Smart sequential multilateration measurement strategy for volumetric error compensation of an extra-small machine tool

Precision Engineering 43 (2016) 178–186

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Smart sequential multilateration measurement strategy for volumetric error compensation of an extra-small machine tool Fabien Ezedine, Jean-Marc Linares ∗ , Jean-Michel Sprauel, Julien Chaves-Jacob Aix-Marseille Université, CNRS, ISM UMR 7287, 13288 Marseille Cedex 09, France

a r t i c l e

i n f o

Article history: Received 26 February 2015 Received in revised form 4 June 2015 Accepted 13 July 2015 Available online 21 July 2015 Keywords: Machine tool Error mapping Compensation Laser tracking interferometer Sequential multilateration

a b s t r a c t Research on machine tools has mainly focused, during these last couple of decades, on methods of error mapping and compensation techniques in aim of improving their geometrical accuracy. Over the last ten years, new measurement methods based on tracking laser (TL) multilateration have appeared. These methods have generally been applied to large machine tools or coordinate measuring machines. However, compact extra-small (XXS) machines have rarely been approached. The aim of this paper is to provide an optimal experimental strategy for estimating volumetric errors and then compensating these machine tools in the best possible manner. The method is based on the sequential multilateration technique, using a TL. This study focuses on the calibration of the translation axes of a five-axis machine tool. In this work, the accuracy of the machine is defined as the mean error vector measured in the entire working volume of the machine. The influence of a great number of factors (TL positions, offset size, acquisition time, temperature, etc.) that could affect the accuracy of the machine tool is then studied. For that purpose, a Design of Experiment (DOE) is carried out to discriminate the effect of these parameters. A Screening process is thus first used to refine the set of factors. This set is then retained to obtain the Response Surface (RS) with its Statistical Confidence Boundaries (SCB). Finally, these factors are optimized to derive the smartest strategy for providing the smallest reduced residual volumetric error after compensation. The results of this optimization are validated by an experiment. © 2015 Elsevier Inc. All rights reserved.

1. Introduction For more than a century, the industrial demand concerning the functionality and lifespan of workpieces has led to the enhancement of machine tool accuracy. During these last couple of decades, research on error mapping of Computer Numerical Control (CNC) machine tools has greatly contributed to improving the volumetric accuracy. This research work targets two fields. The first studies are dedicated to the volumetric error, which involves estimating the error vector in the entire working volume. This error vector is the deviation between the driven nominal tool position and the real cutter location. The second field deals with the compensation techniques, which correct the deviated tool path. The errors of CNC machines with linear axes are linked to the kinematic errors of each axis, which include the positioning error, two straightness errors, and three angular motion deviations, called roll, pitch, and yaw. Squareness defects between axes are also to be

∗ Corresponding author at: Aix Marseille University, 413, Avenue Gaston Berger, 13625 Aix en Provence, France. Tel.: +33 4 42 93 90 96; fax: +33 4 42 93 90 70. E-mail address: [email protected] (J.-M. Linares). http://dx.doi.org/10.1016/j.precisioneng.2015.07.007 0141-6359/© 2015 Elsevier Inc. All rights reserved.

considered. The defects of machine tools with three linear axes are finally represented by twenty-one kinematic errors. These errors come from various sources [1]. Mechanical and geometrical imperfections [2], as well as the misalignment of components in the structure, are the main sources of kinematic errors. Another source of error is the non-uniform expansion of the machine structure, which appears to be due either to different internal heat sources, such as drive motors, controller cards, or friction in the bearings, or to the specific environment of the machine tool. The weights of the different components or moving carriages of the machine tools also have a strong influence on the machine volumetric accuracy. Furthermore, it is well-established that geometric and thermomechanical errors represent the main sources of volumetric errors [2]. Once the sources of error are known, the errors can be defined thanks to an error model. Developing the error model requires different mathematical tools such as homogenous transformation matrices [3–5]. Generally the geometric and thermo-mechanical errors cannot be directly evaluated on CNC machine tools but are estimated using the measurement of distances. Laser interferometers are mostly used [6,7], especially Tracking Lasers (TL) [8–11]. For example, the devices and the software TRAC-CAL developed by Etalon AG,

F. Ezedine et al. / Precision Engineering 43 (2016) 178–186

in collaboration with NPL and PTB [12], for more than ten years is able to provide reliable and accurate error maps. The TL, placed on the table of the machine, is able to track a spherical reflector which is bound to the spindle. The distance measurement resolution is of about 1 nm (Michelson interferometer principle). The coordinates of the reflector can be obtained using four TLs. In this procedure, four distances are measured simultaneously. Then, the location of the reflector in the reference frame of the machine tool can be estimated. This method is called multilateration [13]. However, a TL is quite expensive. In addition, a similar method called sequential multilateration can be used [14]: instead of employing four devices, only one TL is required but it is shifted three times. The measurements are carried out one after another. Once the kinematic errors have been estimated, a compensation matrix can be built. A Compensation matrix is the data which is the most used in order to compensate distorted tool paths. For that purpose, the working volume is meshed. The compensation matrix then provides the kinematic errors at each node of the mesh. The error vector, at any point in the working volume, is then derived from the kinematic model by using interpolated local kinematic errors. Two approaches are finally used to correct the tool path. Either the compensation is internal to the machine tool and the numerical controller compensates the tool path in real time [15], or a post processing procedure is added to the CAM software to define the corrected tool path [16]. This improves the volumetric accuracy. This state of the art intimates that TL techniques are generally applied on any Coordinate Measuring Machine or CNC machine tool. However, the error mapping of a compact extra-small (XXS) machine tools involves some real difficulties. This is due to the small size of the working volume (about 200 × 200 × 200 mm3 ) and its reduced accessibility. In fact, these specific problems were existing for the machine tool used in this study. This machine is a very compact CNC bridge type machining center. First, measuring the tight working volume of this extra small machine tool requires some special fixture. The tracking laser cannot be placed on the cradle of the machine tool which does not support the appliance weight. Then it needs to be placed on a plate bound to the machine tool bed, which avoids collisions between the TL and the reflector. Second, the small size of the working volume (200 × 200 × 200 mm3 ) and its reduced accessibility impose a different approach: the reflector locations cannot be spread equally around the spindle during the sequential measurement process; the same constraint also applies to the positioning of the TL. This reinforces the necessity to improve the calibration process. Of course, other technologies to calibrate a XXS machine tool exist, like the R-test for example. But the advantage of the TL is to be able to estimate the mapping error in the whole working volume and to acquire a large number of data. In consequence, this paper describes the optimization of the experimental calibration strategy of the translation axes for an extra-small CNC bridge type machining centers, using the TL sequential multilateration technique. A Design Of Experiment (DOE) will be used as an analysis tool. This study is organized around three major points: the introduction of the experimental procedure, the analysis of the experimental strategy using DOE and its optimization.

2. Experimental process 2.1. Error mapping procedure for an extra-small machine 2.1.1. Case study environment The studied measurand is the extra-small CNC bridge type machining center, shown in Fig. 1. This machine tool is placed in a temperature-controlled room. In order to avoid the impediments due to the thermal drift, the machine environment has been

179

Fig. 1. CNC machine environment.

customized. Three heat sources are controlled: the drive motors of the axes, the numerical controller cards and the specific environment of the machine. A heat exchanger and a cooling system are used to evacuate the heat. Another heat exchanger is dedicated to controlling the temperature of the coolant. Finally, an air conditioner system regulates the temperature in the room at a precision of ±1.5◦ . The studied machine is a five-axis milling machine. It is dedicated to high precision machining of small workpieces. 5 ␮m is the accuracy specified by the manufacturer. The working volume is limited to 200 × 200 × 200 mm3 and its accessibility is reduced. 2.1.2. Volumetric error estimation using the TL technique The TL used in this study is an ETALON Laser Tracer. A preliminary study, focusing on repeatability, was conducted to validate the use of the TL. A set of length measurements was repeated thirty times. These measurements involved characterizing the distance between the TL and the reflector fixed on the spindle of the machine. After each measurement, the three axes were moved and put back into the initial fixed position. The repeatability was estimated to a tenth of a micron, which is much lower than the order of magnitude required for the compensation. The TL shown in Fig. 2 is a reliable device for measuring the kinematic errors of the machine tools. The measuring technique employed is sequential multilateration. A toolpath trajectory is therefore defined to join up the different nodes of the mesh used to map the working volume of the machine. When a given node of the mesh is reached, the TL characterizes the length between the reference sphere of the TL and the reflector. Due to the dead zone of the interferometer, the acquired data does, however, not match the real measured distances. The same sequence is repeated three times, by shifting the TL position. After computation, an error vector is deduced for any node of the trajectory. The set of error vectors is finally used to determine the compensation which has to be applied to the machine tool. This error vector can be written in function of the kinematic errors of the machine. To deduce these errors, the kinematic model of the machine tool was required. The machine tool is a bridge structure that was assumed to be perfectly rigid. In Fig. 3, which shows the machine tool under study and its structure, M refers to the reference point of the spindle and P represents the center of the reflector. During a machining step, P would refer to the cutting edge location. The error model was deduced from the structure of the machine tool. It is based on general homogeneous transformations and small angle approximations.

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For a given motion on axis L (L = X, Y, Z), the kinematic errors are characterized by three translation shifts (uL , vL , wL ) and three angular deviations (˛L , ˇL ,  L ). (uL ,vL , wL ) define the two straightness errors or the positioning error. (˛L , ˇL ,  L ) are the roll, pitch and yaw motion errors, respectively. The squareness errors of the non-orthogonal machine tool coordinate system M are written as (˛squ , ˇsqu ,  squ ). The error vector dp of a given position M of the spindle is then deduced from these 21 parameters. It is calculated at point P and is expressed by the following equation: dP = A.OM M + AP .MP

⎛

0

⎜ ⎜ y A=⎜ ⎜ −ˇ ⎝ y ⎛

Fig. 2. ETALON Laser Tracer.

0

−squ

ˇy + ˇx + ˇsqu

0

−(˛y + ˛x + ˛squ )

0

0

0

0

0

⎜ ⎜ x + y + z AP = ⎜ ⎜ −(ˇ + ˇ + ˇ ) x y z ⎝ 0

ux + uy + uz

⎞

⎟ ⎟ ⎟ wx + wy + wz ⎠ vx + vy + vz ⎟

1

⎞

−(x + y + z )

ˇx + ˇy + ˇz

0

−(˛x + ˛y + ˛z )

˛x + ˛y + ˛z

0

⎟ ⎟ ⎟ 0⎠

0

0

1

0

0⎟

(1) The homogeneous transformation matrix A defines the defects of the machine structure. Matrix Ap characterizes the global angular errors of the spindle. This type of approach is implemented in the software to derive the kinematic errors from a large set of TL length measurements. The 3.0 version of the TRAC-CAL program has been used in this study. These errors are estimated at any node of the 3D mesh that discretizes the working volume. It is finally used to compute the compensation matrices. The reference location used for that purpose was the center of the working volume, where the compensation is null. Once the compensation matrices are determined, a compensation vector is derived for any node of the 3D mesh. The corrected trajectory can thus be deduced from the nominal one.

Fig. 3. Kinematic model of CNC bridge type machining center.

2.1.3. Characteristics of sequential multilateration applied to extra-small CNC bridge type machining centers The best practice in calibrating machine tools with sequential multilateration leads to equally spread the TL all around the working volume. This cannot be done in the case of XXS machine tools. In fact, due to the small size of the working volume and the limitation in weight, the TL cannot be placed on the cradle of the machine tool. The appliance would be placed inside the measuring space and then would obstruct a large part of it. Our solution was to bind it to the machine tool frame by using a machined support plate, as shown in Fig. 2. Moreover, due to the geometry of the machine tool, the measurement using the TL could be performed from only one side of the working volume, i.e. from the front of the machine tool. The consequence is that the different locations used in the multilateration process cannot be spread equally around the working volume. This last comment can also be made about the locations of the reflector around the spindle: the reflector must always be captured by the laser while the machine tool drives the trajectory. These constraints impact the quality of the data analysis. Consequently, the sequential multilateration process had to be analyzed as subtly as possible. The elaboration of the experimental strategy was partially based on the space requirements listed above. The number of TL positions, the number of reflector offsets and their respective size were criteria to be considered in order to improve the quality of this process. Moreover, as explained in the literature review, the thermal expansion of the measurand and of

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181

TRACKIN NG LASER Expe erimental fa actors

Measure ements

Theoretical trajectory

Kinem matic mod del

Compensattion matrix IN NTERFEROMET TER

Measurementss

Com mpensated tra ajectory

Residuall mapping error in worrking volume Fig. 4. Residual mapping error procedure.

any item used during the measurements was also an important factor to consider. Not only does the room temperature control around the machine tool influence the quality of the measurements, but also any factor inherent to the thermal expansion itself: the preheating of the machine tool or the thermal expansion of the support plate. Thermal expansion is a time-dependent phenomenon: the measurement time and the number of points measured had also to be considered. Moreover, the shape of the measurement trajectory is a factor to be considered too. Then, the optimization of the sequential multilateration process goes back to optimizing this set of factors relative to the measurand, the appliances used and their environment. This set of factors defined the experimental strategy.

Fig. 5. Checking interferometer system.

Straightness measured Fitting process Cosine error Motion axis

2.2. Characterization of the efficiency of the geometrical compensation The optimization of the set of experimental factors requires characterizing the quality of the geometrical compensation. For that purpose, the residual volumetric error, obtained after compensation, was evaluated independently from the TL technique. The experimental procedure, shown in Fig. 4, was therefore developed. The procedure includes two different steps: - The first step is to determine the compensation matrix, using the TL technique. In this step, the parameters characterizing the measurement strategy are firstly selected and the acquisition trajectory is defined. The approximate coordinates of the different TL positions and the components of the offset vector MP, as defined in the machine tool reference frame M shown in Fig. 3, are also entered into the laser acquisition software. This data is used as initial values of the iteration process implemented in the program. It is also used to check that there is no collision between the reflector and the TL when the machine runs the defined trajectory. Finally, the sequential multilateration acquisitions are carried out and the TL software generates the compensation matrix. - The second step is to evaluate the residual volumetric error after compensation. The checking interferometer laser shown in Fig. 5 is used for that purpose. New sets of points were therefore selected, independently from the coordinates of the initial path used to characterize the errors of the machine tool, and were measured by the checking interferometer. Since the aim of these measurements was to characterize residual errors, the trajectories applied to the machine were corrected using the compensation matrices derived from the first step. The points were chosen along a fixed XM or YM direction line. The positioning error of the moving axis and the straightness in both transverse directions were then evaluated for all the points of the

Fig. 6. Fitting process to compensate for the cosine error.

trajectory. This data characterizes the components of the error vector at each measured location. Twenty residual error vectors were evaluated on each measured direction line. However, the straightness measurements needed to be corrected due to some very small misalignments of the laser optics, which led to drifts named “cosine errors”. A linear least squares optimization was used for that purpose (Fig. 6). Finally, the mean value of the norm of the vector was computed over the whole working volume. This is the criterion taken into account to evaluate the residual error of the machine tool. 3. Detection of the parameters influencing the measurement of residual volumetric errors The aim of this study was to provide the best parameters for the experimental strategy to be used to characterize the geometrical errors of an extra-small machine tool and therefore obtain the finest compensation during machining. A Design Of Experiment (DOE) method was used for that purpose [17]. It involves two successive steps. First, a Screening is completed to define the factors that have a significant influence on the mapping error. Second, once these factors have been detected, new experiments are carried out to obtain a precise description of the Response Surface (RS). This approach is used to optimize the set of parameters to be employed during routine measurements. The experimental procedure detailed in Section 2.2 was used in both cases. 3.1. Screening experiments It is important to provide supplementary information about the experimental factors.

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Fig. 7. Screening trajectories.

3.1.1. Room temperature The system of temperature control has been shown in Section 2.1.1. Twenty four hours were necessary to stabilize the required temperature in the laboratory and obtain a constant value around the machine and inside the working volume. 3.1.2. Preheating It consisted in running the machine tool during half an hour using a preheating cycle of all the axes. The temperature of the linear axes was then controlled to check if it had reached the stabilized value. 3.1.3. Sampling step It is the distance between two successive nodes of the measured mesh. This factor is very important in order to provide a sufficient number of measurements and then to obtain a great amount of statistical information about the best-fit errors [18]. 3.1.4. Type of pattern This is the shape of the theoretical trajectory used to drive the machine tool. Concerning the measurement carried out with the TL, cubic (Fig. 7a) or cross (Fig. 7b) shapes were chosen in the Screening. They were used to cover the edge and the inside of the working volume, respectively. Due to the geometrical configuration of the machine tool, the measurements employing the checking interferometer were performed with linear trajectories (Fig. 7c). 3.1.5. Acquisition time This is the time required by the TL acquisition system to collect the measured distance. It integrates the delay necessary for the reflector to stabilize, once the machine has stood, and the range of time to perform the measurement. 3.1.6. Number of TL positions Fig. 8 shows four positions among the six possible positions adopted by the TL. The support plate can be located in three different lateral positions, called “Left”, “Center” and “Right” and at two heights named “Low” and “High”. Three rods were used to place the TL in the “High” positions. The positions which are not represented in Fig. 8 are “High-Center” and “High-Right”. Moreover, the number of TL positions does not exceed six because this would increase the measuring time and therefore the potential for thermal change. 3.1.7. Offset size The size of the offset is defined as the norm of vector MP (Fig. 5). A clamp system was machined in order to bind the reflector to the spindle. As shown in Fig. 9, two types of offsets were used: Small (length between 120 and 220 mm) and Large (length between 350 and 420 mm). 3.1.8. Number of offsets Like the TL positioning, the offsets could not be spread out equally around the spindle because of the reduced working

Fig. 8. TL positions—(a) “Low-Left”; (b) “Down-Center”; (c) “Low-Right”; (d) “HighLeft”.

Table 1 Screening experimental factors with corresponding lower and upper values. Factors X1 X2 X3 X4 X5 X6 X7 X8 X9 X10

Room temperature (◦ C) Preheating Sampling step Type of pattern Feedrate (mm/min) Acquisition time (s) Number of offsets Number of TL positions Offset size Support plate material

Lower value

Upper value

20 No 10 Cube 1000 0.5 3 4 Small Invar

25 Yes 40 Cube + Cross 5000 1.5 6 6 Large Steel

volume. The number of possible offset locations was then limited. A specific solution would involve locating the reflector behind the spindle. However, this would reduce the number of TL positions that prevent laser beam cutting while driving along the entire tool path. 3.1.9. Support plate material Two materials were used, invar and steel. Invar was tested because of its small thermal expansion coefficient. Steel was tested too because it is assumed that the machine tool is mostly composed of this material.

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183

Table 3 Screening results. Experiments

Y (mm)

Coefficients

Value

1 2 3 4 5 6 7 8 9 10 11 12

0.0052 0.0020 0.0035 0.0102 0.0052 0.0117 0.0032 0.0051 0.0092 0.0067 0.0147 0.0030

b0 b1 b2 b3 b4 b5 b6 b7 b8 b9 b10

6.63E − 03 1.73E − 04 −2.41E − 04 2.55E − 04 2.03E − 03 1.06E − 04 −1.19E − 03 1.28E − 06 −1.00E − 03 2.71E − 03 2.38E − 04

3.2. Analysis of the Screening results

Fig. 9. Offset size—(a): Small; (b): Large.

Table 1 summarizes the experimental factors dedicated to the Screening. A linear Screening model was defined as a function of these ten factors, as shown in the following equation:

Y = B · X = b0 +

10 

bi · Xi

Table 3 summarizes the results of the Screening. It shows response Y for all the experiments which have been completed. This data corresponds to the mean norm of the residual error vector calculated from the 60 measured values (3 independent acquisition direction lines, 20 points on each line). The set of estimated coefficients bi , as computed through the pseudo-inverse method (least squares fitting), are also presented. Constant b0 gives the mean measured residual error over the whole set of experiments. Each other coefficient bi represents the influence of factor Xi over response Y. The graph of effects, shown in Fig. 10, provides a graphical representation of this likely relative influence. The corresponding Pareto diagram is also represented in this Figure. A set of four significant factors are identified: the type of pattern, X4 ; the acquisition time, X6 ; the number of TL positions, X8 , and the offset size, X9 . The other factors were found to be strongly insignificant in comparison to these four parameters. The Pareto diagram shows that these four factors represent 87% of the total effect over the response. The other factors were therefore not taken into account in the optimization process.

(2) 4. Optimization of the influential parameters and experimental validation

i=1

Y refers to the response of the model, for a given set X of measurement factors. It is the mean error vector computed in the whole working volume. B refers to coefficient bi of the model. The sequence of experiments was based on a Hadamard matrix and built randomly. It provided a system of twelve experiments, which each mixed the upper and lower values of the factors, as shown in Table 2. These values were normalized to −1 and +1. Once the mean error vector Y had been collected for all the Screening experiments, coefficients bi were estimated, using the least-squares method. The results are presented in next subsection.

Table 2 Experimental random Screening sequence. Experiments

X1

X2

X3

X4

X5

X6

X7

X8

X9

X10

4 8 12 7 1 6 2 5 10 9 11 3

+1 −1 +1 −1 −1 −1 +1 +1 +1 −1 +1 −1

−1 −1 −1 +1 +1 −1 +1 +1 +1 +1 −1 −1

−1 +1 +1 +1 −1 −1 −1 −1 +1 +1 +1 −1

−1 −1 +1 −1 +1 +1 −1 +1 −1 +1 +1 −1

+1 +1 −1 +1 +1 −1 −1 +1 −1 −1 +1 −1

−1 +1 +1 +1 −1 +1 +1 +1 −1 −1 −1 −1

+1 −1 +1 +1 +1 +1 −1 −1 +1 −1 −1 −1

+1 +1 +1 −1 +1 −1 +1 −1 −1 +1 −1 −1

−1 +1 −1 −1 +1 +1 +1 −1 +1 −1 +1 −1

+1 +1 −1 −1 −1 +1 −1 +1 +1 +1 −1 −1

4.1. Characterization of the response surface The Screening process showed that four factors were found to be influential, but only three of them were considered in the determination of the RS (Table 4). The type of pattern, representing Table 4 Design of RS experiments. Experiments

X9

X8

X6

10 16 14 4 8 7 5 3 9 6 17 13 18 1 12 11 15 2

−1 +1 −1 +1 −1 +1 −1 +1 −1 +1 0 0 0 0 0 0 0 0

−1 −1 +1 +1 −1 −1 +1 +1 0 0 −1 +1 0 0 0 0 0 0

−1 −1 −1 −1 +1 +1 +1 +1 0 0 0 0 −1 +1 0 0 0 0

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Fig. 10. Screening results.

Table 5 SR—Insignificant factor settings. Factors X1 X2 X3 X4 X5 X7 X10

Value ◦

Room temperature ( C) Preheating Sampling step Type of pattern Feedrate (mm/min) Number of offsets Plate material

20 Yes 10 Cube 1000 4 Steel

the shape of the measured trajectory, cannot be quantified. It was therefore imposed to the cubic shape that led to lower errors. The experiment carried out at the center of the three-factor domain was repeated four times in order to compute the experimental uncertainty. The Screening factors that proved to be insignificant were set at the level of the smallest response. Table 5 shows the initial setting of each insignificant factor used to characterize the RS. The model of the RS is detailed in Eq. (3). It is a quadratic polynomial in order to describe the non-linear behaviors of response Y and the interaction between the factors. B refers to coefficients bi and bij of the model.

Fig. 11. RS—Variable offset of TL measurements.

Y = B · X = b0 + b6 · X6 + b8 · X8 + b9 · X9 + b69 · X6 · X9 + b68 · X6 · X8 + b89 · X8 · X9 + b66 · X62 + b88 · X82 + b99 · X92 (3) The experimental process used to characterize the RS was identical to the process employed for the Screening, with two exceptions. Concerning the TL measurements, only four offsets were adopted for the entire set of experiments. As demonstrated in the Screening DOE, large offsets (X9 ) lead to increase the residual error vector. Therefore, only three offsets were selected as small as possible. Thus, only one varied and three different lengths (“Short”, “Average” and “Long”) were considered, as shown in Fig. 11. Concerning the checking interferometer measurements, the corrected trajectories were different. They are shown in Fig. 12. Five lines were measured in the YM directions, covering the whole working volume. Twenty residual errors were measured on each line. Furthermore, such as for the Screening process, the TL was used to deduce the compensation matrices. Then, the corrected trajectories were computed and measured by the checking interferometer. Once the measurements along the five lines were collected for a given set of the three experimental factors, response Y was computed. The best estimator, B, of coefficients bi and bij of the RS model was finally derived from the entire results using the pseudoinverse method (least squares fitting). As a result, the RS could be computed and plotted. The related uncertainty U( Y) was also deduced to define the Statistical Confidence Boundaries (SCB) [19]

Fig. 12. RS–Trajectories of checking interferometer measurements.

of the RS. It required identifying the standard deviation of the measured responses that was defined in two manners: First, residues E of the least squares fitting were derived from B, as shown in the following equation: estimated coefficients E = Y − B.X

(4)

The standard deviation ( E ) of these residues was then computed, such as shown in Eq. (5). Ne refers to the number of

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185

σE (mm)

Table 6 SR results. Experiments

Y (mm)

Coefficients

Values

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

0.0029 0.0050 0.0030 0.0028 0.0051 0.0046 0.0019 0.0028 0.0034 0.0021 0.0066 0.0021 0.0030 0.0024 0.0033 0.0029 0.0029 0.0033

b0 b9 b8 b6 b99 b88 b66 b89 b69 b68

3.12E − 03 1.15E − 04 −1.14E − 03 1.35E − 05 −3.95E − 04 1.23E − 03 −4.57E − 04 −8.80E − 05 −1.93E − 04 −3.66E − 04

0.00095

After Student test 3.06E − 03

Insignificant parameters

0.0009

−1.14E − 03

0.00085

1.09E − 03 −5.97E − 04

0.0008 0.00075 Nonee Non

−3.66E − 04

bb96

b68 b

89

b6 b

b69 b

9

b66 b

69

99

b89 b

68

Removed parameters Fig. 13. Reduction of RS parameters.

0,00 Y7 (mm)

X8=-1

0.006 0,00 6 experiments, whereas Nf refers to the number of factors and Em refers to the residue calculated for experiment m.

 1 2 · Em Ne − Nf

0,00 5

Ne

0.004 0,00 4

X8=0

(5)

m=1

0,00 3

Such as pointed out before, the experiment carried out at the center of the three-factor domain was repeated four times. The standard deviation of these four readouts, expressed as  exp , was the second way of evaluating the uncertainty bound to the response. Then, the worse-case scenario was considered. The maximum of these two uncertainties was propagated in order to obtain B, such as shown in the following the variance-covariance matrix of equation: var B = (XT · X)

−1

2 · max(E2 ; exp )

(6)

ˆ of the RS was obtained by Then, the variance of any point Y B. The Jacobian of the quadratic model propagating the variance of was used for that purpose, such as shown in the following equation: var Y = J · var B · JT

(7)

Finally, the uncertainty of a given point of the RS, expressed as U( Y) was obtained using a coverage factor of two.

U( Y) = k.

var Y,

with

k=2

(8)

It permitted to compute and plot the SCB [19]. The SCB is a graphical representation of the error bar of the response. It was used as a verification tool. 4.2. Analysis of the response surface Concerning the characterization of the residual mapping error after compensation, 20 residual error vectors were measured on each of the five lines (Fig. 12). Response Y is the average of the 100 residual error vector norms. Table 6 summarizes the response obtained for each experiment and the deduced set of coefficients bi and bij concerning the RS. A Student’s statistical significance test (t-test) was performed on parameters bi and bij . This statistical test removed the insignificant parameters and reduced the standard deviation of residues ␴E . This result is shown in Fig. 13. The final equation of the RS is detailed in the following equation: Y = b0 + b8 · X8 + b68 · X6 · X8 + b66 · X62 + b88 · X82

(9)

0.002 0,00 2

X8=1

0,00 1

0,7

0,8 0.8 0,9

0,6 0.6

0,3

0,4 0.4 0,5

0,2 0.2

00

0,1

-0,1

-0,3

-0,2 -0.2

-0,5

-0,4 -0.4

-0,7

-0,6 -0.6

-0,9

-1-1

11

X6

00 -0,8 -0.8

  E =

Fig. 14. Surface response with SCB for X8 = −1, X8 = 0 and X8 = 1.

Fig. 14 shows the RS with its SCB. Factor X8 is a discrete variable because it refers to the number of TL positions. The RS has been plotted for each value of X8 (X8 = −1; X8 = 0 and X8 = 1) as a function of X6 (acquisition time). 4.3. Optimization and experimental validation Such as mentioned previously, the aim of this paper is to find the best experimental strategy to calibrate a extra-small CNC bridge type machining center. In this experimental configuration, the statistical study concluded that the offset size (X9 ) does not influence the average residual error vector Y. Therefore, the value of this factor can be set at a value easy to implement. In the current configuration, the offset size was chosen so as to be equal to 0 mm (X9 = −1). This means that only two of the most influential factors, i.e. X6 and X8 , had to be optimized. These optimized factors are expressed as Xiopt , for i = 6 or 8. The optimized response is referred to as Yopt . Our DOE model results are established only inside the tested domain. Such as shown in Eq. (10), the optimization was therefore conducted under constraints due to which the normalized inputs ranged inside this region. Obviously, the number of TL positions is an integer.



Yopt = min( Y) with Xi

i = 6 or 8,

while

−1 ≤ X6opt ≤ 1 X8opt ∈ N

(10)

This optimization provided a mean location error (Yopt ) in the working volume of about 2 ␮m, with an error bar estimated

186

F. Ezedine et al. / Precision Engineering 43 (2016) 178–186

Experimental checking Yopt: X6opt=1, X8opt=1 and X9opt=-1 Without compensation 0

0.005

0.01 (mm)

Fig. 15. Comparison of the results. Table 7 Best experimental strategy to calibrate an extra-small machine tool using a TL.

X1 X2 X3 X4 X5 X6 X7 X8 X9 X10

Factors

Best strategy

Room temperature (◦ C) Preheating Sampling step Type of pattern Feedrate (mm/min) Acquisition time (s) Number of offsets Number of TL positions Offset size (mm) Plate material

20 Yes 10 Cube 1000 1.5 4 6 0 Steel

at ±0.9 ␮m for X6opt = 1, X8opt = 1. This set of parameters corresponds to a configuration that has already been tested in the DOE. Fig. 15 summarizes the real experimental results (1.9 ␮m ± 0.7). Its error bar was derived from the experiments repeated using their standard deviation  exp and a coverage factor of 3.183. This factor was derived from a Student Law with a confidence level of 95% and a degree of freedom of 3. The experimental error bar was thus estimated at ±0.7 ␮m. It includes any variation due to the measurement process (interferometer laser, CNC machine repeatability, temperature changes, user influence, etc.). The optimized result (2 ␮m ± 0.9 ␮m) and a measurement without any compensation (8 ␮m ± 1 ␮m) are shown in Fig. 15. This last value was derived from new independent measurements carried out on the same 5 acquisition lines used in the DOE. The error bar was defined by the standard deviation of the resulting 100 evaluated error vector norms, using a coverage factor of 2. Fig. 15 demonstrates that the experimental checking confirms the DOE result. To conclude, the optimized number of TL positions X8opt was found to be equal to 6 and the acquisition time X6opt to 1.5 s, with a delay required to stabilize the measurement. Moreover, X9opt was chosen as the minimum: 0 mm. Furthermore, Table 7 summarizes the best strategy, in the studied domain of factors, for calibrating an extra-small machine tool by using the multilateration technique applied to a TL. 5. Conclusion This work focused on the best measurement strategy for calibrating the translation axes of a compact extra-small CNC bridge machine-tool, in order to improve its accuracy. For that purpose, a sequential multilateration technique using a tracking laser (TL) was applied to obtain the matrices which could be used to compensate for the geometrical errors of the machine tool. Thereafter, another standard Michelson interferometer was used to determine the residual mapping error after compensation. A design of experiment was then elaborated to optimize the measurement conditions. It was concluded that the multilateration technique cannot be used in the same way for XXS machine tools and standard tools. One difference concerns the number of TL positions to implement

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