Abhilash Nayak
email: abhilash.nayak@irccyn.ec-nantes.fr

Stéphane Caro
email: stephane.caro@ls2n.fr

Philippe Wenger
email: philippe.wenger@ls2n.fr

Local and Full-cycle Mobility Analysis of a 3-RPS-3-SPR Series-Parallel Manipulator

Keywords: series-parallel manipulator, mobility analysis, Jacobian matrix, screw theory, Hilbert dimension

 without any proof, and shown to be five in [4] and [3] with an erroneous proof. Screw theory is used to derive the kinematic Jacobian matrix and the twist system of the mechanism, leading to the determination of its local mobility. I turns out that this local mobility is found to be six in several arbitrary configurations, which indicates a full-cycle mobility equal to six. This full-cycle mobility is confirmed by calculating the Hilbert dimension of the ideal made up of the set of constraint equations. It is also shown that the mobility drops to five in some particular configurations, referred to as impossible output singularities.

Introduction

A series-parallel manipulator (S-PM) is composed of parallel manipulators mounted in series and has merits of both serial and parallel manipulators. The 3-RPS-3-SPR S-PM is such a mechanism with the proximal module being composed of the 3-RPS parallel mechanism and the distal module being composed of the 3-SPR PM. Hu et al. [START_REF] Hu | Analyses of inverse kinematics, statics and workspace of a novel 3-RPS-3-SPR serial-parallel manipulator[END_REF] analyzed the workspace of this manipulator. Hu formulated the Jacobian matrix for S-PMs as a function of Jacobians of the individual parallel modules [START_REF] Hu | Formulation of unified Jacobian for serial-parallel manipulators[END_REF]. In the former paper, it was assumed that the number of local dof of the 3-RPS-3-SPR mechanism is equal to six, whereas Gallardo et al. found out that it is equal to five [START_REF] Gallardo-Alvarado | Mobility and velocity analysis of a limited-dof series-parallel manipulator[END_REF][START_REF] Gallardo-Alvarado | Kinematics of a series-parallel manipulator with constrained rotations by means of the theory of screws[END_REF]. As a matter of fact, it is not straightforward to find the local mobility of this S-PM due to the third-order twist systems of each individual module. It is established that the 3-RPS PM performs a translation and two non pure rotations about non fixed axes, which induce two translational parasitic motions [START_REF] Hunt | Structural kinematics of in-parallel-actuated robot-arms[END_REF]. The 3-SPR PM also has the same type of dof [START_REF] Nayak | Comparison of 3-RPS and 3-SPR parallel manipulators based on their maximum inscribed singularity-free circle[END_REF]. In addition, these mechanisms are known as zero-torsion mechanisms. When they are mounted in series, the axis about which the torsional motion is constrained, is different for a general configuration of the S-PM. Gallardo et al. failed to consider this fact but only those special configurations in which the axes coincide resulting in a mobility equal to five. This paper aims at clarifying that the full-cycle mobility of the 3-RPS-3-SPR S-PM is equal to six with the help of screw theory and some algebraic geometry concepts. Although the considered S-PM has double spherical joints and two sets of three coplanar revolute joint axes, the proposed methodology to calculate the mobility of the manipulator at hand is general and can be applied to any series-parallel manipulator.

The paper is organized as follows : The manipulator under study is described in Section 2. The kinematic Jacobian matrix of a general S-PM with multiple modules is expressed in vector form in Section 3. Section 4 presents some configurations of the 3-RPS-3-SPR S-PM with the corresponding local mobility. Section 5 deals with the full-cycle mobility of the 3-RPS-3-SPR S-PM.

Manipulator under study

The architecture of the 3-RPS-3-SPR S-PM under study is shown in Fig. 1. It consists of a proximal 3-RPS PM module and a distal 3-SPR PM module. The 3-RPS PM is composed of three legs each containing a revolute, a prismatic and a spherical joint mounted in series, while the legs of the 3-SPR PM have these lower pairs in reverse order. Thus, the three equilateral triangular shaped platforms are the fixed base, the coupler and the end effector, coloured brown, green and blue, respectively. The vertices of these platforms are named A i , B i and C i , i = 0, 1, 2. Here after, the subscript 0 corresponds to the fixed base, 1 to the coupler platform and 2 to the end-effector. A coordinate frame F i is attached to each platform such that its origin O i lies at its circumcenter. The coordinate axes, x i points towards the vertex A i , y i is parallel to the opposite side B i C i and by the right hand rule, z i is normal to platform plane. Besides, the circum-radius of the i-th platform is denoted as h i . p i and q i , i = 1, ..., 6 are unit vectors along the prismatic joints while u i and v i , i = 1, ..., 6 are unit vectors along the revolute joint axes.

Kinematic modeling of series-parallel manipulators

Keeping in mind that the two parallel mechanisms are mounted in series, the end effector twist (angular velocity vector of a body and linear velocity vector of a point on the body) for the 3-RPS-3-SPR S-PM with respect to base can be represented as follows:

0 t 2/0 = 0 t PROX 2/0 + 0 t DIST 2/1 =⇒ 0 ω 2/0 0 v O 2 /0 = 0 ω PROX 2/0 0 v PROX O 2 /0 + 0 ω DIST 2/1 0 v DIST O 2 /1 (1)
where 0 t PROX 2/0 is the end effector twist with respect to the base (2/0) due to the proximal module motion and 0 t DIST 2/1 is the end effector twist with respect to the coupler

h 1 h 0 h 2 z 1 y 1 x 1 z 0 y 0 x 0 z 2 y 2 x 2 O 2 O 1 O 0 A 0 B 0 C 0 A 1 B 1 C 1 C 2 B 2 A 2 F 0 F 1 F 2 u 2 u 1 u 3 v 2 v 1 v 3 p 1 p 3 p 2 q 1 q 3 3 q 2 PROXIMAL module

DISTAL module

Fig. 1: A 3-RPS-3-SPR series-parallel manipulator

F 0 F n F 1 Module 1 Module 2 Module n F 2 F n-1
Fig. 2: n parallel mechanisms (named modules) arranged in series (2/1) due to the distal module motion. These twists are expressed in the base frame F 0 , hence the left superscript. The terms on right hand side of Eq. ( 1) are not known, but can be expressed in terms of the known twists using screw transformations. To do so, the known twists are first noted down. If the proximal and distal modules are considered individually, the twist of their respective moving platforms with respect to their fixed base will be expressed as a function of the actuated joint velocities :

A PROX 0 t PROX 1/0 = B PROX ρ13 =⇒         ( 0 r O 1 A 1 × 0 p 1 ) T 0 p T 1 ( 0 r O 1 B 1 × 0 p 2 ) T 0 p T 2 ( 0 r O 1 C 1 × 0 p 3 ) T 0 p T 3 ( 0 r O 1 A 1 × 0 u 1 ) T 0 u T 1 ( 0 r O 1 B 1 × 0 u 2 ) T 0 u T 2 ( 0 r O 1 C 1 × 0 u 3 ) T 0 u T 3         0 ω PROX 1/0 0 v PROX O 1 /0 = I 3×3 0 3×3   ρ1 ρ2 ρ3   (2)
A DIST 1 t DIST 2/1 = B DIST ρ46 =⇒         ( 1 r O 2 A 1 × 1 q 1 ) T 1 q T 1 ( 1 r O 2 B 1 × 1 q 2 ) T 1 q T 2 ( 1 r O 2 C 1 × 1 q 3 ) T 1 q T 3 ( 1 r O 2 A 1 × 1 v 1 ) T 1 v T 1 ( 1 r O 2 B 1 × 1 v 2 ) T 1 v T 2 ( 1 r O 2 C 1 × 1 v 3 ) T 1 v T 3         1 ω DIST 2/1 1 v DIST O 2 /1 = I 3×3 0 3×3   ρ4 ρ5 ρ6   (3)
where, 0 t PROX

1/0
is the twist of the coupler with respect to the base expressed in F 0 and1 t DIST 2/1 is the twist of the end effector with respect to the coupler expressed in F 1 . A PROX and A DIST are called forward Jacobian matrices and they incorporate the actuation and constraint wrenches of the 3-RPS and 3-SPR PMs, respectively [START_REF] Joshi | Jacobian analysis of limited-DOF parallel manipulators[END_REF]. B PROX and B DIST are called inverse Jacobian matrices and they are the result of the reciprocal product between wrenches of the mechanism and twists of the joints for the 3-RPS and 3-SPR PMs, respectively. ρ13 = [ ρ1 , ρ2 , ρ3 ] T and ρ46 = [ ρ4 , ρ5 , ρ6 ] T are the prismatic joint velocities for the proximal and distal modules, respectively. k r PQ denotes the vector pointing from a point P to point Q expressed in frame F k . Considering Eq. ( 1), the unknown twists 0 t PROX 2/0 and 0 t DIST 2/1 can be expressed in terms of the known twists 0 t PROX 1/0 and 1 t PROX 2/1 using the following screw transformation matrices [START_REF] Murray | A Mathematical Introduction to Robotic Manipulation[END_REF][START_REF] Binaud | The kinematic sensitivity of robotic manipulators to joint clearances[END_REF].

0 ω PROX 2/0 0 v PROX O 2 /0 = 2 Ad 1 0 ω PROX 1/0 0 v PROX O 1 /0 (4) with 2 Ad 1 = I 3×3 0 3×3 -0 rO 1 O 2 I 3×3 , 0 rO 1 O 2 =   0 -0 z O 1 O 2 0 y O 1 O 2 0 z O 1 O 2 0 -0 x O 1 O 2 -0 y O 1 O 2 0 x O 1 O 2 0   2 Ad 1 is called the adjoint matrix. 0 rO 1 O 2 is the cross product matrix of vector 0 r O 1 O 2 = [ 0 x O 1 O 2 , 0 y O 1 O 2 , 0 z O 1 O 2 ], pointing from point O 1 to point O 2 expressed in frame F 0 .
Similarly, for the distal module, the velocities 1 ω DIST 2/1 and 1 v DIST O 2 /1 can be transformed from frame F 1 to F 0 just by multiplying each of them by the rotation matrix 0 R 1 from frame F 0 to frame F 1 :

0 ω DIST 2/1 0 v DIST O 2 /1 = 0 R 1 1 ω DIST 2/1 1 v DIST O 2 /1 with 0 R 1 = 0 R 1 I 3×3 I 3×3 0 R 1 (5)
0 R 1 is called the augmented rotation matrix between frames F 0 and F 1 . Consequently from Eqs. ( 4) and (5),

0 t 2/0 = 2 Ad 1 0 t PROX 1/0 + 0 R 1 1 t DIST 2/1 (6)
Note that Eq. ( 6) amounts to the twist equation derived in [START_REF] Hu | Formulation of unified Jacobian for serial-parallel manipulators[END_REF] whereas Gallardo et al. add the twists of individual modules directly without considering the screw transformations. It is noteworthy that Equation [START_REF] Murray | A Mathematical Introduction to Robotic Manipulation[END_REF] in [START_REF] Gallardo-Alvarado | Mobility and velocity analysis of a limited-dof series-parallel manipulator[END_REF] is incorrect, so are any further conclusions based on this equation. Following Eqs. ( 2) and ( 3), with the assumption that the proximal and distal modules are not in a parallel singularity 1 or in other words, matrices A PROX and A DIST are invertible,

0 t 2/0 = 2 Ad 1 A -1 PROX B PROX ρ13 + 0 R 1 A -1 DIST B DIST ρ46 = 2 Ad 1 A -1 PROX B PROX 0 R 1 A -1 DIST B DIST ρ13 ρ46 = J S-PM ρ13 ρ46 (7)
J S-PM is the kinematic Jacobian matrix of the 3-RPS-3-SPR S-PM under study. The rank of this matrix provides the local mobility of the S-PM. Equations ( 6), ( 7) and ( 8) can be extended to a series-parallel manipulator with n number of parallel mechanisms, named modules in this paper, in series as shown in Fig. 2. Thus, the twist of the end effector with respect to the fixed base expressed in frame F 0 can be expressed as follows :

0 t n/0 = n ∑ i=1 0 R (i-1) n Ad i (i-1) t M i i/(i-1) = J 6×3n      ρM 1 ρM 2 . . . ρM n      with 0 R i = 0 R i I 3×3 I 3×3 0 R i , n Ad i = I 3×3 0 3×3 -(i-1) rO i O n I 3×3
and

J 6×3n = n Ad 1 A -1 M 0 B M 0 0 R 1 n Ad 2 A -1 M 1 B M 1 ... 0 R n A -1 M n M n (8)
where, J 6×3n is the 6 × 3n kinematic Jacobian matrix of the n-module hybrid manipulator. M i stands for the i-th module, A M i and B M i are the forward and inverse Jacobian matrices of M i of the series-parallel manipulator, respectively. ρM i is the vector of the actuated prismatic joint rates for the i-th module.

Twist system of the 3-RPS-3-SPR S-PM

Each leg of the 3-RPS and 3-SPR parallel manipulators are composed of three joints, but the order of the limb twist system is equal to five and hence there exist five twists associated to each leg. Thus, the constraint wrench system of the i-th leg reciprocal to the foregoing twists is spanned by a pure force W i passing through the spherical joint center and parallel to the revolute joint axis. Therefore, the constraint wrench systems of the proximal and distal modules are spanned by three zero-pitch wrenches, namely,

0 W PROX = 3 i=1 0 W i PROX = span 0 u 1 0 r O 2 A 1 × 0 u 1 , 0 u 2 0 r O 2 B 1 × 0 u 2 , 0 u 3 0 r O 2 C 1 × 0 u 3 1 W DIST = 3 i=1 1 W i DIST = span 1 v 1 1 r O 2 A 1 × 1 v 1 , 1 v 2 1 r O 2 B 1 × 1 v 2 , 1 v 3 1 r O 2 C 1 × 1 v 3 (9)
Due to the serial arrangement of the parallel mechanisms, the constraint wrench system of the S-PM is the intersection of the constraint wrench systems of each module. Alternatively, the twist system of the S-PM is the direct sum (disjoint union) of the twist systems of each module. Therefore, the nullspace of the 3 × 6 matrix containing the basis screws of 0 W PROX and 1 W DIST leads to the screws that form the basis of the twist system of each module, 0 T PROX = span{ 0 ξ 1 , 0 ξ 2 , 0 ξ 3 } and 1 T DIST = span{ 1 ξ 4 , 1 ξ 5 , 1 ξ 6 }, respectively. The augmented rotation matrix derived in Eq. ( 5) is exploited to ensure that all the screws are expressed in one frame (F 0 in this case). Therefore, the total twist system of the S-PM can be obtained as follows :

0 T S-PM = 0 T PROX 0 T DIST = span{ 0 ξ 1 , 0 ξ 2 , 0 ξ 3 , 0 R 1 1 ξ 4 , 0 R 1 1 ξ 5 , 0 R 1 1 ξ 6 } (10)
The order of the twist system 0 T S-PM yields the local mobility of the whole manipulator. Some general and singular configurations of the 3-RPS-3-SPR S-PM with h 0 = 2, h 1 = 1 and h 2 = 2 are considered and its mobility is listed based on the rank of the Jacobian and the order of the twist system in Table 1. For general configurations like 2 and 3, the mobility is found to be six. The mobility reduces only when some singularities are encountered. For a special configuration when the three platform planes are parallel to each other as shown in the first row of this table, the rotations of the coupler generate translational motions of the end effector. Yet, the torsional axes of both mechanisms coincide and hence, the mechanism cannot perform any rotation about an axis of vertical direction leading to a mobility equal to five. Moreover, a configuration in which any revolute joint axis in the end effector is parallel to its corresponding axis in the fixed base results in a mobility lower than six for the S-PM. For instance, for the 4th configuration in the table, there exists a constraint force f , parallel to the two parallel revolute joint axes resulting in a five dof manipulator locally. Configurations 1 and 4 are the impossible output singularities as identified by Zlatanov et al. [START_REF] Zlatanov | A unifying framework for classification and interpretation of mechanism singularities[END_REF]. It should be noted that if one of the modules is in a parallel singularity, the motion of the moving-platform of the manipulator becomes uncontrollable. A detailed singularity analysis of series-parallel manipulators will be performed in a future work for a better understanding of their behaviour in singular configurations.

Full-cycle mobility of the 3-RPS-3-SPR S-PM

The full cycle mobility can be obtained by calculating the Hilbert dimension of the set of constraint equations of the mechanism [START_REF] Husty | A Proposal for a New Definition of the Degree of Freedom of a Mechanism[END_REF]. Two Study transformation matrices are considered : 0 X 1 from F 0 to F 1 and 1 Y 2 from F 1 to F 2 composed of Study parameters x i and y i , i = 0, 1, ..., 7, respectively. Thus, the coordinates of points A j , B j and C j , j = 0, 1, 2 and vectors u k and v k , k = 1, 2, can be represented in F 0 to yield sixteen constraint equations (six for the 3-RPS PM, six for the 3-SPR Number Study parameters and configuration Rank of J S-PM Order of 0 T S-PM 1

x i = (1 : 0 : 0 : 0 : 0 : 0 : 0 : 0.75) y i = (1 : 0 : 0 : 0 : 0 : 0 : 0 : 0.8)

F 0 F 1 F 2 5 5 2
x i = (0.35 : -0.9 : 0.25 : 0 : 0.57 : 0.27 : -1.76 : -1.33) y i = (1 : 0 : 0 : 0 : 0 : 0 : 0 : -0.8)

F 0 F 1 F 2 6 6 3
x i = (0.99 : 0 : -0.10 : 0 : 0 : 0.21 : 0 : 1.92) y i = (-0.79 : -0.59 : 0.16 : 0 : -0.16 : -0.13 : -1.25 : -2.04)

F 0 F 1 F 2 6 6 4
x i = (0.99 : 0 : -0.10 : 0 : 0 : 0.21 : 0 : 1.92) y i = (-0.39 : 0 : 0.92 : 0 : 0 : -1.88 : 0 : 0.12)

F 0 F 1 F 2 f 5 5
Table 1: Mobility of the 3-RPS-3-SPR S-PM in different configurations PM, Study quadric and normalization equations for each transformations). It was established that the 3-RPS and the 3-SPR parallel mechanisms have two operation modes each, characterized by x 0 = 0, x 3 = 0 and y 0 = 0, y 3 = 0, respectively [START_REF] Schadlbauer | The 3-RPS parallel manipulator from an algebraic viewpoint[END_REF][START_REF] Nayak | Comparison of 3-RPS and 3-SPR parallel manipulators based on their maximum inscribed singularity-free circle[END_REF]. For the S-PM, four ideals of the constraint equations are considered : K 1 , when x 0 = y 0 = 0, K 2 , when x 3 = y 0 = 0, K 3 , when x 0 = y 3 = 0 and K 4 , when x 3 = y 3 = 0. The Hilbert dimension of these ideals over the ring C[h 0 , h 1 , h 2 ] is found to be six 1and hence the global mobility of the 3-RPS-3-SPR S-PM.

dimK i = 6, i = 1, 2, 3, 4. (11)

Conclusions and future work

In this paper, the full-cycle mobility of a 3-RPS-3-SPR PM was elucidated to be six. The kinematic Jacobian matrix of the series-parallel manipulator was calculated with the help of screw theory and the result was extended to n-number of modules. Moreover, the methodology for the determination of the twist system of series-parallel manipulators was explained. The rank of the Jacobian matrix or the order of the twist system gives the local mobility of the S-PM. Global mobility was calculated as the Hilbert dimension of the ideal of the set of constraint equations.

In the future, we intend to solve the inverse and direct kinematics using algebraic geometry concepts and to enlist all possible singularities of series-parallel mechanisms. Additionally, it is challenging to consider n-modules (n > 2) and to work on the trajectory planning of such manipulators the number of output parameters is equal to six and lower than the number of actuated joints, which is equal to 3n.

Parallel singularity can be an actuation singularity, constraint singularity or a compound singularity[START_REF] Nurahmi | Dimensionally homogeneous extended jacobian and condition number[END_REF][START_REF] Maraje | Operation modes comparison of a reconfigurable 3-PRS parallel manipulator based on kinematic performance[END_REF][START_REF] Amine | Classification of 3T1R parallel manipulators based on their wrench graph[END_REF]

The pdf file of the Maple sheet with calculation of Hilbert dimension can be found here : https://www.dropbox.com/s/3bqsn45rszvgdax/Mobility3RPS3SPR.pdf?dl=0

Acknowledgements This work was conducted with the support of both the École Centrale de Nantes and the French National Research Agency (ANR project number: ANR-14-CE34-0008-01).


N Úria Gala
email: nuria.gala@univ-amu.fr

Johannes Ziegler
email: johannes.ziegler@univ-amu.fr

Reducing lexical complexity as a tool to increase text accessibility for children with dyslexia

Lexical complexity plays a central role in readability, particularly for dyslexic children and poor readers because of their slow and laborious decoding and word recognition skills. Although some features to aid readability may be common to many languages (e.g., the majority of 'easy' words are of low frequency), we believe that lexical complexity is mainly language-specific. In this paper, we define lexical complexity for French and we present a pilot study on the effects of text simplification in dyslexic children. The participants were asked to read out loud original and manually simplified versions of a standardized French text corpus and to answer comprehension questions after reading each text. The analysis of the results shows that the simplifications performed were beneficial in terms of reading speed and they reduced the number of reading errors (mainly lexical ones) without a loss in comprehension. Although the number of participants in this study was rather small (N=10), the results are promising and contribute to the development of applications in computational linguistics.

Introduction

It is a fact that lexical complexity must have an effect on the readability and understandability of text for people with dyslexia [START_REF] Hyönä | Eye fixation patterns among dyslexic and normal readers : effects of word length and word frequency[END_REF]. Yet, many of the existing tools have only focused on the visual presentation of text, such as the use of specific dyslexia fonts or increased letter spacing [START_REF] Zorzi | Extra-large letter spacing improves reading in dyslexia[END_REF]. Here, we investigate the use of text simplification as a tool for improving text readability and comprehension.

It should be noted that comprehension problems in dyslexic children are typically a consequence of their problems in basic decoding and word recognition skills. In other words, children with dyslexia have typically no comprehension problems in spoken language. However, when it comes to reading a text, their decoding is so slow and strenuous that it takes up all their cognitive resources. They rarely get to the end of a text in a given time, and therefore fail to understand what they read. Long, complex and irregular words are particularly difficult for them. For example, it has been shown that reading times of children with dyslexia grow linearily with each additional letter [START_REF] Spinelli | Length effect in word naming in reading : role of reading experience and reading deficit in italian readers[END_REF] [START_REF] Ziegler | Developmental dyslexia in different languages : Language-specific or universal[END_REF]. Because children with dyslexia fail to establish the automatic procedures necessary for fluent reading, they tend to read less and less. Indeed, a dyslexic child reads in one year what a normal reader reads in two days [START_REF] Cunningham | What reading does for the mind[END_REF]) -a vicious circle for a dyslexic child because becoming a fluent reader requires extensive training and exposure to written text [START_REF] Ziegler | Modeling reading development through phonological decoding and self-teaching : Implications for dyslexia[END_REF] In this paper, we report an experiment comparing the reading performance of dyslexic children and poor readers on original and simplified corpora. To the best of our knowledge, this is the first time that such an experiment is undertaken for French readers. Our aim was to reduce the linguistic complexity of ten standardized texts that had been developped to measure reading speed. The idea was to identify the words and the structures that were likely to hamper readability in children with reading deficits. Our hypothesis was that simplified texts would not only improve reading speed but also text comprehension.

A lexical analysis of the reading errors enabled us to identify what kind of lexical complexity was particularly harmful for dyslexic readers and define what kind of features should be taken into account in order to facilitate readability.

Experimental Study

Procedure and participants

We tested the effects of text simplification by contrasting the reading performance of dyslexic children on original and manually simplified texts and their comprehension by using multiple choice questions at the end of each text. The children were recorded while reading aloud. They read ten texts, five original and five simplified in a counter-balanced order. Each text was read in a session with their speech therapists. The texts were presented on a A4 sheet printed in 14 pt Arial font. The experiment took place between december 2014 and march 2015.

After each text, each child had to answer the three multiple-choice comprehension questions without looking at the texts (the questions were the same for the original and the simplified versions of the text). Three possible answers were provided in a randomized order : the correct one, a plausible one taking into account the context, and a senseless one. Two trained speech therapists collected the reading times and comprehension scores, annotated the reading errors, and proposed a global analysis of the different errors (cf. 3.1) [START_REF] Brunel | Simplification de textes pour faciliter leur lisibilité et leur compréhension[END_REF].

Ten children aged between 8 and 12 attending regular school took part in the present study (7 male, 3 female). The average age of the participants was 10 years and 4 months. The children had been formally diagnosed with dyslexia through a national reference center for the diagnosis of learning disabilities. Their reading age1 corresponds to 7 years and 6 months, which meant that they had an average reading delay of 2 years and 8 months.

Data set

The corpora used to test text simplification is a collection of ten equivalent standardized texts (IReST, International Reading Speed Texts2 ). The samples were designed for different languages keeping the same difficulty and linguistic characteristics to assess reading performances in different situations (low vision patients, normal subjects under different conditions, developmental dyslexia, etc.). The French collection consists on nine descriptive texts and a short story (more narrative style).

The texts were analyzed using TreeTagger [START_REF] Schmid | Probabilistic part-of-speech tagging using decision trees[END_REF], a morphological analyzer which performs lemmatization and part-of-speech tagging. The distribution in terms of part-of-speech categories is roughly the same in original and simplified texts, although simplified ones have more nouns and less verbs and adjectives. Table 1 shows the average number of tokens per text and per sentence, the average number of sentences per text, the distribution of main content words and the total number of lemmas :

Simplifications

Each corpus was manually simplified at three linguistic levels (lexical, syntactic, discursive). It is worth mentioning that, in previous work, text simplifications are commonly considered as lexical and syntactic [START_REF] Carroll | Simplifying Text for Language Impaired readers[END_REF], little attention is generally paid to discourse simplification with a few exceptions. In this study, we decided to perform three kinds of linguistic transformations because we made the hypothesis that all of them would have an effect on the reading performance. However, at the time being, only the lexical simplifications have been analyzed in detail (cf. section 3.2).

The manual simplifications were made according to a set of criteria. Because of the absence of previous research on this topic, the criteria were defined by three annotators following the recommendations for readers with dyslexia [START_REF] Ecalle | Des difficultés en lecture à la dyslexie : problèmes d'évaluation et de diagnostic[END_REF] for French and [START_REF] Rello | DysWebxia. A Text Accessibility Model for People with Dyslexia[END_REF] for Spanish.

Lexical simplifications. At the lexical level, priority was given to high-frequency words, short words and regular words (high grapheme-phoneme consistency). Content words were replaced by a synonym 3 . The lexical difficulty of a word was determined on the basis of two available resources : Manulex [START_REF] Lété | Manulex : A grade-level lexical database from French elementary-school readers[END_REF] 4 , a grade-level lexical database from French elementary school readers, and FLELex (Franc ¸ois et al., 2014) 5 , a graded lexicon for French as a foreign language reporting frequencies of words across different levels.

If the word in the original text had a simpler synonym (an equivalent in a lower level) the word was replaced. For instance, the word consommer ('to consume') has a frequency rate of 3.55 in Manulex, it was replaced by manger ('to eat') that has 30.13. In most of the cases, a word with a higher frequency is also a shorter word : elle l'enveloppe dans ses fils collants pour le garder et le consommer plus tard > ... pour le garder et le manger plus tard ('she wraps it in her sticky net to keep it and eat it later').

Adjectives or adverbs were deleted if there was an agreement among the three annotators, i.e. if it was considered that the information provided by the word was not relevant to the comprehension of the sentence. To give an example, inoffensives ('harmless') was removed in Il y a des mouches inoffensives qui ne piquent pas ('there are harmless flies that do not sting').

In French, lexical replacements often entail morphological or syntactic modifications of the sentence, in these cases the words or the phrases were also modified to keep the grammaticality of the sentence (e.g. determiner and noun agreement) and the same content (meaning). Example, respectively with number and gender agreement : une partie des plantes meurt and quelques plantes meurent ('some plants die'), or la sécheresse ('drought') and au temps sec ('dry wheather').

Syntactic simplifications. Structural simplifications imply a modification on the order of the constituents or a modification of the sentence structure (grouping, deletion, splitting [START_REF] Brouwers | Syntactic French Simplification for French[END_REF]). In French, the canonical order of a sentence is SVO, we thus changed the sentences where this order was not respected (for stylistic reasons) : ensuite poussent des buissons was transformed into ensuite des buissons poussent ('then the bushes grow'). The other syntactic reformulations undertaken on the IReST corpora are the following : passive voice to active voice, and present participle to present tense (new sentence through ponctuation or coordinate conjunction).

Discursive simplifications. As for transformations dealing with the coherence and the cohesion of the text, given that the texts were short, we only took into account the phenomena of anaphora resolution, i.e. expliciting the antecedent of a pronoun (the entity which it refers to). Although a sentence where the pronouns have been replaced by the antecedents may be stylistically poorer, we made the hypothesis that it is easier to understand. For instance : leurs traces de passage ('their traces') was replaced by les traces des souris ('the mice traces').

The table 2 gives an idea of the transformations performed in terms of quantity. As clearly showed, the majority of simplifications were lexical :

3. The following reference resources were used : the database www.synonymes.com and the Trésor de la Langue Franc ¸aise informatisé (TLFi) http://atilf.atilf.fr/tlf.htm.

4

Results

Two different analyses were performed : one for quantitatively measuring the reading times, the number of errors and the comprehension scores. The second one took specifically into account the lexicon : the nature of the words incorrectly read.

Behavioral data analysis

Reading Times. The significance of the results was assessed with a pairwise t-test (Student) 6 From this table it can be seen that the overall reading times of simplified texts were significantly shorter than the reading times of original texts. While this result can be attributed to the fact that simplified texts were slightly shorter than original texts, it should be emphasized that reading speed (words per minute), which is independent of the length of a text, was significantly greater in simplified texts than in original texts.

Number of errors. The total number of errors included :

-(A) the total number of skipped words, repeated words (words read twice), interchanged words, line breaks, repeated lines (line read twice) -(B) the total number of words incorrectly read for lexical reasons (the word read is a pseudo-word or a different word) -(C) the total number of words incorrectly read for grammatical reasons (the word read has the same grammatical category (part-of-speech) but varies on number, gender, tense, mode, person)

First of all, it should be noted that participants made fewer errors in simplified texts than in original ones (5,5% vs 7,7%) 7 . The table 4 shows the distribution of all the errors : It can be noted that lexical and grammatical errors occurred equally often 8 .

Comprehension scores

6. ** significant results with p < 0.01 7. This difference was significant in a t-test (t = 2,3, p < 0.05) 8. A more detailed analysis of these errors is proposed on section 3.2.

The results of the comprehension questionnaire are better for simplified than for original texts (marginal gain 9 ) as shown on table 5 : These results entail that dyslexic children read the simplified version of the corpus without a significant loss of comprehension. If anything, they showed a marginal increase in comprehension scores for simplified texts.

Lexical analysis

As we were interested in the lexicon of the corpus, an analysis of the content words (i.e. nouns, verbs, adjectives, adverbs) incorrectly read was undertaken in order to better target the reading pitfalls. From our study, we identified 404 occurrences that were incorrectly read, corresponding to 213 different lemmas (to be precise, there were 235 tokens (22 were inflected variants), i.e. arbre and arbres, or restaient, restent, rester). 404 wrong read words corresponds to 26.81 % of the content words of the corpora, which means that more than one word out of four is incorrectly read.

It is worth mentioning that we did not count monosyllabic grammatical words as determiners, pronouns or prepositions, although an important number or errors occurred also on those tokens, i.e. le read la ('the'), ces read des ('these'), pour read par ('for'). We make the hypothesis that the readers concentrate their efforts on decoding content words, and not grammatical ones, because they are those that carry the semantic information and are thus important for text comprehension. Besides, as grammatical words are usually very short and frequent in French, they have a higher number of orthographic neighbours and people with dyslexia tend to confuse short similar words.

We distinguished the words that were replaced by a pseudo-word (29.46%) and those replaced by other existing words on French vocabulary (70.37%). These figures can be compared with those obtained by Rello and collaborators [START_REF] Rello | A First Approach to the Creation of a Spanish Corpus of Dyslexic Texts[END_REF]. Non-word errors are pronunciations that do not result in an existing word, real-word errors are pronunciations that result in an incorrect but existing word. Non-word errors appear to be higher in English (83%) and in Spanish (79%), but not in French where real-word errors were clearly a majority 10 : Grammatical variants concern variations on gender and number for nouns, and for person, tense and mode for verbs. Lexical remplacements are words read as if they were other words with orthographical similarities (lieu > île, en fait > enfin, commun > connu, etc.). Morphological variants are words of 9. p < 0.1 10. This finding will deserve more attention in future work.

the same morphological family (baisse > basse, malchanceux > malchance). As for orthographical neighbours, we specifically distinguish word pairs where the difference is only of one letter (raisins > raisons, bon > don).

Concerning word length for all the mentionned features, 36.88% of the words read were replaced by words of strictly the same length (forment > formant, catégorie > *calégorie), 14.11% were replaced by longer ones (utile > utilisé, suffisant > suffisamment), 49.01% were replaced by shorter ones (nourriture > nature, finie > fine, empilées > empli). The average length of the 404 words incorrectly read is 7.65 characters (the shortest has three characters, bon, and the longest 16, particulièrement).

The average number of orthographical neighbours is 3.24, with eight tokens having more than ten neighbours : bon, bois, basse, foule, fine, fils, garde, sont ('good, wood, low, crowd, thin, thread, keeps, are').

As far as the grammatical categories are concerned, the majority of the errors were on verbs. They concerned grammatical variants of person, tense (past imparfait > present) and mode (present > present participle). The distribution on part-of-speech tags errors is shown on table 8 In French, it is stated that the more frequent (and easier) structure is CV and V. In our results, 58,69% of the words contain this common structure, while 41,31% present a more complex structure (CVC, CVCC, CYC 11 , etc.) We finally analyzed the consistency of grapheme-to-phoneme correspondences which is particularly irregular in French (silent letters, nasal vowels, etc.) 12 . As mentioned above, the average length of the words incorrectly read is 7.65 and their average in number of phonemes is 4.95. This means that the 11. C is a consonant, V is a vowel, Y is a semi-vowel, i.e. [j] in essayait [e-se-je], [w] in doivent [dwav] 12. This is not the case for other languages, e.g. the Spanish writing system has consistent grapheme-to-phoneme correspondences.

average difference between the number of letters and the number of real phonemes is 2.71. Only four tokens were regular (same number of phonemes than letters : existe, mortel, partir, plus ('exists, mortal, leave, plus')). The highest difference is 6 in apparaissent, épargneaient ('appear, saved') with 12 letters and 6 phonemes each, and mangeaient ('ate') with 10 letters and 4 phonemes. All the words incorrectly read were thus irregular as far as grapheme-to-phoneme consistency is concerned.

4 Discussion : determining where complexity is According to the literature, complexity for children with dyslexia should be found on long and less frequent words. More precisely, from the analysis of the reading errors obtained on our first pilot-study, the errors mainly occur on verbs and nouns with complex syllable structure, i.e. irregular grapheme-tophoneme correspondences, words with many orthographic neighbours or many morphological family members which are more frequent. Visual similarity is a source of error, specially for the following pairs 13 : In all the replacements we can observe visual similarities. As shown in table 12 ,the word that is actually read tends to be in most of the cases shorter and more frequent 14 than the original one : To sum up, lexical complexity for dyslexic readers in French is to be found on verbs and nouns longer than seven characters, presenting letters with similar equivalents, with complex syllables and irregular phoneme-to-grapheme consistency. Lexical replacements of words incorrectly read should consider shorter and more frequent words and words with higher grapheme-to-phoneme consistency.

Conclusion

In this paper we have presented the results of a first pilot-study aiming at testing the effects of text simplification on children with dyslexia. From our results, reading speed is increased without a loss of 13. Other possible similar pairs (not found in our corpora) : t/f, u/v, a/o 14. The frequencies have been extracted from the Manulex database (column including the five levels). comprehension. It is worth mentioning that reading errors were lower on simplified texts (in this experiment, simplified texts contained a majority of lexical simplifications). The comprehensive analyses of reading errors allow us to propose a detailed description of lexical complexity for dyslexic children. The causes of lexical complexity were mainly related to word length (words longer than seven characters), irregular spelling-to-sound correspondences and infrequent syllable structures.

The insights obtained as a result of this first pilot-study are currently being integrated into a model aiming at providing better accessibility of texts for children with dyslexia. We are currently working in a new study with children in French schools to refine the features that are to be taken into account in our model. These results will be integrated into a tool that will automatically simplify texts by replacing complex lexical items with simpler ones.

TABLE 1 -

 1 IReST corpora features before and after manual simplifications.

TABLE 2 -

 2 Linguistic transformations on the IReST French corpora.

	Lexical Simplifications	85.91%
	Direct replacements	57.04%
	Removals	13.38%
	Replacements with morphological changes	4.93%
	Replacements with syntactical changes	10.56%
	Syntactic Simplifications	9.86%
	Reformulations	7.75%
	Constituent order	2.11%
	Discursive Simplifications	4.23%
	Total	100 %

. http://www.manulex.com 5. http://cental.uclouvain.be/flelex/

TABLE 3 -

 3 . The results are shown on table 3 : Significance of the results obtained.

	Variables	Original texts Simplified texts T value Significance
	Reading times (sec)	159.94	134.70	-3.528	0.006**
	Reading speed (words per minute)	64.85	71.10	4.105	0.003**

TABLE 4 -

 4 Distribution of the types of errors in original and simplfied texts.

TABLE 5 -

 5 Significance of the results obtained.

TABLE 6 -

 6 Error typology compared accross languages.The overall error typology that we propose is shown on table 7 :

	Type of lexical replacement			Original word	English translations
	Pseudo-word	119 29.46%	grenouille > *greniole	frog, *
	Grammatical variant	135 33.42 %	oubliaient > oublient	forgot, forget
	Lexical replacement	84	20.79%	attendent > attaquent	wait, attack
	Morphological variant	43	10.64% construction > construire	build, to build
	Orthographical neighbour	23	5.69%	jaunes > jeunes	yellow, young
	Total	404	100%

TABLE 7 -

 7 Error typology.

  :

	Part-of-speech tags of tokens incorrectly read
	VERB	196 48.51 %
	NOUN	115 28.47%
	ADJECTIVE	48	11.88%
	ADVERB	25	6.19%
	Other categories (determiners excluded)	20	4.95%

TABLE 8 -

 8 Part-of-speech distribution of the tokens in the corpora.We analyzed the syllabe structure of the 404 tokens. The average number of syllables is 2.09, the distribution is shown on table 9 :

	Number of syllabs
	1 syllab	72	30,64%
	2 syllabs	96	40,85%
	3 syllabs	47	20,00%
	4 syllabs	15	6,38%
	5 syllabs	5	2,13%
		235 100,00%

TABLE 9 -

 9 Syllabs distribution of the tokens in the corpora.

  , as shown on table10 :

	Syllable structure
	CV	230	47,03%
	V	57	11,66%
	CVC	107	21,88%
	CVCC, CCVC, CYVC	47	9,61%
	CYV, CCV, VCC, CVY	34	6,95%
	VC, YV	10	2,04%
	VCCC, CCYV, CCVCC	4	0,82%
		489 100,00%

TABLE 10 -

 10 Syllable structure.

TABLE 11 -

 11 Graphical alternations.

TABLE 12 -

 12 Lexical replacements typology with frequencies of the tokens.

We used standardized reading tests to assess the reading level of each child, i.e. lAlouette[START_REF] Lefavrais | Test de l'alouette[END_REF] and PM47[START_REF] Raven | Pm47 : Standard progressive matrices : Sets a[END_REF] and a small battery of tests to assess general cognitive abilities.

http ://www.vision-research.eu

Acknowledgements

We deeply thank the speech therapists Aurore and Mathilde Combes for collecting the reading data and providing a first analysis of the data. We also thank Luz Rello for her valuable insights on parts of the results.


Guanglei Wu

Shaoping Bai

Stéphane Caro
email: stephane.caro@ls2n.fr

Transmission Quality Evaluation for a Class of Four-limb Parallel Schönflies-motion Generators with Articulated Platforms

Keywords: Schönflies motion, Jacobian, pressure angle, transmission. 1

This paper investigated the motion/force transmission quality for a class of parallel Schönflies-motion generators built with four identical RRΠ RR-type 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, 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 of the robots. Four robots are compared based on the proposed method as illustrative examples.

Introduction

Parallel robots performing Schönflies motions are well adapted to high-speed pickand-place (PnP) operations [START_REF] Pierrot | Optimal design of a 4-dof parallel manipulator: From academia to industry[END_REF][START_REF] Amine | Singularity conditions of 3T1R parallel manipulators with identical limb structures[END_REF], thanks to their lightweight architecture and high stiffness. A typical robot is the Quattro robot [START_REF]Adept Quattro Parallel Robots[END_REF] by Adept Technologies Inc., the fastest industrial robot available. Its latest version can reach an acceleration up to 15 G with a 2 kg payload, allowing to accomplish four standard PnP cycles per second. Its similar version is the H4 robot [START_REF] Pierrot | H4: a new family of 4-dof parallel robots[END_REF] that consists of 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]. Recently, the Veloce. robot [START_REF] Veloce | [END_REF] with a different articulated platform that is connected by a screw pair has been developed. Besides, the four-limb robots with single-platform architecture have also been reported [START_REF] Wu | Architecture optimization of a parallel schönflies-motion robot for pick-and-place applications in a predefined workspace[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]. 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 for the reduced mounting space, 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 analysis of a manipulator, its kinematic Jacobian matrix plays an important role, since the dexterity/manipulability of the robot can be evaluated by the condition number of Jacobians as well as the accuracy/torque capability [START_REF] Merlet | Jacobian, manipulability, condition number, and accuracy of parallel robots[END_REF] be-tween the actuators and end-effector. On the other hand, a problem usually encountered in this procedure 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 conventional performance indices associated with the Jacobian matrix, such as norm or 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 Schönflies-motion generators are concerned, their endeffector generates a mixed motion of three translations and one rotation (3T1R), for which the terms of the kinematic Jacobian matrix do not have the same units. A common approach to overcome this problem is to introduce a characteristic length [START_REF] Altuzarra | Multiobjective optimum design of a symmetric parallel Schönflies-motion generator[END_REF] to homogenize the Jacobian matrix, whereas, the measurement significantly depends on the choice of the characteristic length that is not unique, resulting 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 accommodate 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. The pressure angle based transmission index will be adopted in this work.

This paper presents a uniform evaluation approach for transmission quality of a family of four-limb 3T1R parallel robots with articulated mobile platforms. The pressure angles, derived from the forward and inverse Jacobians straightforward, are used for the evaluation of the transmission quality of the robots. The defined transmission index is illustrated with four robot counterparts for the performance evaluation and comparison. The global 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 base-platform 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 ). Vectors i, j and k represent the unit vectors of x-, yand z-axis, respectively. The axis of rotation of the ith actuated joint is parallel to unit vector u i = R z (α i )i, where R stands for the rotation matrix, and

Manipulator Architecture

α 1 = -α 2 = α -π/2, α 3 = -α 4 = β + π/2
. Moreover, unit vectors v i and w i are parallel to the segments A i B i and B i C i , respectively, namely, the unit vectors along the proximal and distal links, respectively.

C2 C4

H pair

(lead: h) P2 (c)

Kinematics and Jacobian Matrix 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 is the input angle.

Let the mobile platform pose be denoted by χ χ χ = p T φ T , p = x y z T , the Cartesian coordinates of point C i in frame F b are expressed as

c i =    sgn(cos η i )rR z (φ )i + sgn(sin η i )cj + p, Quattro (H4) -sgn(cos η i )rR y (φ )i + sgn(cos η i )cj + p, V4 rR z (η i )i + mod(i, 2)hφ /(2π)k + p, Veloce. (3)
where 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. The inverse geometric problem has been well documented [START_REF] Pierrot | Optimal design of a 4-dof parallel manipulator: From academia to industry[END_REF]. It can be solved from the following the kinematic constraint equations:

(c i -b i ) T (c i -b i ) = l 2 , i = 1, ..., 4 (4)
Differentiating Eq. ( 4) with respect to time, one obtains

φ rw T i s i + w T i ṗ = θi bw T i (u i × v i ) (5)
with

w i = c i -b i l ; s i =    sgn(cos η i )R z (φ )j, Quattro (H4) sgn(cos η i )R y (φ )k, V4 mod(i, 2)hφ /(2π)k, Veloce. (6)
Equation ( 5) can be cast in a matrix form, namely,

A χ χ χ = B θ θ θ (7)
with

A = e 1 e 2 e 3 e 4 T ; χ χ χ = ẋ ẏ ż φ T (8a) B = diag h 1 h 2 h 3 h 4 ; θ θ θ = θ1 θ2 θ3 θ4 T (8b)
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 ) (9)
As along as A is nonsingular, the kinematic Jacobian matrix is obtained as

J = A -1 B (10)
According to the inverse Jacobian matrix, each limb can have two working modes, which is characterized by the sign "-/+" of h i . In order for the robot not to reach any serial singularity, the mode h i < 0, i = 1, ..., 4, is selected as the working mode for all the robots.

Transmission Quality Analysis

Our interests are the transmission quality, which is related to the robot Jacobian. The determinant |B| of the inverse Jacobian matrix B is expressed as

|B| = 4 ∏ i=1 h i = b 4 4 ∏ i=1 w T i (u i × v i ) (11)
sequentially, 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 (12)
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). where w mn = w m × w n . 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:

A i B i C i u i v i w i u i ×v i μ i (a)
σ = cos -1 (w 14 × w 23 ) T s w 14 × w 23 ( 14
)
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)] (15) = w T 23 [w 4 (w 1 • s) -w 1 (w 4 • s)]
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) 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 | (16)
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] (17)
The larger the value of the index η, the better the transmission quality of the manipulator. This index can also be applicable for singularity measurement, where η = 0 means singular configuration.

Transmission Evaluation of PnP Robots

In this section, the transmission index over the regular workspace, for the Quattro, H4, Veloce. and V4 robots, will be mapped to analyzed their motion/force transmission qualities. 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 1, and other parameters are set to R = 275 mm, b = 375 mm and l = 800 mm, respectively.

Table 1 Geometrical parameters of the base and mobile platforms of the four-limb robots. The LTI isocontours of the four robots with different rotation angles of mobile platform are visualized in Fig. 4, from which it is seen that the minimum LTI of the Quattro and Veloce. robots are much higher than those of H4 and V4. Moreover, the volumes of the formers with LTI ≥ 0.7 are larger, to formulate larger operational workspace with high transmission quality. This means that the four-limb robots with a fully symmetrical structure have much better transmission performance than the asymmetric robot counterparts. Another observation is that the transmission performance of the robots decreases with the increasing MP rotation angle.

As displayed in Fig. 4(a), the transmission index of the Quattro robot have larger values in the central region, which admits a singularity-free workspace with rotational capability φ = ±45 • . Similarly, Fig. 4(c) 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 rotational capability of fullcircle rotation. By contrast, the asymmetric H4 and V4 robots result in relatively small operational workspace and relatively low transmission performance, as illustrated in Figs. 4(b) and 4(d), but similar mechanism footprint ratio with same link dimensions and close platform shapes.

Conclusions

This paper presents the transmission analysis for a class of four-limb parallel Schönflies-motion robots with articulated mobile platforms, closely in connection with two pressure angles derived from the forward and inverse Jacobian matrices, wherein the determinant of the forward Jacobian matrices was simplified in an elegant manner, i.e., the scalar product between two vectors, through the Laplace expansion. The cosine function of the pressure angles based indices are defined to evaluate the transmission quality. It appears that the robot with the screw-pair-based mobile platform, namely, the Veloce., is the best in terms of transmission quality for any orientation of the mobile-platform.

Figure 2 (

 2 Figure 2(a) depicts a simplified CAD model of the parallel Schönflies-motion generator, which is composed of four identical RRΠRR 1 -type limbs connecting the base and an articulated mobile platform (MP). The generalized base platform and the different mobile platforms of the four robots are displayed in Figs. 2(b) and 2(c), respectively.The global 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 base-platform 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 ). Vectors i, j and k represent the unit vectors of x-, yand z-axis, respectively. The axis of rotation of the ith actuated joint is parallel to unit vector u i = R z (α i )i, where R stands for the rotation matrix, and α 1 = -α 2 = απ/2, α 3 = -α 4 = β + π/2. Moreover, unit vectors v i and w i are parallel to the segments A i B i and B i C i , respectively, namely, the unit vectors along the proximal and distal links, respectively.

Fig. 1

 1 Fig. 1 The four-limb PnP robots with different base and mobile platforms: (a) Quattro [1]; (b) H4 [9]; (c) Veloce. [2]; (d) "V4" [12].

Fig. 2

 2 Fig. 2 The parameterization of the four-limb robots: (a) simplified CAD model; (b) a generalized base platform; (c) three different mobile platforms for the four robots.

Fig. 3

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

  robots base mobile platform Quattro α = -π/4, β = 3π/4 r = 80 mm, c = 70 mm H4, V4 α = 0, β = π/2 r = 80 mm, c = 70 mm Veloce. α = -π/4, β = 3π/4 r = 100 mm, γ = (2i -1)π/4, h

Fig. 4

 4 Fig. 4 The LTI isocontours of the robots: (a) Quattro, φ = 0 and φ = 45 • ; (b) H4, φ = 0 and φ = 45 • ; (c) Veloce. with φ = 2π, screw lead h = 20 and h = 50; (d) V4, φ = 0 and φ = 45 • .

Acknowledgements The reported work is partly supported by the Fundamental Research Funds for the Central Universities (DUT16RC(3)068) and by Innovation Fund Denmark (137-2014-5).