David Guérin
email: david.guerin@irt-jules-verne.fr

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

Sébastien Garnier
email: sebastien.garnier@irccyn.ec-nantes.fr

Alexis Girin
email: alexis.girin@irt-jules-verne.fr

Optimal Measurement Pose Selection for Joint Stiffness Identification of an Industrial Robot Mounted on a Rail

The paper focuses on the identification of elastostatic properties of an industrial serial robot mounted on a rail. It proposes an identification procedure in order to find the optimal robot configuration to minimize the impact of measurement errors on the identification accuracy of the stiffness parameters. An experimental setup is designed to perform the identification of all stiffness parameters under industrial conditions. The proposed identification procedure is easy to use and takes little time.

I. INTRODUCTION

Nowadays, several industrial companies are interested in analysing the possibility to employ serial robots for industrial machining operations. Indeed, serial robots can be more flexible than computer numerical control (CNC) machines and also cost-effective, but, serial robots are usually not as stiff as a CNC machine and they are not designed specifically to perform machining operations.

The IRT Jules Verne launched in 2013 the project ROBOFIN with several industrial partners (DAHER, DCNS, STX France, Loiretech, Multiplast, Europe Technologies and Coriolis Composites) to investigate machining operations on large size pieces performed by a KUKA KR500 L480-2 robot mounted on a 18 meters long rail.

In order to improve machining capabilities of industrial serial robots it is necessary to detect the main factors that are responsible to the robot inaccuracy. Since the introduction of serial robots in industry, several studies have focused on their geometric parameters identification and calibration [START_REF] Khalil | Modeling, identification and control of robots[END_REF], in order to improve their accuracy. The geometric calibration is mainly focused on the identification of the geometric parameters and joint offsets.

To improve the efficiency of a robot, calibration of non-geometric parameters has to be considered as well. Identification and calibration of robot stiffness parameters (elasto-static calibration) is essential too for a serial machining robot [START_REF] Abele | Modeling and identification of an industrial robot for machining applications[END_REF]. When a serial robot performs machining operations, the efforts applied on the robot end-effector cause deformations on the robot components, reducing the quality of the machined parts. Generally, for an industrial serial robot the compliance of the bodies is assumed to be negligible with respect to the joint compliance [START_REF] Alici | Enhanced stiffness modeling, identification and characterization for robot manipulators[END_REF], [START_REF] Nubiola | Absolute calibration of an ABB IRB 1600 robot using a laser tracker[END_REF] Fig. 1. KR500 L480-2 MT mounted on rail located at IRT Jules Verne and [START_REF] Dumas | Joint stiffness identification of six-revolute industrial serial robots[END_REF]. Nowadays, there are still many research works focusing on the robot stiffness identification. Gautier et al. [START_REF] Gautier | Dynamic identification of flexible joint manipulators with an efficient closed loop output error method based on motor torque output data[END_REF] proposed an off-line joint stiffness identification method based on motor torque output data. In Milutinovic et al. [START_REF] Milutinovic | Cartesian compliance identification and analysis of an articulated machining robot[END_REF] the stiffness identification is applied to a 5-axis serial robot used to perform machining operations, in order to analyse the impact of each joint's compliance on the robot stiffness. In [START_REF] Lightcap | Improved positioning accuracy of the pa10-6ce robot with geometric and flexibility calibration[END_REF], a stiffness identification was performed to a Mitsubishi PA10-6CE, in order to improve static positioning accuracy of the robot.

During machining operations, the robot can be subjected to chattering. Pan et al. performed some studies about the elasto-dynamic identification of serial robots. Their research work aimed at avoiding chatter by selecting the optimal cutting parameters [START_REF] Pan | Chatter analysis of robotic machining process[END_REF]. Other non-geometric parameters can affect robot efficiency. Gong et al. [START_REF] Gong | Nongeometric error identification and compensation for robotic system by inverse calibration[END_REF] studied the impact of thermal effects on serial robot deformations.

In industrial environment, robot parameters identification may be time consuming. Hence, some works have focused on the optimisation of measurement configurations [START_REF] Daney | Choosing measurement poses for robot calibration with the local convergence method and tabu search[END_REF], [START_REF] Zhuang | Optimal selection of measurement configurations for robot calibration using simulated annealing[END_REF]. The study described in [START_REF] Dumas | Joint stiffness identification of industrial serial robots[END_REF] and [START_REF] Klimchik | Optimal selection of measurement configurations for stiffness model calibration of anthropomorphic manipulators[END_REF] is focused on the optimisation of the measurement configurations for robot stiffness identification. In this paper, a method is proposed to decrease the time required for the stiffness identification experiments by providing a simple method easy to use and by minimising the number of measurements configuration. Moreover, this method has been developed, to improve the identification accuracy and is based on the work of Wu et al. [START_REF] Wu | Optimality criteria for measurement poses selection in calibration of robot stiffness parameters[END_REF] and has been adapted to a serial robot mounted on a rail. This paper also presents an experimental setup used to identify all stiffness parameters. Section II presents the track Fig. 2. Experimental setup used to exert an external wrench on the robot end-effector mounted robot and measurement equipment. Section III deals with the kinematic modelling and stiffness modelling of the robot. Section IV details the method used to determine the optimal measurement configurations. Section V provides simulation results. Section VI deals with the conclusion and the future work.

II. ROBOTIC CELL AND EXPERIMENTAL SETUP

The presented work aims to investigate machining operations performed with a serial robot mounted on a rail. The following case study has been taken into consideration. The robotic machining cell located at the IRT Jules Verne and shown in Fig. 1 is composed of a KUKA KR500 L480-2 MT robot and a rail. The robot rated payload is equal to 480 kg. The rail is 18 m long and has been developed by Sud-Ouest Système company. In what remains, the KUKA KR500 L480-2 MT robot will be referred to as KR500. In order to identify the stiffness parameters of the KR500 and the rail, a solicitation effector developed at IRCCyN is mounted on the end-effector. Moreover a chain with an embedded turnbuckle is attached to the solicitation effector at one end and to the floor at the other end, namely, at point P a = [x pa y pa z pa ], as illustrated in Fig. 2.

The aim of this experimental setup is to be cost effective with no extra infrastructure and easy to integrated in industrial environment. Moreover this setup can be used to identify all stiffness parameters of the presented robotic cell.

An identification procedure will be developed in Section IV in order to find the robot configurations associated with the optimal measurement poses and the optimal location of the attachment point, P a , such that the effect of measurement errors on the identification accuracy of the stiffness parameters is a minimum. In this identification procedure, the

j α j [rad] d j [mm] θ j [rad] r j [mm] 1 -π/2 0 0 r 1 2 π/2 0 θ 2 1048 3 π/2 500 θ 3 0 4 0 1300 θ 4 0 5 π/2 55 θ 5 1525 6 -π/2 0 θ 6 0 7 π/2 0 θ 7 290
Cartesian coordinates of points P 1 , P 2 , P 3 , P l1 and P l2 will be measured by a Leica Laser Tracker and retroreflectors.

III. ROBOT STIFFNESS MODELLING

A. Kinematic modelling and parametrization

Fig. 3 illustrates a kinematic drawing of the KR500 mounted on a rail. The modified Denavit Hartenberg (mDH) parameters are used to parametrize the KR500 robot and the rail.

• the z j axis is along the axis of joint j • the x j axis is aligned with the common normal between z j and z j+1 . • the origin O j is the intersection point of z j and x j The mDH parameters of the KR500 mounted on a rail are given in Table I.

The direct kinematic model gives us the relation between the end-effector twist, t, and the joint rates vector q, expressed in the base frame F b :

t = J(q) q (1)
where q = [r 1 θ 2 θ 3 θ 4 θ 5 θ 6 θ 7 ] T is the joint coordinate vector, q = [ ṙ1 θ2 θ3 θ4 θ5 θ6 θ7 ] T is the joint velocity vector, r 1 is the prismatic joint length and [θ 2 ... θ 7 ] are the revolutes joint angles. J is the kinematic Jacobian matrix of the KR500 mounted on a rail.

B. Stiffness model

Performing machining operations with a serial robot generates an external wrench on the robot end-effector. This external wrench may produce elastic deformations in the robot. The robot links are assumed to be rigid. The elasticity of the rail is modelled by a linear elastic spring along the direction of the rail. The revolute joint stiffness is characterized by a linear elastic torsional spring. The Cartesian stiffness matrix K X of the KR500 mounted on a rail is expressed as:

K X = J -T K q J -1 (2)
where K q is the diagonal joint stiffness matrix expressed as:

K q =           k r1 0 0 0 0 0 0 0 k θ2 0 0 0 0 0 0 0 k θ3 0 0 0 0 0 0 0 k θ4 0 0 0 0 0 0 0 k θ5 0 0 0 0 0 0 0 k θ6 0 0 0 0 0 0 0 k θ7           (3)
Zhang et al. [START_REF] Zhang | Stiffness identification for serial robot manipulator based on uncertainty approach[END_REF] showed that the Cartesian stiffness matrix K X can be rewritten as follows:

K X = J -T l K q J -1 m (4)
where J m is the Jacobian matrix expressed at the measurement point and J l is the Jacobian matrix expressed at the loading point. Therefore, the relationship between the endeffector displacement δd and the external wrench w can be expressed as:

δd = J m K -1 q J T l w (5)
where δd = [δp φ] T is the vector of the end-effector displacement, with δp and φ being respectively the pointdisplacement and the rotational error of the robot endeffector. w = [f m] T is the vector of external wrench (forces and moments).

In order to consider the measurement errors and the pointdisplacement of the measurement points Eq. ( 5) is rewritten as:

δp i = A i x + i (i = 1, ..., n) (6)
where δp i is the point-displacement of the ith measurement point, x is the vector of joint compliances, x = [1/k r1 , 1/k θ2 , ..., 1/k θ7 ] T and i denotes the measurement error due to the Leica laser tracker. The matrix A i is composed of the columns of matrix J l and J m , and the external force f applied on the loading point, namely,

A i = [J m1,i J T l1,i f i |...|J m7,i J T l7,i f i ](i = 1, ..., n) (7)

A. Problem statement

A method defined in Wu et al., in 2012 [START_REF] Wu | Optimality criteria for measurement poses selection in calibration of robot stiffness parameters[END_REF] proposed a criterion to obtained optimal measurement poses to identify stiffness parameters of an industrial serial robot. This methodology is focused on the test-pose (TP) base approach. The test-pose is a particular robot configuration in which very good accuracy is required after compensation due to the machining operation to be realized. For experimental validation a vertical wrench was applied on a Kuka KR270-2 robot [START_REF] Klimchik | Design of calibration experiments for identification of manipulator elastostatic parameters[END_REF]. In this case, the first joint stiffness cannot be identified, because it is not solicited. Therefore, some other experiments are required for the identification of this first joint stiffness.

In the case of the robot mounted on a rail, an experiment with a vertical wrench leads to similar results for the identification of the first prismatic joint stiffness and the first revolute joint stiffness. Then, the experimental setup presented in Section II is used in order to solicitate all the stiffness of all joints.

Note that the method developed in [START_REF] Wu | Optimality criteria for measurement poses selection in calibration of robot stiffness parameters[END_REF] is used to find the optimal measurement poses for the joint stiffness identification of the KR500 mounted on a rail shown in Fig. 1. However, the optimisation problem at hand has been enhanced in order to find the optimal location of the base attachment point P a .

B. Adapted methodology

The criterion ε T P ac is employed to evaluate the root-meansquared error of the robot end-effector deflection for a given test pose, as illustrated in Fig 4.

ε T P ac = σ 2 trace   A 0 n i=1 A T i A i -1 A T 0   (8)
where the matrix A 0 amounts to matrix A i , defined in Eq.( 7), evaluated at the test-pose, σ = 0.03 mm is the standard deviation of the measurement noise as used in [START_REF] Klimchik | Design of calibration experiments for identification of manipulator elastostatic parameters[END_REF] for the same measurement device. To minimize the effect of measurement errors on the robot end-effector pose error evaluation after error compensation, we should find optimal measurement configurations

q opt i = [r opt 1i θ opt 2i θ opt 3i θ opt 4i θ opt 5i θ opt 6i θ opt 7i ]
T of the KR500 robot and find the optimal of the base attachment point P a . The z-coordinate, z pa , of point P a expressed in the base frame F b , is equal to -0.891 mm due to the height of the rail. The optimisation of coordinate y pa is redundant with the optimisation of joint length r 1 because this axis is assumed to be perfectly collinear with axis y b . Therefore, y pa is set to 6 m. As a consequence, the following optimisation is solved in order to find the optimum measurement poses and the optimum location of the base attachment point P a :

Thus, the optimisation problem can be formulated as follows min ε T P ac over x = [q opt 1 , ..., q opt m0 , x pa ] T subject to where m 0 is the number of measurement poses.

P b                0 m ≤ r opt 1i ≤ 14 m -70 ˚≤ θ opt 2i ≤ 70 zθ opt 7 i ≥ 500 mm f 2 = 4000 N 2 m ≤ x pa ≤ 6 m (9)
In our case, the direction of the external force applied on the end-effector is defined by the position of the attachment point and the position of the robot end-effector. In the experimental setup the direction of external forces is obtained from the locations of points P l2 and P l1 located at the both ends of the spring (as shown in Fig. 2). Thus, the direction of external forces is expressed as follows :

u = P l2 -P l1 ||P l2 -P l1 || 2 (10)
The external force is expressed as:

f = ||f ||.u (11)

C. Optimisation results

The optimisation problem has been launched with a genetic algorithm for a single test-pose and for several measurement poses. It is noteworthy that the measurements can be repeated in the measurement poses in order to ease the experimentations.

The test-pose is defined by the following robot joint coordinate vector q T P and the force f T P applied on the robot end-effector

q T P =[8.5m -10 • 35 • 39 • 81 • 22 • -10 • ] T (12) f T P =[-124.5 N -175.4 N -209.1 N] T (13)
Table II gives the values of criterion ε T P ac as a function of measurement poses and number of experiment. Figure 5 illustrates the value of ε T P ac as a function of the number of measurement poses, m 0 , and the the number of measurements, n. It is apparent that ε T P ac is a minimum when n = 12 and m 0 = 12. To improve the time required to achieve the stiffness identification for a same number of experiments, a minimum number of measurement configurations is needed. As a conclusion in order to minimize the number of robot configurations to make the experiments, twelve experiments will be performed with three measurement poses. It means that four measurements are made in each measurement pose.

Equations ( 14) to [START_REF] Zhang | Stiffness identification for serial robot manipulator based on uncertainty approach[END_REF] give the joint angles vectors associated with the three optimal measurement poses.

q opt 1 =[5.3 m -37.3˚57.6˚80.8˚101.2˚-27˚-82.2˚] T (14) q opt 2 =[14 m 69˚33.9˚-33.8˚107.5˚29.2˚116˚] T (15) q opt 3 =[4.7 m 19˚12.2˚75.4˚89.4˚39.7˚117.1˚] T (16)
kr 1 k θ 2 k θ 3 k θ 4 k θ 5 k θ 6 k θ 7
8.10 7 3.8.10 6 6.6.10 6 3.9.10 6 5.6.10 5 6.6.10 5 4.7.10 5

Equation ( 17) gives the optimal Cartesian coordinates of the base attachment point P a expressed in the base frame F b .

x pa =5.04 m, y pa = 6 m, z pa = -0.891 m (

Fig. 6 presents value of the criterion ε T P ac as a function of x pa for twelve experiments and three measurement poses.

V. SIMULATION

To simulate the stiffness identification we consider the measurement poses defined by Eqs. ( 14) to ( 17) for the testpose defined by Eq. ( 12) and ( 13). This section aims to show the impact of the stiffness identification on the estimation of the robot end-effector deflection around the defined test-pose and for poses defined in the global workspace.

A. Evaluation of the joint stiffness values

The joint stiffness identification is performed by using the experimental setup presented in Section II. It is assumed that the geometric calibration has been realized. The spring scale is used to measure the magnitude of the external forces. Thus, to obtain the direction of external forces f , a set of two SMR (Spherical Mounted Retro-reflector) are placed at the ends of the Spring scale at points P l1 and P l2 . To measure the displacement of the robot end-effector, three SMR are set above (P 1 , P 2 and P 3 ).A measurement error is considered for each measurement. Moreover, the measurement system is a Leica Laser Tracker. Measurement errors are taken in a spherical coordinate system. Angles θ and φ are respectively taken in the intervals [0, π/2] and [-π, π] with a uniform distribution, the radius r follows a normal distribution.

r ∼ N (µ, σ) (18)
with µ = 0 and σ = 0.03 mm. Eq. ( 6) expressed the displacement of the end effector as a function of the vector of nominal joint compliances presented in Table III and measurement errors. This equation can be computed as follows to take into account the number of measurement points, p, and the number of optimal measurement poses, m 0 :

B =         A 1 1 . . . A j i . . . A p m0         i = 1, ..., m 0 j = 1, ..., p (19)
and c =         δp 1 1 . . . δp j i . . . δp p m0         i = 1, ..., m 0 j = 1, ..., p (20)
In that case, the estimates value of joint compliances x qe can be obtained with the generalized inverse of B as follows:

x qe r = (B T r B r ) -1 B T r c r r = 1, ..., 200 (21)
where r is the number of repetitions of measurements.

Table IV presents the standard deviation, σ kq e , and the bias, b kq e , for the joint stiffness identification performed 200 times, the percentage of estimation error is also expressed in the Table IV to highlight the joints that are mainly responsible for the stiffness measurement inaccuracy.

These results are used further to evaluate the impact of the stiffness identification results on the estimation of the endeffector displacement in the global workspace and around the test-pose when an external wrench is applied on it.

B. Estimation of the robot end-effector displacement

Repetitive machining operations do not necessarily require a high accuracy in terms of stiffness identification. However, a correct estimation of the robot deflection is essential. The criterion illustrated hereafter aims to represent the accuracy of the estimation of the robot end-effector displacement. Thus, the smaller µ j pi represent the better displacement compensation.

µ j pi = ||δp j ni -δp j ei || 2 ||δp j ni || 2 i = 1, 2, 3 j = 1, ..., 1000 (22)
where j is the number of validation measurement, i is the number of measurement point, δp j ni is the nominal displacement of the robot end-effector, which is calculated with the nominal stiffness values presented in Table III. δp j ei is the estimation of the robot end-effector displacement calculated with the joint stiffness values given in Table IV. The joint stiffness estimation is computed as: where µ kq e is the average value of k qe and σ kq e is the standard deviation of k qe . Moreover, the compensation of the end-effector pose error should be better around the test-pose defined in Eqs. ( 12) and ( 13).

k qe ∼ N (µk qe , σk qe ) (23)
Accordingly, a simulation is performed with a set of 1000 validation measurement configurations taken around the test-pose, and an other set of 1000 validation measurement configurations taken randomly in the global workspace of the KR500 mounted on a rail. For each measurement, the displacement of the robot end-effector is measured at points P 1 , P 2 and P 3 . Then, the criterion µ p j i is calculated with respect to the nominal displacement δp j ni . Table V provides the results of those simulations.

It is apparent that the estimation of the robot end-effector displacement is identified with an accuracy of 0.16% to 0.2% around the test-pose and with an accuracy of 0.68% in the global workspace. Note that only the effect of the measurement errors on the evaluation of the robot endeffector pose error, while subjected to external wrenches, has been considered in the scope of this paper.

VI. CONCLUSION

This paper dealt with a methodology to identify all stiffness parameters of an industrial serial robot mounted on a rail. Moreover, an existing test-pose based approach was used to identify the joint stiffness values of the KUKA KR500 MT robot mounted a rail, while minimizing the effect of measurement errors on the end-effector pose error evaluation after compensation. This method provides optimal measurement configurations adapted to the experimental setup.

In order to obtain the optimal measurement configurations, we firstly had to define the robot kinematic model. Then, we used the optimisation measurement configurations to run the simulations. Outcomes show the interest of using the optimisation method with defined machining configurations, called test pose.

In future work, the proposed methodology will be improved to consider the effect of the masses of the bodies on the robot deflection and the geometric errors. Moreover, a enhanced stiffness model of rail will be developed and some experimental validations will be performed in order to identify the geometric and stiffness parameters of the KUKA KR 500 MT robot mounted on a rail shown in Fig. 1.

Fig. 3 .

 3 Fig. 3. Kinematic drawing of the KR500 L480-2 MT mounted on a rail

Fig. 4 .

 4 Fig. 4. Drawing representation of the criterion ε T Pac[START_REF] Klimchik | Design of calibration experiments for identification of manipulator elastostatic parameters[END_REF]

Fig. 5 .

 5 Fig. 5. ε T P ac as a function of the number of measurement poses, m 0 , and the number of measurements, n.

Fig. 6 .

 6 Fig. 6. ε T P ac as a function of xp a for n = 12 and m 0 = 3

TABLE I MDH

 I PARAMETERS OF THE KR500 ROBOT MOUNTED ON A RAIL

TABLE II CRITERION

 II

ε T P ac

ACKNOWLEDGMENT

This study is part of the ROBOFIN project managed by IRT Jules Verne (French Institute in Research and Technology in Advanced Manufacturing Technologies for Composite, Metallic and Hybrid Structures). The authors wish to associate the industrial and academic partners of this project; respectively DAHER, DCNS, STX France, Loiretech, Multiplast, Europe Technologies, Coriolis Composites, IRCCyN (UMR CNRS-ECN-Nantes University 6597).

This work is supported by the


St Claire Dumas
email: claire.dumas@irccyn.ec-nantes.fr

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

S Ébastien Garnier
email: sebastien.garnier@irccyn.ec-nantes.fr

Benoît Furet
email: benoit.furet@irccyn.ec-nantes.fr

WORKPIECE PLACEMENT OPTIMIZATION OF SIX-REVOLUTE INDUSTRIAL SERIAL ROBOTS FOR MACHINING OPERATIONS

Roboticists are faced with new challenges in robotic-based manufacturing. Up to now manufacturing operations that require both high stiffness and accuracy have been mainly realized by using computer numerical control machine tools. This paper aims to show that manufacturing finishing tasks can be performed with robotic cells knowing the process cutting phenomena and the robot stiffness throughout its Cartesian workspace.

It makes sense that the finishing task of large parts would be cheaper with robots. However, machining robots have not been adapted for such operations yet. As a consequence, this paper introduces a methodology that aims to determine the best placement of the workpiece to be machined knowing the cutting forces exerted on the tool and the elastostatic model of the robot. In this vein, a machining quality criterion is proposed and an optimization problem is formulated. The KUKA KR270-2 robot is used as an illustrative example throughout the paper. NOMENCLATURE a p Depth of cut f z Feed per tooth F t Tangential component of the cutting force F r Radial component of the cutting force N Rotational speed of the tool K t Coefficient for tangential component of the cutting force K r Coefficient for radial component of the cutting force V f Feed speed

INTRODUCTION

Industrial robots were originally dedicated to "pick and place" operations. They start to be used for machining operations such as trimming, deflashing, degating, sanding and sawing. Machining robots are currently developed either to help the operator realize machining operations, which have been performed with Computer Numerical Control (CNC) machines so far. First, industrial robots can be used to reduce scrap rates and production cost and to increase the volume and the flexibility of production lines. The second case deals with some applications that require CNC machine tools because of the expected machining quality. As a matter of fact, industrial robots are not as accurate as CNC machine tools. They are three to four times cheaper than their counterparts for a given workspace. Therefore, it makes sense that the larger the parts to be machined, the more interesting robot machining.

Some research works on robotic-based manufacturing have been conducted the last two decades. Those research works also considered parallel robots [START_REF] Terrier | Machinesoutils à structure parallèle et usinage à grande vitesse[END_REF], which turn to have a smaller footprint ratio than industrial six-axis robots. The larger the part to be machined, the better the robots with large footprint ratio. The main drawback of industrial serial robots for machining operations turns to be their lack of stiffness and as a result, the large manufacturing errors [START_REF] Matsuoka | High-Speed End Milling of an Articulated Robot and its Characteristics[END_REF][START_REF] Zhang | Machining with Flexible Manipulator: Toward Improving Robotic Machining Performance[END_REF][START_REF] Abele | Modeling and Identification of an Industrial Robot for Machining Applications[END_REF][START_REF] Olabi | Planification de l'avance sur trajectoire pour un robot 6 axes usinant[END_REF]. Indeed, a conventional machine tool is about fifty times stiffer than an industrial serial robot [START_REF] Zhang | Machining with Flexible Manipulator: Toward Improving Robotic Machining Performance[END_REF].

Matsuoka et al. [START_REF] Matsuoka | High-Speed End Milling of an Articulated Robot and its Characteristics[END_REF] studied the robotic-based milling. They claimed that the machining quality can be improved by decreasing the cutting forces, namely, by decreasing the diameter of the tool and by increasing the spindle speed. Zhang et al. [START_REF] Zhang | Machining with Flexible Manipulator: Toward Improving Robotic Machining Performance[END_REF] proposed a real-time compensation of the robot end-effector displacement, based on the evaluation of each joint deflection. To this end, they developed an elastostatic model of the robot at hand and used a wrench sensor to measure the forces and moments applied on the robot end-effector at any time. Besides, Olabi et al. [START_REF] Olabi | Planification de l'avance sur trajectoire pour un robot 6 axes usinant[END_REF] showed that the robot manufacturing errors can be reduced with jerk-limited trajectories.

The vibration of the robot itself is another issue to deal with in robotic machining [START_REF] Abele | Modeling and Identification of an Industrial Robot for Machining Applications[END_REF][START_REF] Pan | Chatter Analysis of Robotic Machining Process[END_REF]. As a matter of fact, the first natural frequency of an industrial robot is about 10 to 100 times smaller than the one of conventional machine tools [START_REF] Pan | Chatter Analysis of Robotic Machining Process[END_REF]. Pan et al. also noticed that the vibrations occurring during robotic machining operations are not regenerative but structural [START_REF] Pan | Chatter Analysis of Robotic Machining Process[END_REF]. Therefore, they came up with a stability criterion from the elastodynamic modeling of the robot and the cutting forces. Then, they formulated some rules for robotic machining from several experimental tests. The placement of the workpiece to be machined may also affect its machining quality and/or the robot energy consumption. In this vein, Ur-Rehman et al. focused on the multiobjective path placement optimization of a parallel kinematics machines based on energy consumption, shaking forces and maximum actuators torques [START_REF] Ur-Rehman | Multiobjective Path Placement Optimization of Parallel Kinematics Machines Based on Energy Consumption, Shaking Forces and Maximum Actuators Torques: Application to the Orthoglide[END_REF].

It makes sense that the finishing task of large parts would be more reliable and cheaper with robots. However, machining robots have not been adapted for such operations yet. As a consequence, this paper introduces a methodology that aims to determine the best placement of the workpiece to be machined knowing the cutting forces exerted on the tool and the elastostatic model of the robot. In this vein, a machining quality criterion is proposed and an optimization problem is formulated. The KUKA KR270-2 robot is used as an illustrative example throughout the paper.

The paper is organized as follows: Section 2 presents the model used to predict the cutting forces applied by the tool on the workpiece. Section 3 deals with the joint stiffness identification of the KUKA KR270-2 robot. Section 4 shows the influence of the workpiece placement and the kinematic redundancy on the machining quality of a test workpiece through some experimental results. Section 5 introduces a methodology that aims to determine the best placement of the workpiece to be machined knowing the cutting forces exerted on the tool and the elastostatic model of the robot. The proposed methodology is explained through the formulation of a mono-objective optimization problem. Section 6 shows some results of the optimization of the optimization problem at hand. Section 7 is about the conclusions of the paper and the future works.

CUTTING FORCE MODEL

The tool is composed of several teeth. We consider in this paper a tool composed of four teeth. Figure 1 presents the chip and the directions of the tangential force F t and the radial force F r applied by each tooth on the workpiece.

Feed Feed direction direction ϕ in ϕ out
Piece to mill

F y F x F r F t f z sin(ϕ) f z ϕ FIGURE 1.
Chip geometry and definition of ϕ in and ϕ out

The instantaneous forces exerted by the tooth on the workpiece are defined in [START_REF] Tlusty | Dynamics of cutting forces in end milling[END_REF] and are expressed as follows:

δ F t = K t a p f z sin(ϕ) dϕ δ F r = K r δ F t (1)
The mean forces F x and F y along the x and y axes associated with one tooth are expressed as:

F x = ϕ in ϕ out ( F t cos(ϕ) + F r sin(ϕ)) dϕ F y = ϕ in ϕ out (-F t sin(ϕ) + F r cos(ϕ)) dϕ (2)
Therefore, if ϕ ∈ [ϕ in ϕ out ],

F x = K t a p f z (- cos(2ϕ out ) 4 + K r ϕ out 2 -K r sin(2ϕ out ) 4 ) + K t a p f z ( cos(2ϕ in ) 4 -K r ϕ in 2 + K r sin(2ϕ in ) 4
)

F y = K t a p f z ( sin(2ϕ out ) 4 - ϕ out 2 -K r cos(2ϕ out ) 4 ) + K t a p f z (- sin(2ϕ in ) 4 + ϕ in 2 + K r cos(2ϕ in ) 4 ) (3)
else,

F x = 0 F y = 0 ( 4
)
f z is the feed rate and a p is the depth of cut. Then, the global cutting force exerted by the entire tool on the piece can be calculated. Coefficients K t and K r were determined from cutting experiments with a force measurement system.

IDENTIFICATION OF THE JOINT STIFFNESS VAL-UES

The translational and rotational displacements of the robot end-effector can be evaluated if the forces and moments applied on it are known. To this end, the stiffness model of the robot defined by Eq. ( 5) was determined based on the CCT Theory [START_REF] Chen | The 6x6 Stiffness Formulation and Transformation of Serial Manipulators via the CCT Theory[END_REF]. The links of the robot are supposed to be quite stiffer than the joints and not known as it happens usually for industrial robots [START_REF] Dumas | Développement de méthodes robotisées pour le parachèvement de pièces métalliques et composites[END_REF]. Then, the relationship between the wrench applied on the robot end-effector and its small displacement screw is defined as follows:

w w w = K X δ X X X (5)
where K X is the Cartesian stiffness matrix of the robot. w w w is the 6-dimensional vector of forces and moments exerted on the robot end-effector and derived from Sec. 2. δ X X X is the obtained 6-dimensional small displacement screw of the end-effector. The Cartesian stiffness matrix K X that depends on the robot configuration, the joint stiffness values and the wrench applied on the end-effector takes the form:

K X = J -T (K θ -K C ) J -1 (6)
with

K θ =         k θ 1 0 0 0 0 0 0 k θ 2 0 0 0 0 0 0 k θ 3 0 0 0 0 0 0 k θ 4 0 0 0 0 0 0 k θ 5 0 0 0 0 0 0 k θ 6         (7)
and

K C = ∂ J T ∂ θ 1 w w w ∂ J T ∂ θ 2 w w w ∂ J T ∂ θ 3 w w w ∂ J T ∂ θ 4 w w w ∂ J T ∂ θ 5 w w w ∂ J T ∂ θ 6 w w w (8)
J is the kinematic Jacobian matrix of the robot. K θ is its diagonal joint stiffness matrix and K C is the complementary stiffness matrix, which depends on the wrench applied on the end-effector.

k θ 1 k θ 2 k θ 3 k θ 4 k θ 5 k θ 6
0.237 3.32 2.79 0.486 0.521 0.38

An identification procedure was developed in [START_REF] Dumas | Joint Stiffness Identification of Six revolute Industrial Serial Robots[END_REF] in order to determine all joint stiffness values of the robot. Accordingly, the translational and rotational displacements of the end-effector can be predicted knowing the wrench exerted on it. It is noteworthy that the flexibility of the links is considered through the identified joint stiffness values. The joint stiffness values of the KUKA KR270-2 robot are given in Table 1. They were obtained by using the experimental setup shown in Fig. 2 and described in [START_REF] Dumas | A Methodology for Joint Stiffness Identification of Serial Robots[END_REF][START_REF] Dumas | Joint Stiffness Identification of Industrial Serial Robots[END_REF].

INFLUENCE OF THE WORKPIECE PLACEMENT AND THE KINEMATIC REDUNDANCY ON THE MA-CHINING QUALITY

This section aims to show the influence of the workpiece placement and the kinematic redundancy on the machining quality of a test workpiece through some experimental results.

Evaluation of the tool displacement during a

milling operation Knowing the joint stiffness values of the robot from Sec. 3 and the forces exerted on the robot end-effector from Sec. 2, it is possible to predict the tool displacement using the stiffness matrix defined with Eq. ( 5). Four machining operations were realized in order to validate the theoretical elastostatic model of the KUKA KR270-2 robot: (i) a workpiece made up of a resin material called LAB was milled along a 300 mm linear path. It is noteworthy that the LAB material exerts a small force on the tool; (ii) a workpiece made up of an Aluminium alloy was milled along a 300 mm linear path with the cutting conditions P3 given in Table 2; (iii) a workpiece made up of an Aluminium alloy was milled along a 300 mm linear path with the cutting conditions P4 given in Table 2; (iv) a workpiece made up of an Aluminium alloy was milled along a 300 mm linear path with the cutting conditions P5 given in Table 2. N, a p and V f denote the rotational speed of the tool, the depth of cut and the feed speed, respectively. Figure 3 illustrates the corresponding measured and calculated paths. The inherent straightness of the robot explains the difference between the calculated and measured paths in the LAB material. We can notice that the difference between the calculated and measured paths for the other machining operations is about 5% and is also partly due to the inherent straightness of the robot. Consequently, it is possible to evaluate the path that will be followed by the tool considering its displacement due to the cutting forces from the elastostatic model of the robot defined with Eq. ( 5).

Influence of the task placement on the machining quality FIGURE 4. Test workpiece

In order to show the influence of the task placement on the machining quality, a test workpiece shown in Fig. 4 was milled in two different placements within the robot Cartesian workspace. The two placements of the test workpiece are illustrated in Fig. 5. Four machining operations were realized with the same cutting condition, namely, the cutting condition P4 described in Table 2: (i) the milling of the test workpiece made up of a LAB material located in Placement 1; (ii) the milling of the test workpiece made up of an Aluminium alloy located in Placement 1; (iii) the milling of the test workpiece made up of a LAB material located in Placement 2; (iv) the milling of the test workpiece made up of an Aluminium alloy located in Placement 2.

Figure 6 shows the measures of the four resulting workpieces, namely, the milling quality of the four resulting workpieces. It is apparent that the resulting workpieces made up of a LAB material are closed to the desired workpiece represented with the black straight line in Fig. 6, while there is a non negligible gap betweeen the resulting workpieces made up of an Aluminium alloy and the desired workpiece. Moreover, the difference between the two resulting workpieces made up of an Alu- minium alloy is important. As a consequence, we can claim that the placement of the workpiece really affects its milling quality.

Influence of the kinematic redundancy on the machining quality

The KUKA KR270-2 robot has six degrees of freedom. The milling operations sets only five degrees of freedom because the rotation of the spindle about the tool axis is not fixed. Therefore, the robot is redundant with respect to the task, the robot redundancy being equal to one as explained in [START_REF] Conkur | Clarifying the definition of redundancy as used in robotics[END_REF][START_REF] Wenger | Curve-following for redundant manipulators with obstacles : feasibility analysis and solutions[END_REF]. Here, the robot redundancy is characterized by the angle β , which denotes the rotation angle of the spindle about the tool axis and is depicted in Fig. 7. Figure 8 illustrates the quality of two test workpieces made up of an aluminium alloy and milled in the same place- ment, but with two different redundancy planning schemes. It is apparent that the milling quality also depends on the redundancy planning scheme chosen for the machining operation.

OPTIMUM PLACEMENT OF THE WORKPIECE

This section presents a methodology that aims to determine the best placement of the workpiece to be machined knowing the cutting forces exerted on the tool and the elastostatic model of the robot. The proposed methodology is explained through the definition of a machining quality criterion and the formulation of a mono-objective optimization problem.

Machining quality criterion

The tool displacement along the normal to both the feed direction and the tool axis is mainly responsible for the machining errors shown in Figs. 3, 6 and8.

Let us discretize the path to be followed during the machining operation into n points. Let c i be the tool displacement along the normal to both the feed direction V f ,i and the tool axis z t,i at the ith point of the discretized path. c i can be expressed as follows:

c i = δ d T i V f ,i × z t,i (9)
where δ d i is the point-displacement of the tool at the ith point obtained from Eq. ( 5) and × denotes the cross product. Therefore, the machining quality criterion f MQC , which is the objective function of the optimization problem at hand is defined as the root mean square of the c i values, namely,

f MQC = 1 n c 2 1 + c 2 2 + • • • + c 2 n → min (10)
The smaller f MQC , the better the machining quality.

Decision variables

The decision variable vector of the optimization problem contains the Cartesian coordinates of the workpiece geometric center and its orientation parameters expressed in the robot base frame. Let x p , y p and z p be the Cartesian coordinates of the workpiece geometric center expressed in the robot base frame along the x-, yand z-axes, respectively. The orientation of the workpiece can be fully represented with the variables (Q 2 , Q 3 , Q 4 ), a subset of the quaternions coordinates. Indeed, the quaternions represent the rotations of the workpiece with a rotation axis u = [u x u y u z ] T and an angle θ . The relation between the quaternions and the axis and angle representation can be found in [START_REF] Khalil | Modeling, Identification and Control of Robots[END_REF]:

Q 1 = cos(θ /2) (11) Q 2 = u x sin(θ /2) (12) Q 3 = u y sin(θ /2) (13) Q 4 = u z sin(θ /2) ( 14
)
where u 2 x + u 2 y + u 2 z = 1 and 0 ≤ θ ≤ π. The decision variable vector also contains the redundancy planning scheme. Let β i be the rotation angle of the spindle about the tool axis, namely, the angle β shown in Fig. 7, at the ith point of the discretized path, i = 1, . . . , n. Accordingly, the redundancy planning scheme is defined by vector β β β expressed as:

β β β = β 1 β 2 • • • β n T (15)
The KUKA KR270-2 robot may have eight solutions to the inverse geometric model. Let µ denote the solution number to the inverse geometric model of the robot. Therefore, µ is also a decision variable of the optimization problem.

Constraints

Four types of constraints arise in the optimization problem at hand: 1. The joint angles of the KUKA KR270-2 robot should be bounded between its joint limits, namely,

-185 deg ≤ θ 1i ≤ 185 deg (16) 0 ≤ θ 2i ≤ 146 deg (17) -245 deg ≤ θ 3i ≤ 29 deg (18) -350 deg ≤ θ 4i ≤ 350 deg (19) -125 deg ≤ θ 5i ≤ 125 deg (20) -350 deg ≤ θ 6i ≤ 350 deg (21)
θ ji being the jth joint angle of the robot, j = 1, . . . , 6, when the tool is located at the ith point of the discretized path, i = 1, . . . , n. Note that the robot was parameterized by using the modified Denavit-Hartenberg convention [START_REF] Khalil | Modeling, Identification and Control of Robots[END_REF]. 2. The tool displacement c i expressed with Eq. ( 9), i = 1, . . . , n, should be smaller than a given value c max , which is defined with regard to the expected machining quality, i.e., max

(|c 1 |, |c 2 |, • • • , |c n |) ≤ c max (22)
where |.| denotes the absolute value. 3. The workpiece should be located within an area of the robot cell free of obstacles, namely,

x pmin ≤ x p ≤ x pmax (23) y pmin ≤ y p ≤ y pmax (24) z pmin ≤ z p ≤ z pmax (25)
where x pmin , y pmin , z pmin and x pmax , y pmax , z pmax are the lower and upper bounds of the Cartesian coordinates x p , y p , z p of the workpiece geometric center, respectively. It is noteworthy that those lower and upper bounds are defined with regard to the obstacles within the robot cell in order to be sure that the workpiece does not collide any obstacle. 4. The robot should not meet any singularity while the tool is following the path, namely,

|det (J i ) | > 0, i = 1, . . . , n (26)
J i being the kinematic Jacobian matrix of the robot defined in [START_REF] Dumas | Joint Stiffness Identification of Industrial Serial Robots[END_REF] and evaluated at the ith point of the path. 5. The robot joint trajectory should be continuous.

Formulation of the optimization problem

From Eqs. ( 10) to (26), the optimization problem to be solved to find the best workpiece placement within the robot cell can be formulated as follows:

minimize f MQC over x = x p y p z p Q 2 Q 3 Q 4 β β β T µ T subject to max (|c 1 |, |c 2 |, • • • , |c n |) ≤ c max -185 deg ≤ θ 1i ≤ 185 deg 0 ≤ θ 2i ≤ 146 deg -245 deg ≤ θ 3i ≤ 29 deg -350 deg ≤ θ 4i ≤ 350 deg -125 deg ≤ θ 5i ≤ 125 deg -350 deg ≤ θ 6i ≤ 350 deg |det (J i ) | > 0 x pmin ≤ x p ≤ x pmax y pmin ≤ y p ≤ y pmax z pmin ≤ z p ≤ z pmax µ ∈ [1, . . . , 8] i = 1, . . . , n (27)
Therefore, optimization problem (27) aims to find the optimum workpiece placement and the corresponding optimum redundancy planning scheme that minimize the machining quality criterion f MQC defined by Eq. [START_REF] Dumas | Joint Stiffness Identification of Six revolute Industrial Serial Robots[END_REF] while respecting the set of the constraints expressed by Eqs. ( 16) to (26).

An illustrative example is presented in the next section in order to explain how optimization problem (27) can be solved.

ILLUSTRATIVE EXAMPLE

This illustrative example aims to find the optimal placement of the workpiece illustrated in Fig. 9. The 2-norms of the tangential cutting force F t,i and radial cutting force F r,i applied by the workpiece on the tool at the ith point are:

F t,i 2 = 1500 N (28) F r,i 2 = 500 N (29)
The milling path is a rectangle of size:

w W P = 0.6 m (30) h W P = 0.3 m (31)
Besides, the workpiece is supposed to stay in a horizontal configuration within the robot base frame, i.e.,

Q 2 = Q 3 = 0 (32)
The workpiece is free to rotate about the vertical axis, namely,

-1 ≤ Q 4 ≤ 1 (33)
Nevertheless, the geometric center of the workpiece is bounded, namely,

x pmin = 0.4 m (34)

x pmax = 1.1 m (35) y pmin = 0.4 m (36) y pmax = 1.1 m (37) z pmin = 0.3 m (38) z pmax = 1.1 m (39)
Moreover, the maximum point-dispacement c max of the tool should be smaller than 3 mm because of the expected machining quality, i.e., c max = 3 mm (40) A hybrid optimization algorithm was used to solve optimization problem (27) with the foregoing data. As a matter of fact, a genetic algorithm [START_REF] Goldberg | Genetic Algorithms in Search, Optimization & Machine Learning[END_REF] was used to find the initial guess of an interior-point algorithm for large-scale nonlinear programming [START_REF] Byrd | An Interior Point Algorithm for Large-Scale Nonlinear Programming[END_REF]. In this vein, the Matlab ga function combined with point number and β angle for the optimum milling path place-ment. The red curve characterizes the optimum redundancy planning scheme, i.e., the β β β opt vector. The white areas can not be reached by the robot because of its joint limits expressed by Eqs. ( 16) to (21). Besides, the point-displacement of the tool is higher than c max in the red areas. It is noteworthy that x p , y p , z p , Q 2 , Q 3 , Q 4 and µ are the only decision variables considered by the ga and fmincon functions in this optimization problem solving. As a matter of fact, an optimal β β β vector is searched at each iteration of the genetic algorithm and at each iteration of the interior-point algorithm. This vector is obtained by minimizing the point displacement of the tool at each point of the milling path. Figure 12 depicts the optimum milling path placement and the decision space defined with Eqs. (34) to (39). An animation of the corresponding optimum milling operation can be downloaded 1 .

O W P X W P Y W P F P u W P v W P F t,i F r,i V f ,i z t,i
The optimum redundancy planning scheme, i.e., the red curve in Fig. 11, appears to be piecewise continuous. It should be considered during the cutting strategy to avoid signs of wear on the workpiece after the milling operation. As a consequence, a cutting strategy is illustrated in Fig. 13 to avoid signs of wear in the workpiece corners.

CONCLUSIONS

This paper introduced a methodology to determine the optimum placement of any workpiece to be machined knowing the cutting forces exerted on the tool and the elastostatic model of the KUKA KR270-2 robot. First, a cutting force model was presented to predict the cutting forces applied by the tool on the workpiece. Then, a joint stiffness identification procedure was explained and the joint stiffness values of the KUKA KR270-2 robot were given. The influence of the workpiece placement and the kinematic redundancy on the machining quality of a test workpiece was highlighted through some experimentations. A mono-objective optimization problem was formulated to determine the optimum placement of a workpiece to be machined. Finally, an illustrative example was developed in order to show the contributions of the paper.

The proposed methodology will help the robot user find the optimum placement of any workpiece to be machined within the Cartesian workspace of any six-revolute industrial serial robot. It will also help him/her deal with the kinematic redundancy of the robot with regard to the task.

The experimental validations of the results and the extension of the proposed methology to robot machining with a higher kinematic redundancy of the robot with regard to the task are part of the future works.

FIGURE 2 .

 2 FIGURE 2. Experimental setup for the joint stiffness identification of the KUKA KR270-2 robot

FIGURE 3 .

 3 FIGURE 3. Calculated and measured paths associated with four milling conditions

FIGURE 5 . 7 FIGURE 6 .

 576 FIGURE 5. Two placements of the test workpiece

FIGURE 9 .

 9 FIGURE 9. Workpiece and cutting forces

FIGURE 10 . 4 FIGURE 11 .

 10411 FIGURE 10. Convergence of the genetic algorithm

FIGURE 12 .

 12 FIGURE 12. Optimum milling path placement

FIGURE 13 .

 13 FIGURE 13. Modification of the cutting strategy to avoid signs of wear on the workpiece

TABLE 1 .

 1 Apparent

joint stiffness values in [MNm/rad] of the KUKA KR270-2 robot

TABLE 2 .

 2 Cutting conditions Reference N [tr/min] a p [mm] V f [mm/min]

	P3	20 000	5	3 600
	P4	10 000	5	3 600
	P5	6 670	5	3 600

FIGURE 7 .

 7 Kinematic redundancy characterized with angle β

							Spindle
				β
								z t
			V f
	y piece	[mm] -49,6 -49,4 -49,2 -49 -48,8 -48,6 -48,4 -48,2 -48 -47,8 -47,6 -47,4	-110	-60	-10	40	Redundancy 1 Redundancy 2 LAB 90 140	190
		x piece					[mm]

FIGURE 8. Quality of the test workpiece milled in a given placement, but with two different redundancy planning schemes

http://www.irccyn.ec-nantes.fr/ ˜caro/ASME_ ESDA2012/OptimSolKR270.avi

ACKNOWLEDGMENT

The authors would like to acknowledge the financial support of the ANR, France (Project ANR-2010-SEGI-003-02-COROUSSO).