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

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

Global Identification of Joint Drive Gains and Dynamic Parameters of Robots

Keywords: Industrial robots, Drive gains, Dynamic parameters identification

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 input reference of the motor current loop) by the joint drive gains.

Then it is essential to get accurate values of joint drive gains to get accurate estimation of the motor torques and accurate identification of dynamic parameters. The previous works proposed to identify the gain of one joint at a time using data of each joint separately. This is a sequential procedure which accumulates errors from step to step. To overcome this drawback, this paper proposes a global identification of the drive gains of all joints and the dynamic parameters of all links. They are calculated altogether in a single step using all the data of all joints at the same time. The method is based on the total least squares solution of an over-determined linear system obtained with the inverse dynamic model calculated with available input reference of the motor current loop and joint position sampled data while the robot is tracking some reference trajectories without load on the robot and some trajectories with a known payload fixed on the robot. The method is experimentally validated on an industrial Stäubli TX-40 robot.

I. INTRODUCTION

Several schemes have been proposed in the literature to identify the dynamic parameters of Paper DS- robots [START_REF] Canudas De Wit | Parameters identification of robots manipulators via sequential hybrid estimation algorithms[END_REF]- [START_REF] Lu | Experimental determination of dynamic parameters of robotic arms[END_REF]. Most of the dynamic off-line identification methods:

-use an Inverse Dynamic Identification Model (IDIM) that gives the linear relations between the joint forces/torques and the dynamic parameters, -build an over-determined linear system of equations obtained by sampling the IDIM while the robot is tracking some trajectories in position closed-loop control, -estimate the parameter values using least squares techniques (LS). This procedure is called the IDIM-LS technique.

Good experimental results can be obtained if:

-a well-tuned derivative band-pass filtering of joint position is used to calculate the joint velocities and accelerations, -accurate values for joint drive gains are known to calculate the joint force/torque as the product of the input references of the motor current loop by the joint drive gains [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].

This requires the calibration of the drive train constituted by a current controlled voltage source amplifier with gain t G which supplies a permanent magnet DC or a brushless motor with torque constant t K coupled to the link through direct drive or gear train with gear ratio N . Because of large values of the gear ratio for industrial robots, (N>50), the total joint drive gain, t 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 amplifiers and motors, which require opening the drive chain of each joint [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]. This sensitivity to errors directly affects the accuracy of the force/torque computation as well as the interaction force between the robot and its environment or payload estimations that are required in many modern robotic applications.

More recent works [START_REF] Gautier | New Method for Global Identification of the Joint Drive Gains of Robots using a Known Inertial Payload[END_REF], [START_REF] Gautier | New Method for Global Identification of the Joint Drive Gains of Robots using a Known Payload Mass[END_REF] have proposed to apply sequential procedures to identify the total joint drive gains j g  for each actuated joint separately by using a known payload fixed on the Paper DS-13-1373 end-effector. In both methods, the estimation of the drive gain of one joint was done using only data coming from the corresponding joint equation which implies the loss of information about the coupled data on the other joints. With these sequential approaches, leading to error accumulation, good results were obtained only for the first four robot joints.

In this paper, a new method is proposed for the global identification of all robot dynamic parameters, including joint drive gains, using the input reference of the motor current loop and the joint 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. Inertial payload parameters are measured or calculated with a CAD software. Contrary to the previous works, all dynamic parameters and drive gains are calculated in one step as the Total LS solution (TLS) of an over-determined system that takes into account the coupling between the robot axes.

Such a method avoids the cumulative errors of the previous sequential procedures. In order to show the method efficiency, it is experimentally validated on an industrial robot manufactured by Stäubli: the TX-40.

It should be mentioned that this method is easy to implement, versatile and suitable for the automatic calibration of the drive gains of any industrial robots.

A first condensed version of this work has been proposed in [START_REF] Gautier | Global Identification of Drive Gains Parameters of Robots Using a Known Payload[END_REF]. The present paper contains detailed explanations on the TLS procedure to enlighten the theoretical understanding of the method, especially in terms of statistical properties, and gives additional experimental results that show the interest of the method in terms of force/torque estimation and parameter identification. Paper DS-13-1373

II. USUAL IDENTIFICATION PROCEDURE WITH INVERSE DYNAMIC MODEL AND LEAST SQUARES (IDIM-LS)

A. Inverse Dynamic Identification Model (IDIM)

It is known that, using the modified Denavit-Hartenberg description of moving multibody systems [START_REF] Khalil | Modeling, identification and control of robots[END_REF], the dynamic model of any serial manipulator composed of n links and n actuators can be linearly written in term of a   1 st n  vector of standard parameters st χ [5], [6]:

( , ) ( ) idm st st st , , , ,      τ q q q χ IDM q q q χ (1)
where idm τ is the motor torque vector, q ,  q and  q are respectively the   ...

T T T n T st st st st      χ χ χ χ .
For a rigid robot, the link j and joint j can be parameterized by 14 standard parameters regrouped into the vector j st χ such that

j T j 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 Fs       χ
, where:

 , , , , , j j j j j j
XX XY XZ YY YZ ZZ are the six components of the inertia matrix j I of link j w.r.t.

frame j at its origin, i.e.

j j j j j j j j j j XX XY XZ XY YY YZ XZ YZ ZZ            I  j
M is the mass of link j , j Ia is a total inertia moment for rotor of actuator j and gears of the  is an offset parameter which regroups the amplifier offset and the asymmetrical Coulomb friction coefficient [START_REF] Hamon | New Dry Friction Model with Load-and Velocity-Dependence and Dynamic Identification of Multi-DOF Robots[END_REF].

The identifiable parameters are the base parameters, which are the minimal number of dynamic parameters from which the dynamic model can be calculated. They are obtained from the standard parameters by eliminating those which have no effect in [START_REF] Canudas De Wit | Parameters identification of robots manipulators via sequential hybrid estimation algorithms[END_REF] and by regrouping some of the others by means of linear relations [START_REF] Mayeda | Base parameters of manipulator dynamic models[END_REF], using simple closed-form rules [START_REF] Khalil | Modeling, identification and control of robots[END_REF], [START_REF] Gautier | Direct calculation of minimum set of inertial parameters of serial 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 denoted as χ as follows:

( ) idm , ,    τ IDM q q q χ (2)
where IDM is a subset of independent columns of st IDM which defines the identifiable parameters.

Because of perturbations due to noise measurement and modeling errors, the actual force/torque τ differs from idm τ by an error, e , such that: Paper DS-13-1373

( ) idm , ,       τ τ e IDM q q q χ e (3)
where τ is calculated with the drive chain relations:

1 1 0 0 n n v g v g                              τ v g (4)
 v is the ( ) n n  matrix of the actual motor 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 j j j t t g N G K  
corresponds to actuator j, where j t G is the gain of the current controlled voltage source amplifier of the motor j which supplies a permanent magnet DC or a brushless motor with torque constant j t K coupled to the link through direct drive or gear train with gear ratio j N ) that is given by a priori manufacturer's data or measured with special time-consuming heavy tests on amplifiers and motors separately [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].

Equation (3) represents the Inverse Dynamic Identification Model (IDIM).

B. IDIM with a payload

The payload is considered as a link 1 n  fixed to the link n of the robot [START_REF] Khalil | Identification of the payload inertial parameters of industrial manipulators[END_REF]. Only kL n among its ten inertial parameters are considered to be known (i.e. there is 10

uL kL n n   unknown parameters). The model (3) becomes:   uL kL uL tot tot kL              χ τ IDM IDM IDM χ e IDM χ e χ (5)
where aL χ (a = u or k) is a ( 1)

aL n  vector containing the unknown ( uL χ ) or known ( kL χ ) inertial Paper DS-13-1373 parameters of the payload; aL IDM is the ( ) aL n n  Jacobian matrix of idm τ , w.r.t. the vector aL χ .

C. Least Squares Identification of the Dynamic Parameters with IDIM

The off-line identification of the robot base dynamic parameters χ can be achieved given measured or estimated off-line data for τ and   , ,

 

q q q , collected while the robot is tracking some trajectories. The model ( 3) is sampled, low-pass filtered and decimated (parallel decimation of Y and each column of W ) in order to get an over-determined linear system of ( ) n r  equations and n b unknowns:

    ˆ    Y τ W q,q,q χ ρ (6)
where ˆ( )

 

q, q, q is an estimation of ( )

 

q, q, q , obtained by sampling, band-pass filtering the measure of q with zero-phase non causal butterworth filter and central difference algorithm [START_REF] Hollerbach | Model Identification[END_REF], ρ is the ( 1) r  vector of errors, and   ˆˆ   W q, q, q is the ( ) b r n  observation matrix.

In [START_REF] Gautier | Extended Kalman filtering and weighted least squares dynamic identification of robot[END_REF], [START_REF] Gautier | Dynamic identification of robots with power model[END_REF], practical rules for tuning these filters and for avoiding a biased estimation of the velocities and accelerations [START_REF] Kavanagh | Performance analysis and compensation of M/T-type digital tachometers[END_REF] are given, taking advantage of non-causal off-line pass-band filtering.

Using the base parameters and tracking "exciting" reference trajectories, i.e. optimized trajectories that can be computed by nonlinear minimization of a criterion function of the condition number of the W matrix [START_REF] Swevers | Optimal robot excitation and identification[END_REF], [START_REF] Presse | New criteria of exciting trajectories for robot identification[END_REF] a well-conditioned matrix W can be obtained. In this work, the motion generator of the industrial controller which is a point-to-point trapezoidal acceleration generator is used. Some trajectories are tested covering the whole robot workspace Paper DS-13-1373 until a good criterion is obtained [START_REF] Presse | New criteria of exciting trajectories for robot identification[END_REF], which is an easy and fast procedure that gives good identification results. The LS solution χ of ( 6) is given by:

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

Standard deviations ˆi   , 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 [2]:

T 2 ( ) r E      C 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 ( ) b ˆˆr n      Y Wχ (9)
The covariance matrix of the estimation error is given by [START_REF] Gautier | Extended Kalman filtering and weighted least squares dynamic identification of robot[END_REF]:

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

100 ri i ˆˆi %       (11) for 0 i   .
The ordinary LS can be improved by a weighted LS procedure (IDIM-WLS) where data in Y and W are sorted w.r.t. each joint j equation and weighted with the inverse of the standard deviation of the error calculated from ordinary LS solution of the equations of joint j [START_REF] Gautier | Dynamic identification of robots with power model[END_REF]. Paper DS-13-1373

D. Identification with a payload

In order to identify both the robot and the payload dynamic parameters, it is necessary that the robot carried out two types of trajectories: (a) trajectories without the payload and (b) trajectories 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 IDIM (5) is then written as:

a a uL b b u L k L kL                          χ Y W 0 0 Y χ ρ Y W W W χ (12)
where a Y ( b Y , resp.) is the vector of sampled input torques of the robot in the unloaded (loaded, resp.) case, a W ( b W , resp.) is the observation matrix of the robot in the unloaded (loaded, resp.) case and uL W ( kL W , resp.) is the observation matrix of the robot corresponding to the unknown (known, resp.) payload inertial parameters.

III. GLOBAL IDENTIFICATION OF THE ROBOT DYNAMIC PARAMETERS AND THE DRIVE GAINS

A. Least Square Identification of the Robot Dynamic Parameters and the Joint Drive Gains

In the usual IDIM-WLS, accurate values of the drive gains are necessary to compute vector Y.

However, the manufacturer's data give drive gains parameters  g with an uncertainty of about 10%, thus leading to parameter identification and torque estimation errors. Therefore, it is preferable to introduce the drive gains into the base parameters.

Taking into account that parameters kL χ are known, (12) can be written as: Paper DS-13-1373

1 a a uL b b u L k Lk L                             χ V W 0 0 Y g χ ρ V W W W χ (13)
where

a  V ( b  V , resp.
) is the block-diagonal matrix of  v samples in the unloaded (loaded, resp.)

,

j j a b j r v v                                   V 0 V V 0 V , (14)
, j k v  is the k-th sample of current reference for actuator j, and j  V regroups all the current reference samples for actuator j.

A simple approach to identify the drive gains is to take into account that the vector kL kL W χ is known. Then ( 13) can be rewritten as:

a a L r r kL kL b b uL uL                                  g 0 V W 0 Y χ ρ W χ ρ W χ V W W χ (15)
with r χ the vector of the unknown inertial parameters of the robot and the payload plus the drive gain parameters.

As a result, the LS solution ˆr χ of ( 15) is given by:

    1 ˆT T r r r r kL kL r L     χ W W W W χ W Y (16)
Because L Y and r W depend on the same data containing perturbations ˆ( )

 

q, q, q (due to the Total Least Squares Identification (IDIM-TLS) procedure. This procedure is detailed below.

use of kL W in L Y and of uL W in r W ),

B. Total Least Square Identification of the Robot Dynamic Parameters and the Joint Drive

Gains (IDIM-TLS)

Details on the TLS identification method can be found in [START_REF] Rao | Linear Models: Least Squares and Alternatives, Second Edition[END_REF], [START_REF] Van Huffel | The Total Least Squares Problem: Computational Aspects and Analysis[END_REF] and many other papers (see also [START_REF] Markovsky | Overview of total least-squares methods[END_REF]- [START_REF] Van Huffel | The Generalized Total Least Squares Problem: Formulation, Algorithm and Properties[END_REF]). Eq. ( 13) can be rewritten as:

tot tot  W χ ρ, (17)
where ) depending on a scale coefficient  . However because of the measurement perturbations, tot W is a full rank matrix. Therefore, the system ( 17) is changed to the compatible system closest to (17) w.r.t. the Frobenius norm:

a a tot b b u L k L k L              V W 0 0 W V W W W χ is a ( ) r c  matrix (with + 1 b uL c n n n    ), and 1
T T T T tot uL       χ g χ χ is a   1 c  vector.
ˆtot tot  W χ 0 , ˆˆˆ1 T T T T tot uL       χ g χ χ (18)
where ˆtot

W is the ( ) r c  rank deficient matrix,
ˆT tot tot c c c s   W W U V , ( 20
)
where c s is the smallest singular value of tot W and c U ( c V , resp.) the last columns of U (V, resp.) corresponding to c s . Then, the normalized optimal solution ˆn tot χ is given by the last column

V c of V, ˆn tot c  χ
V , which belongs to the kernel of ˆtot W .

There is infinity of vectors ˆˆn It must be mentioned that the TLS 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 (dof) robot but gave arguable results due to the lack of accurate reference parameters (in [START_REF] Gautier | Identification of inertial and drive gain parameters of robots[END_REF], the reference parameter was the drive gain of actuator 1 that was not accurately known). As a result, the authors were not able to correctly identify payload masses added on the end-effector. In this paper, a major improvement is to scale the parameters using the accurate weighed value of an additional payload mass. Paper DS-13-1373

C. Statistical analysis

Standard deviations ˆi   on the dynamic and drive gains parameters, are estimated assuming that all errors in data matrix tot W are independently and identically distributed with zero mean and

common covariance matrix WW C such that 2 ˆW WW W r   C I , (21)
where

W r I is the identity matrix of dimension ( ) ( ) r c r c    .
An unbiased estimation of the standard deviation ˆW  is [START_REF] Van Huffel | The Total Least Squares Problem: Computational Aspects and Analysis[END_REF]:

ˆ/ W c s r c    (22)
The covariance matrix of the estimation error is approximated by [START_REF] Van Huffel | The Total Least Squares Problem: Computational Aspects and Analysis[END_REF]:  is given by: ˆˆ% 100

   1: 1 1: 1 1 2 2 ˆˆ1: 1 2 ˆ1 c c T W c t o t t o t         C χ W W (23)
ri i i       , for ˆi  ≠ 0.
In order to improve the estimation of * ˆtot χ , the rows of tot W are weighted taking into account the confidence on the measures. As proposed in IDIM-WLS (Section II.B), to improve the TLS solution, each row corresponding to joint j equation is weighted by the inverse of ˆj W  , i.e. the standard deviation corresponding to the data of the joint j equations. Paper DS-13-1373

IV. CASE STUDY

A. The Stäubli TX-40 Robot

The Stäubli TX-40 robot (Fig. 1) has a serial structure with six rotational joints. Details on its kinematics can be found in [START_REF] Gautier | Global Identification of Drive Gains Parameters of Robots Using a Known Payload[END_REF]. It is to be noted that the payload is numbered as the link 7, rigidly fixed on the last robot joint numbered as the link 6. The TX-40 robot is characterized by a coupling between the joints 5 and 6 such that:

qr N q qr N N q                        , 5 5
6 6 5 6 6 0 c r c r N N N                           (24)
where j qr  is the velocity of the rotor of motor j, j q  is the velocity of joint j, N j is the transmission gain ratio of axis j, τ cj is the motor torque of joint j, taking into account the coupling effect on the motor side, τ rj is the electro-magnetic torque of motor j.

The model for the coupled geared drive chain of joints 5 and 6 adds 3 supplementary inertia, Ia q q q q Ia q q q q Fv Fs sign Fv Fs sign

                     (25)
where τ j already contains the terms of the uncoupled joints ( ( ) )

j j j j j j j o f f Ia q Fv q Fc sign q        .
The TX-40 IDIM is automatically computed using the software Symoro+ [START_REF] Khalil | Symoro+: a system for the symbolic modeling of robots[END_REF] which applies a recursive and optimized Newton-Euler algorithm that gives the model expression with the minimal number of operations [START_REF] Khalil | Modeling, identification and control of robots[END_REF].

B. Identification Results

To validate the proposed method, a calibrated payload is used (Fig. 1). Its mass has been measured with an accurate weighing machine (M 7 = 4.59 kg ± 0.005 kg). The other parameters have been calculated using CAD software. Their values are set in bold font in Table I.

Before presenting the identification result, it is to be noticed that during the identification process, some small base parameters remain poorly identifiable because they have no significant contribution in the joint torques [START_REF] Khalil | Identification of the payload inertial parameters of industrial manipulators[END_REF]. These essential parameters, which are a subset of the base parameters, can be cancelled in order to simplify the dynamic model. They are calculated using an iterative procedure starting from the base parameters estimation. At each step the base parameter which has the largest relative standard deviation is cancelled. A new TLS parameter estimation of the simplified model is carried out with new relative error standard deviation The obtained results are shown in Tables I to IV. The parameters with the subscript R stand for the regrouped parameters [START_REF] Khalil | Modeling, identification and control of robots[END_REF]. The results show in all cases that the error on the identified drive gains grows up to 30%! In the case where the payload mass is used only, some identified payload parameters are far from the a priori CAD values. These results could be explained by the fact that parameters extracted from CAD data are not accurate due to differences between CAD and reality (errors on material properties, payload shape, etc.). It can also be noticed in Tables I and III that the relative difference between the identified parameters is generally small with a mean value less than 10%. Paper DS-13-1373

C. Cross-Validations

To validate the methods used in Cases 1 to 4, obtained results are cross-validated. Three trajectories, different from each other and from the one used for identification (condition number of the corresponding observation matrices is given in Table V), are performed with the robot on which is fixed the payload of 4.59 kg. The positions and current measured are recorded during the robot displacements. Then, the observation matrix is computed for each trajectory.

First, the actuator torques calculated with the relation (4)    τ v g (where  v is the measured motor current reference and ˆ g the vector of the identified drive gains) are compared with Paper DS-13-1373 torques computed using the IDIM (2)  τ IDMχ ( χ are the identified dynamic parameters). Five sets of parameters are chosen:

1. drive gains given by the manufacturer and robot and payload dynamic parameters identified using a classical IDIM-WLS procedure [18],

2. drive gains and robot/payload dynamic parameters identified with the IDIM-WLS method introduced in Section III.A, with the knowledge of the ten payload parameters (Case 1, Section IV.B, Tables I andII), 3. drive gains and robot/payload dynamic parameters identified with the same IDIM-WLS method, with the knowledge of the payload mass only (Case 2, Section IV.B, Tables I andII), 4. drive gains and robot/payload dynamic parameters identified with the IDIM-TLS method introduced in Section III.B, with the knowledge of the ten payload parameters (Case 3, Section IV.B, Tables III andIV), 5. drive gains and robot/payload dynamic parameters identified with the same IDIM-TLS method, with the knowledge of the payload mass only (Case 4, Section IV.B, Tables III andIV), Paper DS-13-1373 measured motor current reference and ˆ g the vector of the identified drive gains) are compared with torques computed using the IDIM (2)  τ IDMχ (where χ are the identified dynamic parameters). It can be observed that the torques are well calculated using the identified IDIM. It should be mentioned that the relative errors for joints 3 and 5 are a bit higher due to unmodelled phenomena (backlash in gearboxes and non-linear friction terms).

Then, using the data collected on each trajectory, the payload is estimated using a classical IDIM-WLS [START_REF] Khalil | Identification of the payload inertial parameters of industrial manipulators[END_REF] presented in Sections II.C and II.D, in which the actuator torques are calculated with the relation (4)    τ v g which requires the knowledge of the drive gains. Five different values of ˆ g are thus considered, which are the ones defined in the five previous cases (a priori gains or gains identified using the techniques presented in Section III). The results are given in Table VII. Only the payload mass is shown, as it is the only payload parameter value which we can trust, because it is accurately weighed.

It can be observed that the best payload estimation is obtained for drive gains identified in Case 4 (%e=0.65%), i.e. with IDIM-TLS techniques considering the knowledge of the payload mass only. The worst results are obtained for the gains identified in Cases 1 and 2 (%e=8.35%), i.e. the gains obtained with the modified IDIM-WLS procedure. Indeed, as mentioned previously, the noises in Y L and W r of ( 16) are correlated. Such correlation introduces a bias in the results and leads to wrong estimation of the parameters. In order to definitely validate our method, a second payload of 1.686±0.005 kg is attached on the end-effector and the same experiments are performed. Then, using the data collected on each trajectory, the payload is estimated using a classical IDIM-WLS [START_REF] Khalil | Identification of the payload inertial parameters of industrial manipulators[END_REF], in which the actuator torques are calculated with the relation (

   τ v g . The same five different values of ˆ g are considered. The results are given in the payload mass only. This result is very close to the one obtained with the mass used for the identification (%e=0.65% in Table VII), i.e. the identified gains make the identification not sensitive to the added mass. And the worst results are obtained for the gains identified in Cases 1 and 2, i.e. the gains obtained with the modified IDIM-WLS procedure which lead to a biased parameter estimation.

All these results show the effectiveness of this approach: for calibrating the drive gains, it is only necessary to the payload mass and to carry out standard trajectories of industrial robot. And the calibration of the drive gains improves the torques estimation and parameters identification.

To conclude this section, and in order to strengthen the method validation, we would like to mention that the proposed method has been experimentally tested on two other industrial robots (the Staübli RX-90 robot (about 10kg of payload) and the Kuka KR270 robot (270kg of payload)) and on two prototypes of parallel robots developed in French laboratories (the Orthoglide from the IRCCyN of Nantes [START_REF] Chablat | Architecture Optimization of a 3-DOF Parallel Mechanism for Machining Applications, the Orthoglide[END_REF] and the DualV from the LIRMM of Montpellier [START_REF] Van Der Wijk | Generic Method for Deriving the General Shaking Force Balance Conditions of Parallel Manipulators with Application[END_REF]). Experimental results have shown significant improvements of the identification of the drive gains values leading to better payload estimations for all these robots compared to the manufacturer's values.

V. CONCLUSION

This paper has presented a new method for the global identification of the robot dynamic parameters including the gains of the total drive chain. This method is easy to implement and does not need any special test or measurement on the components of the joint drive train. It is based on an IDIM-TLS technique using motor current reference and joint position sampled data Paper DS-13-1373

Four methods have been tested: (i) a modified IDIM-WLS using ten payload parameters calculated with CAD, (ii) the same modified IDIM-WLS using only accurately weighed payload mass, (iii) the IDIM-TLS using ten payload parameters calculated with CAD and (iv) the IDIM-TLS using only accurately weighed payload mass. Then, the manufacturer's drive gains and the identified ones, results have been compared in terms of torque reconstruction and payload parameter estimation. The best results have been obtained when the IDIM-TLS is performed with the weighed payload mass only while modified IDIM-WLS techniques gave the poorest results due to noise correlations between the observation matrix and the measurement vector.

This approach is very simple to perform and the experimental results have shown its  is the i-th parameter of IDIM-WLS using the knowledge of the ten payload parameters and 2 ˆi  is the i-th parameter of IDIM-WLS using the knowledge of the payload mass only. Paper DS-13-1373

List of figures

Fs

  joint j drive chain, Paper DS-13-1373 are the 3 components of the first moment of link j , i.e. is the position of the center of mass of the link j expressed in the frame attached at the origin of the considered link[START_REF] Khalil | Modeling, identification and control of robots[END_REF] are the viscous and Coulomb friction coefficients of the drive chain, respectively, and j off

Without perturbation,  ρ 0 and tot W

 tot must be rank deficient to get the non-null solutions ˆˆn ˆn tot χ is a vector of unit norm, i.e. ˆ1 n tot  χ

  solutions of (18) depending on a scale factor  . The unique solution * ˆˆn tot tot   χ χ for the robot can be found by taking into account that the last value * ˆc tot  of * ˆtot χ must be equal to 1 according to (18), i.e. ˆ1/

Fig. 1 .

 1 Fig. 1. TX-40 robot and its calibrated payload of 4.59 Kg.

Fs

  to both τ c5 and τ c6 : Paper DS-13-

 Case 2 : Case 3 : Case 4 :

 234 where r  is a ratio ideally chosen between 10 and 30 depending on the level of perturbation in Y and W.The robot joint drive gains and dynamic parameters are identified using four different methods: Case 1: with the IDIM-WLS method introduced in Section III.A, taking advantage of the Paper DS-13-1373 knowledge of the ten payload parameters, to calculate the vector L Y of(15), similar to Case 1, but considering the knowledge of the payload mass only, with the IDIM-TLS method introduced in Section III.B, taking advantage of the knowledge of the ten payload parameters, similar to Case 3, but considering the knowledge of the payload mass only.

Case 4 ,

 4 i.e. with parameters identified with IDIM-TLS techniques considering the knowledge of the payload mass only. In Figure2, using the parameters identified in Case 4 of Section IV.B, the actuator torques along the trajectory 1 calculated with the relation[START_REF] Kozlowski | Modelling and identification in robotics[END_REF]    τ v g (where  v is the Paper DS-13-1373

  effectiveness: for calibrating the joint drive gains, it is only necessary to accurately weigh the payload mass and to carry standard trajectories of industrial robot. Paper DS-13-1373 to a Redundant Planar 4-RRR Parallel Manipulator," Proceedings of the 13th World Congress in Mechanism and Machine Science, Guanajuato, Mexico, June 19-25. Paper DS-13-1373List of tables

Fig. 1 .Fig. 2 .Fig. 1 .

 121 Fig. 1. TX-40 robot and its calibrated payload of 4.59 Kg Fig. 2. Motor torques (joint side units) calculated with identified gains (red) and with IDIM (blue) of the TX-40 with the payload of 4.59kg.



  is the i-th parameter of IDIM-TLS using the knowledge of the ten payload parameters and 2 ˆi  is the i-th parameter of IDIM-TLS using the knowledge of the payload mass only. Paper DS-

TABLE I THE

 I ESSENTIAL DYNAMIC PARAMETERS OF THE TX-40 IDENTIFIED WITH IDIM-WLS.

		parameter	with 10 cad param. id.val. % ri    id.val. with 1 weighed mass m 7 % ri   %e i
		ZZ 1R Fv 1 Fs 1 XX 2R XZ 2R ZZ 2R MX 2R Fv 2 Fs 2 XX 3R ZZ 3R MX 3 MY 3R Ia 3 Fv 3 Fs 3 XY 4 MX 4 Ia 4 Fv 4 Fs 4 MY 5R Ia 5 Fv 5 Fs 5 Ia 6 Ia m6 Fv 6 Fvm 6 Fsm 6  off6 XX 7 XY 7 XZ 7 YY 7 YZ 7 ZZ 7 MX 7 MY 7 MZ 7	1,17e+0 6,35e+0 5,18e+0 -4,54e-1 -1,43e-1 9,85e-1 1,97e+0 4,17e+0 7,28e+0 9,56e-2 1,19e-1 3,63e-2 -5,19e-1 7,52e-2 1,46e+0 5,81e+0 --1,53e-2 2,50e-2 6,54e-1 1,72e+0 -2,40e-2 4,69e-2 1,24e+0 2,81e+0 9,80e-3 8,49e-3 4,57e-1 4,30e-1 1,52e+0 -0.64e-1 -1.80e-2 2.60e-2 0.64e-1 2.60e-2 4.40e-2 -2.90e-1 -2.90e-1 4.10e-1	2,28 2,37 3,80 3,53 5,85 1,36 1,57 2,09 2,60 13,42 5,50 24,39 2,69 7,29 3,22 2,87 -18,62 4,63 2,49 3,11 14,60 8,11 3,63 3,87 6,29 19,92 2,07 3,21 2,95 ----------	1,14e+0 6,21e+0 4,98e+0 -4,43e-1 -1,43e-1 9,60e-1 1,92e+0 4,09e+0 7,09e+0 9,14e-2 1,16e-1 3,77e-2 -5,02e-1 7,39e-2 1,42e+0 5,64e+0 -5,27e-3 -1,34e-2 2,36e-2 6,37e-1 1,68e+0 -2,26e-2 4,24e-2 1,21e+0 2,71e+0 8,33e-3 8,95e-3 4,34e-1 4,11e-1 1,53e+0 7,91e-2 8,39e-2 -1,19e-2 2,22e-2 8,20e-2 3,18e-2 5,21e-2 -2,43e-1 -2,54e-1 3,91e-1	2,66 2,70 4,10 3,81 5,88 1,92 2,21 2,52 2,91 13,84 5,75 23,00 3,30 7,44 3,60 3,29 35,04 21,25 5,55 3,76 4,26 15,67 9,28 4,56 4,83 7,77 18,72 3,51 4,31 4,11 24,72 5,79 17,55 9,69 5,82 5,33 3,92 3,29 3,41 3,90	2,56 2,20 3,86 2,42 0,00 2,54 2,54 1,92 2,61 4,39 2,52 3,86 3,28 1,73 2,74 2,93 -12,42 5,60 2,60 2,33 5,83 9,59 2,42 3,56 15,00 5,42 5,03 4,42 0,66 -31,09 33,89 14,62 28,13 22,31 18,41 16,21 12,41 4,63
		M 7 =4.59±0.005		M 7 =4.59±0.005	-
		rel. err. norm ||ρ||/||Y|| =12.40%	rel. err. norm ||ρ||/||Y||=12.43%
	) : 7.86%, 2 1 ˆˆ% i mean e 1 / i i i i e     	, (% a b  Y V g ,   % ., where 1 ˆi  is the i-th parameter of IDIM-WLS using the knowledge of the ten payload
	parameters and

2 ˆi

 is the i-th parameter of IDIM-WLS using the knowledge of the payload mass only. Paper DS-

TABLE II THE

 II DRIVE GAINS OF THE TX-40: A PRIORI STAÜBLI'S DATA AND IDENTIFIED VALUES WITH IDIM-WLS.

		A priori	with 10 CAD parameters		with 1 weighed mass M 7
	i	g 	i	0	ˆi g 		% ri 	%e i			ˆi g 			% ri 	%e i
	1	32.96	30,20		2,41	8,37		31,10		2,02	5,64
	2	32.96	29,80		1,72	9,59		30,50		1,09	7,46
	3	25.65	21,80		2,36	15,01		22,40		1,74	12,67
	4	-11.52	-8,06		3,33	30,03		-8,27		1,76	28,21
	5	18.48	14,00		3,10	24,24		14,40		1,36	22,08
	6	7.68	5,88		3,15	23,44		6,08		1,44	20,83
						(% ) :18.45% i mean e		(% ) :16..5% i mean e
				% ri   : relative standard deviation.	i e		% i g g   	i	0	/	g 	i	0	%	.

TABLE III THE

 III ESSENTIAL DYNAMIC PARAMETERS OF THE TX-40 IDENTIFIED WITH IDIM-TLS.

										par.	with 10 cad param. id.val. % ri    id.val. with 1 weighed mass m 7 % ri   %e i
										ZZ 1R Fv 1 Fs 1  off1 XX 2R XZ 2R ZZ 2R MX 2R Fv 2 Fs 2 XX 3R ZZ 3R MX 3 MY 3R Ia 3 Fv 3 Fs 3 XY 4 MX 4 Ia 4 Fv 4 Fs 4 MY 5R Ia 5 Fv 5 Fs 5 Ia 6 Ia m6 Fv 6 Fvm 6 Fsm 6  off6 XX 7 XY 7 XZ 7 YY 7 YZ 7 ZZ 7 MX 7 MY 7 MZ 7	1,28e+0 6,93e+0 5,61e+0 --4,95e-1 -1,54e-1 1,02e+0 2,04e+0 4,29e+0 7,56e+0 1,07e-1 1,24e-1 4,02e-2 -5,53e-1 8,03e-2 1,52e+0 6,12e+0 -1,53e-2 2,51e-2 6,58e-1 1,73e+0 -2,52e-2 4,73e-2 1,24e+0 2,84e+0 9,92e-3 8,51e-3 4,61e-1 4,32e-1 --0.64e-1 --1.80e-2 2.60e-2 0.64e-1 2.60e-2 4.40e-2 -2.90e-1 -2.90e-1 4.10e-1	-	2,00 2,08 3,18 -2,91 4,86 1,34 1,55 2,10 2,48 9,79 5,25 18,36 2,39 6,77 3,14 2,72 21,74 5,48 2,87 3,62 18,08 11,60 4,65 4,87 9,38 20,93 2,73 2,87 -----------	1,28e+0 7,01e+0 5,46e+0 3,01e-1 -5,01e-1 -1,58e-1 1,05e+0 2,11e+0 4,45e+0 7,72e+0 1,05e-1 1,27e-1 4,14e-2 -5,71e-1 8,31e-2 1,57e+0 6,28e+0 -5,61e-3 -1,50e-2 2,71e-2 7,27e-1 1,92e+0 -2,37e-2 4,90e-2 1,41e+0 3,15e+0 9,69e-3 1,04e-2 5,02e-1 4,75e-1 1,77e+0 9,33e-2 9,59e-2 -1,43e-2 2,46e-2 9,29e-2 3,73e-2 6,03e-2 -2,75e-1 -2,84e-1 4,48e-1	2,04 2,07 3,20 27,13 2,84 4,42 1,51 1,73 2,02 2,28 9,80 4,53 16,79 2,55 6,14 2,93 2,62 26,20 19,45 5,02 3,22 3,74 17,29 8,67 3,88 3,93 6,79 17,29 2,78 2,88 3,07 17,46 5,13 21,79 7,88 4,92 4,48 3,41 2,65 2,75 3,11	0,00 1,15 2,67 -1,21 2,60 2,94 3,43 3,73 2,12 1,87 2,42 2,99 3,25 3,49 3,29 2,61 -1,96 7,97 10,49 10,98 5,95 3,59 13,71 10,92 2,32 22,21 8,89 9,95 --49,84 20,56 5,38 45,16 43,46 37,05 5,17 2,07 9,27
										M 7 =4.59±0.005	M 7 =4.59±0.005	-
										rel. err. norm ||ρ||/||Y|| =11.95% rel. err. norm
										||ρ||/||Y||=11.15%
	i mean e	) : 8.28%,	, (% a b  Y V g  
	i e		1 ˆˆ% i   	2 i	/		1 i	%	., where 1 ˆi  is the i-th parameter of IDIM-TLS using the knowledge of the ten payload
	parameters and 2 ˆi

 is the i-th parameter of IDIM-TLS using the knowledge of the payload mass only. Paper DS-

TABLE IV THE

 IV DRIVE GAINS OF THE TX-40: A PRIORI STAÜBLI'S DATA AND IDENTIFIED VALUES WITH IDIM-TLS.

		A priori	with 10 CAD parameters		with 1 weighed mass M 7
	i	g 	i	0	ˆi g 		% ri 	%e i		ˆi g 			% ri 	%e i
	1	32.96	33,8
					0 1,83	2,55 33,90 1,87	2,85
	2	32.96	31,5
					0 1,10	4,43 32,40 1,36	1,70
	3	25.65	23,5
					0 1,70	8,38 24,20 1,87	5,65
	4	-11.52	-8,31 1,99	27,86 -9,21 2,75	20,05
	5	18.48	14,5
					0 1,71	21,54 16,30 2,53	11,80
	6	7.68	6,12 1,76	20,31		6,79 2,57	11,59
						(% ) :14.18% i mean e				(% ) : 8.94% i mean e
		% ri   : relative standard deviation.	i e		% i g g   	i	0	/	g 	i	0	%	.
	For each experiment, the relative error norms || || / || j	j	||

ρ

Y computed on each joint j equation are given in Table

VI

. Results show that the best reconstruction is achieved for parameters of

Table VIII .

 VIII Paper DS-13-1373

TABLE VII ESTIMATION

 VII OF THE PAYLOAD MASS OF 4.59 KG ± 0.005 KG

		ĝ 	Manuf Case 1 Case 2 Case 3 Case 4
	Traj. 1	7  M	% ri 	4,44 1,68	4,32 1,54	4,22 1,54	4,53 1,54	4,57 1,56
		%e		3,27	5,88	8,06	1,31	0,44
	Traj. 2	7  M	% ri 	4,44 1,18	4,29 1,09	4,19 1,09	4,50 1,08	4,54 1,10
		%e		3,27	6,54	8,71	1,96	1,09
	Traj. 3	7  M	% ri 	4,46 1,03	4,31 0,94	4,21 0,94	4,53 0,94	4,57 0,95
		%e		2,83	6,10	8,28	1,31	0,44
	Mean(%e)			3,12	6,17	8,35	1,53	0,65
				7 e M M M 7 % 7 / %  

  Once again, it can be observed that the best payload estimation is obtained for drive gains identified in Case 4 (%e=0.67%), i.e. with IDIM-TLS techniques considering the knowledge of Paper DS-

				TABLE VIII
	ESTIMATION OF THE PAYLOAD MASS OF 1.686 KG ± 0.005 KG
		ĝ 	Manu f	Case 1 Case 2 Case 3 Case 4
	Traj. 1	7  M	% ri 	1,67 2,61	1,58 2,51	1,55 2,51	1,66 2,49	1,68 2,53
		%e		0,95	6,29	8,07	1,54	0,36
	Traj. 2	7  M	% ri 	1,70 1,92	1,60 1,87	1,56 1,87	1,68 1,86	1,70 1,88
		%e		0,83	5,10	7,47	0,36	0,83
	Traj. 3	7  M	% ri 	1,70 1,94	1,59 1,89	1,56 1,89	1,67 1,88	1,70 1,89
		%e		0,83	5,69	7,47	0,95	0,83
	Mean(%e)			0.87	5,69	7,67	0,95	0,67
				7 e M M M 7 % 7 / %  

TABLE I :

 I THE ESSENTIAL DYNAMIC PARAMETERS OF THE TX-40 IDENTIFIED WITH IDIM-WLS.

TABLE II :

 II THE DRIVE GAINS OF THE TX-40: A PRIORI STAÜBLI'S DATA AND IDENTIFIED VALUES WITH IDIM-WLS.

TABLE III :

 III THE ESSENTIAL DYNAMIC PARAMETERS OF THE TX-40 IDENTIFIED WITH IDIM-TLS.

TABLE IV :

 IV THE DRIVE GAINS OF THE TX-40: A PRIORI STAÜBLI'S DATA AND IDENTIFIED VALUES WITH IDIM-TLS.

TABLE V :

 V CONDITION NUMBER OF THE OBSERVATION MATRIX

TABLE VI :

 VI RELATIVE ERROR NORMS ON TORQUE ESTIMATION FOR CROSS-VALIDATION TRAJECTORIES

TABLE VII :

 VII ESTIMATION OF THE PAYLOAD MASS OF 4.59 KG ± 0.005 KG

TABLE VIII :

 VIII ESTIMATION OF THE PAYLOAD MASS OF 1.686 KG ± 0.005 KG

TABLE I THE

 I ESSENTIAL DYNAMIC PARAMETERS OF THE TX-40 IDENTIFIED WITH IDIM-WLS.

	parameter	with 10 cad param. id.val. % ri    id.val. with 1 weighed mass m 7 % ri   %e i
	ZZ 1R Fv 1 Fs 1 XX 2R XZ 2R ZZ 2R MX 2R Fv 2 Fs 2 XX 3R ZZ 3R MX 3 MY 3R Ia 3 Fv 3 Fs 3 XY 4 MX 4 Ia 4 Fv 4 Fs 4 MY 5R Ia 5 Fv 5 Fs 5 Ia 6 Ia m6 Fv 6 Fvm 6 Fsm 6  off6 XX 7 XY 7 XZ 7 YY 7 YZ 7 ZZ 7 MX 7 MY 7 MZ 7	1,17e+0 6,35e+0 5,18e+0 -4,54e-1 -1,43e-1 9,85e-1 1,97e+0 4,17e+0 7,28e+0 9,56e-2 1,19e-1 3,63e-2 -5,19e-1 7,52e-2 1,46e+0 5,81e+0 --1,53e-2 2,50e-2 6,54e-1 1,72e+0 -2,40e-2 4,69e-2 1,24e+0 2,81e+0 9,80e-3 8,49e-3 4,57e-1 4,30e-1 1,52e+0 -0.64e-1 -1.80e-2 2.60e-2 0.64e-1 2.60e-2 4.40e-2 -2.90e-1 -2.90e-1 4.10e-1	2,28 2,37 3,80 3,53 5,85 1,36 1,57 2,09 2,60 13,42 5,50 24,39 2,69 7,29 3,22 2,87 -18,62 4,63 2,49 3,11 14,60 8,11 3,63 3,87 6,29 19,92 2,07 3,21 2,95 ----------	1,14e+0 6,21e+0 4,98e+0 -4,43e-1 -1,43e-1 9,60e-1 1,92e+0 4,09e+0 7,09e+0 9,14e-2 1,16e-1 3,77e-2 -5,02e-1 7,39e-2 1,42e+0 5,64e+0 -5,27e-3 -1,34e-2 2,36e-2 6,37e-1 1,68e+0 -2,26e-2 4,24e-2 1,21e+0 2,71e+0 8,33e-3 8,95e-3 4,34e-1 4,11e-1 1,53e+0 7,91e-2 8,39e-2 -1,19e-2 2,22e-2 8,20e-2 3,18e-2 5,21e-2 -2,43e-1 -2,54e-1 3,91e-1	2,66 2,70 4,10 3,81 5,88 1,92 2,21 2,52 2,91 13,84 5,75 23,00 3,30 7,44 3,60 3,29 35,04 21,25 5,55 3,76 4,26 15,67 9,28 4,56 4,83 7,77 18,72 3,51 4,31 4,11 24,72 5,79 17,55 9,69 5,82 5,33 3,92 3,29 3,41 3,90	2,56 2,20 3,86 2,42 0,00 2,54 2,54 1,92 2,61 4,39 2,52 3,86 3,28 1,73 2,74 2,93 -12,42 5,60 2,60 2,33 5,83 9,59 2,42 3,56 15,00 5,42 5,03 4,42 0,66 -31,09 33,89 14,62 28,13 22,31 18,41 16,21 12,41 4,63
	M 7 =4.59±0.005		M 7 =4.59±0.005	-
	rel. err. norm ||ρ||/||Y|| =12.40%	rel. err. norm ||ρ||/||Y||=12.43%
	, (% a b  Y V g ,   1 / % i e    ) : 7.86%, i mean e 1 2 ˆˆ% i i i   ., where 1 ˆi

TABLE II THE

 II DRIVE GAINS OF THE TX-40: A PRIORI STAÜBLI'S DATA AND IDENTIFIED VALUES WITH IDIM-WLS.

		A priori	with 10 CAD parameters		with 1 weighed mass M 7
	i	g 	i	0	ˆi g 		% ri 	%e i			ˆi g 			% ri 	%e i
	1	32.96	30,20		2,41	8,37		31,10		2,02	5,64
	2	32.96	29,80		1,72	9,59		30,50		1,09	7,46
	3	25.65	21,80		2,36	15,01		22,40		1,74	12,67
	4	-11.52	-8,06		3,33	30,03		-8,27		1,76	28,21
	5	18.48	14,00		3,10	24,24		14,40		1,36	22,08
	6	7.68	5,88		3,15	23,44		6,08		1,44	20,83
						(% ) :18.45% i mean e		(% ) :16..5% i mean e
				% ri   : relative standard deviation.	i e		% i g g   	i	0	/	g 	i	0	%	.
													Paper DS-13-1373

TABLE III THE

 III ESSENTIAL DYNAMIC PARAMETERS OF THE TX-40 IDENTIFIED WITH IDIM-TLS.

	par.		with 10 cad param. id.val. % ri    id.val. with 1 weighed mass m 7 % ri   %e i
	ZZ 1R Fv 1 Fs 1  off1 XX 2R XZ 2R ZZ 2R MX 2R Fv 2 Fs 2 XX 3R ZZ 3R MX 3 MY 3R Ia 3 Fv 3 Fs 3 XY 4 MX 4 Ia 4 Fv 4 Fs 4 MY 5R Ia 5 Fv 5 Fs 5 Ia 6 Ia m6 Fv 6 Fvm 6 Fsm 6  off6 XX 7 XY 7 XZ 7 YY 7 YZ 7 ZZ 7 MX 7 MY 7 MZ 7	1,28e+0 6,93e+0 5,61e+0 --4,95e-1 -1,54e-1 1,02e+0 2,04e+0 4,29e+0 7,56e+0 1,07e-1 1,24e-1 4,02e-2 -5,53e-1 8,03e-2 1,52e+0 6,12e+0 -1,53e-2 2,51e-2 6,58e-1 1,73e+0 -2,52e-2 4,73e-2 1,24e+0 2,84e+0 9,92e-3 8,51e-3 4,61e-1 4,32e-1 --0.64e-1 --1.80e-2 2.60e-2 0.64e-1 2.60e-2 4.40e-2 -2.90e-1 -2.90e-1 4.10e-1	-	2,00 2,08 3,18 -2,91 4,86 1,34 1,55 2,10 2,48 9,79 5,25 18,36 2,39 6,77 3,14 2,72 21,74 5,48 2,87 3,62 18,08 11,60 4,65 4,87 9,38 20,93 2,73 2,87 -----------	1,28e+0 7,01e+0 5,46e+0 3,01e-1 -5,01e-1 -1,58e-1 1,05e+0 2,11e+0 4,45e+0 7,72e+0 1,05e-1 1,27e-1 4,14e-2 -5,71e-1 8,31e-2 1,57e+0 6,28e+0 -5,61e-3 -1,50e-2 2,71e-2 7,27e-1 1,92e+0 -2,37e-2 4,90e-2 1,41e+0 3,15e+0 9,69e-3 1,04e-2 5,02e-1 4,75e-1 1,77e+0 9,33e-2 9,59e-2 -1,43e-2 2,46e-2 9,29e-2 3,73e-2 6,03e-2 -2,75e-1 -2,84e-1 4,48e-1	2,04 2,07 3,20 27,13 2,84 4,42 1,51 1,73 2,02 2,28 9,80 4,53 16,79 2,55 6,14 2,93 2,62 26,20 19,45 5,02 3,22 3,74 17,29 8,67 3,88 3,93 6,79 17,29 2,78 2,88 3,07 17,46 5,13 21,79 7,88 4,92 4,48 3,41 2,65 2,75 3,11	0,00 1,15 2,67 -1,21 2,60 2,94 3,43 3,73 2,12 1,87 2,42 2,99 3,25 3,49 3,29 2,61 -1,96 7,97 10,49 10,98 5,95 3,59 13,71 10,92 2,32 22,21 8,89 9,95 --49,84 20,56 5,38 45,16 43,46 37,05 5,17 2,07 9,27
	M 7 =4.59±0.005			M 7 =4.59±0.005	-
	rel. err. norm ||ρ||/||Y|| =11.95% rel. err. norm
						||ρ||/||Y||=11.15%
	) : 8.28%,	, (%

TABLE IV THE

 IV DRIVE GAINS OF THE TX-40: A PRIORI STAÜBLI'S DATA AND IDENTIFIED VALUES WITH IDIM-TLS.

		A priori	with 10 CAD parameters		with 1 weighed mass M 7
	i	g 	i	0	ˆi g 		% ri 	%e i		ˆi g 			% ri 	%e i
	1	32.96	33,8
					0 1,83	2,55 33,90 1,87	2,85
	2	32.96	31,5
					0 1,10	4,43 32,40 1,36	1,70
	3	25.65	23,5
					0 1,70	8,38 24,20 1,87	5,65
	4	-11.52	-8,31 1,99	27,86 -9,21 2,75	20,05
	5	18.48	14,5
					0 1,71	21,54 16,30 2,53	11,80
	6	7.68	6,12 1,76	20,31		6,79 2,57	11,59
						(% ) :14.18% i mean e			(% ) : 8.94% i mean e
		% ri   : relative standard deviation.	i e		% i g g   	i	0	/	g 	i	0	%	.
													Paper DS-13-1373

TABLE VII ESTIMATION

 VII OF THE PAYLOAD MASS OF 4.59 KG ± 0.005 KG

		ĝ 	Manuf Case 1 Case 2 Case 3 Case 4
	Traj. 1	7  M	% ri 	4,44 1,68	4,32 1,54	4,22 1,54	4,53 1,54	4,57 1,56
		%e		3,27	5,88	8,06	1,31	0,44
	Traj. 2	7  M	% ri 	4,44 1,18	4,29 1,09	4,19 1,09	4,50 1,08	4,54 1,10
		%e		3,27	6,54	8,71	1,96	1,09
	Traj. 3	7  M	% ri 	4,46 1,03	4,31 0,94	4,21 0,94	4,53 0,94	4,57 0,95
		%e		2,83	6,10	8,28	1,31	0,44
	Mean(%e)			3,12	6,17	8,35	1,53	0,65
				7 e M M M 7 % 7 / %  
								Paper DS-13-1373

TABLE VIII ESTIMATION

 VIII OF THE PAYLOAD MASS OF 1.686 KG ± 0.005 KG

		ĝ 	Manu f	Case 1 Case 2 Case 3 Case 4
	Traj. 1	7  M	% ri 	1,67 2,61	1,58 2,51	1,55 2,51	1,66 2,49	1,68 2,53
		%e		0,95	6,29	8,07	1,54	0,36
	Traj. 2	7  M	% ri 	1,70 1,92	1,60 1,87	1,56 1,87	1,68 1,86	1,70 1,88
		%e		0,83	5,10	7,47	0,36	0,83
	Traj. 3	7  M	% ri 	1,70 1,94	1,59 1,89	1,56 1,89	1,67 1,88	1,70 1,89
		%e		0,83	5,69	7,47	0,95	0,83
	Mean(%e)			0.87	5,69	7,67	0,95	0,67
				7 e M M M 7 % 7 / %  

ACKNOWLEDGEMENTS

This work was supported in part by the French FUI project IRIMI (F1004026 Z) and by the French ANR project ARROW (ANR 2011BS3 006 01).

Paper DS- while the robot is tracking some reference trajectories without load fixed on the robot and some trajectories with a known payload fixed on the robot end-effector. The ten inertial parameters are measured or calculated CAD software. The method has been successfully experimentally validated on a Stäubli TX-40 robot.


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

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

Complete Shaking Force and Shaking Moment Balancing of In-line Four-bar Linkages by adding a Class-two RRR or RRP Assur Group

Keywords: Shaking force, shaking moment, balancing, planar linkage, Assur group

This paper deals with the complete shaking force and shaking moment balancing of planar in-line four-bar linkages with constant input speed. A new solution is discussed, which is carried out without counter-rotations. The complete shaking force and shaking moment balancing is obtained by using a coupler link, which is a physical pendulum and by adding a class-two Assur group with prescribed geometrical and mass parameters. Two types of Assur group are examined: two links group with three revolute joints (RRR) and two links group with two revolute and one prismatic joints (RRP). Numerical simulations carried out using ADAMS software validate the proposed balancing technique and illustrate that the obtained six-bar mechanism is reactionless, i.e. the sum of all ground bearing forces and their moments are eliminated.

Introduction

When an unbalanced linkage must run at high speeds, or contains massive links, considerable shaking force and shaking moment are transmitted to its surroundings. These disturbances cause vibrations and therefore limit the full potential of many machines. Many high-speed machines contain planar four-bar linkages and the problem of their mass balancing is of continuing interest to machine designers. The previous works on the balancing of planar fourbar linkages may be arranged in the following groups.

(i) Complete shaking force balancing [START_REF] Fischer | Über die reduzierten Systeme und die Hauptpunkte der Glieder eines Gelenkmechanismus[END_REF]- [START_REF] Hilpert | Weight balancing of precision mechanical instruments[END_REF]. In general, it is carried out by counterweights mounted on the movable links of the linkage. With regards to the several approaches employed for the redistribution of movables masses, the developed methods could be divided into three principal groups. a) The method of «principal vectors» [START_REF] Fischer | Über die reduzierten Systeme und die Hauptpunkte der Glieder eines Gelenkmechanismus[END_REF]; The aim of this approach was to study the balancing of the mechanism relative to each link and in the determination of those points on the links relative to which a static balance was obtained. These points were called «principal points». Then, from the condition of similarity of the vector loop of the principal points and the structural loop of the mechanism, the necessary conditions of balancing were derived. b) The method of «static substitution of masses»; its aim was to statically substitute the mass of the coupler by concentrated masses, which are balanced thereafter together with the rotating links. Such an approach changes the problem of mechanism balancing into a simpler problem of balancing rotating links. This method was illustrated for four-bar linkage in [START_REF] Artobolevskii | Methods of inertia calculation for mechanisms of agricultural machines[END_REF]- [START_REF] Smith | Inertia forces in a four-bar linkage[END_REF].

c) The method of «linearly independent vectors» [START_REF] Berkof | A new method for completely force balancing simple linkages[END_REF], in which the vector equation describing the position of the center of total mass of the mechanism is treated in conjunction with the closed equation of its kinematic chain. The result is an equation of static moments of moving link masses containing single linearly independent vectors. Thereafter following the conditions for balancing the mechanism by reducing the coefficients, which are time-dependent to zero.

It should be noted that the addition of a counterweighted pantograph device to the planar four-bar linkage has also been used for its complete shaking force balancing [START_REF] Hilpert | Weight balancing of precision mechanical instruments[END_REF].

(ii) Complete shaking force and partial shaking moment balancing [START_REF] Lowen | Theory of Shaking Moment Optimization of Force-Balanced Four-Bar Linkages[END_REF]- [START_REF] Arakelian | Complete Shaking Force and Partial Shaking Moment Balancing of Planar Four-bar Linkages[END_REF]. Two principal approaches may be distinguished: a) the shaking moment minimization of fully force balanced linkages [START_REF] Lowen | Theory of Shaking Moment Optimization of Force-Balanced Four-Bar Linkages[END_REF]- [START_REF] Haines | Minimum r.m.s. shaking moment or driving torque of a force-balanced four-bar linkage using feasible counterweights[END_REF], in which it is shown that the optimum conditions of partial moment balance can be obtained by certain link mass distribution ratios. b) the minimization of the unbalance of shaking moment by transferring the rotation axis of the counterweight mounted on the input crank [START_REF] Schcepetilnikov | The determination of the mass centers of mechanisms in connection with the problem of mechanism balancing[END_REF]- [START_REF] Arakelian | Complete Shaking Force and Partial Shaking Moment Balancing of Planar Four-bar Linkages[END_REF]. In the study [START_REF] Schcepetilnikov | The determination of the mass centers of mechanisms in connection with the problem of mechanism balancing[END_REF], the first harmonic of the shaking moment is eliminated by attaching the required input link counterweight, not to the input shaft itself, but to a suitable offset one which rotates with the same angular velocity. This approach is original in that, while maintaining the force balance of the mechanism, it is possible to create an additional balancing moment, thereby reducing the shaking moment. This approach has been further developed in works [START_REF] Arakelian | Partial shaking moment balancing of fully shaking force balanced linkages[END_REF], [START_REF] Arakelian | Complete Shaking Force and Partial Shaking Moment Balancing of Planar Four-bar Linkages[END_REF].

It should be noted that optimization algorithms are also widely used in partial balancing of four-bar linkages [START_REF] Zhang | A constitutive method of objective function for the dynamic optimum balance of shaking force in linkage[END_REF]- [START_REF] Chaudhary | Balancing of four-bar linkages using maximum recursive dynamic algorithm[END_REF].

(iii) Complete shaking force and shaking moment balancing [START_REF] Kamenskii | On the question of the balancing of planar linkages[END_REF]- [START_REF] Arakelian | Shaking force and shaking moment balancing of mechanisms: an historical review with new examples[END_REF]. The first method of complete shaking force and shaking moment balancing was proposed in study [START_REF] Kamenskii | On the question of the balancing of planar linkages[END_REF], which was extended in [START_REF] Berkof | Complete force and moment balancing inline four-bar linkages[END_REF]. In this approach, the mass of the connecting coupler is substituted dynamically by concentrated masses located at the coupler joints. Thus, the dynamic model of the coupler becomes a weightless link with two concentrated masses. This transforms the problem of fourbar linkage shaking force and shaking moment balancing into a problem of balancing rotating links carrying concentrated masses. The parallelogram structure has also been applied for complete shaking force and shaking moment balancing of four-bar linkages [START_REF] Bagci | Complete shaking force and shaking moment balancing of link mechanisms using balancing idler loops[END_REF]. In the studies [START_REF] Ye | Complete balancing of planar linkages by an equivalence method[END_REF]- [START_REF] Dresig | Vollständiger und harmonischer Ausgleich ebener Mechanismen[END_REF], the authors have proposed methods for complete shaking force and shaking moment balancing by counterweights with planetary gear trains. In [START_REF] Esat | A theory of complete force and moment balancing of planar linkage mechanisms[END_REF] a toothed-belt transmission is used to rotate counterweights intended for shaking force balancing, which also allows shaking moment balancing. The disadvantage of these methods is the need for the connection of gears to the oscillating links. The oscillations of the links of the mechanism will create noise unless expensive anti-backlash gears are used.

Another solution using the copying properties of the pantograph was developed [START_REF] Arakelian | Patent FR2817008[END_REF], [START_REF] Arakelian | Shaking force and shaking moment balancing of mechanisms: an historical review with new examples[END_REF], in which the gears driven by the coupler suffer no such sudden reversals so that this problem is almost eliminated. However, it should be noted that the use of the gears for the balancing of four-bar linkages is a drawback for the industrial applications and a fully shaking force and shaking moment balanced four-bar linkage without any gears is more appealing.

The shaking moment balancing of fully force balanced linkages using a prescribed input speed fluctuation was proposed in [START_REF] Kochev | Full shaking moment balancing of planar linkages by a prescribed input speed fluctuation[END_REF]. However, such balancing is complicated because it is necessary to use a special type of drive generator. Moreover, it cannot be used for balancing of linkages, which generate the prescribed motions of the output links.

In this paper a new solution is discussed, which allows the complete shaking force and shaking moment balancing of in-line1 four-bar linkages with constant input speed by adding a class-two Assur group, i.e. a group which does not add any supplementary degree of freedom into the mechanism [START_REF] Leinonen | Terminology for the theory of machines and mechanisms[END_REF]. It should be noted that the balancing of the shaking moment without counter-rotations of three particular classes of four-bar linkages was discussed in the studies [START_REF] Ricard | On the development of reactionless parallel manipulators[END_REF]- [START_REF] Gosselin | Dynamic balancing of planar mechanisms using toric geometry[END_REF]. However, such a method cannot be extended to general four-bar linkages. In this paper it is proposed to take advantages of the use of the properties of the four-bar linkage with prescribed geometric parameters [START_REF] Ricard | On the development of reactionless parallel manipulators[END_REF]- [START_REF] Gosselin | Dynamic balancing of planar mechanisms using toric geometry[END_REF] and to combine it with (i) the principle of the dynamic substitution of link mass by concentrated masses and (ii) with the prescription of constant input speed. It should be mentioned here that the suggested balancing approach can be efficiently applied on the cyclic high-speed machines executing motions in the steady-state regime when the input speed is constant [START_REF] Demeulenaere | Improving machine drive dynamics: a structured design approach towards balancing[END_REF], [START_REF] Berkof | The input torque in linkages[END_REF]. Before considering the suggested balancing concept, let us recall basic notions concerning the balanced four-bar linkages with prescribed geometrical parameters.

In the paper of Berkof and Lowen [START_REF] Lowen | Theory of Shaking Moment Optimization of Force-Balanced Four-Bar Linkages[END_REF], the angular momentum H and the shaking moment M, expressed at point O, of a force balanced in-line four-bar linkage (Fig. 1) were expressed as:

   3 1 i i i I H   ,       3 1 i i i I H dt d M    (1) with   i i i i i i l r r k m I    2 2
, (i = 1 and 3)

(2)

  2 2 2 2 2 2 2 2 l r r k m I    , (3)
where r i is the length of vector r i which connects the pivot P i to the centre of mass of link i, l i is the length of vector l i which connects the proximal revolute joint P i to the distal joint on the same link, d is the length of the base which is the distance between the two fixed joints on the base, and k i is the radius of gyration with respect to the centre of mass of link i, m i is the mass of link i.

Moreover,  i is the angular position of link i with respect to the x-axis. It should be carefully noted that the terms I i of equations ( 2) and (3) should not be mistaken for the moment of inertia of a link, even if the units are the same.

With regard to the shaking force balancing, the following expressions were obtained: (5) In the works [START_REF] Ricard | On the development of reactionless parallel manipulators[END_REF], [START_REF] Wu | Synthesis of reactionless spatial 3-dof and 6-dof mechanisms without separate counter-rotations[END_REF], it has been shown that it is possible to cancel the shaking moment of a four bar mechanism by associating mentioned geometric constraints with an optimal distribution of masses. Three kinds of shaking force and shaking moment balanced four-bar mechanisms were found, which are shown in Fig. 2. In order to illustrate the shaking moment balancing, let us consider the mechanism shown in Fig. 2b. The geometrical constraints of this mechanism are the following:

  1 1 2 2 2 1 2 / m r m l r l l  , (4)
3 1 l l  , (6)
2 l d  .

(7) This leads to the following kinematic relationships: 0

3 2 1          . ( 8
) Thus, from expressions (1) and ( 8), it is easy to see that the shaking moment will be cancelled if 2) and ( 3)). For this purpose the following relationships must be established [START_REF] Leinonen | Terminology for the theory of machines and mechanisms[END_REF]:

I 1 =-I 2 =I 3 (see Eqs. (
  2 1 2 2 2 2 2 2 2 m I r r l m k    , (9)
 

3 1 2 3 3 3 3 2 3 m I r r l m k     , (10)
where

  1 1 2 1 2 1 1 1 l r r k m I    . ( 11
) It should be noted that similar results have been obtained for the mechanisms shown in Figs. 2a and2c.

Statement of the problem

The aim of the suggested balancing approach consists of adding a two-link kinematic chain with prescribed geometrical parameters to an in-line four-bar linkage with arbitrary geometrical parameters. It is important to note that the added structure must be an Assur group, i.e. a group which does not add any supplementary degree of freedom into the mechanism [START_REF] Leinonen | Terminology for the theory of machines and mechanisms[END_REF]. This allows for the modification of the mass redistribution of the obtained six-bar mechanism without perturbation of the kinematic properties of the initial four-bar linkage. We would like to restate that this technique allows for the complete shaking force and shaking moment balancing without counter-rotating masses. Now let us consider the shaking force and shaking moment balancing of an in-line four-bar mechanism using class-two Assur groups with RRR kinematic chain.

Shaking force balancing

Fig. 3 shows an in-line four-bar linkage with the added class-two RRR Assur group. Let us denote the following vectors as:

l 1 = d OA , l 2 = d AB , l 3 = d CB , l' 3 = d CP2 , l 4 = d P2P'2 , l 5 = d P3P'2 , r 1 = d OS1 , r 2 = d AS2 , r 3 = d CS3 , r 4 = d P2S4 , r 5 = d P3S5 , r CP1 = d OSCP1 , r CP2 = d CSCP2 , r CP3 = d P3SCP3 .

The notation d

PQ represents the vector starting from from P to point Q. The added class-two RRR Assur group has the above mentioned properties, i.e. it is designed such as:

5 3 ' l l  , ( 12
)
4 el  , (13) where l i (i = 3 to 5) is the norm of vector l i and e the distance between C and P 3 . As it is shown in Fig. 3, the Assur group P 2 P' 2 P 3 is attached to the initial linkage OABC in such a way that it forms a four-bar linkage with link (BC), discussed in paragraph 2.1. Let us now derive the expression of the shaking force F sh of the obtained mechanism:

5 1 sh i Si i m    Fd ( 14
)
where m i is the mass of the link i and Si d   the translational acceleration of the centre of mass S i .

Developing and simplifying, one obtains:

                      1 1 2 1 1 1 1 1 sin cos cos sin            l A d , ( 16a
)                       3 3 2 3 3 3 3 3 sin cos cos sin            l B d , ( 16b
)                            ) sin( ) cos( ) cos( ) sin( 3 3 2 3 3 3 3 3 3                r S d , ( 16c
)
P l                              d , ( 16d
)                       5 5 2 5 5 5 5 5 2 ' sin cos cos sin            l P d . ( 16e
)
The shaking force F sh may be cancelled through the addition of three counterweights positioned at points S CPi (Fig. 3), with masses m CPi (i = 1, 2, 3). With such counterweights, the expression of the shaking force becomes:

CP CP r                              d . ( 17b
)
Thus the shaking force is cancelled if the distribution of the masses is as follows: (18d)

            2 2 2 2 1 1 1 1 1 1 l r l m l r m r l m CP CP , (18a)
    3 3 4 4 4 3 2 3 2 2 3 3 4 4 4 3 sin 1 / ' sin tan cos 1 / ' cos m r m r l l m l r l m r m r l l         , ( 18b
)         22 3 3 4 4 4 3 2 3 2 2 3 3 4 4 4 3 2 2 2 sin 1 / ' sin cos 1 / ' cos CP CP m r m r l l m l r l m r m r l l m r            (18c)           

Shaking moment balancing

Let us now derive the expression of the shaking moment M, expressed at point O, of such a mechanism: are the position and accelerations along x and y axes of points S CPi , respectively, k i is the radius of gyration of link i. Now, let us consider that link 2 is a physical pendulum2 [START_REF] Berkof | Complete force and moment balancing inline four-bar linkages[END_REF], i.e. it can be replaced dynamically by two point masses located at joint centres A and B. This implies that:

             3 1 5 1 2 i SCPi SCPi SCPi SCPi CPi i i i Si Si Si Si i x y y x m k x y y x m M            (19)
  2 2 2 2 2 r l r k   . ( 20
) Then, considering that the input speed is constant, i.e. 20) and ( 21), one can simplify Eq. ( 19) as:

   5 3 i i i I M    ( 22
)
where ()

  2 2 2 2 '2 2 4 4 3 3 3 3 2 3 2 2 4 3 24 CP CP r l r I m k r m l m r m l ll       , (23a)
I m k r r l    , (23b)
   

Thus, this new six-bar mechanism has the same shaking moment as the four-bar mechanism composed of links (P 1 P 2 ), (P 2 P' 2 ) and (P' 2 P 3 ). Therefore, the initial four-bar linkage balancing problem is transformed in the balancing of the four-bar linkage formed by the added Assur group. Note that the latter has specific geometry and its balancing conditions have been examined in section 2.1.

Applying these results to the considered mechanism, we obtain:

  22 4 3 4 4 4 4 / k I r l r m     , (24)
   

     . ( 25
)
Substitution of Eqs. ( 24)-( 25) into eq. ( 22) leads to:

  5 4 3 3          I M , (26)
Taking into account relations [START_REF] Elliot | The theory of torque, shaking force and shaking moment balancing of four link mechanisms[END_REF] and ( 13), we have 0

5 4 3             . ( 27
) and consequently 0  M . ( 28
)
The proposed balancing technique has been illustrated using the four-bar linkage shown in Fig. 2b. However, it can also be achieved via the mechanism of Fig. 2a or Fig. 2c.

Complete shaking force and shaking moment balancing by adding a class-two RRP Assur group

Shaking force balancing

The second solution, which is proposed for the cancellation of the shaking moment of a four-bar linkage is carried out by adding a class-two RRP Assur group (Fig. 4). Let us denote the following vectors as: In this case, the lengths of added links are the following: (29) Thus, the new part created by added links is a Scott-Russell mechanism, which is attached to the initial linkage at an angle of .

The relations between link accelerations are the following:

4 3       . ( 30
)
Let us now derive the expression of the shaking force F sh of such a mechanism:

   5 1 i Si i sh m d F   ( 31
)
where m i is the mass of the link i and Si d   the translational acceleration of the centre of mass S i .

Developing and simplifying, one obtains: [START_REF] Arakelian | Shaking force and shaking moment balancing of mechanisms: an historical review with new examples[END_REF] where r i and l i are the algebraic values of the norm of vectors r i , and l i , respectively,

  d d d d d F                                    5 4 4 4 5 4 3 3 2 2 2 2 2 2 2 1 1 1 m l r m m m m l r m l r l m l r m D S B A sh
                     
           l d , ( 33
)                            ) sin( ) cos( ) cos( ) sin( 3 3 2 3 3 3 3 3 3                r S d . ( 34
)
and D d   represent the acceleration of point D. Its expression is:

                           ) sin( ) cos( ) cos( ) sin( ' 3 3 2 3 3 3 3 3                l D d . ( 35
)
The shaking force F sh may be cancelled through the addition of three counterweights positioned at points S CPi (Fig. 4), with masses m CPi (i = 1, 2, 3). With such counterweights, the expression of the shaking force becomes:

       F F F F d d d d , ( 36
)
where r CPi is the algebraic values of the norm of vectors r CPi , and

CP CP r                              d . ( 37
)
Thus the shaking force is cancelled if the distribution of the masses is as follows:

            2 2 2 2 1 1 1 1 1 1 l r l m l r m r l m CP CP , ( 38a
)            5 4 4 4 3 4 3 m l r m r l m CP CP , ( 38b
)          cos ' cos sin ' sin tan 3 5 4 3 3 2 2 3 2 3 5 4 3 3 l m m r m l r l m l m m r m       , (38c)
        2 2 2 3 5 4 3 3 2 2 3 2 2 3 5 4 3 3 2 cos ' cos sin ' sin CP CP r l m m r m l r l m l m m r m m            . (38d)

Shaking moment balancing

With regard to the shaking moment M, expressed at point O, we have: are the positions and accelerations along x and y axes of points S CPi , respectively, k i is the radius of gyration of link i. Now, let us consider as in the previous case, that link 2 is a physical pendulum and that the input speed is constant.

             3 1 5 1 2 i SCPi SCPi SCPi SCPi CPi i i i Si Si Si Si i x y y x m k x y y x m M            (39)
Taking into account Eqs. ( 20) and ( 21), one can simplify Eq. (39) as:

3 3 4 4 M I I   (40) where 
   2 2 2 2 3 3 5 4 2 3 2 2 2 2 3 2 3 3 3 ' CP CP CP r m l m m m l l r m r k m I        , (41)
  (42) Introducing eq. ( 30) in this expression, we find:

  3 4 3    I I M   . ( 43
) Thus, the mechanism will be moment balanced if:

I 3 = I 4 , (44)
which can be obtained using a design of link 4 for which the radius of gyration should be equal to

It should be mentioned that, in order to avoid the singular configurations of the added structure, the value of angle  should be chosen carefully during the design process.

The next section presents two illustrative examples of the proposed balancing technique.

Illustrative examples and numerical simulations

Balancing by adding a class-two RRR Assur group

Let us carry out the complete shaking force and shaking moment balancing of a four-bar linkage with parameters given in table 1. The simulations of the proposed mechanism have been carried out using ADAMS software and the obtained results are shown in Fig. 5 (full line). Now we add the RRR Assur group with prescribed distribution of the center of masses and inertia. Its geometric and mass properties are given in table 2 and the location and mass of the added counterweights are given in table 3. The radii of gyration of elements 4 and 5 and the mass of the counterweight m CP2 are not expressed in these tables as they depend on the value of angle (Fig. 3). Their variations as a function of  are shown in Fig. 6. In these figures, the values of  are bounded between -60 and 180 deg in order to avoid the RRR Assur group to cross a singularity during the motion. In Fig. 5 (dotted line), it is shown that after the addition of the Assur group, the shaking force and shaking moment are cancelled. As angle  is a free parameter that has only influence on the values k 4 , k 5 and m CP2 , it can be chosen so that it minimizes one supplementary criterion. In the remainder of the paper, this criterion is chosen to be the linkage input torque [START_REF] Demeulenaere | Improving machine drive dynamics: a structured design approach towards balancing[END_REF], [START_REF] Berkof | The input torque in linkages[END_REF]. It should however be mentioned that angle  could be used to minimize another criterion such as the power consumption, the energy, etc.

It should also be noted that the input torque , i.e. the torque requested by the actuator to move the mechanism, is computed in the two illustrative examples by using the Lagrange equations [START_REF] Berkof | The input torque in linkages[END_REF]:

1 1                  L L dt d  , (46)
where L = T -V is the Lagrangian of the system, V is the potential energy (equal to 0 in absence of gravity) and T is the kinetic energy:

  2 2 2 11 22      i Si Si j j ij T m x y I , (47)

Si

x  , Si y  being the velocities along x and y axes of any centres of masses (for links and counterweights).

In Fig. 7, the maximum of the input torque absolute value of as a function of angle  is shown. Thus, it is possible to see that if the value of  is chosen arbitrarily, the input torques can grow up to 2140 Nm (for  = 0 deg). It also appears that the input torque will be minimal if  = 164 deg. In this case, the value of the input torque is 1010 Nm, i.e. about 2 times less than in the first case.

Balancing by adding a class-two RRP Assur group

We now propose obtaining the complete shaking force and shaking moment balancing of the same mechanism by adding a class-two RRP Assur group. Its geometric and mass properties are given in table 4 and the location and mass of the added counterweights are given in table 5. The radii of gyration of element 4 and the mass of the counterweight m CP2 are not expressed in these tables as they depend on the value of angle  (Fig. 4). Their variations as a function of  are shown in Fig. 8. In these figures, the values of  are bounded between 25 and 100 deg or 205 and 280 deg in order to avoid the RRP Assur group to cross a singularity during the motion. The simulations of the proposed mechanism have been carried out using ADAMS software and the results are similar to the previous case shown in Fig. 5. As it was mentioned above, the angle  is a not fixed design parameter and it can be found from minimization of the input torque of the mechanism. In Fig. 9, maximum of the input torque absolute value as a function of angle  is shown. It is possible to see that if the value of  is chosen arbitrarily, the input torques can grow up to 2300 Nm (for  = 25 deg). It also appears that the input torque will be minimal if  = 205 deg. In this case, the value of the input torque is 1380 Nm, i.e. about 1.7 times less than in the first case.

Conclusion

In this study, a new balancing method is proposed, which allows the shaking force and shaking moment cancellation in the in-line four-bar linkages with constant input speed. It is carried out without counter-rotating masses because, in this balancing approach, the properties of the self-balanced four-bar linkage with prescribed geometric parameters are combined with the principle of the dynamic substitution of link by concentrated masses. The suggested balancing is obtained by adding a class-two RRR or RRP Assur groups with prescribed geometrical parameters to the initial mechanism. The simulations carried out using ADAMS software have shown that the obtained six-bar mechanism transmits no inertia loads to its surroundings, i.e. the sum of all ground bearing forces and their moments are eliminated.

Figures captions

Tables captions

Table 1. Four-bar linkage's parameters. Table 2. Parameters defining the RRR Assur group. Table 3. Parameters defining the counterweights. Table 4. Parameters defining the RRP Assur group. Table 5. Parameters defining the counterweights.

2 . 2 . 1 .

 221 Complete shaking force and shaking moment balancing by adding a class-two RRR Assur group Theoretical background related to the balanced four-bar linkages with prescribed geometrical parameters.

Fig. 1 .

 1 Fig. 1. A general in-line four-bar mechanism.

4

 4

Fig. 2 .

 2 Fig. 2. The three kinds of shaking force and shaking moment balanced four-bar mechanisms.

Fig. 3 .

 3 Fig. 3. The balanced mechanism with the class-two RRR Assur group.

l 1 =

 1 d OA , l 2 = d AB , l 3 = d CB , l' 3 = d CD , l 4 = d DE , r 1 = d OS1 , r 2 = d AS2 , r 3 = d CS3 , r 4 = d DS4 , r 5 = d ES5 , r CP1 = d OSCP1 , r CP2 = d CSCP2 , r CP3 = d DSCP3 .

Fig. 4 .

 4 Fig. 4. The balanced four-bar linkage with the class-two RRP Assur group.

  positions and accelerations along x and y axes of points S i , respectively, and x SCPi , y SCPi ,

10

 10

Fig. 5 .

 5 Fig. 5. Shaking force and shaking moment of the four-bar mechanism before (full line) and after (dotted line) balancing.

Fig. 6 .

 6 Fig. 6. Variation of the values of parameters k 4 , k 5 and m CP2 as a function of angle .

Fig. 7 .

 7 Fig. 7. Variation of the maximal input torque absolute value as a function of angle .

Table 4 .Fig. 8 .

 48 Fig. 8. Variation of the values of parameters k 4 and m CP2 as a function of angle .

Fig. 9 .

 9 Fig. 9. Variation of the maximal input torque absolute value as a function of angle .

Fig. 1 .Fig. 2 .Fig. 3 .Fig. 4 .Fig. 5 .Fig. 6 .Fig. 7 .Fig. 8 . 9 .

 123456789 Fig. 1. A general in-line four-bar mechanism. Fig. 2. The three kinds of shaking force and shaking moment balanced four-bar mechanisms. (a) Case I: l 1 = d and l 2 = l 3 . (b) Case II: l 1 = l 3 and l 2 = d. (c) Case III: l 1 = l 2 and l 3 = d. Fig. 3. The balanced mechanism with the class-two RRR Assur group. Fig. 4. The balanced four-bar linkage with the class-two RRP Assur group. Fig. 5. Shaking force and shaking moment of the four-bar mechanism before (full line) and after (dotted line) balancing. (a) Shaking force along x-axis. (b) Shaking force along y-axis. (c) Shaking moment along z-axis. Fig. 6. Variation of the values of parameters k 4 , k 5 and m CP2 as a function of angle . (a) variation of k 4 and k 5 . (b) variation m CP2 . Fig. 7. Variation of the maximal input torque absolute value as a function of angle . Fig. 8. Variation of the values of parameters k 4 and m CP2 as a function of angle . (a) variation of k 4 for   [25 100] deg. (b) variation of k 4 for   [205 280] deg. (c) variation m CP2 for   [25 100] deg. (d) variation m CP2 for   [205 280] deg. Fig. 9. Variation of the maximal input torque absolute value as a function of angle . (a) for   [25 100] deg. (b) for   [205 280] deg.

  , P 2 and P' 2 respectively. Their expressions are:

	sh		1 1 r l   2 2 2 4 1 2 2 3 3 4 2 2 4 A B S l r r l r 4 m m m m m l l l      	P	2		5 5 r l  4 4 5 4 r m m l   	P	'2	(15)
	F where r i is the algebraic values of the norm of vectors r i , and A d d d d d , B d , 3 S d , 2 P d and the acceleration of points A, B, S 3	 d 	d 2 ' P	represent

Table 1 .

 1 Four-bar linkage's parameters.

	Parameters	link 1	link 2	link 3 ( = 0 deg)
	l	i (m)	0.2	0.27	0.25
	m i (kg)	1	1	1
	r	i (m)	0.1	0.135	0.125
	k i (m)	0.056	0.135	0.086

Table 2 .

 2 Parameters defining the RRR Assur group.

	Parameters	link 4	link 5
	l	i (m)	0.8	0.25
	m i (kg)	1.5	1
	r	i (m)	0.4	0.125

Table 3 .

 3 Parameters defining the counterweights.

	Parameters	Counterweight 1	Counterweight 2	Counterweight 3
	r	CPi (m)	-0.1	0.25	-0.125
	m i (kg)	2	-	2.5

The term 'in-line' means that the centres of mass of the links must lie on the line connecting the pivots (which can be extended beyond the pivots). The links need not be symmetrical in any way[START_REF] Berkof | Complete force and moment balancing inline four-bar linkages[END_REF].

A 'physical pendulum' is a link which has such a distribution of masses that it allows the dynamic substitution of link's mass and inertia by two concentrated masses.