Using a double ball bar to identify position ... - Springer Link

Int J Adv Manuf Technol (2014) 70:2071–2082 DOI 10.1007/s00170-013-5432-9

ORIGINAL ARTICLE

Using a double ball bar to identify position-independent geometric errors on the rotary axes of five-axis machine tools Sitong Xiang & Jianguo Yang & Yi Zhang

Received: 24 April 2013 / Accepted: 15 October 2013 / Published online: 12 November 2013 # Springer-Verlag London 2013

Abstract A measuring method using a double ball bar (DBB) is proposed for identifying the eight position-independent geometric errors (PIGE) on the rotary axes of five-axis machine tools. Three measuring patterns are used, in which the translational axes are kept stationary and only two rotary axes move to obtain a circular trajectory. In this way, the effects of translational axes are totally excluded, and the measurement accuracy is improved. Motion equations, describing how the A-axis and C-axis move simultaneously to realize a circular trajectory, are presented. The influence of each deviation on the measurement patterns is simulated, and analytical solutions for the eight PIGEs are demonstrated. Finally, the measuring method is verified in a five-axis CNC machine tool. Experimental results confirm that the method provides precision results for the eight PIGEs. Keywords Five-axis CNC machine tool . Rotary axes . Double ball bar . Position-independent geometric error . Error measurement

1 Introduction Five-axis CNC machine tools provide greater productivity, better flexibility, and less fixture time than threeaxis machining centers, because the cutting tool can approach the workpiece from any direction. However, the two rotary axes bring in additional geometric errors, such as squareness and parallelism between a rotary axis and a translational axis [1]. S. Xiang (*) : J. Yang : Y. Zhang School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China e-mail: [email protected]

To improve the cutting performance and processing accuracy of the machine tools, error measurement, error identification, and tool condition monitoring are of great importance. Dimla et al. reviewed the methods of tool wear monitoring in metal cutting [2]. Stavropoulos et al. developed a processmonitoring system that is capable of simultaneously monitoring the spindle’s and tool’s condition [3]. Methods of measuring and identifying the geometric errors on rotary axes of fiveaxis CNC machine tools are widely reported, as will be discussed below. The geometric errors of a controlled axis can be categorized as position-dependent geometric errors (PDGEs) and position-independent geometric error (PIGEs) [4]. PDGEs are the errors caused mainly by the defects in the components of the controlled axis itself. Conversely, PIGEs are generally caused by imperfections in the assembly process [5]. In ISO 230-7 [6], a PDGE is considered a component error. Many other terms for a PIGE can be found in the literature, such as location error [6], link error parameter [7], kinematic error [8, 9], and systematic deviation [1]. In this study, the measuring and identification methods for the eight PIGEs of rotary axes are considered. In the literature, various measuring methods for calibrating the geometric errors of rotary axes have been proposed. Direct measurement methodologies of geometric errors in machine tools were reviewed by Schwenke [10], and indirect methods were introduced by Ibaraki [11]. The research group of Ibaraki proposed various measurement methodologies for geometric errors of five-axis CNC machine tools, such as R -test [12–14], cone frustum [9, 15], machining test [16], touch-trigger probe [17], and double ball bar (DBB) [9]. DBB is a mature measuring instrument and is effective for measuring the relative displacement to the reference length. A lot of researchers have proposed different measuring patterns of DBB for identifying the geometric errors of rotary axes. Tsutsumi et al. [1] proposed a classical measuring method, in

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which three types of motion patterns in the radial, tangential, and axial directions were designed for each rotary axis, and the deviations were approximately determined by calculating the eccentricities of the measured circular trajectories. Based on Tsutsumi’s measuring method, Wang [8] provided a reliable algorithm for the separation of the eight PIGEs. The setup errors of DBB were modeled to increase the estimation accuracy by Kwang et al. [5, 18] and Zargarbashi et al. [19]. However, in the measuring methodologies mentioned above, to get a circular trajectory, the translational axes and rotary axes need to move simultaneously. Therefore, assumptions must be made, such as the geometric errors caused by the translational axes have been compensated perfectly. There is no doubt that the errors of translational axes cannot be fully compensated and the residuals actually still have some influence on the measuring results of rotary axes. Zhang et al. [20] proposed a novel DBB measuring method, in which only the C-axis rotated. Unfortunately, it could only evaluate five PDGEs of the C-axis. Similarly, the DBB methodology developed by Khan et al. is capable of evaluating five errors out of the six error components in rotary axes [21]. Lei et al. [22] presented a particular circular test path, which caused the two rotary axes only to move simultaneously and kept the other three linear axes stationary. In his study, interference analysis for the ball bar installation and simulations for DBB measuring patterns were conducted. In this study, a new measuring method for identifying the eight PIGEs of rotary axes using a DBB is developed. It keeps translational axes stationary and causes the rotary axes to move simultaneously to obtain three different motion patterns. The structure of this paper is as follows: the kinematic model of five-axis CNC machine tools is described in Section 2. Section 3 discusses the three measuring patterns in detail. The simulation and the analytical solutions for the eight PIGEs are presented in Section 4. Experiments and verifications are described in Section 5, and finally, conclusions are drawn in Section 6.

Int J Adv Manuf Technol (2014) 70:2071–2082

Fig. 1 Configuration of the five-axis CNC machine tool

below CCS. The workpiece coordinate system (WCS) overlaps CCS, and the machine coordinate system (MCS) overlaps YCS. All the coordinate systems mentioned above are coaxial. Z CA is the distance between CCS and ACS, and Z AY is the distance between ACS and YCS.

2.2 Definition of the eight PIGEs There are two types of definition for the eight PIGEs, absolute notation and relative notation. The former is described in

2 Kinematic model of five-axis CNC machine tools 2.1 Establishment of coordinate systems For the five-axis CNC machine tools whose configurations are depicted in Fig. 1, the establishment of coordinate systems is shown in Fig. 2. The C -axis coordinate system (CCS) is defined at the center of the rotary table. The A-axis coordinate system (ACS) is set at the intersection of the axis lines of the A-axis and C-axis. The Z-axis coordinate system (ZCS) is located at the spindle nose which is just above CCS. The Yaxis coordinate system (YCS) is on the Y-axis and is just

Fig. 2 Establishment of coordinate systems


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ISO230-7 [6], and the latter is introduced by Inasaki [23]. The difference and relationship between these two notations have been summarized in detail by Soichi [11]. Relative notation has the advantage of simplifying the description of the kinematic model, particularly for five-axis CNC machine tools [11]. In this paper, relative notation is used. For the five-axis CNC machine tools, whose configurations are depicted in Fig. 1, there are eight PIGEs of rotary axes, namely α AY, β AY, γ AY, β CA, δ xAY, δ yAY, δ zAY, δ yCA [23], shown in Fig. 3. α AY, β AY, and γ AY are the angular errors of the A-axis with respect to the Y-axis about X-, Y-, and Z-axes. β CA is the angular error of the C-axis with respect to the A-axis about the Y-axis. δ xAY, δ yAY, and δ zAY are linear shifts of the A-axis with respect to the Yaxis in the X, Y, and Z directions. δ yCA is the linear shift of the C-axis with respect to the A-axis in the Y direction. Because MCS overlaps YCS, the subscript AY actually means the Aaxis with respect to MCS. 2.3 Kinematic model of five-axis CNC machine tools Tsutsumi [1] and Soichi [9] have established a well-defined kinematic model of the five-axis CNC machine tool whose configuration is shown in Fig. 1. The kinematic modeling process is briefly summarized by Eqs. 1–6. M

A

PO2 ¼M TY ⋅Y TA ⋅A TC ⋅C PO 2

b TC ¼ D2 δyCA D5 ðβCA ÞD3 ð−Z CA ÞD6 C

ð1Þ ð2Þ

TA ¼ D1 ðδxAY ÞD2 δyAY D3 ðδzAY þZ AY Þ

ð3Þ

M

TY ¼ I

ð4Þ

M

PO1 ¼M TY ⋅Y TX ⋅X TZ ⋅Z PO 1 ¼ ½x; y; z; 1

ð5Þ

Y

D4 ðαAY ÞD5 ðβAY ÞD6 ðγ AY ÞD4 Ab

L2 ¼ M PO 2 − M PO 1 k

ð6Þ

where UT V means the transfer matrix from the V coordinate system to the U coordinate system. x, y, z are the nominal displacement of the machine tool linear axes. O1 refers to the ball clamped on the spindle nose, and O2 is the ball mounted b and C b are the tool orientation angles on the rotary table. A about the X- and Z-axes. L stands for the measured distance by DBB. D 1–6(*) represents 4×4 homogeneous transformation matrices (HTMs); D 1(*), D 2(*), and D 3(*) represent the HTMs for linear motions in the X, Y, and Z directions; and D 4(*), D 5(*), and D 6(*) represent the HTMs for angular motions about the X-, Y-, and Z -axes. Some examples of D 1–6(*) are presented in Eqs. 7–12. 2

1 60 1 D ðδxAY Þ ¼ 6 40 0 2

1 6 0 D2 δyCA ¼ 6 40 0

0 1 0 0

0 0 1 0

3 δxAY 0 7 7 0 5 1

ð7Þ

0 1 0 0

3 0 0 0 δyCA 7 7 1 0 5 0 1

ð8Þ

2

1 60 3 D ðδzAY þ Z AY Þ ¼ 6 40 0 2

1 6 0 D4 ðαAY Þ ¼ 6 40 0 2

Fig. 3 The eight PIGEs of rotary axes

0 1 0 0

0 cosðαAY Þ sinðαAY Þ 0

cosðβ AY Þ 6 0 D5 ðβAY Þ ¼ 6 4 −sinðβ AY Þ 0

0 1 0 0

3 0 0 7 0 0 7 1 δzAY þ Z AY 5 0 1

ð9Þ

0 −sinðαAY Þ cosðαAY Þ 0

3 0 07 7 05 1

ð10Þ

sinðβAY Þ 0 cosðβ AY Þ 0

3 0 07 7 05 1

ð11Þ

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2

cosðγ AY Þ −sinðγ AY Þ 6 sinðγ AY Þ cosðγ AY Þ 6 D ðγ AY Þ ¼ 6 4 0 0 0 0

0 0 1 0

3 0 07 7 05 1

ð12Þ

3 DBB measuring patterns 3.1 Pattern 1: only the C-axis rotates As shown in Fig. 4, we precisely locate the ball O1 on the axis line of the C-axis rotary table and locate the ball O2 on the worktable. The A-axis and all the translational axes are kept stationary, and we implement the movement about the C -axis from 0° to 360°. The process of defining the location of the C-axis line is depicted in Fig. 5. Step 1 A gauge block is installed on the C rotary table, and a dial gauge is used to ensure that the gauge block is precisely parallel to the X -axis. Step 2 Move the tool tip to the point P 1 and obtain its Y coordinate, whose value is R 1. Step 3 Rotate the C-axis by 180° and reversely move the Yaxis until the tool tip touches P 2, and then obtain its Y coordinate, which equals R 2. If the starting point of the tool tip is exactly on the axis line of the C table, R 1 should equal R 2. ΔR, the difference between R 1 and R 2, is used to revise the starting point from C 1 to C 2. The steps described above are repeated at several testing points with different X coordinates, so as to minimize the positioning error. In this way, the Y coordinate of the C-axis line is obtained.

Fig. 4 Pattern 1: only the C-axis rotates

Fig. 5 Definition of the C-axis line location

Similarly, the X coordinate of the C -axis line can be obtained. With the X and Y coordinates, the location of the C-axis line in the XY plane is defined. Finally, the steps described above are repeated at several testing points with different Z coordinates, so as to determine the whole axis line in the volumetric space. Depending on the accuracy of the dial gauge and the average value of various testing results, the error margin of defining the C-axis line is less than 1.5 μm. 3.2 Pattern 2: only the A-axis rotates As shown in Fig. 6, we clamp a fixture on the worktable and install the ball O2 on the fixture. The ball O1 is precisely located on the axis line of the A-axis. The C -axis is kept stationary, and the A -axis rotates from −45° to +45°, so that the motion trajectory of the ball O2 is part of the dashed circle depicted in Fig. 6. The process of defining the location of the A -axis line is depicted in Fig. 7.

Fig. 6 Pattern 2: only the A-axis rotates


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Fig. 7 Definition of the A-axis line location

Step 1 A gauge block is installed on the C rotary table, and a dial gauge is used to ensure that the gauge block is precisely parallel to the X -axis. Step 2 Locate the tool tip on the C -axis line, move the tool tip to point P 1, and obtain its Z coordinate, whose value is R 1. Step 3 Rotate the A-axis by 90° and move the Y-axis until the tool tip touches P 2, and then obtain its Y coordinate, which equals R 2. If the starting point of the tool tip is exactly on the A-axis line, R 1 should equal R 2. ΔR, the difference between R 1 and R 2, is used to revise the starting point from A 1 to A 2. The steps described above are repeated at several testing points with different X coordinates, so as to determine the whole axis line in the volumetric space. Consequently, the location of the A-axis line is defined. The error margin of defining the A-axis line is depending on the accuracy of defining the C -axis line, so it is about 2 μm.

We simply let the Z coordinate be r/3, namely z =r sinθ sinα =r/3; the DBB measuring circular trajectory will then be in the plane r/3 above the C-axis rotary table. This being so, θ and α will satisfy the following relationship expressed in Eq. 14. sinθsinα ¼ 1=3

ð14Þ

Substituting Eq. 14 into Eq. 13, we easily obtain the circular trajectory equation of the ball O2 shown in Eq. 15, whose pffiffiffiffiffiffiffiffiffiffiffiffi radius is 8r2 =9 . x2 þ y2 ¼ 8r2 =9

ð15Þ

Because the motion equations are estimated in the CCS coordinates, after the rotation of the A- and C -axes, the coordinate components x, y, and z are still expressed in the

3.3 Pattern 3: A- and C-axes rotate simultaneously Z

We precisely locate the ball O1 on the axis line of the C -axis rotary table. As shown in Fig. 8, the initial position of the ball O2 is the point P. It moves to point P′ when the C-axis has a movement of θ . It then reaches point P″ when the A-axis has rotated α degrees. Figure 8 also depicts the coordinate decomposition of P″ in the X, Y, and Z directions. Assuming that the initial position of the ball O2 is P(r, 0, 0) in the CCS coordinates, the motion equations of the ball O2 in the CCS coordinates are estimated using Eq. 13. Here, counterclockwise is set as the positive direction. 8 < x ¼ rcosθ y ¼ rsinθcosα : z ¼ rsinθsinα

r sin sin Y

os rc P

r sin cos

r

P

X

ð13Þ

P Fig. 8 Pattern 3: A- and C-axes rotate simultaneously and coordinate decomposition

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CCS coordinates. Consequently, the coordinate transfer matrix in pattern 3 should be revised as follows: A b ð16Þ TC ¼ D2 δyCA D5 ðβCA ÞD6 C Y

TA ¼ D1 ðδxAY ÞD2 δyAY D3 ðδzAY þZ AY −Z CA Þ D4 ðαAY ÞD5 ðβAY ÞD6 ðγ AY ÞD4 Ab

ð17Þ

4 Simulation and solutions for the deviations 4.1 Influences of deviations on the measuring patterns Simulation is conducted to demonstrate the influences of the eight PIGEs on the measuring patterns. Some dimensions of the machine tool configuration and the deviations for simulation are given in Table 1. Simulation results are shown in Table 2. The blanks in the table mean that the simulated curve coincides with the reference circular arc, namely the influence of the corresponding deviation does not appear. Pattern 1: β AY and β CA have the same effect on the simulated trajectory, so do δ yAY and δ yCA. β AY and β CA have significant effects on the eccentricity in the X direction, while α AY influences the eccentricity in the Y direction. The circular trajectory influenced by δ zAY is concentric to the standard circle. Pattern 2: β AY and δ zAY have the opposite effect on the simulated trajectory; so do β CA and δ xAY. There are only two deviations, γ AY and δ yAY, that lead to the eccentricity in the Z direction. Because the rotation range of the A-axis is only from −45° to +45°, it is hard to calculate the eccentricity in the Y direction from the DBB measuring results, but the eccentricity in the Z direction can still be calculated. Pattern 3: Deviations except for γ AY and δ z AY have irregular influences on the simulated circular arc. The influence of γ AY on pattern 3 does not appear, and the circular trajectory influenced by δ z AY is concentric to the standard circle.

4.2 Solutions for the eight PIGEs Pattern 1: The C-axis rotates from 0° to 360°, when θ =π/2, π, 3π/2, 2π, and we record the DBB measuring length as L 1, L 2, L 3, and L 4. Pattern 2: The A-axis rotates from −45° to +45°, when θ = 0, π/4, −π/4, and we record the DBB measuring length as L 5, L 6, and L 7. Pattern 3: The ball O2 is at the initial position when θ = 28°7′32″, α =45°, and it reaches the terminal position when θ =77°3′33″, α =20°. We record the measuring length as L 8 and L 9 at these two positions. L8 2 ¼ 82; 582sinðβ AY Þ þ 374δxAY þ 200δyCA −441δzAY þ 46;820 ða1 −a2 þ a2 sinðβCA Þ þ a1 sinðβCA ÞÞ þ 212 a4 δzAY −a1 δzAY −a4 δyAY þ a3 δyAY − 441 δyCA ⋅ sinð45 þ αAY Þþ 93 ;713 ð18Þ where a1 a2 a3 a4

¼ cosðπ=4 þ αAY þ 0:4909Þ ¼ cosðπ=4 þ αAY −0:4909Þ ¼ sinðπ=4 þ αAY þ 0:4909Þ ¼ sinðπ=4 þ αAY −0:4909Þ

L9 2 ¼ 20970sinðβAY Þ þ 95δxAY þ 413δyCA −441δzAY þ46820 ðb1 −b2 þ b2 sinðβ CA Þ þ b1 sinðβCA ÞÞ þ 212 b4 δzAY −b1 δzAY −b4 δyAY þ b3 δyAY − 441δyCA ⋅ sinðπ=9 þ αAY Þ þ 93713

ð19Þ

ð20Þ

where b1 b2 b3 b4

¼ cosðπ=9 þ αAY þ 1:3449Þ ¼ cosðπ=9 þ αAY −1:3449Þ ¼ sinðπ=9 þ αAY þ 1:3449Þ ¼ sinðπ=9 þ αAY −1:3449Þ

ð21Þ

Table 1 Dimensions and deviations for simulation

The equations L 1 to L 9 are derived from the kinematic models in the three different measuring patterns. The equations for L 1, L 2, L 3, L 4 and L 5, L 6, L 7 are relatively brief, while those for L 8 and L 9 are lengthy. Therefore, we start with the brief equations to solve the majority of the deviations, and we then obtain the solutions for the few remaining deviations based on L 8 and L 9.

Dimensions and deviations

Value

Step 1 δ zAY can be obtained from L 1, L 2, L 3, and L 4.

Measuring radius Distance between CCS and ACS Distance between ACS and YCS Positional deviation Angular deviation

R 1, R 3 =250 mm, R 2 =150 mm Z CA=120 mm Z AY =500 mm δ xAY, δ yAY, δ zAY, δ yCA=±10 μm α AY, β AY, γ AY, β CA=±0.005° (18″)

δzAY ¼ 250; 000− L1 2 þ L2 2 þ L3 2 þ L4 2 =1; 600 ð22Þ Step 2 Equation 23 and β AY can be obtained from L 5, L 6, and L 7.


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Table 2 Influences of eight deviations on three measuring patterns Deviation

AY=

±0.005°

AY =

±0.005°

AY =

±0.005°

CA =

±0.005°

xAY =

±10 m

yAY =

±10 m

zAY =

±10 m

yCA =

±10 m

Pattern 1 C-axis only

Pattern 2 A-axis only

Pattern 3 A and C-axes

Deviation

Pattern 1 C-axis only

Pattern 2 A-axis only

Pattern 3 A and C-axes

1div.:10 m

L6 2 −L7 2 ¼ 62; 400sinðγ AY Þ⋅sin45∘ þ 360δyAY ⋅sin45∘

ð23Þ

Step 3 Substituting δ zAY and β AY, determined by steps 1 and 2, into L 4 and L 5, we can then calculate β CA and δ xAY. 1 1:25L5 2 þ L4 2 −232; 500sinðβ AY Þ C B þ 625 δzAY − 90 ; 625 C ¼ sin−1 B A @ 232; 500 0

βAY ¼ sin−1

L5 2 −0:5ðL6 2 þ L7 2 Þ þ 180δzAY ð1−cos45∘ Þ 32; 400⋅ð1−cos45∘ Þ

ð24Þ

β CA

ð25Þ

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Fig. 11 Measuring procedure of pattern 3 Fig. 9 Measuring procedure of pattern 1

δxAY

L4 2 −192; 000ðsinðβ CA Þ þ sinðβAY ÞÞ þ 400 δzAY − 62;500 ¼ 300

Step 6 Finally, γ AY is obtained by substituting δ yAY into Eq. 23. 2 ∘ 2 −1 L6 −L7 −360δ yAY ⋅sin45 γ AY ¼ sin ð28Þ 62; 400sin45∘ ð26Þ

Step 4 The relationship between α AY, δ yCA, and δ yAY can be obtained using L 1 and L 3. αAY ¼ sin−1

L3 2 −L1 2 þ 600δyCA þ 600δyAY ð27Þ 384; 000

As mentioned above, all the eight PIGEs can be calculated, but the derivation process is not presented here because of limited space.

5 Experimental verification and compensation Step 5 Substituting α AY, δ x AY, β AY, δ z AY, and β CA into Eqs. 18 and 20, we can then calculate δ yCA and δ yAY by the method of least squares. α AY can then be obtained using Eq. 27.

To validate the effectiveness of this measuring method, experiments are conducted on a five-axis machine tool. Before conducting the experiments, some preparation is needed. The ambient temperature is controlled to around 20 °C, so that the impact of thermal error is excluded. There are two extremely important steps for the experiment, namely, carefully install

Table 3 Deviations preand postcompensation

Fig. 10 Measuring procedure of pattern 2

Deviation

Value (pre)

Value (post)

α AY β AY γ AY β CA δ xAY δ yAY δ zAY δ yCA

−8.6″ −7.9″ −24.1″ 15.6″ −20 μm −9 μm 17 μm 7 μm

−1″ 3.2″ −6.1″ −2.2″ −4 μm −1 μm 3 μm 2 μm


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Fig. 12 DBB result of measuring pattern 1 pre- and postcompensation Fig. 14 DBB result of measuring pattern 3 pre- and postcompensation

the DBB to minimize the setup errors and precisely find the locations of the axis lines of the A- and C -axes. The total measuring procedures include three measuring tests, the details of which follow: The measuring procedure of pattern 1 is depicted in Fig. 9. 1. Move the rotary axes to the initial position where θ =0 and α =0. 2. Clamp the ball O1 on the spindle nose and move the center of O1 to the position (0, 0, 640) in the MCS coordinates. Install the ball O2 on the worktable and adjust it to the initial position (150, 0, 60) in the CCS coordinates. The height of the magnetic fixture of DBB is 60. 3. Keep the A -axis stationary, let the C-axis rotate from 0° to 360°, and conduct the first measuring test.

Fig. 13 DBB result of measuring pattern 2 pre- and postcompensation

The measuring procedure of pattern 2 is depicted in Fig. 10. 1. Move the rotary axes to the initial position where θ =0 and α =0. 2. Precisely locate the ball O1 at the position (180, 0, 500) in the MCS coordinates, which is also exactly on the axis line of the A -axis. Locate the center of the ball O2 at (60, 0, 30) in the CCS coordinates. 3. Keep the C -axis stationary, rotate the A-axis from −45° to +45°, and conduct the second measuring test. The measuring procedure of pattern 3 is depicted in Fig. 11. 1. Precisely locate the ball O1 at the initial position (0, 0, 660.711) in the MCS coordinates, and locate the ball O2 at the initial position (212.132, 0, 60) in the CCS coordinates.

Fig. 15 Setup errors of DBB

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Table 4 Influence of 1 μm setup error on measuring results Pattern

Δt s i,x

Δt s i,y

Δt s i,z

Δw s i,x

Δw s i,y

Δw s i,z

1 2 3

0 0.8 μm 0

0 0 0

0.8 μm