrices for this class of parallel robots can be expressed as the scalar product of two vectors, the first vector being the cross product of the four unit vectors along the parallelograms, and the second one being related to the rotation of the mobile platform. The pressure angles, derived from the determinants of forward and inverse Jacobians, respectively, are used for the evaluation of the transmission quality and the detection of robot singularities. Five robots are compared based on the proposed indices as an illustrative example.

Introduction

Parallel robots performing Schönflies motions or SCARA (Selective Compliance Assembly Robot Arm) motion [START_REF] Krut | A high-speed parallel robot for SCARA motions[END_REF], namely, three independent translations and one rotation about an axis of fixed direction (3T1R), are well adapted to high-speed pick-and-place (PnP) operations, thanks to their advantages in terms of lightweight structure and high stiffness. Amongst those parallel Schönflies-motion generators (SMGs) [START_REF] Pierrot | Optimal design of a 4-dof parallel manipulator: From academia to industry[END_REF][START_REF] Kim | Kinematic analysis and optimal design of a 3T1R type parallel mechanism[END_REF][START_REF] Altuzarra | A symmetric parallel Schönflies-motion manipulator for pick-and-place operations[END_REF][START_REF] Liu | Optimal design of a 4-DOF SCARA type parallel robot using dynamic performance indices and angular constraints[END_REF][START_REF] Wu | On the stiffness of three/four degree-of-freedom parallel pickand-place robots with four identical limbs[END_REF], most of them inherit the architecture of the H4 [START_REF] Pierrot | H4: a new family of 4-dof parallel robots[END_REF] robot with four identical limbs and an articulated traveling plate [START_REF] Company | Internal singularity analysis of a class of lower mobility parallel manipulators with articulated traveling plate[END_REF]. Its similar version is the I4 robot [START_REF] Krut | I4: A new parallel mechanism for SCARA motions[END_REF] and the symmetrical Par4 [START_REF] Nabat | Par4: very high speed parallel robot for pick-and-place[END_REF]. The commercial version of Par4, namely, the Quattro robot [START_REF]Adept Quattro Parallel Robots[END_REF] by Adept Technologies Inc., is the fastest industrial robot available. The rotation is realized by the relative movement in parallel between the two sub-platforms in this double-platform architecture, and the rotation can also be amplified by the pulley-belt mechanism [START_REF]Adept Quattro Parallel Robots[END_REF]. This articulated system enhances the rotational capability of the robot end-effector. However, the manufacturing cost is getting more expensive. Therefore, if the applications do not require a high rotational capability, the end-effector of the robot free of the amplification mechanism is preferable to simplify the robot architecture. Later on, the Veloce.

robot [START_REF] Veloce | [END_REF] with a different articulated traveling plate that is connected by a screw pair, was developed.

Besides, the four-limb robots with single-platform architecture have also been reported [START_REF] Salgado | Synthesis and design of a novel 3T1R fully-parallel manipulator[END_REF][START_REF] Xie | Design and development of a high-speed and high-rotation robot with four identical arms and a single platform[END_REF][START_REF] Wu | Architecture optimization of a parallel Schönflies-motion robot for pick-and-place applications in a predefined workspace[END_REF]. Other parallel Schönflies-motion robots are also noticeable for their high performance [START_REF] Kong | Type synthesis of 3T1R 4-dof parallel manipulators based on screw theory[END_REF][START_REF] Gogu | Structural synthesis of fully-isotropic parallel robots with Schönflies motions via theory of linear transformations and evolutionary morphology[END_REF][START_REF] Richard | Kinematic analysis and prototyping of a partially decoupled 4-DOF 3T1R parallel manipulator[END_REF][START_REF] Li | Parallel mechanisms with bifurcation of Schoenflies motion[END_REF][START_REF] Briot | PAMINSA: A new family of partially decoupled parallel manipulators[END_REF], such as the Delta based robots [START_REF] Pierrot | DELTA: a simple and efficient parallel robot[END_REF] and the two-limb SMGs [START_REF] Angeles | Kinetostatic design of an innovative Schönfliesmotion generator[END_REF][START_REF] Lee | Singularity and workspace analysis of three isoconstrained parallel manipulators with schoenflies motion[END_REF][START_REF] Harada | Kinematics and singularity analysis of a CRRHHRRC parallel Schönflies motion generator[END_REF][START_REF] Kim | Development of a novel two-limbed parallel mechanism having schönflies motion[END_REF], etc.. The four-limb parallel robots with an articulated mobile platform are displayed in Fig. 1. It is noteworthy that the H4 robot with the modified mobile platform can be mounted vertically instead of the horizontal installation with reduced mounting space [START_REF] Wu | Kinematic analysis and optimal design of a wall-mounted four-limb parallel Schönfliesmotion robot for pick-and-place operations[END_REF], in order to provide a rotation around an axis of vertical direction, which is named as "V4" for convenience in the following study.

In the design and application of a manipulator, its kinematic Jacobian matrix plays an important role, since the dexterity/manipulability of the robot can be evaluated through the conditioning number of the kinematic Jacobian matrix. This quantity also reflects the accuracy/wrench capability [START_REF] Merlet | Jacobian, manipulability, condition number, and accuracy of parallel robots[END_REF] between the actuators and end-effector. On the other hand, in the performance evaluation using the Jacobian matrix, a problem usually encountered is that the parallel manipulators with mixed input or/and output motions, i.e., compound linear and angular motions, will result in dimensionally inhomogeneous Jacobians. Thus, the performance indices of the conventional Jacobian matrix, such as condition number, will lack in physical significance [START_REF] Kim | New dimensionally homogeneous Jacobian matrix formulation by three end-effector points for optimal design of parallel manipulators[END_REF]. As far as SMGs are concerned, their endeffector generates a mixed motion, i.e., the Schönflies motions, for which the terms of the kinematic Jacobian matrix do not have the same units. The common approach to overcome this problem is to introduce a characteristic length [START_REF] Ranjbaran | The mechanical design of a seven-axes manipulator with kinematic isotropy[END_REF][START_REF] Altuzarra | Multiobjective optimum design of a symmetric parallel Schönflies-motion generator[END_REF] to homogenize the Jacobian matrix or the formulation of the point-based Jacobian matrices [START_REF] Pond | Formulating Jacobian matrices for the dexterity analysis of parallel manipulators[END_REF][START_REF] Wei | Design of planar parallel robots with preloaded flexures for guaranteed backlash prevention[END_REF] by means of vertex velocity transformation of the mobile platform. On the other hand, the measurement significantly depends on the choice of the characteristic length or the selection of the vertices that is not unique, which results in biased evaluation, although a "best" one can be found by optimization technique [START_REF] Angeles | Is there a characteristic length of a rigid-body displacement?[END_REF]. Alternatively, an efficient approach to ac-

(a) (b) (c) (d)
Figure 1: The four-limb parallel PnP robots with different base and mobile platform shapes: (a) Quattro [START_REF]Adept Quattro Parallel Robots[END_REF]; (b) H4 [START_REF] Pierrot | H4: a new family of 4-dof parallel robots[END_REF]; (c) Veloce. [START_REF] Veloce | [END_REF]; (d) "V4" [START_REF] Wu | Kinematic analysis and optimal design of a wall-mounted four-limb parallel Schönfliesmotion robot for pick-and-place operations[END_REF]. commodate this dimensional inhomogeneity is to adopt the concept of the virtual coefficient, namely, the transmission index, which is closely related to the transmission/pressure angle [START_REF] Dresner | Definition of pressure and transmission angles applicable to multi-input mechanisms[END_REF][START_REF] Bawab | Rectified synthesis of six-bar mechanisms with welldefined transmission angles for four-position motion generation[END_REF][START_REF] Balli | Transmission angle in mechanisms (Triangle in mech)[END_REF][START_REF] Takeda | A transmission index for in-parallel wire-driven mechanisms[END_REF]. The screw theory based transmission indices have been widely used to evaluate the motion/force transmissibility of various parallel manipulators [START_REF] Wang | Performance evaluation of parallel manipulators: Motion/force transmissibility and its index[END_REF][START_REF] Liu | A new approach for singularity analysis and closeness measurement to singularities of parallel manipulators[END_REF][START_REF] Liu | A dual space approach for force/motion transmissibility analysis of lower mobility parallel manipulators[END_REF][START_REF] Zhao | Constant motion/force transmission analysis and synthesis of a class of translational parallel mechanisms[END_REF][START_REF] Brinker | Kinematic performance evaluation of high-speed delta parallel robots based on motion/force transmission indices[END_REF]. Moreover, the analysis of singularities is a central issue due to their major effect onto the robot performance. The transmission index being bounded between 0 and 1 and characterized by the sine/cosine function of the transmission/pressure angle, the transmission analysis can be used to identify singularities numerically and to evaluate the closeness to a singularity [START_REF] Liu | A generalized approach for computing the transmission index of parallel mechanisms[END_REF], compared to the Grassmann geometry, Grassman-Cayley algebra and the Jacobian method [START_REF] Gosselin | Singularity analysis of closed-loop kinematic chains[END_REF][START_REF] Merlet | Singular configurations of parallel manipulators and Grassmann geometry[END_REF][START_REF] Amine | Singularity conditions of 3T1R parallel manipulators with identical limb structures[END_REF][START_REF] Amine | Classification of 3T1R parallel manipulators based on their wrench graph[END_REF]. Accordingly, the pressure angle is adopted as the transmission index in this work. This paper presents a uniform evaluation approach for transmission quality and singularity analysis of a family of four-limb 3T1R parallel robots with an articulated mobile platform. We show that the determinant of the forward Jacobian matrices for this class of parallel robots can be expressed as the scalar product of two vectors, the first vector being the cross product of the four unit vectors along the parallelograms, the second one being related to the rotation of the mobile platform. The pressure angles, derived from the forward and inverse Jacobians straightforward, are used for both the evaluation of the transmission quality and the singularity identification of the robots. Finally, five parallel Schönflies motion generators are compared based on the proposed index as an illustrative example.

Manipulator Architecture

Figure 2(a) depicts the simplified CAD model of a Schönflies-motion robot, which is composed of four identical RRΠRR 1 -type limbs to connect the base and mobile platform (MP). For this type of robots, the universal (U) joints composed of Π joints is kinematically equivalent to parallelogram with spherical (S) joints, hence, the U joints are adopted to depict the robot linkage. The general base platform and the different mobile platforms are displayed in Figs. 2(b) and 2(c), respectively.

The fixed coordinate frame F b is built with the origin located at the geometric center of the base platform. The x-axis is parallel to the segment A 2 A 1 (A 3 A 4 ), and the z-axis is normal to the baseplatform plane pointing upwards. The moving coordinate frame F p is attached to the mobile platform and the origin is at the geometric center, where X-axis is parallel to segment C 2 C 1 (C 3 C 4 ). Hereafter, vectors i, j and k represent the unit vectors along x-, yand z-axis, respectively. The axis of rotation

η i )rs, s = R z (φ)j V4 -sgn(cos η i )rR y (φ)i + sgn(cos η i )cj + p sgn(cos η i )rs, s = R y (φ)k Veloce. rR z (η i )i + mod(i, 2)hφ/(2π)k + p mod(i, 2)hφ/(2π)k
a sgn(•) stands for the sign function of (•), and mod stands for the modulo operation, h being the lead of the screw pair of the Veloce. robot of the ith actuated joint is parallel to unit vector u i = R z (α i )i, where R stands for the rotation matrix around an axis, and α 1 = -α 2 = α, α 3 = -α 4 = β. Moreover, unit vectors v i and w i are parallel to segments A i B i and B i C i , respectively, namely, the unit vectors along the active link and parallelogram in the ith limb.

Kinematics and Jacobian Matrices of the Robots

The Cartesian coordinates of points A i and B i expressed in the frame F b are respectively derived by

a i = R [cos η i sin η i 0] T ( 1
)
b i = bv i + a i ; v i = R z (α i )R x (θ i )j (2)
where η i = (2i -1)π/4, i = 1, ..., 4, and θ i are the input angles.

Let the mobile platform pose be denoted by χ = p T φ T , p = [x y z] T . The inverse geometric problem of the manipulators is presented in detail in [START_REF] Pierrot | Optimal design of a 4-dof parallel manipulator: From academia to industry[END_REF]. It can be solved from the following kinematic constraint equations:

(c i -b i ) T (c i -b i ) = l 2 , i = 1, ..., 4 ( 3 )
where c i are the Cartesian coordinates of point C i in frame F b , as listed in Table 1.

Let w i = (c ib i )/l, upon differentiating Eq. ( 3) with respect to time, one obtains the following equation

w T i ( ċi -ḃi ) = w T i ( φrs i + ṗ -θi bu i × v i ) = 0 (4)
that can be expressed as

w T i ṗ + φrw T i s i = θi bw T i (u i × v i ), i = 1, ..., 4 ( 5 )
where ṗ = [ ẋ ẏ ż] T , and s i are unit vectors that depend on the rotation angle of the mobile platform given in Table 1. Equation ( 5) can be cast in a matrix form:

A χ = B θ (6)
with

A = [e 1 e 2 e 3 e 4 ] T ; χ = ẋ ẏ ż φ T (7a) B = diag [h 1 h 2 h 3 h 4 ] ; θ = θ1 θ2 θ3 θ4 T ( 7b
)
where A and B are the forward and inverse Jacobian matrices, respectively, and

e i = w T i rw T i s i T ; h i = bw T i (u i × v i ) ( 8 )
As long as A is nonsingular, the kinematic Jacobian matrix of the manipulators is expressed as

J = A -1 B (9)
For a given pose of the moving-platform, each limb can have two postures, which are characterized by the sign "-/+" of h i . It means that the robot can have up to sixteen working modes, i.e., sixteen solutions to the inverse kinematics problem. In order for the robot not to reach any serial singularity and to remain in one working mode, the working mode defined by h i < 0, i = 1, ..., 4, is selected. The robot is supposed not to switch from this working mode to another one.

Performance Evaluations

Our interests are the transmission quality and also the singularity, both are related to the robot Jacobian [START_REF] Wu | Kinematic analysis and optimal design of a wall-mounted four-limb parallel Schönfliesmotion robot for pick-and-place operations[END_REF].

Transmission Quality

Here, two pressure angles, corresponding to the motion (input) and force (output) transmissions, are first derived from the forward and inverse Jacobians, respectively.

|B| = 4 i=1 h i = b 4 4 i=1 w T i (u i × v i ) (10)
subsequently, the pressure angle μ i associated with the motion transmission in the ith limb, i.e., the motion transmitted from the actuated link to the parallelogram, is defined as:

μ i = cos -1 w T i (u i × v i ) , i = 1, ..., 4 (11)
namely, the pressure angle between the velocity of point B i along the vector of u i × v i and the pure force applied to the parallelogram along w i , as shown in Fig. 3(a).

Likewise, while using the Laplace expansion, the determinant |A| of the forward Jacobian matrix

|A| = w T 1 rw T 1 s w T 2 -rw T 2 s w T 3 -rw T 3 s w T 4 rw T 4 s = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ -rw T 1 s w T 2 w T 3 w T 4 + rw T 4 s w T 1 w T 2 w T 3 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ + ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ -rw T 2 s w T 1 w T 3 w T 4 + rw T 3 s w T 1 w T 2 w T 4 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (12) = r(w 1 × w 4 ) × (w 2 × w 3 ) • s -r(w 2 × w 3 ) × (w 1 × w 4 ) • s = 2r(w 1 × w 4 ) × (w 2 × w 3 ) • s = 2r(w 14 × w 23 ) T s
where w mn = w m × w n , and s are given in Table 1. By the same token, the expressions of |A| for the robots under study can be calculated as listed in Table 2, where the corresponding vectors s are listed in Table 1.

Taking the Quattro robot as an example, the pressure angle σ amongst limbs, namely, the force transmitted from the end-effector to the passive parallelograms in the other limbs, provided that the actuated joints in these limbs are locked, is derived below:

σ = cos -1 (w 14 × w 23 ) T s w 14 × w 23 (13)
wherefrom the geometrical meaning of angle σ can be interpreted as the angle between the minus Y -axis (s is normal to segment P 1 P 2 ) and the intersection line of planes B 1 P 1 B 4 and B 2 P 2 B 3 , where plane

B 1 P 1 B 4 (B 2 P 2 B 3
) is normal to the common perpendicular line between the two skew lines along w 1 and w 4 (w 2 and w 3 ), as depicted in Fig. 3(b). To illustrate the angle σ physically, (w 14 × w 23 ) T s can be rewritten in the following form:

(w 14 × w 23 ) T s = w T 14 [w 3 (w 2 • s) -w 2 (w 3 • s)] = w T 23 [w 4 (w 1 • s) -w 1 (w 4 • s)] (14)
The angle σ now can be interpreted as the pressure angle between the velocity in the direction of w 1 × w 4 and the forces along w 2 × w 3 imposed by the parallelograms in limbs 2 and 3 to point P , under the assumption that the actuated joints in limbs 1 and 4 are locked simultaneously. The same explanation is applicable for the case when the actuated joints in limbs 2 and 3 are locked.

By the same token, the pressure angle for the remaining robot counterparts can be defined. Consequently, the motion κ and force ζ transmission indices (TI) in a prescribed configuration are defined as the minimum value of the cosine of the pressure angles, respectively,

κ = min(| cos μ i |), i = 1, ..., 4; ζ = | cos σ| ( 15
)
To this end, the local transmission index (LTI) [START_REF] Wang | Performance evaluation of parallel manipulators: Motion/force transmissibility and its index[END_REF] is defined as

η = min{κ, ζ} = min{| cos μ i |, | cos σ|} ∈ [0, 1] (16)
The larger the value of η, the better the transmission quality of the manipulator.

Singularity Analysis

It is known that the manipulator reaches a singularity when the determinant of its forward or/and the determinant of its inverse Jacobian matrix is (are) equal to zero [START_REF] Gosselin | Singularity analysis of closed-loop kinematic chains[END_REF]. Henceforth, the manipulator singularities are analyzed by resorting to the motion/force transmission, as this approach can effectively evaluate the closeness to a singularity, compared to the Jacobian method [START_REF] Gosselin | Singularity analysis of closed-loop kinematic chains[END_REF][START_REF] Wu | Dynamic modeling and design optimization of a 3-DOF spherical parallel manipulator[END_REF] and Grassmann

Geometry [START_REF] Merlet | Singular configurations of parallel manipulators and Grassmann geometry[END_REF] or Grassmann-Cayley Algebra [START_REF] Amine | Singularity conditions of 3T1R parallel manipulators with identical limb structures[END_REF]. For the Quattro robot, apparently, Eqs. [START_REF]Adept Quattro Parallel Robots[END_REF] and [START_REF] Salgado | Synthesis and design of a novel 3T1R fully-parallel manipulator[END_REF] show that:

• the type-1 singularity, also named limb singularity, occurs when μ i = 90 • , namely, |B| = 0;

• the type-2 singularity, also named actuation singularity [START_REF] Amine | Classification of 3T1R parallel manipulators based on their wrench graph[END_REF][START_REF] Maraje | Operation modes comparison of a reconfigurable 3-PRS parallel manipulator based on kinematic performance[END_REF], occurs when σ = 90 • , leading to |A| = 0.

The pressure angles μ i and σ are now employed to describe the type-1 and type-2 singularities.

Figure 4 shows the type-1 singularities, i.e., the actuated link A i B i and the segment of parallelogram B i C i being coplanar. Type-1 singularities usually exist at the reachable workspace boundaries, outside the regular operational workspace, which can be readily identified. In this work, we will focus on the type-2 singularity. • Figure 5(d) shows another parallel singularity in the case of the general linear complex. The wrench exerting on the end-effector is generated by one of the transmission wrenches that is proportional to those that are generated by the other transmission wrenches, thus, the robot loses 1-DOF motion in this singular configuration.

Based on the above analysis, the transmission index defined in Eq. ( 16) can be used for the evaluation of singularity and closeness to a singularity, where a large value for η means that the robot is far away from singular configurations. The singularity occurs when η = 0. When η ≤ , 0 < 1 being a computational tolerance, the robot is considered to be in a singular configuration.

It can also effectively measure the closeness to a singularity that cannot be found by the Grassmman algorithm based singularity analysis, with the index's value bounded by [0, 1]. Moreover, this method can efficiently evaluate the closeness to a singularity numerically, compared to the conditioning of the normalized Jacobian matrix by a characteristic length due to the dimensional inhomogeneity, as the latter significantly depends on the selection of the characteristic length that is not unique.

Case Studies-Performance Evaluation of PnP Robots

In this section, the motion/force transmission and singularities of the Quattro, H4, Veloce. and V4 robots are analyzed. According to the technical parameters of the Quattro robot [START_REF]Adept Quattro Parallel Robots[END_REF], the parameters of the robots' base and mobile platforms are given in Table 3, and other parameters are set to R = 275 mm, b = 375 mm and l = 800 mm, respectively. The operational WS of the robots are shown in Fig. 6.

Pressure Angles Along a Spiral Trajectory

To illustrate the previously defined pressure angles, the Jacobian determinants and the pressure angles of the Quattro robot along a spiral trajectory within the operational workspace are plotted in Fig. 7.

The spiral path is expressed as:

x = 650ρ cos ϑ; y = 650ρ sin ϑ; z = -975 + 500ρ; φ = πρ -π/2 (17) It is noteworthy that the robot encounters Type-2 singularity when |φ| = π/2 at the end points of the path, as shown in Fig. 7(b). On the other hand, the pressure angle σ is equal to zero, i.e., the transmission quality of force transmission is the best, when φ = 0. Moreover, the pressure angles μ i are in the range of [0, 65 • ], meaning that the robot is far from the type-1 singularity, which is consistent with the varying of the determinants of inverse Jacobian.

Isocontours of the Transmission Index

The LTI isocontours of the four robots under consideration are visualized in Figs. 8 to 11, together with the max-/minimum LTI within the operational workspace listed in Table 4. It is seen that the minimum LTI of the Quattro and Veloce. robots are much higher than those of H4 and V4 robots. Moreover, the volume of the working area where LTI is greater than 0.7, namely, where the transmission quality is good, is larger with the Quattro and Veloce. robots than with the H4 and V4 robots. This means that the four-limb robots with a fully symmetrical architecture have better transmission performance than their counterparts with an asymmetric one. It should be also noted As displayed in Fig. 8, the transmission index of the Quattro robot has larger values in the central region, which admits a singularity-free workspace with rotational capability φ = ±45 • . Similarly, Fig. 9 shows that the Veloce. robot can also have a high-transmission workspace free of singularity with smaller lead of screw pair, which means that this type of mobile platform allows the robot to have high performance in terms of transmission quality and a large orientation workspace. On the contrary, the asymmetric H4 and V4 robots result in relatively small operational workspace and relatively low transmission performance, as shown in Figs. 10 and11, but with similar mechanism footprint ratio.

On the basis of the previous comparison, let's investigate transmission performance of the recently reported 'Ragnar' robot [START_REF] Wu | Architecture optimization of a parallel Schönflies-motion robot for pick-and-place applications in a predefined workspace[END_REF] with the mobile platform of Veloce. robot. This four-limb robot adopts an asymmetrical architecture, as depicted in Fig. 12, which admits a long and narrow workspace for the efficient use of the shop-floor space. The Cartesian coordinates of points A i , i = 1, ..., 4, are the corresponding LTI isocontours over the workspace are displayed in Fig. 13 sequentially with the max-/minimum LTI in Table 4, from which it is seen that this platform can also make the Ragnar robot have a high-transmission workspace free of singularity. This new robot configuration performs high transmission quality with high rotational capability, particularly, a rectangular workspace suitable for pick-and-place operations.

Closeness to Actuation Singularity

With aid of the transmission analysis, for instance, the LTI distributions in Figs. 8 to 11, numerical evaluation of singularity or closeness to a singularity is applicable. Based upon the analysis, this type of robots is free of constraint singularity under the predefined working mode in Sec. robots with the articulated mobile platforms are limited to ±90 • in practice. Moreover, the Quattro robot still admits a large singularity-free workspace with rotational capability φ = ±75 • , while the V4 robot has the smallest dexterous workspace with the increasing rotation angle. On the other hand, the V4 robot can be mounted vertically to save space.

Conclusions

This paper presents a uniform evaluation method for the transmission quality and singularity analysis for a class of four-limb parallel Schönflies-motion robots with articulated mobile platforms, which are closely in connection with two pressure angles derived from the forward and inverse Jacobian matrices.

The cosine function of the pressure angles based indices are defined to evaluate the transmission quality.

Moreover, the two pressure angles can be used efficiently to measure the transmission quality of four- The proposed method is applied to five robot counterparts, where critical rotation angles, i.e., 75 • , 60 • and 45 • , are identified for three of them with the end-effectors connected to the two subplatforms by two revolute joints, showing their constant-orientation translational workspace with different rotational capabilities. Finally, it is shown that robots with the screw-pair-based mobile platform, i.e., the Veloce. robot and the new configuration of the asymmetrical Ragnar robot, lead to high transmission quality and large operational workspace free of singularity.

1Figure 2 :

 2 Figure 2: The parameterization of the four-limb 3T1R parallel robots: (a) simplified CAD model; (b) a general base platform; (c) different mobile platforms.

Figure 3 :

 3 Figure 3: The pressure angles of the four-limb robots in the motion/force transmission: (a) μ i for all robots; (b) σ for Quattro.

Figure 5

 5 Figure 5 shows four typical Type-2 singularities with σ = 90 • or ζ = | cos σ| = 0 [5, 50]:

Figure 4 :

 4 Figure 4: The type-1 singularity of the ith limb with φ ∈ [-π/2, π/2]: (a) limb in a singular configuration with coplanar vectors u i , v i , w i ; (b) singularity loci.

Figure 5 :Table 3 :

 53 Figure 5: Four cases of the Type-2 singularities of the Quattro robot: (a) w 1 w 4 (w 2 w 3 ); (b) w 14 w 23 ; (c) w 14 (w 23 ) s; (d) (w 14 × w 23 ) ⊥ s.

Figure 6 :

 6 Figure 6: The operational workspace of the robots with rotational capability of φ = ±90 • .

Figure 7 :

 7 Figure 7: The Jacobian determinants and pressure angles along a spiral line: (a) trajectory; (b) determinants and pressure angles.

Table 4 :

 4 The min-/maximum LTI within the operational workspace.robots φ = 0 φ = 45 • φ = 360 • (h = 20 mm)LTI η min η max η min η max η min η max the rotation angle of the moving-platform, the lower the transmission performance of the robots.

Figure 8 :Figure 9 :

 89 Figure 8: The LTI isocontours of the Quattro robot: (a) φ = 0; (b) φ = 45 • .

Figure 10 :

 10 Figure 10: The LTI isocontours of the H4 robot: (a) φ = 0; (b) φ = 45 • .

Figure 11 :T, a 2 =

 112 Figure 11: The LTI isocontours of the V4 robot: (a) φ = 0; (b) φ = 45 • .

Figure 12 :

 12 Figure 12: The asymmetrical Ragnar robot: (a) CAD model; (b) parameterization of the ith limb.

Figure 13 :

 13 Figure 13: The LTI isocontours of the Ragnar robot with the screw-paired mobile platform of lead h = 20: (a) φ = 0; (b) φ = 2π.

Figure 14 :

 14 Figure 14: Actuation singularity loci of Quattro: (a) φ = ±75 • ; (b) φ = ±90 • .

Figure 15 :

 15 Figure 15: Actuation singularity loci of H4: (a) φ = ±60 • ; (b) φ = ±90 • .

Figure 16 :

 16 Figure 16: Actuation singularity loci of V4: (a) φ = ±45 • ; (b) φ = ±90 • .

Table 1 :

 1 Cartesian coordinates of points C i and unit vectors of s i

	a

Table 2 :

 2 The determinant of the forward Jacobian matrix for the four-limb robots.

	robots Quattro (H4)	V4	Veloce.
	|A|	2r(w 14 × w 23 )

T s 2r(w 14 × w 23 ) T s hφ/(2π)(w 13 × w 24 ) T k A for the Quattro robot is expressed as:

Acknowledgement

This work was partly supported by the Doctoral Scientific Research Foundation of Liaoning Province (No. 20170520134) and Innovation Fund Denmark (137-2014-5). The support from Aalborg University and Blue Workforce A/S is gratefully acknowledged.


Jiuchun Gao
email: jiuchun.gao@ls2n.fr

Anatol Pashkevich
email: anatol.pashkevich@ls2n.fr

Marco Cicellini
email: marco.cicellini@eleves.ec-nantes.fr

Stéphane Caro
email: stephane.caro@ls2n.fr

Optimal Coordination of Robot Motions with Positioner and Linear Track in a Fiber Placement Workcell

Keywords: Redundant Robotic System, Composite Fiber Placement, Optimal Motion Coordination

The paper proposes a methodology for optimal coordination of motions in robotic systems with multiple redundant actuators. In contrast to our previous results dealing with a single redundant axis, the extended technique is proposed allowing the robot, positioner and linear track to be actuated simultaneously in order to reduce the total processing time. The developed technique transforms the original continuous problem into a discrete one where the desired time-optimal motions are presented as a shortest path on the task graph satisfying the problem-specific acceleration and velocity constraints imposed on the joint coordinates. The desired time optimal motions are generated using enhanced dynamic programming algorithm that considers both of these constraints. Two case studies are presented to demonstrate efficiency of the approach and evaluate benefits of simultaneous actuation of all robotic system axes.

INTRODUCTION

Currently, composite materials have been increasingly used in aerospace and automotive industries because of their good strength-to-weight ratio and durability [START_REF] Pham | A review on design, manufacture and mechanics of composite risers[END_REF][START_REF] Garoushi | Fiber-Reinforced Composites[END_REF]. For fabricating complex composite parts, automated fiber placement technique is widely used [START_REF] Gay | Composite materials: design and applications[END_REF][START_REF] Frketic | Automated manufacturing and processing of fiberreinforced polymer (FRP) composites: An additive review of contemporary and modern techniques for advanced materials manufacturing[END_REF]. The relevant technological process can be implemented by using either specifically designed CNC machines or robotic systems. Such machines have no limitations on the component size, but they are expensive and usually require large work-floor areas [START_REF] Gallet-Hamlyn | Multiple-use robots for composite part manufacturing[END_REF]. In contrast, the robotic systems are relatively cheap and flexible, allowing changing the product type easily. However, they are usually kinematically redundant because of excessive number of actuated axes that are provided by a 6-dof robot, a 1-dof positioner and a 1-dof linear track. For this reason, in robotic fiber placement the optimal coordination of the manipulator motions with the positioner/track movements is an important issue. In literature, there are a number of works that deal with the redundancy resolution in robotic systems. Relevant techniques are usually based on the pseudo inverse of the kinematic Jacobian [START_REF] Flacco | Discrete-time redundancy resolution at the velocity level with acceleration/torque optimization properties[END_REF]. However, they can be hardly applied to the considered problem because they do not allow generating optimal trajectories satisfying real-life industrial requirements [START_REF] Kazerounian | Redundancy resolution of serial manipulators based on robot dynamics[END_REF]. Alternatively, there are also several techniques based on conversion of the original continuous problem to a discrete one. The simplest one is able to generate time-optimal trajectories for point-to-point motions and was applied to the spot-welding [START_REF] Gueta | Coordinated motion control of a robot arm and a positioning table with arrangement of multiple goals[END_REF][START_REF] Gueta | Compact design of a redundant manipulator system and application to multiple-goal tasks with temporal constraint[END_REF]. A slightly different method was proposed in [START_REF] Pashkevich | Multiobjective optimization of robot motion for laser cutting applications[END_REF] for the laser cutting and arc-welding applications where the motion amplitude for the actuated axes was minimized but the tool speed was assumed to be constant. For the considered process, where the tool speed variations are allowed in certain degree, a discrete optimization based methodology was proposed in our previous work [START_REF] Gao | Optimization of the robot and positioner motion in a redundant fiber placement workcell[END_REF]. It allows the user to convert the original problem to the combinatorial one taking into account particularities of the fiber placement technology and to generate time-optimal coordinated motions for the robot and positioner. However, the technique was applied to a planar benchmark example only, with a single redundant variable. In this work, an extension of the previous results is proposed allowing dealing with the optimal motion coordination for robotic systems with higher degree of redundancy, which arises when the robotic manipulator, the positioner and linear track actuated simultaneously.

ROBOTIC FIBER PLACEMENT PROBLEM

A typical robotic fiber placement workcell is presented in Figure 1. Here, the workpiece to be covered by the composite fiber reinforcement is manipulated by the positioner, which is able to change its orientation in order to improve accessibility of certain zones by the technological tool. This tool is attached to the robot flange and ensures placement of the fiber reinforcement in desired locations. The robot is installed on a translational linear track allowing adjusting its base location while processing relative large products.

Figure 1: Robotic fiber placement workcell.

In robot-based composite manufacturing, preparation of the manufacturing process includes a number of stages presented in Figure 2, where the motion coordination of all robotic system components is one of the most difficult procedures. Within this process, the desired fiber placement path is generated using a dedicated CAM system and it is presented in a discrete form. Further, the obtained set of task points is transformed into the task graph that describes all possible combinations of the robot, positioner and linear track coordinates. Then, the motion generator produces the optimal trajectories that correspond to the "shortest" path on the task graph. Finally, the obtained motions are converted into the program for the robot control system.

SYSTEM KINEMATIC MODEL

To describe the fiber placement task, let us present it as a sequence of the frames

 () { , 1,2,... } i task F i
n , in such a way that the X-axis is directed along the path direction and Z-axis is normal to the workpiece surface pointing outside of it. Using these notations, the task locations can be described by a set of 4×4 homogenous transformation matrices and the considered task is formalized as follows:

  
(1) ( ) ( ) ; 1,2,... (1) where all vectors of positions and orientations are expressed with respect to the workpiece frame (see superscript "w"). To execute the given fiber placement task, the technological tool must visit the frames defined by (1) as fast as possible. In any task location, the spatial configurations of the robot, positioner and linear track can be described by the joint coordinates R q , P q and L q . So, the task locations can be expressed using the direct kinematic functions of these components, which are further denoted as () RR g q , () PP gq and () LL gq . This allows us to write the kinematic equations describing the given fiber placement task in the following form

        ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1,2,... ; World i i Tool Lbase L L R R task World i w i Pbase P P task g q g g q i n T q T TT (2)
where all notations are defined in Figure 1. It is clear that the above equations cannot be solved for R q , P q and L q in unique way because the robotic system is kinematically redundant. On the other side, it gives some freedom for optimizing the coordinated motions of the robotic manipulator with the positioner and linear track movements.

GENERATION OF OPTIMAL COORDINATED MOTIONS

To present the problem in a formal way, let us define the function

)) ( ( )) ( ( )) () () R L P R i P i L i t q t q t World Tool Lbase L L i R R i task World w i Pbase P P i task R P L R P L i R P L R P L R P L i find t q t q t i n such that T st g q t g t g q t t t q q T q T TT q q q qq                                    max ,, (
( ), ( ), ( ) 0 0, ; 1, 2, ...

R P L R P L R P L i R P L Ri R i P i L i n t cond t C colls t q t q t where t t T i n q q q q Jq q (3)
where the main objective is to minimize the total motion time using full capacities of the redundant robotic system, which are limited by the maximum velocity/acceleration values for the actuated joints.

Besides, the collision constraints () colls  as well as the distance to singularities () cond  are also taken into account. Because of specific constraints, the above presented continuous optimization problem cannot be solved in a straightforward way. For this reason, the considered problem is converted into a discrete form by sampling the redundant variables corresponding to the positioner and the linear track. Then, using ideas proposed in our previous work [START_REF] Gao | Optimization of the robot and positioner motion in a redundant fiber placement workcell[END_REF] and applying sequentially the direct kinematics of the positioner and linear track as well as the inverse kinematics of the robot, one can get a configuration state for the robotic system in joint space (see Figure 3). This allows generating an extended task graph where all task locations are ordered in time. This graph contains all possible configuration states of the considered robotic system for executing the given fiber placement path, and the desired time-optimal solution of the relevant optimization problem is presented as the shortest path connecting the initial and the final layers. An outline of the task graph generation algorithm is presented in Appendix A. Using such presentation, the original continuous problem (1) is converted into a specific shortest-path problem on the graph, where all three successive nodes satisfy the acceleration constraints and the distances between two nodes 12 ( , , )

k k i task L and  12 ( , , 1) k k i task L
are equal to the technological tool displacement time from the i th to the (i+1) th task point, which is restricted by the maximum velocities and accelerations of the robot, the positioner and the linear track. It should be also noted that some of the nodes are excluded from the graph because of violation of the collision or singularity constraints as well as the joint limits. These nodes are marked as "inadmissible" ones in Figure 4, and they are not connected to any neighbour. So, the objective function to be minimized (robot motion time)

) min i i i i k k i k k i task task n i T dist LL (4)
that is computed as Corresponding optimal solution is represented by the sequence

            11 1 2 1 2 11 1 2 1 2 ( , , ) ( , , 1) ( , , ) ( , , 1)
 1 1 2 2 1 2 1 2 1 2 ( , , 1) ( , , 2) ( , , ) { } { } { } nn k k k k k k n task task task L L L
that contains the actuated coordinates of the robot, positioner and linear track. It worth mentioning that the above expression straightforwardly takes into account the velocity constraints, while the acceleration constraints are verified by means of the second order approximation applied to the corresponding functions . It allows us to present the acceleration constraints on the desired trajectory of the considered robotic system in the following form:

                ( 1) ( ) 1 max 11 2 ii i j i j j i i i i t q t q q t t t t (5)
where

     11 1 2 1 2 ( , , ) ( , , 1) () i i i i k k i k k i i j j j q q q
and the time intervals To find the desired optimal path, conventional optimization techniques can be hardly applied because of extremely high computing time [START_REF] Gao | Optimization of the robot and positioner motion in a redundant fiber placement workcell[END_REF]. Besides, these techniques are not able to take into account the acceleration constraints that very essential here. For these reasons, a dedicated problem-oriented algorithm has been developed for this problem. This algorithm is based on the dynamic programming principle, aiming at sequentially finding the shortest paths for the problems of lower dimensions, i.e. from 

              1 2 1 2 1 2 1 2 , 1 2 ( , , ( , ) , , 1 , , min
,

k k k k i k k , i k k i k k i task task d d dist L L (6)
This expression is applied recursively, starting from the second layer of the task graph ( 2 i  ) and finishing by the last one ( in  ). So, the desired optimal path can be obtained after selection of the minimum length

 12 , , 1 k k i d
corresponding to the last layer. An outline of this path planning algorithm is presented in Appendix B. In fact, this algorithm is generalization of our previous technique that was developed for motion coordination of the robotic manipulator and positioner (without linear track). As follows from relevant study, this algorithm is rather time efficient in this more complicated case; which deals with two redundant axes.

COMPARISON OF MOTION COORDINATION STRATEGIES

To demonstrate advantages of the proposed technique, let us apply it to an industrial problem that deals with fabrication of a high-pressure composite vessel. Relevant robotic fiber placement workcell (see Figure 5) is composed of 6-axis serial robot KUKA KR210 R3100, 1-axis translational linear track KUKA KL2000 and 1-axis rotational positioner AFPT 550.

For comparison purposes, two cases will be considered where the robot base is assumed either fixed or movable by means of the linear track. These two cases correspond to different degrees of redundancy provided by the positioner only or by the positioner together with the linear track. For the first case, the technique described in our previous work [START_REF] Gao | Optimization of the robot and positioner motion in a redundant fiber placement workcell[END_REF] will be applied while the second case is based on the technique proposed in this paper.

Figure 5: Robot-based fiber placement workcell and arrangement of coordinate frames.

Optimal Motion Coordination for Fixed Robot Base

A composite vessel considered in this case study is relatively small compared to the robot workspace. It is composed of a cylindrical part and two elliptical domes at both ends of the cylinder. The cylinder is 168 mm in diameter and 1200 mm in length. The laying task includes a single circuit placement of helical lamina. This allows executing the manufacturing task with fixed robot base, which simplifies the motion coordination but obviously leads to some increase of the total motion time.

Nevertheless, here also arises another optimization problem that deals with optimal robot placement relative to the workpiece mounted on the positioner.

To find the optimal location of the robot base, the space of linear track coordinate (defining the robot placement) was sampled and the proposed motion planning technique was applied several times, assuming that the robot and the positioner coordination is required only. This yields the 2D task graph corresponding to a single redundant variable (positioner rotation angle), which was used to generate the time-optimal motions for each robot base location. Relevant results are presented in Table 1. As follows from the obtained results, the optimal robot location corresponds to the linear track coordinate 2300 mm, which ensures the smallest motion time of 4.13 sec to execute the desired technological task using capacities of the robot and the positioner only. It worth mentioning that this value is about 50% less compared to the non-optimal robot positioning when the desired path is located very close to the border of the robot workspace. In the following section, the obtained motion time will be compared with the similar value obtained for the case of movable robot base (when the actuation capacities of the linear track are also used).

Optimal Motion Coordination for Movable Robot Base

Another alternative for manufacturing the considered composite vessel is to actuate both positioner and linear track simultaneously, which corresponds to movable robot base. To coordinate motions of all mechanical components in this case, the joint coordinate spaces of both the positioner and the linear track were sampled with the step of 5° and 15 mm respectively. Then, the proposed extension of the time-optimal motion generation technique that deals with two redundant variables was applied. Relevant computational results are presented in Figure 6. As follows from them, using movable robot base allows reducing the total motion time down to 3.87 sec, which is about 6% less compared to the case of the fixed robot base. The obtained trajectories are shown in Figures 7 and8, which show the displacement and velocity profiles for all eight actuators of the 6-axis robot, 1-axis positioner and 1-axis linear track. It should be mentioned that after discrete optimization, a dedicated smoothing technique may be also used to locally improve the velocity and acceleration profiles and ameliorate the actuator working conditions.

CONCLUSIONS

This paper presents an extended technique dealing with the optimal motion coordination in redundant robotic fiber placement systems. In contrast to previous results dealing with a single redundant axis only, this technique allows the robot, positioner and linear track to be actuated simultaneously in order to reduce the total motion time.

First, the original continuous optimization problem is converted into a combinatorial one where the desired timeoptimal motions are presented as a specific shortest path on the task graph. Then, the desired time-optimal motions are generated using enhanced dynamic programming algorithm that takes into account the actuator capabilities (coordinate limits, velocities and accelerations) as well as the kinematic and geometric constraints allowing avoiding collisions and singular configurations of the manipulator. The proposed technique is illustrated by two case studies confirming simultaneous actuation of all robotic system axes. In future, the proposed technique will be generalized for the robotic systems with higher degrees of redundancy.

APPENDIX A TASK GRAPH GENERATION

As mentioned in Section 4, here the pseudo-codes of the 3D task graph generation are presented. Its input includes is a sequence of 4×4 location matrices {

( ) | 1, 2,... } Task i i n 
, the discretization densities m1 and m2, and the upper/lower limits of the redundant variables denoted as max P q , min P q and max L q , min L q respectively. The algorithm transforms the task locations () Task i into the joint space. This procedure contains two steps. Firstly, the redundant variables are uniformly discretized in the ranges of ] , [ max min P P q q and ] , [ max min L L q q , and m1×m2×n matrix 1 2 1 2 { ( , , ), ( , , )} PL q k k i q k k i is obtained, where { ( , , ), ( , , ) | , , } PL q k k i q k k i k k i    are computed. After checking with the joint limits, collision and the distance to singularities, the task graph is finally generated with nodes:

1 2 1 2 1 2 1 2
( , , ) { ( , , ), ( , , ), ( , , )}

P L R k k i q k k i q k k i k k i  Lq

Figure 2 :

 2 Figure 2: Manufacturing process preparation for robotic fiber placement processes.

  corresponding to the cases when the technological tool visits the task locations (1). So, the considered motion coordination problem can be presented as

Figure 3 :

 3 Figure 3: Transformation of the original continuous problem into discrete form.The structure of this 3D graph is presented in Figure4where the nodes  12( , , )

Figure 4 :

 4 Figure 4: Task graph corresponding to the motion generation problem with two redundant variables. can be presented as the sum of the edge weights  

  between the task locations with the indices i and i+1,

Figure 6 :

 6 Figure 6: Total motion time for the cases of non-actuated and actuated linear track.

Figure 7 :

 7 Figure 7: Displacement profiles of the coordinated timeoptimal motions of the robot, positioner and linear track.

Figure 8 :

 8 Figure 8: Velocity profiles of coordinated time-optimal motions of the robot, positioner and linear track.

Table 1 :

 1 Total motion time for different robot locations.

	Robot location [mm]	Motion time [sec]
	2000	4.99
	2100	4.80
	2200	4.88
	2300	4.13
	2400	4.16
	2500	4.19
	2600	4.75
	2700	8.51

  max (i), qP min (i)|i=1…n}upper/lower limits of positioner coordinate.{qL max (i), qL min (i)|i=1…n}upper/lower limits of linear track coordinate. Collision test function; cond(.)-Condition number calculation; Tran(.)-Transformation from robot base to positioner base.

	where 1 k	1 1, 2,... ; m k  2	2 m 1, 2,...	and	1, 2,... in 	.
	Input:
	{Task(i)|i=1…n} -matrices of task locations.
	Output:
	{L(k1,k2,i)|k1=1…m1;k2=1…m2;i=1…n}
	-3D matrices of task locations:
	Notations:
	qL, qP, qR -Linear track, positioner and robot joint
	coordinates;

{qP m1discretization density for positioner. m2discretization density for linear track. urobot configuration index. P Ttask -Transformation from positioner base to task locations; R Ttask -Transformation from robot base to task locations; Invoked functions: gP(.) -Positioner direct kinematic function; gL(.) -Linear track direct kinematic function; gR -1 (.) -Robot inverse kinematic function; coll(.)-

ACKNOWLEDGEMENTS

This work is supported by the China Scholarship Council (Grant N°210404490018). The author also acknowledged CETIM for the motivation of this research work.

APPENDIX B SHORTEST PATH SEARCH

Here, the pseudo-codes of the shortest path search are presented. The input is the 3D matrix of the locations 12 { ( , , )} k k i L , which contains information on the configuration states satisfying the equality constraint, the collision constraint and the singularity constraint. The algorithm operates with two tables ( , ) D k i and ( , ) P k i that include the minimum distances for the sub-problem of lower size (for the path 1 i  ) and the pointers to the previous locations respectively. The procedure is composed of four basic steps. The step (1) reshapes the 3D graph to the one with m1×m2 lines and n columns for simplifying the programming. In step (2), the recursive formula (6) is implemented. For the admissible configuration nodes, the acceleration constraints are examined using the expression (5) for each candidate path connecting the nodes with the indices i, i-1 and i-2. It is worth mentioning that the function () accl  requires three inputs corresponding to i, i-1 and i-2, but the location for i-2 is determined using the pointer ( , 1) P j i  to the previous location in the current path. Then, it finds the minimum path from the current node to the first column and records the reference into the pointer matrix. In steps (3) and (4), the optimal solution is finally obtained and corresponding path is extracted by means of the backtracking.

4) Backtracking:

for i=n to 2 k 0 (i-1):=P(k 0 (i),i); end

Input:

{L(k1,k2,i)|k1=1…m1;k2=1…m2;i=1…n} -3D matrix of task locations

Output:

DMINminimum path length; {k 0 (i)|i=1…n}optimal path indices; Notations:

-2D matrix of task locations;

Invoked functions: reshape(.)-transform m1×m2 matrix to one column array; dist(.) -distance between nodes; accl(.) -Acceleration test on nodes.

Positioner and linear track discretization:

for i=1 to n dqP(i):=(qP max (i)-qP min (i))/(n-1); dqL(i):=(qL max (i)-qL min (i))/(n-1); for k1=1 to m1 for k2=1 to m2 qP(k1,k2,i):=qP min (i)+(k1-1)•dqP(i); qL(k1,k2,i):=qL min (i)+(k2-1)•dqL(i); end end end

Location matrix creation:

for i=1 to n for k1=1 to m1 for k2=1 to m2 The above presented algorithm has been tested using Matlab 2016b environment (running at Intel ® i5 CPU @2.67GHz 2.67GHz). In the case of the fixed robot base, it took about one minute to generate the timeoptimal motions for each sampled robot location. In the case of the movable robot base, the computation required about one hour. It is clear that in other programming environment, the computing time would be essentially smaller.