Sébastien Briot
email: sebastien.briot@irccyn.ec-nantes.fr

Victor Glazunov

Vigen Arakelian
email: vigen.arakelyan@insa-rennes.fr

Investigation on the Effort Transmission in Planar Parallel Manipulators

In the design of a mechanism, the quality of effort transmission is a key issue. Traditionally, the effort transmissivity of a mechanism is defined as the quantitative measure of the power flowing effectiveness from the input link(s) to the output link(s). Many researchers have focused their work on the study of the effort transmission in mechanisms and diverse indices have been defined. However, the developed indices have exclusively dealt with the studies of the ratio between the input and output powers and they do not seem to have been devoted to the studies of the admissible reactions in passive joints. However, the observations show that is possible for a mechanism to reach positions in which the transmission indices will have admissible values but the reaction(s) in passive joint(s) can reach excessively high values leading to the breakdown of the mechanism. In the present paper, a method is developed to ensure the admissible values of reactions in passive joints of planar parallel manipulators. It is shown that the increase of reactions in passive joints of a planar parallel manipulator depends not only on the transmission angle but also the position of the instantaneous centre of rotation of the platform. It allows the determination of the maximal reachable workspace of planar parallel manipulators taking into account the admissible reactions in its passive joints. The suggested method is illustrated vie a 5R planar parallel mechanism and a planar 3-RPR parallel manipulator.

I Introduction

Parallel manipulators have many advantages in terms of acceleration capacities and payload-to-weight ratio [START_REF] Merlet | Parallel robots[END_REF], but one of their main drawbacks concerns the presence of singularities [START_REF] Bonev | Singularity analysis of 3-DOF planar parallel mechanisms via screw theory[END_REF]- [START_REF] Zlatanov | Constraint singularities of parallel mechanisms[END_REF]. It is known that in the neighbourhood of the singular positions the reactions in joints of a manipulator considerably grow up.

In order to have a better understanding of this phenomenon, many researchers have focused their works on the analysis of the effort transmission in parallel manipulators. One of the evident criterions for evaluation of effort transmission is the transmission angle (or pressure angle which is equal to 90 degrees minus the transmission angle) [START_REF] Arakelian | Increase of singularity-free zones in the workspace of parallel manipulators using mechanisms of variable structure[END_REF]- [START_REF] Balli | Transmission angle in mechanisms[END_REF]. The pressure angle is well known for characterizing the transmission quality in lower kinematic pairs, such as cams [START_REF] López-Cajún | Optimization of cam mechanisms[END_REF], but this idea was also used for effort transmission analysis in the parallel manipulators [START_REF] Arakelian | Increase of singularity-free zones in the workspace of parallel manipulators using mechanisms of variable structure[END_REF], [START_REF] Balli | Transmission angle in mechanisms[END_REF].

To evaluate the effort transmission quality, several indexes have been introduced. One of the first attempts was proposed in [START_REF] Sutherland | A transmission index for spatial mechanisms[END_REF]. This paper presents a criterion named the Transmission Index (TI) that is based on transmission wrench screw. The TI varies between 0 and 1. If it is equal to 0, the considered link is in a dead position, i.e. it cannot move anymore. If it is equal to 1, this link has the best static properties.

In the same vein as [START_REF] Sutherland | A transmission index for spatial mechanisms[END_REF], the study [START_REF] Chen | Generalized transmission index and transmission quality for spatial linkages[END_REF] generalizes the TI for spatial linkages and defines the Global TI (GTI). The authors also prove that the GTI is equal, for prismatic and revolute joints, to the cosine of the pressure angle.

The conditioning index was also proposed [START_REF] Gosselin | A global performance index for the kinematic optimization of robotic manipulators[END_REF] for characterizing the quality of transmission between the actuators and the end-effector. This index is based on the Jacobian matrix or its "norm", which relate the actuator velocities (efforts, resp.) to the platform twist (wrench, resp.) by the following relations:  t Jq  and T   w J τ , where J is the Jacobian matrix, t the platform twist, q  the input velocities,  the actuator efforts, and w the wrench applied on the platform.

All these indices have been used in many works for design and analysis of parallel mechanisms [START_REF] Rakotomanga | Kinetostatic performance of a planar parallel mechanism with variable actuation[END_REF]- [START_REF] Takeda | Kinematic and static characteristics of in-parallel actuated manipulators at singular points and in their neighborhood[END_REF]. However, it is also known that because of the non homogeneity of the terms of the Jacobian matrix, the conditioning index is not well appropriated for mechanisms having both translational and rotational degrees of freedom (DOF) [START_REF] Merlet | Jacobian, manipulability, condition Number, and accuracy of parallel robots[END_REF]. Moreover, all the previously mentioned indices do not take into account the real characteristics of the actuators, i.e. the fact that their input efforts are bounded between [-max i  , max i  ] [13].

In order to solve this problem, in study [START_REF] Hubert | Static of parallel manipulators and closeness to singularity[END_REF] a numerical analysis method has been developed. It has been proposed to characterize the force workspace of robots taking into account a given fixed wrench applied on the platform and actuator efforts comprised in the boundary interval [-max i  , max i  ]. However, this workspace depends on the given direction and norm of the external force/moment and will change with the variation of the applied wrench. Moreover, for many robot applications, the external wrench direction is not known, contrary to its norm. Therefore, in [START_REF] Briot | Optimal technology-oriented design of parallel robots for highspeed machining applications[END_REF], a way to compute the maximal workspace taking into account the actuator effort limitations for a given norm of the external force and moment was proposed. This approach is based on the computation of the transmission factors of matrix J - T , which are obtained geometrically through the mapping of a cube by the Jacobian matrix.

All the previously mentioned approaches analyse the quality of the effort transmission by taking into account the input torque limitations only. However, there are such positions of parallel manipulators for which the limitations of input torques can be satisfied and the limitation of reactions in passive joints not at all. To have a better understanding of this phenomenon, let us consider a simple example via a planar 5R manipulator (Fig. 1). Close to such singularity a small effort w applied on the endeffector of the manipulator will create a large reaction R 1 in the passive joint B. But in this pose, the actuator torques  1 will stay under acceptable values, as it depends only on the small component F 1 of the reaction R 1 . Thus, for the mentioned case, the transmission indices will have acceptable values but they will not give any information about the inadequately high values of reactions in passive joints of the mechanism.

Therefore, the development of criteria for limitation of the passive joints' reactions is also an important consideration in effort transmission field. It is a complementary condition to the transmission indices for characterizing the quality of effort transmission especially in the neighbourhood of singularity.

This paper focuses on the study of the effort transmission in planar parallel mechanisms (PPM) from the above point of view. It aims at proposing a new criterion for taking into account the passive joint reactions and at finding the relationships between this criterion and the previously developed indices, especially the transmission angle.

First, the expressions of the maximal platform joint reactions as a function of the parameters of the wrench applied on the platform is presented, i.e. the maximal force norm and the absolute value of the moment. Then, it is disclosed that the maximal values of platform joint reactions depend not only the value of the transmission angle but also the position of the instantaneous centre of rotation of the platform. Moreover, the obtained results allow one to define the ranges for the admissible values of the transmission angles and distance to the instantaneous centre of rotation that ensure a good effort transmission quality. Finally, two illustrative examples and simulation results are presented.

II. A criterion for evaluation of the passive joints reactions

For any PPM, the reaction forces R i in the platform passive joints (denoted as B i in Fig. 2, i = 1, …, 3) can be related to the external wrench w T = [f T , C] T (f is the external force, and C the scalar value of the external moment applied on the platform) applying the Newton-Euler equations at any point Q:

3 1 i i    f R and 3 1 T QBi i i C     d R (1)
where

[ , ]

T QBi QBi QBi y x   d
, with x QBi and y QBi are the coordinates of vector QB i along x and y axes (the position of point B i may vary if the passive joints linked to the platform are prismatic joints). It should be mentioned that in Fig. 2 and the followings, the double arrows indicates the direction of the platform reaction forces, not their norm. Considering that R i = R i r i , where r i is a dimensionless unit vector and ||R i || = R i , and applying the Newton-Euler equations at point B 1 , it comes that: Matrix A used in equation ( 2) is defined in [START_REF] Merlet | Parallel robots[END_REF][3] as the parallel Jacobian matrix than can be found through the differentiation of the loop closure equations of the robot with respect to the platform coordinates. As a result, this matrix is always invertible if the robot is not in a Type 2 singularity.

1 2 1 3 1 2 3 0 T T B B B B T R R R                            
The reactions R of the passive joints can be found from (2):

T    R A w (3
) where matrix A -1 can be expressed in the form [START_REF] Bonev | Singularity analysis of 3-DOF planar parallel mechanisms via screw theory[END_REF]:

1       t1 t2 t3 A s s s ,with [ , ] T T i i i   t s v (4
) where s ti is a screw corresponding to the direction of the platform twist when leg i is disconnected. Moreover, it should be noted that: v i is a dimensionless vector parallel to the platform translational velocity vector v Bi expressed at point B i , i.e. v Bi = λv i where λ is a scalar of which dimension is in m/s (Fig. 2); - i is a scalar that is related to the platform rotational velocity  i by  i = λ i [START_REF] Arakelian | Increase of singularity-free zones in the workspace of parallel manipulators using mechanisms of variable structure[END_REF] (Fig. 2). Therefore, the dimension of  i is in m -1 . On Fig. 2, point I 1 corresponds to the position of the instantaneous centre of rotation of the platform when leg 1 is disconnected.

Without loss of generality, let us consider the norm R 1 of the reaction force in the joint attaching the leg 1 to the platform (located at B 1 ). Developing (3), it comes that:

    1 1 1 1 1 1 ( ) ( ) T T T T B P B P R C C            1 1 v f d f v d f .
(5) For a given norm f of the external force f and a given value C of the external moment, and for any direction of vector f, the maximal value R 1max of R 1 appears when

  1max 1 , 2 2 1 1 1 1 1 1 max( ) 2 c o s C R R f b b C          f 1 1 v v (6)
where b 1 is the distance between the application point of the external wrench, denoted as P, and point B 1 .  1 is the angle between vectors v 1 and 3). Generalizing the approach to the other legs (i = 1, 2, 3):

1 1 B P   d (Fig.
  2 2 max , max( ) 2 cos i i i i i i i i i i C R R f b b C          f v v . (7)
Equation ( 7) characterizes the effort transmission between the external wrench applied on the platform and the reaction of the platform joint located at B i . For a given mechanism configuration and a given value of R imax , it is thus possible to find the admissible ranges for f and C, i.e. for the parameters of the external wrench applied on the platform. Moreover, in order to avoid the breakdown of the platform joint located at B i , technological requirements must imply that the admissible value of R i should not be superior to a given value R adm , i.e.

max i adm R R 
. It is shown in the following section that (7) can be related to the value of the pressure angle (the pressure angle is equal to 90 deg minus the transmission angle) but, also, to the value of the distance of the platform instantaneous centre of rotation I i when leg i is disconnected.

III. Relationships between the maximal passive joint reaction and the pressure angle

By combining (2) and ( 4), it comes that 1

T T   w1 t1 1 1
s s r v . Moreover, it was shown in [START_REF] Arakelian | Increase of singularity-free zones in the workspace of parallel manipulators using mechanisms of variable structure[END_REF] that the pressure angle of the leg 1, depicted as  1 (see Fig. 2), may be expressed as the acute angle between the directions of vectors r 1 and v 1 [START_REF]Terminology for the mechanism and machine science[END_REF]. Therefore, if B 1 does not coincide with I 1 :

1 1 c o s T    1 1 1 1 r v r v , i.e. 1 cos T   1 1 1 1 r v r v (8)
By definition, r 1 is a unit vector. As a result,  

1 1 cos   1 v .
Moreover, from the definition of a planar displacement for a rigid body,

1 1 d   1 v
, where d 1 is the distance from the platform instantaneous centre of rotation I 1 to point B 1 (Fig. 2). Introducing these expressions into (6), it comes that Fig. 3. Instantaneous system equivalent to the planar 3-RPR robot platform.

 

2 1 1 1 1 1 1 1max 1 1 / 2cos / / cos b d b d f C d R       , for d 1 >0 (9a) 1max 1 1 ( ) R f b C    , for d 1 =0 (9b)
Generalizing the approach to the other legs (i = 1, 2, 3):

max / cos i i i i f C d R     , for d i > 0 (10a) max ( ) i i i R f b C    , for d i = 0 (10b) where   2 1 / 2cos / i i i i i i b d b d      , d i is the distance
between point B i and the instantaneous centre of rotation of the platform when leg i is disconnected,  i is the pressure angle of the leg i, b i is the distance between P and B i , and  i is the angle between vectors v i and i BiP   d (and as a result between vectors B i P and B i I i -Fig. 3), BiP d , v i and  i being defined at (1) and ( 4).

It should be mentioned that the distance between point P and I i is equal to  i d i .

Equation [START_REF] López-Cajún | Optimization of cam mechanisms[END_REF] shows that, for a given set of external force and moment applied on the platform and for d i > 0, the reactions in passive joints depend not only on the pressure angle but also the position of the instantaneous centre of rotation of the platform when one of the leg is disconnected. To the best of our knowledge, this property has not been rigorously formulated and demonstrated before. It allows not only giving a qualitative evaluation of the effort transmission in PPM but also disclosing the geometrical interpretation of the problem: from the equation (10a), it can be shown that the mechanical system under study can be instantaneously replaced by a virtual cantilever attached to the ground at I i (Fig. 3) lying on a virtual point contact at B i , of which direction is parallel to the vector R i . The external force f is applied on the point P of the cantilever.

As  i is a criterion used for the kinetostatic design of robots [START_REF] Balli | Transmission angle in mechanisms[END_REF], [START_REF] Rakotomanga | Kinetostatic performance of a planar parallel mechanism with variable actuation[END_REF], [START_REF] Frisoli | A two degrees-of-freedom planar haptic interface with high kinematic isotropy[END_REF], [START_REF] Takeda | Kinematic and static characteristics of in-parallel actuated manipulators at singular points and in their neighborhood[END_REF], it is of interest to find its boundaries with respect to the technological requirements that imply that the admissible reaction R i in passive joints should not be superior to a given value R adm , i.e. max

i adm R R  .
The following of the paper focused on finding the ranges on  i and d i , d i > 0, i.e. the parameters of the equivalent cantilever system, for which this inequality max

i adm R R 
is respected. The case d i = 0 is discarded as the pressure angle cannot be defined. However, it can be shown that the joint reaction stays under acceptable value if and only if ( )

i i a d m f b C R    .
Introducing (10a) into the inequality max

i adm R R  leads to:   2 1 / 2cos / cos / i i i i i a d m i i f b d b d R C d       . ( 11
)
Please note that, as by definition, f, cos  i , d i and R adm have positive values, a necessary condition for the existence of ( 11) is:

0 c o s / adm i i R C d     cos i adm i C d R   . ( 12
)
If not, the left term of (11) will always be superior to R adm . Let us now square the left and right sides of equation ( 11) and multiply them by

2 i d (from (10a), d i > 0):     2 2 2 2 2 2 2cos cos 2 c o s i i i i i i a d m i i a d m i f d b bd d R C d C R         (13)
The obtained expression can be rewritten as: 0 ( )

i i p d  , (14) where, 2 ( )
i i i i i i i p d u d v d w    , 2 2 2 cos i adm i u R f    ,   2 2 c o s c o s i a d m i ii v CR bf      , 2 2 2 i i w C f b   .
The inequality [START_REF] Rakotomanga | Kinetostatic performance of a planar parallel mechanism with variable actuation[END_REF] has different solutions, depending on the vanishing and signs of terms u i , v i and w i . There are three main cases u i > 0, u i < 0 and u i = 0. Let us consider these cases.

A. u i > 0 u i > 0 implies that cos

adm i f R   .
In this case, the polynomial p i has got two roots but only one corresponds to the real mechanism, i.e. to a solution of [START_REF] Chen | Generalized transmission index and transmission quality for spatial linkages[END_REF]. The other root is solution of:

    2 2 2 2 2 2 2cos cos 2 c o s i i i i i i a d m i i a d m i f d b bd d R C d C R          (15)
In order to obtain a root of ( 14) with physical values, it is necessary that the condition 2 4 0

i i i v uw   is respected, i.e     2 2 2 2 2 2 2 4 2 2 cos 2 cos cos sin 0 adm i i adm i i i i i R f b C R b f b f f C         (16)
Developing and simplifying, it can be shown that this polynomial in cos  i has roots with real values if and only if:

2 2 2 4
sin 0

adm i i i R b f w   . ( 17
)
For the analysis of this inequality, two following cases must be considered: w i ≤ 0 and w i > 0.

A1. w i > 0

The condition w i > 0 implies that C > f b i . Here, (16) has no real roots, i.e. ( 16) is true for any value of  i . Thus, the condition for [START_REF] Chen | Generalized transmission index and transmission quality for spatial linkages[END_REF] to be true is that

      1 max , / cos i i adm i d d C R   . ( 18
)
where     2 1 [START_REF] Gosselin | Singularity analysis of closed-loop kinematic chains[END_REF] 2

i i i i i i d v v uw u     is the root of (14)
solution of [START_REF] Chen | Generalized transmission index and transmission quality for spatial linkages[END_REF].

A2. w i ≤ 0

The condition w i ≤ 0 implies that C ≤ f b i . ( 16) is true if its roots are bounded by 2 2 2 cos sin cos

i i i i adm i C fb C R b       (19a) or 2 2 2 cos sin cos i i i i adm i C fb C R b       (19b)
After mathematical simplifications, it can be proven that if (19b) is true, then,

cos i adm i adm C f R b R    . ( 20
)
which is in contradiction with u i > 0. Therefore, the only condition for ( 16) to be true is that 2 2 2 cos sin cos max ,

i i i i adm i adm C fb C f R b R               (21)
However, it can be demonstrated, by studying the sign of the function 2 2 2 cos sin

i i i i g C fb C f b       , that 2 2 2
cos sin

i i i adm i adm C fb C f R b R     
, for u i > 0 and w i ≤ 0. ( 22)

Thus [START_REF] Takeda | Kinematic and static characteristics of in-parallel actuated manipulators at singular points and in their neighborhood[END_REF] implies that u i > 0, which is true. Then, ( 16) is always true is this section. As a result, the only condition for (11) to be true is [START_REF] Frisoli | A two degrees-of-freedom planar haptic interface with high kinematic isotropy[END_REF].

B. u i < 0 u i < 0 implies that cos

adm i f R   .
Introducing this into [START_REF] Chen | Generalized transmission index and transmission quality for spatial linkages[END_REF] leads to 0 ≤  i ≤ 1, i.e. the cantilever allows decreasing the applied force f. Here also, two cases of coming from the analysis of (17) should be studied, i.e.: w i ≤ 0 and w i > 0.

B1. w i > 0

If w i > 0, i.e. C > f b i , from (12) it can be shown that:

cos cos i i adm i adm i C fb d R R     . ( 23
)
As in section II.B u i < 0, which is equivalent to

R adm cos  i < f, from (23) it comes cos i i adm i fb b R   , thus i i b d  . ( 24
)
If ( 24) is true, the expression of  i at (10a) is bounded by

0 i i i i i i i b d d b d d       . ( 25
)
Introducing ( 25) into [START_REF] Chen | Generalized transmission index and transmission quality for spatial linkages[END_REF], and as C > f b i , it comes that:

    cos i i i i i adm i i i f d b C f d b fb f R d d         . ( 26
)
Thus, cos

adm i f R   , or equivalently u i > 0, (27)
which is impossible in section B.

B2. w i ≤ 0

If w i ≤ 0, i.e. C ≤ f b i , it could be shown after several mathematical simplifications and looking at the results of section A2 that ( 16) is true if and only if 2 2 2 cos sin cos

i i i i adm adm i C fb C f R Rb        . ( 28
)
Once this condition is achieved, the condition for ( 11) to be true is that

  2 max , cos 2cos
i i i adm i i C b d d R             . ( 29
)
where     2 2 [START_REF] Gosselin | Singularity analysis of closed-loop kinematic chains[END_REF] 2

i i i i i i d v v uw u     is the root of (14)
solution of [START_REF] Chen | Generalized transmission index and transmission quality for spatial linkages[END_REF].

C. u i = 0

We have to analyze one last case:

u i = 0, i.e. cos / i adm f R   .
Here also, this condition leads to 0 ≤  i ≤ 1, i.e. the cantilever allows decreasing the applied force f. In such a case, the solution of ( 14) is solution of

0 i i i v d w   with   2 cos i i i v f b f C    and 2 2 2 i i w C f b  
. Three cases will appear: v i > 0, v i < 0 and v i = 0.

C1. v i > 0

In this case, max

i adm R R  can be satisfied if and only if   2 2 2 2 cos i i i i i i w f b C d v f bf C       . ( 30
) C2. v i < 0 In this case, max i adm R R  can be satisfied if and only if   2 2 2 0 2 cos i i i i i i w f b C d v f b f C        . ( 31
)
As in section II.B1, the condition w i > 0 is not compatible with the fact that u i = 0. For w i ≤ 0:

  2 2 2 0 2 cos i i i f b C f bf C     . ( 32
)
Therefore, if v i < 0, the only condition is that ( 12) should be respected.

C3. v i = 0

Condition ( 14) can be satisfied if and only if w i ≥ 0. But, as previously, w i > 0 is not compatible with the fact that u i = 0. Therefore, it exist only one possible case, w i = 0, i.e. i fb C  .

In table 1 are summarized all conditions for obtaining

max i adm R R 
, for d i > 0 (for d i = 0, the solution is directly given at (9b)). It should be mentioned that for planar parallel robots, the reactions in the other joints that are not linked to the platform can be found using linear relationships with respect to R imax (Table 2 1 ).

Let us now consider two illustrative examples.

IV. Illustrative examples

Let us now consider, for given external wrenches, the evolution of the maximal joint reactions within the workspace of two given planar robots: the DexTAR robot, which is a planar five-bar mechanism developed at the ETS [START_REF] Campos | Development of a five-bar parallel robot with large workspace[END_REF], and a 3-RPR robot, which is the planar model of the PAMINSA manipulator developed at the INSA of Rennes [START_REF] Briot | Design and prototyping of a partially decoupled 4-DOF 3T1R parallel manipulator with high-load carrying capacity[END_REF].

A. The DexTAR robot

The DexTAR is a five-bar mechanism [START_REF] Chablat | The isoconditioning loci of a class of closed-chain manipulators[END_REF] (Fig. 4) of which dimensions are:

-

l AB = l DE = l AB = l DE = 0.23 m
1 In this table, the joints depicted in grey correspond to the actuated ones.

Table 1. Conditions for limiting the maximal values of the revolute joint linked to the platform, for

d i > 0. i C fb  i C fb  i C fb  cos i adm f R     1 max , cos i i adm i C d d R           cos i adm f R   N/A 2 2 2 cos sin cos i i i i adm i C fb C R b       ,   2 max , cos 2cos
i i i adm i i C b d d R             cos i adm f R   N/A v i > 0 v i < 0 v i = 0 v i > 0 v i < 0 v i = 0 max , cos i i adm i i C w d R v            N/A N/A cos i adm i C d R   N/A i i d b  , for 0 i  
Table 2. Norm of the reaction effort inside of the leg joints of planar parallel robots.

Type of legs Joint reactions Type of legs Joint reactions RPR

T Ai w = [ T Ai f ,] T , T Ci w = [ T Ci f , C Ci ] T with Ai Ci i R   f f , i i R    , Ci i i C lR  RPR T Ai w = [ T Ai f ,] T , T Ci w = [ T Ci f , 0] T with Ai Ci i R     f f RRR T Ai w = [ T Ai f ,] T , RRR T Ai w = [ T Ai f , 0] T , T Ci w = [ T Ci f ,] T with Ai Ci i R   f f , 2 sin i i i l R    T Ci w = [ T Ci f , 0] T with Ai Ci i R   f f , 1 cos i i i l R    PRR T Ai w = [ T Ai f ,C Ai ] T , T Ci w = [ T Ci f , 0] T with Ai Ci i R   f f , 1 sin Ai i i i C l R   cos i i R    PRR T Ai w = [ T Ai f , 0] T , T Ci w = [ T Ci f ,] T with Ai Ci i R   f f , 2 sin i i i l R    PPR T Ai w = [ T Ai f ,C Ai ] T , T Ci w = [ T Ci f , C Ci ] T with Ai Ci i R   f f , 1 cos Ai i i i i C R l     2 Ci i i C Rl  , sin i i R    PPR T Ai w = [ T Ai f ,C Ai ] T , Ci w = [ T Ci f , C Ci ] T with Ai Ci i R   f f , 1 cos Ai i i i i C R l     2 cos Ci i i i C Rl   , sin i i R    PRP T Ai w = [ T Ai f ,C Ai ] T , T Ci w = [ T Ci f , C Ci ] T with Ai Ci i R   f f , 1 sin Ai i i i C Rl   , Ci i i C R  , RRP T Ai w = [ T Ai f ,] T , T Ci w = [ T Ci f , C Ci ] T with Ai Ci i R   f f , Ci i i C R  , 1 sin i i i R l    Fig. 4
. Kinematic chain of the planar five-bar robot.

a = 0.1375 m In the following of the paper, it is considered that the leg 1 is composed of the segments AB and BP and that leg 2 is composed of segments ED and DP. The active joints are located at points A and E.For five-bar mechanisms, it can be shown that the matrix A T of ( 2) is equal to [START_REF] Merlet | Parallel robots[END_REF]:

1 2 1 2 cos cos sin sin T                 1 2 A r r . ( 33
)
Moreover, disconnecting leg 1 (2, resp.) from the end-effector, for a fixed position of the actuator 2 (1, resp.), the direction v 1 (v 2 , resp.) of the translational velocity vector of the end-point of leg 2 (1, resp.) is orthogonal to the direction of the segment DP (BP, resp.) (Fig. 4). Therefore,

2 2 sin cos           1 v . ( 34
)
As a result,

      1 1 1 2 1 1 1 2 cos cos sin cos sin cos T T                               1 1 1 1 2 2 2 2 r v r v r v r v . ( 35
)
where  is the angle between segments BP and DP (Fig. 4). Please note that: - is denoted as the transmission angle of the robot [START_REF] Balli | Transmission angle in mechanisms[END_REF]; -fixing angle  is equivalent to fixing the shape of the triangle BPD and the robot can be shown as a 1-DOF planar four-bar mechanism (Fig. 5). Taking only into account the each revolute joint can admit a maximal force R adm , and as for such mechanisms no moments are applied on the controlled point, the only condition for non breakdown of the mechanism under a force f applied at P is given by cos  

1 1 m a x sin cos     , 2 1      , 3 1     , 4 1       . ( 36
)
Therefore, four four-bar mechanisms can be studied, depending on the assembly mode of the five-bar robot (Fig. 5). Therefore, the constant pressure angle loci, and as a result, the constant joint reaction loci, can be found algebraically by studying the displacement of points P of the four-bar mechanisms. These borders are portions of sextic curves [START_REF] Merlet | Parallel robots[END_REF].

The variations of the joint reaction within the workspace of the DexTAR robot, on which is applied a force equal to 100 N, are presented in Fig. 6 for the four working modes of the robot. On this picture, the dotted black lines correspond to the Type 1 singularities and the full black lines to the Type 2 singularities [START_REF] Gosselin | Singularity analysis of closed-loop kinematic chains[END_REF], i.e. the maximal workspace boundaries. It can be shown that, the closer the robot from Type 2 singular configurations, the higher the joint reactions.

In [START_REF] Campos | Development of a five-bar parallel robot with large workspace[END_REF], it is shown that the DexTAR is able to pass through Type 1 singularities [START_REF] Gosselin | Singularity analysis of closed-loop kinematic chains[END_REF]. For one given assembly mode, a position of the end-effector is able to be attained by two working modes. Taking this result into account, the borders of the force workspace for a given assembly mode are computed. As previously, these borders can also be found algebraically, by studying the displacement of points P of the four-bar mechanisms. For one given assembly mode, a point of the sextic curves will belong to the border of the force workspace if the pressure angle of the mechanism at this end-effector position is always superior or equal to  max for any of the working modes. If not, this point can be reached by at least one of the working modes, i.e. it is not a workspace boundary. Some examples of the force workspace, for several values of , are presented in Fig. 7. Obviously, the smaller  max , the smaller the workspace. When  max is large, the workspace has only one aspect. For  max small, several aspects will appear.

B. The 3-RPR robot

The PAMINSA (Fig. 8) is a parallel manipulator with 4 DOF (Schoenflies motions) of which translation along the vertical axis is decoupled from the displacement in the horizontal plane. When the vertical translational motion is locked up, the PAMINSA is fully equivalent to a 3-RPR manipulator (Fig. 8b) with equilateral platform and base triangles, of which circumcircles have the following radii:

for the base, R b = 0.35 m for the platform, R p = 0.1 m For 3-RPR robots, it can be shown that the matrix A T of (2) expressed at point B i is equal to [START_REF] Briot | Design and prototyping of a partially decoupled 4-DOF 3T1R parallel manipulator with high-load carrying capacity[END_REF]:

1 2 3 T C C C             1 2 3 w1 w2 w3 r r r A s s s . ( 37
)
with sin cos

T i i i q q       r and 1 0 C  , 2 sin 3 cos T p C R           2 r ,     3 sin / 3 3 cos / 3 T p C R                 3 r , if B i = B 1 (38a)     1 sin 3 cos T p C R                 1 r , 2 0 C  , (
T p C R                 3 r , if B i = B 2 (38b)     1 sin 4 / 3 3 cos 4 / 3 T p C R                 1 r ,     2 sin / 3 3 cos / 3 T p C R                   2 r , 3 0 C  , if B i = B 3 (38c)
For this mechanism, the way to compute the pressure angle and the distance between the observed joint and the platform instantaneous centre of rotation is explained in [START_REF] Arakelian | Increase of singularity-free zones in the workspace of parallel manipulators using mechanisms of variable structure[END_REF].

The variations of the joint reaction at point B 1 within the workspace for several platform orientations  are presented in Fig. 9 for f = 100 N and C = 5 Nm. The dotted lines correspond to the Type 2 singularities that appear if [START_REF] Briot | Design and prototyping of a partially decoupled 4-DOF 3T1R parallel manipulator with high-load carrying capacity[END_REF]: -for a given orientation  of the platform, the point P is

located on a circle C() centred in O, of radius equal to 2 2 2 cos b p b p R R R R    - for orientations 1 cos ( / ) p b R R    
, the robot is in singular configuration for any position of P. It can be observed that, the closer from Type 2 singularities, the higher the joint reaction. Moreover, the lowest values of the joint reactions appear in the centre of the workspace.

Let us analyze the force workspace of the robot. On the contrary of the DexTAR for which the obtained expressions are symbolic, and the force workspace boundaries can be obtained algebraically, for this robot, a numerical method has to be used. The method consists in discretising the workspace using polar coordinates (r, ). For one given angle , the algorithm tests for all rising values of r that the manipulator can support the applied wrench (see tables 1 and 2). In the case where the manipulator cannot support the applied wrench, the boundary of the force workspace is defined by the previous computational point.

In the remainder of this example, we take only into account the maximal admissible value R adm of the reactions of the revolute joints located at B i . It is applied on the platform a force of norm f = 100 N, and a moment of norm C = 5 Nm. The shape of the force workspace, for one given assembly mode and for several values of R max and platform orientation  is shown in Fig. 10.

It can be observed that, the greater the value of , i.e. the closer from the orientation for which the robot is in singular configuration for any position of the platform centre, the smaller the workspace.

V. Conclusions

This paper extends the previous works on the quality of the effort transmission in iso-static planar closed-loop mechanisms. The traditional transmission indices only show the ratio between input and output powers but they don't show an unacceptable high increase of the reactions in the passive joints. In this study, it is disclosed that the increase of reactions in passive joints depends not only on the transmission angle but also the position of the instantaneous centre of rotation of the platform. This is the first time that such a kinetostatic property is rigorously formulated and clearly demonstrated. The obtained results allow expanding the knowledge about the effort transmission quality. They are complementary conditions to the traditional transmission indices and allow avoiding a breakdown close to the singularity. In this aim, the boundaries for the admissible values of the transmission angle and of the distance to the instantaneous centre of rotation are computed. The Dextar and the 3-RPR robot have been studied as illustrative examples. The effort transmission in these manipulators has been studied as well as their reachable workspaces taking into account the limitation of the efforts in passive joints.

Finally, it should be noted that in this paper, the disclosed properties have been only devoted to the study of planar parallel mechanisms. It is quite possible that such a concise and accurate criterion can be also obtained for spatial case. In our future work we will try to extend this approach to spatial parallel mechanisms, but it is a real challenge.

Fig. 1 .

 1 Fig. 1. Planar 5R manipulator close to a Type 2 singular pose.

Fig. 2 .

 2 Fig. 2. Determination of the pressure angle for the planar 3-RPR robot.

  being unit screws corresponding to the direction of the platform joint reaction wrenches.

(Fig. 5 .

 5 Fig. 5. The four equivalent 4-bars mechanisms, for a fixed value of  max .

4 Fig. 6 .

 46 Fig. 6. Variation of the robot joint reaction (in Newton) within the workspace for f = 100 N.

Fig. 7 .

 7 Fig. 7. Workspace shape as a function of the maximal pressure angle  max .

Fig. 10 .

 10 Fig. 10. Constant orientation workspace as a function of R max and the platform orientation .


Maxime Gautier
email: maxime.gautier@irccyn.ec-nantes.fr

Sébastien Briot
email: sebastien.briot@irccyn.ec-nantes.fr

Global Identification of Robot Drive Gains Parameters Using a Known Payload and Weighted Total Least Square Techniques

Off-line robot dynamic identification methods are based on the use of the Inverse Dynamic Identification Model (IDIM), which calculates the joint forces/torques that are linear in relation to the dynamic parameters, and on the use of linear least squares technique to calculate the parameters (IDIM-LS technique). The joint forces/torques are calculated as the product of the known control signal (the current reference) by the joint drive gains. Then it is essential to get accurate values of joint drive gains to get accurate identification of inertial parameters. In the previous works, it was proposed to identify each gain separately. This does not allow taking into account the dynamic coupling between the robot axes. In this paper the global joint drive gains parameters of all joints are calculated simultaneously. The method is based on the weighted total least squares solution of an over-determined linear system obtained with the inverse dynamic model calculated with available current reference and position sampled data while the robot is tracking one reference trajectory without load on the robot and one trajectory with a known payload fixed on the robot. The method is experimentally validated on an industrial 6 joint Stäubli TX-40 robot.



INTRODUCTION

Several schemes have been proposed in the literature to identify the dynamic parameters of robots [START_REF] Gautier | Direct calculation of minimum set of inertial parameters of serial robots[END_REF], [START_REF] Hollerbach | Model identification -Chapter 14[END_REF], [START_REF] Khalil | Modeling, identification and control of robots[END_REF], [START_REF] Khosla | Parameter identification of robot dynamics[END_REF], [START_REF] Lu | Experimental determination of dynamic parameters of robotic arms[END_REF]. Most of the dynamic identification methods have the following features: -the use of an Inverse Dynamic Identification Model (IDIM) which calculates the joint force/torque linear in relation to the dynamic parameters, -the construction of an over-determined linear system of equations obtained by sampling IDIM while the robot is tracking some trajectories in closed-loop control, -the estimation of the parameter values using least squares techniques (LS). This procedure is called the IDIM-LS technique. The experimental works have been carried out either on prototypes in laboratories or on industrial robots and have shown the benefits in terms of accuracy in many cases. Good results can be obtained provided two main conditions are satisfied: -a well-tuned derivative band-pass filtering of joint position is used to calculate the joint velocities and accelerations, -the accurate values of joint drive gains g  are known to calculate the joint force/torque as the product of the known control signal calculated by the numerical controller of the robot (the current references) by the joint drive gains [START_REF] Restrepo | Calibration of drive chain of robot joints[END_REF]. This needs to calibrate the drive train constituted by a current controlled voltage source amplifier with gain i G which supplies a permanent magnet DC or a brushless motor with torque constant t K coupled to the link through direct or gear train with gear ratio N . Because of large values of the gear ratio for industrial robots, ( N >50), joint drive gain, i t g NG K   , is very sensitive to errors in i G and t K which must be accurately measured from special, time consuming , heavy tests, on the drive chain [START_REF] Restrepo | Calibration of drive chain of robot joints[END_REF], [START_REF] Corke | In situ measurement of robot motor electrical constants[END_REF].

Several papers on the topic of the joint drive gains identification have been published in the past [START_REF] Corke | In situ measurement of robot motor electrical constants[END_REF], (Gautier and Briot 2011a,b), [START_REF] Restrepo | Calibration of drive chain of robot joints[END_REF], but all of them propose to identify each joint gain separately. This does not allow taking into account the dynamic coupling between the robot joint force/torque.

In this paper it is proposed a new method for the global identification of the joint drive gains, using current reference and position sampled data measured while the robot is tracking one reference trajectory without load fixed on the robot and one trajectory with a known payload fixed on the robot. Contrary to the previous works, all drive gains are calculated in one step by the weighted total LS solution (WTLS) of an over-determined system in order to take into account the coupling between the robot axes. The method is experimentally validated on a 6 joint industrial Stäubli TX-40 robot.

USUAL DYNAMIC IDENTIFICATION METHOD

Inverse Dynamic Identification Model (IDIM)

It is known that the dynamic model of any manipulator can be linearly written in term of a   q,q,q, q,q,q

         (1)
where:

idm  is the   ...

T T T n T st st st st         
q,q,q   are the vectors of the joint positions, velocities and accelerations, respectively.

For rigid robots, there are 14 standard parameters by link and joint. For the joint and link j, these parameters can be regrouped into the (14×1) vector j st  [START_REF] Khalil | Modeling, identification and control of robots[END_REF]:

j j T s t j j j j j j j j j j j j j o f f XX XY XZ YY YZ ZZ MX MY MZ M Ia Fv Fc        (2)
where:

j j j j j j
XX , XY , XZ , YY , YZ , ZZ are the 6 components of the inertia matrix of link j at the origin of frame j . The identifiable parameters are the base parameters which are the minimum number of dynamic parameters from which the dynamic model can be calculated. They are obtained from the standard inertial parameters by regrouping some of them by means of linear relations [START_REF] Mayeda | Base parameters of manipulator dynamic models[END_REF]), which can be determined for the serial robots using simple closed-form rules [START_REF] Gautier | Direct calculation of minimum set of inertial parameters of serial robots[END_REF], [START_REF] Khalil | Modeling, identification and control of robots[END_REF], or by numerical method based on the QR decomposition [START_REF] Gautier | Numerical calculation of the base inertial parameters[END_REF].

The minimal dynamic model can be written using the b n base dynamic parameters  as follows:

( ) idm q,q,q       (3)
where  is obtained from st  by eliminating the columns corresponding to the non identifiable parameters.

Because of perturbations due to noise measurement and modelling errors, the actual force/torque  differs from idm τ by an error, e , such that: ( )

idm τ e q,q,q e          (4)
where

1 1 0 0 0 0 0 0 n n v g v g v g                                (5) v  is the ( ) n n
 matrix of the actual current references of the current amplifiers ( j v  corresponds to actuator j) and g  is the ( 1) n  vector of the joint drive gains ( j g  corresponds to actuator j). Equation (4) represents the IDIM.

Least Squares Identification of the Dynamic Parameters (IDIM-LS)

The off-line identification of the base dynamic parameters  is considered, given measured or estimated off-line data for τ and   q, q, q   , collected while the robot is tracking some planned trajectories. The model ( 4) is sampled and low pass filtered in order to get an over-determined linear system of ( ) n r  equations and b n unknowns:

    ˆŶ τ W q,q,q χ ρ     (6)
where ( ) ˆq, q, q   are an estimation of ( ) q, q, q   , obtained by band- pass filtering and sampling the measure of q (Gautier 1997),

ρ is the ( 1) r  vector of errors,   ˆŴ q, q, q   is the ( ) b r n  observation matrix.
Using the base parameters and tracking "exciting" reference trajectories, a well conditioned matrix W is obtained. The LS solution χ of ( 6) is given by:

    1 T T χ W W W Y W Y     (7)
It is computed using the QR factorization of W .

Standard deviations

i   , are estimated assuming that W is a deterministic matrix and  , is a zero-mean additive independent Gaussian noise, with a covariance matrix C  , such that:

T 2 ( ) r C E ρρ I      (8)
E is the expectation operator and I r , the ( ) r r  identity matrix. An unbiased estimation of the standard deviation   is:

2 2 ( ) Ŷ -W r b      (9)
The covariance matrix of the estimation error is given by:

T 2 T 1 [( )( ) ] ( ) ˆˆĈ E χ χ χ χ W W         . ( ) i 2 ˆĈ i,i      is the i th diagonal coefficient of ˆĈ  ( 10
)
The relative standard deviation ri %   is given by:

ri i ˆˆi %       , for i  ≠ 0 (11)
The ordinary LS can be improved by taking into account different standard deviations on joint j equations errors [START_REF] Gautier | Dynamic identification of robots with power model[END_REF]. Data in Y and W of ( 6) are sorted and weighted with the inverse of the standard deviation of the error calculated from OLS solution of the equations of joint j [START_REF] Gautier | Dynamic identification of robots with power model[END_REF].

This weighting operation normalises the errors in ( 6) and gives the weighted LS estimation of the parameters (IDIM-WLS).

GLOBAL IDENTIFICATION OF THE JOINT DRIVE GAINS

IDIM Including a Payload and Drive Gains

The payload is considered as a link 1 n  fixed to the link n of the robot. Only kL n of its parameters are considered known. The model (4) becomes:

T T T T uL kL uL kL v g e                     ( 12
)
where:

Weighted Total Least Squares Identification of the Drive Gains (IDIM-WTLS)

Details on the Total LS (TLS) identification method can be found in (Van Huffel and Vandewalle 1991) and many papers of the same authors. This method has been applied in [START_REF] Gautier | Identification of inertial and drive gain parameters of robots[END_REF] for the identification of the drive gains and the dynamic parameters on a two degrees of freedom robot (dof) but gives arguable results due to the lack of an accurate scale factor. In this paper three major improvements are proposed:

the accurate scaling of parameters using the precise weighed value of an additional payload mass; -a weighting procedure of rows and columns of the observation matrix taking into account an a priori confidence on the measures. -an experimental validation on a 6 dof industrial robot which shows the efficiency of these approaches. In order to identify the payload parameters, it is necessary that the robot carried out two trajectories: (a) without the payload and (b) with the payload fixed to the end-effector [START_REF] Khalil | Identification of the payload inertial parameters of industrial manipulators[END_REF]). The sampling and filtering of the model IDIM ( 12) can be then written as:

0 T a a T T T uL kL b b u L k L V W 0 Y g V W W W                           (13)
where:

a V  is the matrix of v  samples in the unloaded case, b V  is the matrix of v  samples in the loaded case, 1 1 2 2 0 0 0 0 0 0 j i , i j i, j i i i j n i,r / n i v V v V V ,V v V                                               , with i =a, b (14) j i,k
v  is the k-th sample of current reference for actuator j, a W is the observation matrix of the robot in the unloaded case, b W is the observation matrix of the robot in the loaded case, uL W is the observation matrix of the robot corresponding to the unknown payload inertial parameters, kL W is the observation matrix of the robot corresponding to the known payload inertial parameters. Eq. ( 13) becomes:

tot tot W    , (15)
where 0 0

a a tot b b u L k L k L W V W W V W W             is a ( + +11 ) b k L r n n n   matrix,
and

T T T T tot uL g           is a ( + +11 ) b k L n n n  vector
and  is a scalar which should be equal to 1.

Without perturbation, 0

  and tot W should be rank deficient to get the solutions 0 tot   depending on a scale coefficient  . However because of the measurement perturbations, tot W is a full rank matrix. Therefore, the system (15) is replaced by the compatible system closest to (15) with respect to the Frobenius norm:

0 tot tot ˆŴ   , ( 16
)
where tot Ŵ is the rank deficient matrix, with the same dimension as tot W , which minimizes the Frobenius norm

tot tot F Ŵ W  , T T T T tot uL ĝ        
 is the solution of the compatible system closest to (15). tot Ŵ can be computed thanks to the Singular Value Decomposition (SVD) of tot W [START_REF] Golub | Matrix computation[END_REF]: 16) is given by: In order to improve the estimation of * tot  , the rows and columns of tot W are weighted taking into account the confidence on the measures. Two types of weighting factors are used: 1. As proposed in IDIM-WLS [START_REF] Gautier | Dynamic identification of robots with power model[END_REF]) (section 2.2), to improve the ordinary LS, each row corresponding to joint j equation is weighted by the inverse of j ˆ  ;

i T tot diag( s ) W U V 0        , ( 17
)
11 11 11 b k L b k L b k L T tot tot n n n n n n n n n Ŵ W s U V            , ( 18
2. It is also proposed to weight the columns of the observation matrix in order to take into account the a priori relative confidence between the columns. Indeed, as explained in section 2.2, the coefficients of matrices a W , b W , uL W and kL W are calculated with the values of ( ) ˆq, q, q   estimated by band-pass filtering and sampling the measure of q. Therefore, the coefficients of these columns are considered less accurate than a V  and b V  obtained with the direct measures of v  . In order to increase the confidence on these columns, a V  and b V  are weighted by a factor and the new system becomes:

w w tot tot W    ( 19
)
where 0 0

a a w tot b b u L k L k L W V W W V W W               ,
and

T w T T T tot uL g /            .

Discussion on the A Priori Knowledge of the Payload Parameters and on the Choice of the Weighting Factors

The accuracy of  depends on the accuracy of kL  , depending on the knowledge of the payload parameters.

The most accurate payload parameter is the mass value L M that can be accurately measured using a weighing machine.

The choice of j ˆ

 to weight the rows of p tot W is physically meaningful but the choice of parameter  is not straightforward. In the remainder, it is proposed to minimize the relative norm of error

( ) ( ) ( ) Y W       with
respect to . The value ˆ5   gives the best results which are presented in the following section.

CASE STUDY

Description of the TX-40 Kinematics

The Stäubli TX-40 robot (Fig. 1) has a serial structure with six rotational joints. Its kinematics is defined using the modified Denavit and Hartenberg notation (MDH) [START_REF] Khalil | Modeling, identification and control of robots[END_REF]. In this notation, the link j fixed frame is defined such that the j z axis is taken along joint j axis and the j x axis is along the common normal between j z and j 1 z  (Fig. 1). The geometric parameters defining the robot frames are given in Table 1. The payload is denoted as the link 7. The parameter 0 j   , means that joint j is rotational, j  and j d parameterize the angle and distance between j 1 z  and j z along j 1 x  , respectively, whereas j  and j r parameterize the angle and distance between j 1 x  and j x along j z , respectively. For link 7, 2 j   means that the link 7 is fixed on the link 6. Since all the joints are rotational then j  is the joint position value q j given by the CS8C controller of the TX-40 robot, except for joints 2 and 3 where the MDH notation differs the Staübli variables (Table 1):

 j  j d j  j r j 1 0 0 0 q 1 0 2 0  0 q 2  0 3 0 0 d 3 = 0.225 m q 3  rl 3 = 0.035 m 4 0  0 q 4 rl 4 = 0.225 m 5 0  0 q 5 0 6 0  0 q 6 0 7 2 0 0 0 0 2 2 2 q /      2 3 3 q /     .
The TX40 robot is characterized by a coupling between the joints 5 and 6 such that:

5 5 6 6 qr q K 5 0 qr q K6 K6                        , 5 5 6 6 c r c r K 5 K6 0 K6                            (20)
where

j qr
 is the velocity of the rotor of motor j, j q  is the velocity of joint j, K5 is the transmission gain ratio of axis 5 and K6 is the transmission gain ratio of axis 6, τ cj is the motor torque of joint j, taking into account the coupling effect, τ rj is the electro-magnetic torque of the rotor of motor j. With the coupling between joints 5 and 6, ( 5) and ( 14) becomes:

1 1 0 0 0 0 0 0 0 5 6 5 6 6 v g v v g v g                                           
, and

1 0 0 0 0 0 0 0 i i 5 6 i i 6 i V V V V V                          (21)
The coupling between joints 5 and 6 also adds the effect of the inertia of rotor 6 and new viscous and Coulomb friction parameters fv m6 and fc m6 , to both τ c5 and τ c6 .

We can write: sign( )

5 c 5 6 6 6 6 6 6 Ia q fvm q fcm q          and   sign( + ) sign( ) 6 c 6 6 5 6 5 6 5 6 6 Ia q fvm q fcm q q q             .
where τ 5 , τ 6 already contain the terms j j j j j j ( Ia q fv q fc sign( q ))  

  

, for j=5 and 6 respectively, Ia K Ja K Ja   and

2 6 6 6 Ia K Ja  ( 22
)
Ja j is the moment of inertia of rotor j .

The TX40 has N s =86 standard dynamic parameters given by the 14×6 usual standard parameters, plus fv m6 and fc m6 .

Identification of the Drive Gains

The proposed method is validated using a calibrated payload (Fig. 2).

Its mass has been measured with a weighing machine ( L M = 4.59 Kg± 0.05 Kg). The other parameters have been calculated using CAD software. They are given in table 2. Their values are accurate due to the simplicity of the payload shape (Fig. 2).

Two different identifications of the payload inertia parameters are achieved: -Case 1: the payload parameters are identified using the manufacturer's drive gains -Case 2: the drive gains are first identified with the base parameters with IDIM-WTLS using the knowledge on the payload mass. They are then used in order to achieve a new identification of the new the payload parameters and the robot dynamic parameters with IDIM-WLS. Finally, in order to validate the new drive gain values a new payload is identified (Table 5). The parameters are very close to the a priori ones in all cases. The torques calculated with the model ( 12) identified with the gains of Case 2 are presented in Fig. 3. It is possible to conclude that the drive gains have been well identified with the IDIM-WTLS.

CONCLUSION

This paper has presented a new method for the global identification of the total drive gains for robot joints. This method is easy to implement and does not need any special test or measurement on elements inside the joint drive train. It XX L 0.64e-1 1.12e-1 1.99e-3 1.8 8.14e-2 1.54e-3 1.9 XY L -1.80e-2 -1.83e-2 7.21e-4 3.9 -1.53e-2 6.61e-4 4.3 XZ L 2.60e-2 2.93e-2 8.88e-4 3.0 2.87e-2 5.87e-4 2.0 YY L 0.64e-1 1.15e-1 1.80e-3 1.6 8.71e-2 1.41e-3 1.6 YZ L 2.60e-2 4.15e-2 6.32e-4 1.5 3.74e-2 5.16e-4 1.4 ZZ L 4.40e-2 7.02e-2 5.78e-4 0.8 4.51e-2 4.37e-4 1.0 MX L -2.90e-1 -3.02e-1 2.47e-3 0.8 -2.73e-1 2.05e-3 0.8 MY L -2.90e-1 -3.20e-1 2.55e-3 0.8 -2.85e-1 2.11e-3 0. is based on a IDIM-WTLS technique using current reference and position sampled data while the robot is tracking one reference trajectory without load fixed on the robot and one trajectory with a known payload fixed on the robot, whose inertial parameters are measured. The method has been experimentally validated on an industrial Stäubli TX-40 robot. Using the identified drive gains, the identification of the total dynamic model has been improved and another payload has been accurately identified. This shows the effectiveness of the method.

  of inertial parameters of the payload which are estimated with a CAD software or measured with a balance,

Fig. 1 .

 1 Fig. 1. Link frames of the TX-40 robot

Fig. 2 .

 2 Fig. 2. The 4.59 Kg payload

Fig. 3 .

 3 Fig. 3. Measured and computed torques of the TX-40 with the payload of 4.59 kg.

Table 1 .

 1 Geometric parameters of the TX-40 robot with the payload

	j

Table 2 .

 2 Identification of the payload dynamic parameters.

			Case 1	Case 2
	Par.	Nom. val.	Id.val.

Table 3 .

 3 Identified drive gains.

		 Joint 1 Joint 2 Joint 3 Joint 4 Joint 5 Joint 6
	A priori val.	j g 		32.96 32.96 25.65 -11.52 18.48 7.68
	Case 2	j g 		33.7	32.2	24.1 -8.80 16.50 6.82
		2 		i	0.04	0.03	0.03	0.02 0.03 0.01
			ri % 	0.61	0.45	0.60	0.83 0.77 0.78

Table 4 .

 4 Quality of identification.

		Error norm  Relative Error norm ˆ/ Y 	ˆ 
	Case 1	33.8366		0.043245	0.495164
	Case 2	31.9702		0.0409025	0.4667879
		ˆŶ W   		is the minimal norm of error.

Table 5 .

 5 Identification of the new payload dynamic parameters.

	Par. Nom. val.	Id.val.	Case 1 2    i		ri % 	Id. val.	Case 2 2 i   		ri % 
	XX L	1.51e-2	1.57e-2 8.75e-4 5.6	1.11e-2 8.07e-4	7.2
	XY L XZ L YY L	-9.06e-4 3.61e-3 1.51e-2	-2.41e-3 3.16e-4 13.1 -2.08e-3 2.79e-4 13.4 4.21e-3 3.34e-4 7.9 4.11e-3 2.94e-4 7.1 1.40e-2 8.95e-4 6.4 1.13e-2 7.57e-4 6.7
	YZ L ZZ L MX L	3.61e-3 3.44e-3 -4.00e-2	1.93e-3 3.00e-4 15.5 1.49e-3 2.66e-4 17.9 3.88e-3 2.88e-4 7.4 1.34e-3 2.88e-4 21.5 -2.37e-2 1.59e-3 4.7 -2.74e-2 1.30e-3 4.8
	MY L MZ L M L	-3.99e-2 0.15 1.686	-4.18e-2 1.41e-3 3.4 -3.90e-2 1.31e-3 0.192 2.51e-3 1.3 0.166 2.22e-3 1.66 1.74e-2 1.0 1.66 1.73e-2	3.3 1.3 1.0

i   is the standard deviation and ri %   its relative value

ACKNOWLEDGEMENTS

This work is supported by the French grant FUI IRIMI F1004026 Z.