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The detection of rotary axis of NC machine tool based on multi-station and time-sharing measurement Jindong Wang a,⇑, Junjie Guo a, Baoqing Zhou b, Jian Xiao c a

State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an, China Dalian Machine Tool Group, Dalian, China c Qinchuan Machine Tool Group Co. Ltd., Baoji, China b

a r t i c l e

i n f o

Article history: Received 24 September 2011 Received in revised form 2 April 2012 Accepted 21 April 2012 Available online 28 April 2012 Keywords: Laser tracker Rotary axis Multi-station and time-sharing measurement Error separation

a b s t r a c t At present, the detection of rotary axis is a difﬁcult problem in the errors measurement of NC machine tool. In the paper, a method with laser tracker on the basis of multi-station and time-sharing measurement principle is proposed, and this method can rapidly and accurately detect the rotary axis. Taking the turntable measurement for example, the motion of turntable is measured by laser tracker at different base stations. The redundant equations can be established based on the large amount of measured data concerning the distance or distance variation between measuring point and base station. The coordinates of each measuring point during turntable rotation can be accurately determined by solving the equations with least square method. Then according to the error model of rotary axis, the motion error equations of each measuring point can be established, and each error of turntable can be identiﬁed. The algorithm of multi-station and time-sharing measurement is derived, and the error separation algorithm is also deduced and proved feasible by simulations. Results of experiment show that a laser tracker completes the accuracy detection of the turntable of gear grinding machine within 3 h, and each error of the turntable are identiﬁed. The simulations and experiments have veriﬁed the feasibility and accuracy of this method, and the method can satisfy the rapid and accurate detecting requirements for rotary axis of multi-axis NC machine tool. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction With the development of modern manufacturing, the demand of multi-axis NC machine tool is increasing. The multi-axis NC machine tool can achieve the efﬁcient and accurate processing for large and complex parts, which has been widely used in the manufacturing of large wing, blade and other complex mechanical parts. The machining accuracy is an important index to evaluate the machine characteristics. Quick and accurate detecting the error of machine tool and making error compensation have become the important ways to improve the machining accuracy ⇑ Corresponding author. Address: School of Mechanical Engineering, Institution of Precision Engineering, 28 Xianning West Road, Xi’an, China. E-mail address: [email protected] (J.D. Wang). 0263-2241/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.measurement.2012.04.015

[1–4]. Currently, there are many methods for detecting the linear axes of machine tool, however, the methods for detecting rotary axis are relatively less [5–7]. Meanwhile, detecting the error of rotary axis is also a difﬁcult problem. By using of the autocollimator and polygon can only evaluate the position error of rotary axis, and the other errors cannot be measured. Each error can be detected by using of the laser interferometer, Renishaw rotary measuring system RX10 and some assistant measurement tools, but the detecting period is long. When the ball bar is used to detect the rotary axis, the multi-axis movement is needed. Meanwhile, the measurement should also be carried out on different modes, and the measurement process is relatively complex. The error of rotary axis directly affect the machining accuracy of multi-axis NC machine tool, so how to quickly and accurately detect the error of rotary axis is

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particularly important. Taking the turntable measurement for example, a laser tracker is adopted the multi-station and time-sharing measurement principle to rapidly and accurately detect the error of turntable in the paper.

2. The principle for turntable measurement Since 1980s, with the development of robot technology and large workpiece measurement [8], three-dimensional and dynamic tracking measurement has developed rapidly. Laser tracking measurement system has fast, high-precision features, and it satisﬁes the requirements of largescale, ﬁeld and no-guild measurement, so it has become an irreplaceable tool in many areas, such as: aerospace, shipbuilding, automotive and other ﬁelds. Currently, laser tracker has also been applied to the detection of errors in machine tool [9,10]. According to the number of laser tracker in the measurement, it can be divided into single station measurement and multi-station measurement [11]. Single station measurement adopts spherical coordinate measurement to calculate the space coordinates of moving target by means of measuring azimuth ui, elevation hi and length Li. The distance measurement of laser tracker is based on the laser interference principle, and the measurement accuracy of distance is higher. The measurement accuracy of angle is generally limited, and the measurement uncertainty of angle will be larger with the distance increasing, which affects the overall accuracy of spatial coordinates. The distance measurement accuracy can be generally guaranteed within 1 106 mm, however, the measurement uncertainty of coordinate is ±1 105 considering the impact of angular error. So single station measurement is not suitable for the situations requiring high measurement accuracy. The multi-station measurement adopts the GPS principle. Only distance is involved in the measurement, therefore this method has high accuracy. However, the method requires multiple laser trackers to measure the target point simultaneously, so the cost is relatively high. In order to save cost and maintain high measurement accuracy, a laser tracker measures the target points successively at different base stations, and only distance is involved in the measurement, which is called ‘‘multistation and time-sharing measurement’’. The target mirror (cat eye) is installed in the vicinity of the cutter, and the position of cat eye’s center is deﬁned as the location of base station in the measurement. The position of base station can be changed by controlling the motion of cutter. The laser tracker is installed on the turntable, and follows the rotation of turntable. The rotating mirror’s center in the laser tracker is deﬁned as the target point. Fig. 1 shows the measurement principle of turntable by multi-station and time-sharing measurement, and Fig. 2 shows its mathematical model. P1, P2, P3 and P4 represent four positions of base station respectively. The measurement area is given by a circle, and A0 is deﬁned as the initial measuring point. When the turntable rotates a preset angle, it is controlled to stop, and the distance reading of laser tracker will be recorded. When the turntable rotates through 360°, the measurement at the ﬁrst base station is com-

Fig. 1. The principle of multi-station and time-sharing measurement.

z

P1 P3

O

P2 P4 y

A0

x Fig. 2. The mathematical model.

pleted, and the cat eye is moved to the second base station by the motion of cutter. Then, repeat the above measurement process until the turntable measurement is ﬁnished at all base stations. Based on the GPS principle, the actual coordinates of each measuring point during the turntable rotation can be determine by the measured data at different base stations. Then the motion error of turntable at different rotational angle can be determined by comparing the actual coordinates of each measuring point with its theoretical coordinates.

3. The Algorithm for turntable measurement Taking four stations and time-sharing measurement for example, the algorithm is derived as follows. The positions of four base stations in machine coordinate system are deﬁned as: P1(xp1, yp1, zp1), P2(xp2, yp2, zp2), P3(xp3, yp3, zp3), P4(xp4, yp4, zp4), and these positions are known in practical measurement. Meanwhile, the coordinates of turntable center is deﬁned as O0 (px, py, pz) in machine coordinate system. Currently, the laser trackers of Leica, FARO and other companies have used the bird nest technology, and the reference distance is known, so these laser trackers can

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Through above calculation, the coordinates of each measuring point Ai in the machine coordinate system can be obtained during the turntable rotation. In order to easily separate various errors of turntable motion, the coordinates of each measuring point are transformed in the turntable coordinate system.

measure the absolute distance between the cat eye and the center of tracking mirror [12]. Meanwhile, the ETALON laser tracker can only measure the relative distance variation between the cat eye and the center of tracking mirror. In view of the above two different types of laser trackers, the measurement algorithms are derived respectively. (1) The laser tracker being able to measure absolute.

(2) The laser tracker only being able to measure relative distance.

When the turntable rotate per h, the coordinates of each measuring point are as Ai(xi, yi, zi), (i = 1, 2, . . . , n). The distances between the measuring point Ai and four base stations P1, P2, P3 and P4 are deﬁned as L1i, L2i, L3i and L4i respectively [13]. Based on the distance formula between two points, the following equations can be obtained.

8 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ > ðxp1 xi Þ2 þ ðyp1 yi Þ2 þ ðzp1 zi Þ2 ¼ L1i > > > > q ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ > > > ðx x Þ2 þ ðy y Þ2 þ ðz z Þ2 ¼ L < p2 p2 i i 2i p2 i qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ > 2 2 2 > > > ðxp3 xi Þ þ ðyp3 yi Þ þ ðzp3 zi Þ ¼ L3i > > qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ > > : ðx x Þ2 þ ðy y Þ2 þ ðz z Þ2 ¼ L p4

i

p4

p4

i

i

When this type of laser tracker is used, we should ﬁrstly calibrate the distance between the initial measurement point and each base station. When the laser tracker is aimed at the initial measuring point, the distance reading of laser tracker is assumed to be set 0. The distance between base station P1 and initial measuring point A0 is deﬁned as L1, and the variation of relative distance between measuring point Ai(i = 1, 2, . . . , n) and P1 is deﬁned as l1i. The coordinates of the initial measuring point A0 is assumed as A00 x00 ; y00 ; z00 in the turntable coordinate system. When the turntable rotate hi around the z axis, the coordinates of each measuring point in the turntable coordinate system can be obtained from Eq. (7).

ð1Þ

4i

A0i x0i ; y0i ; z0i

The ﬁrst equation is squared on both sides, then we can obtain

x2p1 2xp1 xi þ x2i þ y2p1 2yp1 yi þ y2i þ z2p1 2zp1 zi þ z2i L21i ¼ 0

ð2Þ

Eq. (2) is a nonlinear equation concerning xi, yi, zi, and it can be linearized by letting C ¼ x2i þ y2i þ z2i .

x2p1 2xp1 xi þ y2p1 2yp1 yi þ z2p1 2zp1 zi L21i þ C ¼ 0

ð3Þ

2 "

j¼1

2

4 X xpj ypj j¼1

4 X 2 y2pj j¼1 4 X 2 ypj zpj j¼1

2

4 X xpj zpj j¼1

4 X 2 ypj zpj j¼1

2

4 X z2pj j¼1

4 X ypj

4 X zpj

j¼1

j¼1

1

#

6 60 6 ¼6 60 4

x00

y00

ð7Þ

qx

32

x00 cos hi y00 sin hi

3

7 76 0 0 7 6 1 0 qy 7 76 x0 sin hi þ y0 cos hi 7 7 76 7 6 0 1 qz 7 z00 5 54 0

0

1

1 3 ð8Þ

1

ð5Þ

The equations concerning measuring point Ai(xi, yi, zi) and base station P1(xp1, yp1, zp1) can be established as follows:

Then, we can deduce 4 X 6 2 x2pj 6 6 j¼1 6 4 6 X 6 6 2 xpj ypj 6 j¼1 6 6 4 6 X 62 x z pj pj 6 6 j¼1 6 6 4 6 X 4 x pj

¼ T1

A0i

0

cos hi sin hi þ qx 7 6 0 6 x0 sin hi þ y00 cos hi þ qy 7 7 6 ¼6 7 7 6 z00 þ qz 5 4

It should satisfy following conditions to make F minimum.

2

"

1 0

0

ð4Þ

@F ¼0 @C

#

2

j¼1

@F ¼ 0; @zi

Ai 1

Fðxi ; yi ; zi ; CÞ 4 2 X ¼ x2pj þ y2pj þ z2pj 2xpj xi 2ypj yi 2zpj zi L2ji þ C

@F ¼ 0; @yi

32 0 3 x0 cos hi sin hi 0 4 ¼ sin hi cos hi 0 54 y00 5 z00 0 0 1 2 0 3 x0 cos hi y00 sin hi ¼ 4 x00 sin hi þ y00 cos hi 5 z00

Then, the coordinates of each measuring point in the machine coordinate system can be determined from Eq. (8).

According to least squares principle, the objective function is deﬁned as:

@F ¼ 0; @xi

2

3 3 2 4 4 X X xpj 7 xpj x2pj þ y2pj þ z2pj L2ji 7 6 7 7 6 j¼1 7 7 6 j¼1 7 7 6 4 4 7 7 6 X 72 x 3 6 X 2 7 2 2 2 i ypj 7 ypj xpj þ ypj þ zpj Lji 7 6 76 y 7 6 j¼1 7 j¼1 76 i 7 6 7 ¼6 7 7 6 7 4 4 7 4 zi 5 6 X 7 X 6 zpj 7 zpj x2pj þ y2pj þ z2pj L2ji 7 7 C 7 6 7 7 6 j¼1 j¼1 7 7 6 7 7 6 4 X 7 7 6 2 5 41 2 x2 þ y 2 þ z 2 L 5 pj

2

j¼1

pj

pj

ji

ð6Þ

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J.D. Wang et al. / Measurement 45 (2012) 1713–1722

8 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 > > xp1 x00 cos h1 þ y00 sin h1 qx þ yp1 x00 sin h1 y00 cos h1 qy þ zp1 qz z00 ¼ L1 þ l11 > > > q ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ > > 2 2 2ﬃ > < xp1 x00 cos h2 þ y00 sin h2 qx þ yp1 x00 sin h2 y00 cos h2 qy þ zp1 qz z00 ¼ L1 þ l12 > .. > > > . > q ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ > > 2 2 2ﬃ > : 0 0 0 xp1 x0 cos hi þ y0 sin hi qx þ yp1 x0 sin hi y00 cos hi qy þ zp1 qz z00 ¼ L1 þ l1i

Let xp1 qx = a, yp1 ay = b, zp1 qz = c, sin hi = mi, coshi = ni. Then, the ﬁrst equation in Eq. (9) can be simpliﬁed as Eq. (10).

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 a n1 x00 þ m1 y00 þ b m1 x00 n1 y00 þ c z00

Here, D is a 5 5 matrix.

D11 ¼ 2

N X ðani þ bmi Þ2 ; i¼1

D13

D15

2

2

a þb þc

2 l11

2cz00 þ x02 0 þ x02 0

ð2an1 þ 2bm1 Þx00 þ ð2am1 2 02 y02 0 þ z0 L1 2L1 l11 ¼ 0

y02 0

z02 0

ð11Þ

L21 ,

Let W ¼ þ þ Eq. (11) can be linearized. According to least squares principle, the objective function is deﬁned as: N X 2 2 F x00 ; y00 ; z00 ; L1 ; W ¼ ða2 þ b þ c2 l1i

2cz00 þ W 2L1 l1i Þ2

D22 ¼ 2

@F ¼ 0; @z00

@F ¼ 0; @L1

@F ¼0 @W

N X @2F ¼ 8 C 2 > 0; @z02 0 i¼1

@W 2

ðami bni Þ2 ;

@2F @L21

2

D24

D31

N X ¼ 2 ðani þ bmi Þc;

@ F ¼ 8 ðami bni Þ2 > 0 @y02 0 i¼1

N X 2 ¼ 8 l1i > 0;

N X D25 ¼ ðbni ami Þ; i¼1

D32

N X ¼ 2 ðbni ami Þc;

D33 ¼ 2

i¼1

N X

c2 ;

D34 ¼ 2

i¼1

D41

N 3 X 2 2 2 2 ðani þ bmi Þ a þ b þ c l1i 7 6 7 6 i¼1 7 6 7 6X N 7 2 2 2 3 6 2 2 7 6 ðbn bm Þ a þ b þ c l i i 1i x0 7 6 7 6 i¼1 6 7 6 7 6 y0 7 6 N 7 X 6 7 6 7 2 2 2 2 7 6 D 6 z0 7 ¼ 6 c a þ b þ c l1i 7 7 6 7 6 i¼1 7 6 4 L1 5 6 7 N X 7 6 2 2 W 7 6 l1i a2 þ b þ c2 l1i 7 6 7 6 i¼1 7 6 7 6 N X 5 4 2 2 2 2 a þ b þ c l1i

ð15Þ

N X D35 ¼ c;

i¼1

i¼1

D42

N X ¼ 2 ðbni ami Þl1i ; i¼1

N X ¼ 2 cl1i ;

D44

N X 2 ¼ 2 l1i

N X

D51 ¼

N X D45 ¼ l1i ;

i¼1

ðani þ bmi Þ;

i¼1

D52 ¼

i¼1

N X

ðbni ami Þ;

i¼1 N X D54 ¼ l1i ;

i¼1

2

cl1i ;

i¼1

i¼1

Then, we can deduce

N X

N X ¼ 2 ðani þ bmi Þl1i ;

N X D53 ¼ c;

ð14Þ

N X ðbni ami Þc;

i¼1

D43

N X

¼N>0

i¼1

D23 ¼ 2

i¼1

i¼1

@ F ¼ 8 ðani þ bmi Þ2 > 0; @x02 0 i¼1

@2F

i¼1

N X i¼1

ð13Þ

Meanwhile, N X

D21

N X ¼ 2 ðbni ami Þl1i ;

ð12Þ

It should satisfy following conditions to make F minimum.

@F ¼ 0; @y00

i¼1 N X ¼ 2 ðani þ bmi Þðbni ami Þ;

i¼1

ð2ani þ 2bmi Þx00 þ ð2ami þ 2bni Þy00

2

D14

N X ¼ ðani þ bmi Þ;

i¼1

@F ¼ 0; @x00

i¼1

i¼1

2bn1 Þy00

N X ðani þ bmi Þðbni ami Þ;

N X ¼ 2 ðani þ bmi Þl1i ;

i¼1

Eq. (10) is squared on both sides, then we can obtain 2

D12 ¼ 2

N X ¼ 2 ðani þ bmi Þc;

¼ L1 þ l11 ð10Þ

ð9Þ

D55 ¼

i¼1

N : 2

Through the calculation, the initial distance L1 between the base station P1 and initial measuring point A0 can be determined. Repeat the above process, the initial distances between other base stations and the initial measuring point can also be obtained. When the initial distances between the four base stations and initial measuring point are known, the equations concerning initial measuring point A0(x0, y0, z0) and base station Pj(xpj, ypj, zpj) can be established as follows:

8 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 > > > ðxp1 x0 Þ þ ðyp1 y0 Þ þ ðzp1 z0 Þ ¼ L1 > > q ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ > > > < ðxp2 x0 Þ2 þ ðyp2 y0 Þ2 þ ðzp2 z0 Þ2 ¼ L2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ > > > ðxp3 x0 Þ2 þ ðyp3 y0 Þ2 þ ðzp3 z0 Þ2 ¼ L3 > > > qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ > > : ðx x Þ2 þ ðy y Þ2 þ ðz z Þ2 ¼ L p4

0

p4

0

p4

0

4

ð16Þ

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J.D. Wang et al. / Measurement 45 (2012) 1713–1722

can be determined. In the measurement, the cat eye is installed in the vicinity of the cutter, and the position change of base station is realized by the motion of cutter. By using of the coordinate relations in cutter motion, the position of each base station can be directly determined, and the calibration process of each base station can be omitted, which brings conveniences in calculation. In addition, the position accuracy of base station directly affects measurement accuracy. To ensure measurement accuracy, the linear axis of machine tool should have certain accuracy guarantee in the measurement. In order to improve the measurement accuracy, the errors of linear axis can be ﬁrstly measured and compensated before turntable measurement, then the motion accuracy of linear axis can be further improved. Meanwhile, the position deviations of each base station can also be reduced.

Then, we can deduce

2

4 X 2 6 2 xpj 6 j¼1 6 6 X 6 4 6 2 xpj ypj 6 6 j¼1 6 4 6 X 6 2 xpj zpj 6 6 j¼1 6 4 6 X 4 x pj

4 X 2 xpj ypj j¼1 4 X

2

j¼1

y2pj

2

4 X 2 xpj zpj 4 X 2 ypj zpj

j¼1 4 X

j¼1 4 X

ypj zpj j¼1 4 X

j¼1 4 X

ypj

j¼1

z2pj

2

zpj

j¼1

3 4 X xpj 7 7 j¼1 7 72 3 4 X 7 ypj 7 x0 76 7 76 y0 7 j¼1 74 5 4 X 7 z0 zpj 7 7 C1 7 j¼1 7 7 5 2

j¼1

4 3 X 2 2 2 2 xpj xpj þ ypj þ zpj Lj 7 6 7 6 j¼1 7 6 7 6X 4 7 6 2 6 ypj x2pj þ y2pj þ z2pj Lj 7 7 6 7 6 j¼1 ¼6 4 7 7 6X 2 2 2 2 6 zpj xpj þ ypj þ zpj Lj 7 7 6 7 6 j¼1 7 6 4 7 6 X 41 2 5 2 2 2 x þy þz L

2

pj

2

pj

ð17Þ

j

pj

j¼1

From Eq. (17), the coordinates of the initial measuring point in the machine coordinate system can be obtained. When the turntable rotates, the coordinates of each measuring point Ai(xi, yi, zi) in the machine coordinate system can be obtained by solving Eq. (18).

8 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ > ðxp1 xi Þ2 þ ðyp1 yi Þ2 þ ðzp1 zi Þ2 ¼ L1 þ l1i > > > > q ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ > > > ðx x Þ2 þ ðy y Þ2 þ ðz z Þ2 ¼ L þ l < p2 p2 2 i i 2i p2 i qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ > 2 2 2 > > > ðxp3 xi Þ þ ðyp3 yi Þ þ ðzp3 zi Þ ¼ L3 þ l3i > > qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ > > : ðx x Þ2 þ ðy y Þ2 þ ðz z Þ2 ¼ L þ l p4

i

p4

p4

i

4

i

ð18Þ

4i

4. The principle for turntable error separation When the turntable rotates around z axis, Fig. 3 shows the errors of turntable. When the turntable does not rotate, the coordinates of a point at the turntable is deﬁned as M1(xm1, ym1, zm1) in the turntable coordinate system. When the turntable rotate h, the point M1 should arrive at the pointN1 without considering the motion error of turntable. However, the point M1 actually arrives at the point N 01 due to the error. The position deviation between the pointN1 and N 01 is the motion error of the point M1 with turntable rotating h [14]. When the turntable rotate h, the ideal homogeneous transformation matrix is [15].

3 cos h sin h 0 0 6 sin h cos h 0 0 7 7 6 T¼6 7 4 0 0 1 05 2

0

The solving result is

2

4 X 2 6 2 xpj 6 j¼1 6 6 X 6 4 6 2 xpj ypj 6 6 j¼1 6 4 6 X 6 2 xpj zpj 6 6 j¼1 6 4 6 X 4 x pj

2

4 X xpj ypj j¼1 4 X

2

j¼1 4 X

ypj zpj j¼1 4 X ypj

j¼1

4 X xpj zpj j¼1

y2pj

2

2

4 X 2 ypj zpj j¼1 4 X

z2pj

2

j¼1 4 X

zpj

j¼1

3

pj

2

3 1 0 0 0 6 0 cos e ðhÞ sin e ðhÞ 0 7 x x 7 6 T1 ¼ 6 7 4 0 sin ex ðhÞ cos ex ðhÞ 0 5 2

0 2

pj

j

j ji

0

0

cos ey ðhÞ

0

ð21Þ

1

sin ey ðhÞ

0

3

6 0 1 0 07 7 6 T2 ¼ 6 7 4 sin ey ðhÞ 0 cos ey ðhÞ 0 5

j¼1

pj

0 1

In the motion, the transformation matrixes caused by ex (h), ey(h) and ez(h) are respectively:

4 X xpj 7 7 j¼1 7 72 3 4 X 7 ypj 7 xi 76 7 76 yi 7 j¼1 74 5 4 X 7 zi zpj 7 7 C2 7 j¼1 7 7 5 2

4 3 X 2 2 2 2 2 x x þ y þ z L 2L l l 7 6 pj j ji ji j pj pj pj 7 6 j¼1 7 6 7 6X 4 7 6 2 2 2 2 2 6 ypj xpj þ ypj þ zpj Lj 2Lj lji lji 7 7 6 7 6 j¼1 ¼6 4 7 7 6X 2 6 zpj x2pj þ y2pj þ z2pj L2j 2Lj lji lji 7 7 6 7 6 j¼1 7 6 4 7 6 4 1 X x2 þ y2 þ z2 L2 2L l l2 5

2

0

ð20Þ

0

0

0

ð22Þ

1 z

δ z (θ )

y

ð19Þ

ji

j¼1

When the measurement is adopted the principle of GPS, the position of each base station should be ﬁrstly calibrated, and then the coordinates of each measuring point

ε z (θ ) ε y (θ )

ε x (θ )

δ y (θ )

δ x (θ )

Fig. 3. The errors of turntable.

x

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J.D. Wang et al. / Measurement 45 (2012) 1713–1722

2

cos ez ðhÞ sin ez ðhÞ 0

6 sin e ðhÞ z 6 T3 ¼ 6 4 0

3

0

07 7 7 1 05

0

0 1

cos ez ðhÞ

0

0

0

base stations, the turntable is controlled to stop, and the laser tracker is moved to another position as shown in Fig. 4. Then, repeat the previous measurement. When the turntable does not rotate, the ﬁrst and second initial positions of laser tracker on the turntable are assumed as E1(xe1, ye1, ze1), E2(xe2, ye2, ze2). When the turntable rotate h, the motion errors of laser tracker at different initial positions can be obtained according to Eq. (28). Each motion error is written in matrix form, then we obtain

ð23Þ

The transformation matrixes caused by dx(h), dy (h) and dz(h) is

3

3

3 D x1 6 sin h cos h 0 z cos h z sin h y sin h þ x cos h 7 6 d ðhÞ 7 6 Dy 7 e1 e1 e1 7 6 y 7 6 17 6 e1 7 6 7 6 7 6 7 6 dz ðhÞ 7 6 Dz1 7 6 0 0 1 ye1 xe1 0 7¼6 76 7 6 7 6 7 6 6 cos h sin h 0 z sin h ze2 cos h xe2 sin h ye2 cos h 7 e2 7 6 ex ðhÞ 7 6 Dx2 7 6 7 6 7 6 7 6 4 sin h cos h 0 ze2 cos h ze21 sin h ye2 sin h þ xe2 cos h 5 4 ey ðhÞ 5 4 Dy2 5 ez ðhÞ Dz 2 0 0 1 ye2 xe2 0 2

cos h sin h 0

2

1 0 0

ze1 sin h

dx ðhÞ

ze1 cos h

xe1 sin h ye1 cos h

3

6 0 1 0 d ðhÞ 7 y 7 6 T4 ¼ 6 7 4 0 0 1 dz ðhÞ 5 0

0

0

1

The total homogeneous transformation matrix is

ð25Þ

When ex (h), ey (h) and ez (h) are very small, the simpliﬁcations can be obtained.

cos ex ðhÞ 1;

sin ex ðhÞ ex ðhÞ;

ey ðhÞ;

cos ey ðhÞ 1 sin ey ðhÞ

cos ez ðhÞ 1;

sin ez ðhÞ ez ðhÞ:

ð26Þ

Then, we can deduce

2

1

6 e ðhÞ 6 z Q ¼6 4 ey ðhÞ 0

ez ðhÞ 1

ex ðhÞ 0

3

ey ðhÞ dx ðhÞ ex ðhÞ dy ðhÞ 7 7 7 dz ðhÞ 5 1

1 0

dx ðhÞ

2

ð27Þ

The motion error of point M1 can be obtained from Eq. (28).

3

3 Dx1 7 6 sin h cos h 0 z cos h z sin h y sin h þ x cos h 7 6 dx ðhÞ 7 6 6 Dy 7 e1 e1 e1 e1 7 6 d ðhÞ 7 6 1 7 6 y 7 6 7 6 Dz 1 7 6 0 0 1 ye1 xe1 0 7 6 7 7 6 6 7 6 dz ðhÞ 7 6 . 7 6 . .. .. .. .. .. 76 7¼6 . 7 6 . . . . . . . . 7 6 e ðhÞ 7 6 7 6 7 6 x 7 7 6 6 7 7 6 6 cos h sin h 0 zen sin h zen cos h xen sin h y cos h 7 6 en 7 4 ey ðhÞ 5 6 Dxn 7 6 7 7 6 6 4 sin h cos h 0 zen cos h zen sin h yen sin h þ xen cos h 5 4 Dy n 5 ez ðhÞ Dzn 0 0 1 yen xen 0 2

cos h sin h 0

ze1 sin h

ze1 cos h xe1 sin h ye1 cos h

1 1 1 0 0 xm1 xm1 Dx B Dy C By C By C C B B m1 C B m1 C C ¼ TQ B C TB C B @ Dz A @ zm1 A @ zm1 A

ð29Þ

If the motion errors of laser tracker at different initial positions with turntable rotating h are known, each error of turntable can be easily identiﬁed by Eq. (29). In calculation, the initial position of laser tracker and the actual position of laser tracker in the turntable rotating can be determined by the multi-station and time-sharing measurement. When the turntable rotate h, the theoretical position of laser tracker in the turntable coordinate system can be obtained by the rotation transformation of its initial position. Then, we can calculate the motion errors of laser tracker with turntable rotating. Eq. (29) has six error parameters, and the number of equations is also 6. Therefore, when the matrix is full rank, it has a unique solution. However, when there are only two initial positions of laser tracker on the turntable, the singular matrix will easily occur in calculation. To increase the amount of redundant data and avoid the singular matrix in calculation, the number of laser tracker’s initial position on the turntable is usually 3–4. Then, we can obtain Eq. (30) is solved by least square method, then each error of turntable can be separated.

ð24Þ

Q ¼ T1T 2T3T4

2

2

3

2

ð30Þ

0

1

1

ð28Þ

1

In order to separate each error of turntable, when the measurement of turntable motion has been completed at all

5. Simulation for error measurement and error separation of turntable Taking the detection of turntable by four-station and time-sharing measurement as example, the simulations are carried out as follows.

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J.D. Wang et al. / Measurement 45 (2012) 1713–1722

Fig. 4. The initial position change of laser tracker.

5.1. Simulation for turntable measurement The four positions of base station are arranged in a plane, and the positions of each base station in the machine coordinate system are assumed as follows: P1(400, 300, 100), P2(600, 500, 100), P3 (700, 750, 100), P4(900, 800, 100). The position of turntable center and the initial position of laser tracker in the machine coordinate system are O0 (100, 150, 50) and A0(300, 150, 50) respectively. In the measurement, the turntable rotates and stops at each interval of 30°. Some pﬃﬃﬃ measuring points are presented pﬃﬃﬃas follows: A1 ¼ ð100 3 þ 100; 250; 50Þ,pA ﬃﬃﬃ2 ¼ ð200; 100 3þ 150; 50Þ, A3 = (100, 350, 50), A4 ð0; 100 3 þ 150; 50Þ. While the simulations are carried out under two different conditions: without considering the position deviations of base station and considering the position deviations of base station. (1) Without considering the position deviations of base Station. By using of the four base stations and the distance reading of laser tracker, each measuring point can be determined. Table 1 shows the calibration deviations of part measuring points. From Table 1, the calibration deviations for x and y coordinates of four measuring points are very small, but the calibration results of z coordinate are distorted. It is analyzed that, the four positions of base station in the above simulation are arranged in the XY plane. Each term in the third row of 4 4 matrix in Eq. (6) is linearly correlated with the term in the fourth row, so the singular matrix occurs in calculation, which leads to identiﬁcation

Table 1 The calibration deviations of part measuring points mm. Measuring point A1 A2 A3 A4

Dx 1.5745 1011 7.9296 1012 3.9790 1013 4.5194 1012

Dy 1.3386 1011 6.93489 1012 5.11590 1013 4.2064 1012

distortion of z coordinate. To avoid the singular matrix, the positions of base stations should not be arranged in a plane. Then, the position of the fourth base station is assumed as P4(900, 800, 400), and the four base stations are not in a plane this time. Table 2 shows the calibration deviations of part measuring points under the new base station layout. It can be seen from Table 2, the calibration deviations of each measuring point are very small, so the algorithm for multi-station and time-sharing measurement is feasible. To improve the measurement accuracy, the number of base station is increased to obtain more redundancy data at each measuring point. The new positions of the base station are assumed as P5 = (1000, 900, 300), P6 = (1100, 1000, 500), P7 = (1300, 1200, 700). Table 3 gives the calibration deviations of measuring point A1 with different number of base station. It can be seen from Table 3, the calibration accuracy of measuring point is somewhat improved by increasing the number of base station. Taking into account the measurement efﬁciency, the suitable number of base station is 4–6. (2) Considering the position deviations of base station. Due to the motion error of linear axis, the position deviations of each base station necessarily exits. The effect of different position deviations on calibration of each measuring point is analyzed [16,17]. In order to simplify the calculation, a random error is added to the theoretical position of each base station. The random error obeys random normal distribution within [0, 1 lm], [0, 3 lm] and [0, 5 lm] respectively. Meanwhile, the length measurement error of laser tracker obeys random normal distribution within [0, 1 lm] in the simulations. The simulations are carried out with different errors. Table 4 shows the calibration deviations of some measuring points with the random error within [0, 1 lm], and Table 5 gives the calibration deviations of measuring point A1 with different random errors.

Table 2 The calibration deviations of part measuring points mm. Measuring point

Dx

Dy

Dz

A1 A2 A3 A4

5.3205 1011 1.2192 1012 1.6171 1012 2.0894 1011

3.3992 1011 8.8107 1012 1.1198 1012 1.4495 1011

2.6624 1011 5.6630 1012 7.0627 1012 1.0317 1011

Table 3 The calibration deviations of measuring point A1 mm. Number of base station

Dx

Dy

Dz

Dz 1.9232 103 3.8192 102 1.4567 102 55.8906

4 5 6 7

5.3205 1011 1.6711 1012 4.2064 1013 5.3195 1013

3.3992 1011 1.2192 1012 4.6327 1013 1.7104 1013

2.6624 1011 8.5833 1012 3.0553 1012 1.61386 1012

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J.D. Wang et al. / Measurement 45 (2012) 1713–1722

Table 4 The calibration deviations of part measuring points mm. Measuring point

Dx

Dy

Dz

A1 A2 A3 A4

0.00117 0.00157 0.00123 0.00145

0.00069 0.00108 0.00156 0.00161

0.00137 0.00132 0.00082 0.00089

Table 5 The calibration deviations of measuring point A1 mm. Different errors

Dx

Dy

Dz

1 2 3

0.00117 0.00211 0.00334

0.00069 0.00124 0.00226

0.00137 0.00246 0.00439

It can be seen from Table 5, the calibration deviations of measuring points are becoming larger with the increasing position deviations of base station. 5.2. Simulation for error separation In order to separate each error of turntable, when the measurement of turntable motion has been completed at all base stations, the laser tracker is moved to another position to repeat the previous measurement. According to the number of laser tracker’s initial position on the turntable, it is divided into three conditions: (1) there are two initial positions,pﬃﬃﬃand the initial positions pﬃﬃﬃare assumed as C 1 ð100; 100 3; 0Þ and C 2 ð100; 100 3; 0Þ in the turntable coordinate system respectively; (2) there are three initial positions, and the p initial positions are assumed ﬃﬃﬃ pﬃﬃﬃ as D1(200,0,0), D2 ð100; 100 3; 0Þ and D3 ð100; 100 3; 0Þ respectively; (3) there are four initial positions, and the initial positions are assumed as E1(200,0,0), E2(0, 200, 0), E3 (200, 0, 0) and E4(0, 200, 0) respectively. By using of the motion errors of measuring points, each error of turntable can be easily identiﬁed. To verify the accuracy of error separation algorithm, when the turntable rotate 30°, the errors of turntable are assumed as follows:

8 > < DxE1 ¼ 0:00166 ð3Þ DyE1 ¼ 0:01289 > : DzE1 ¼ 0:00700 8 D > < xE3 ¼ 0:00233 DyE3 ¼ 0:00596 > : DzE3 ¼ 0:01300

While the simulations are carried out under two different conditions: without considering the effect of random error during rotary rotation and considering the effect of random error during rotary rotation. When there are only two initial positions of laser tracker on the ﬁrst condition, the identiﬁcation results are distorted due to generating the singular matrix in calculation, which is consistent with the previous analysis. When there are three and four initial positions of laser tracker on the ﬁrst condition, Table 6 shows the identiﬁcation deviations of each error. From Table 6, each error can be accurately identiﬁed, and the identiﬁcation deviations are very small, so the error separation algorithm is feasible. On the second condition, a random error is added to the theoretical motion error of each measuring point. The random error obeys random normal distribution within [0, 5 lm] and [0, 10 lm] respectively. When there are three and four initial positions of laser tracker on the turntable, Table 7 shows the identiﬁcation results with random error within [0, 5 lm]. When there are four initial positions of laser tracker, Table 8 shows the identiﬁcation results with different random error distributions. It can be seen from table Tables 7 and 8, each error of turntable can be accurately identiﬁed even considering the impact of random error. From Table 8, the maximum deviation is 0.001051 mm for dx(h) on the ﬁrst occasion, and the maximum deviation is 0.002316 mm for dx(h) on the second occasion. The deviations are small, which further veriﬁes the feasibility of error separation algorithm. Table 6 The identiﬁcation deviation of each error. Identiﬁcation deviations

dx ðhÞ ¼ 0:005 mm dy ðhÞ ¼ 0:008 mm dz ðhÞ ¼ 0:010 mm 5

5

5

ex ðhÞ ¼ 1 10 rad ey ðhÞ ¼ 1:5 10 rad ez ðhÞ ¼ 2 10 rad When the laser tracker rotate 30° with different initial positions on the turntable, the theoretical motion errors of each measuring point are as follows:

8 > < DxC1 ¼ 0:00166 ð1Þ DyC1 ¼ 0:00596 > : DzC1 ¼ 0:01323

8 > < DxC2 ¼ 0:00433 DyC2 ¼ 0:00942 > : DzC2 ¼ 0:00976

8 > < DxD1 ¼ 0:00166 ð2Þ DyD1 ¼ 0:01289 > : DzD1 ¼ 0:00700 8 D > < xD3 ¼ 0:00433 DyD3 ¼ 0:00942 > : DzD3 ¼ 0:00976

8 > < DxD2 ¼ 0:00166 DyD2 ¼ 0:00596 > : DzD2 ¼ 0:01323

8 > < DxE2 ¼ 0:00313 DyE2 ¼ 0:00742 > : DzE2 ¼ 0:01200 8 D > < xE4 ¼ 0:00379 DyE4 ¼ 0:01142 > : DzE4 ¼ 0:00800

Ddx(h)/mm Ddy(h)/mm Ddz(h)/mm Dex(h)/rad Dey(h)/rad Dez(h)/rad

Number of initial position 3

4

2.6020 1018 3.4694 1018 3.4694 1018 6.7762 1021 0 3.3881 1021

1.7347 1018 1.9432 1018 0 3.3881 1021 1.6940 1021 0

Table 7 The identiﬁcation results of each error. Identiﬁcation results

dx(h)/mm dy(h)/mm dz(h)/mm ex(h)/rad ey(h)/rad ez(h)/rad

Number of initial position 3

4

0.00658 0.00763 0.01126 0.000008 0.000039 0.000023

0.00615 0.00893 0.01012 0.000013 0.000033 0.000020

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J.D. Wang et al. / Measurement 45 (2012) 1713–1722 Table 8 The identiﬁcation results of each error. Identiﬁcation results

Different error distributions

dx(h)/mm dy(h)/mm dz(h)/mm ex(h)/rad ey(h)/rad ez(h)/rad

1

2

0.00615 0.00893 0.01012 0.000013 0.000033 0.000020

0.00731 0.00986 0.01023 0.000027 0.000052 0.000021

6. Experimental veriﬁcation A laser tracker is adopted the multi-station and timesharing measurement principle to detect the turntable of gear grinding machine as shown in Fig. 5. At each base station, the turntable rotates and stops at each interval of 10°, and the distance reading of laser tracker is written down at the corresponding position. When the measurement is completed at the ﬁrst base station, the cat eye is moved to other base station. The four positions of base station in the machine coordinate system are P1(100, 300, 200), P2(100, 900, 200), P3 (200, 600, 250) and P4(150, 1150, 200). In order to separate each error of turntable, when the turntable does not rotate, there are three initial positions of laser tracker on the turntable. The three initial positions are A0(450.324, 450.079, 150.721), B0 (550.213, 150.944, 150.317) and C0(200.815, 600.054, 150.241) by calculations. The total measuring time is within 3 h. Table 9 gives the calibration results of part measuring points with laser tracker at the ﬁrst initial position in the measurement. Fig. 6 shows the part error curves of turntable. To verify the feasibility and accuracy of multi-station and time-sharing measurement, the position error of

turntable detected by laser interferometer and Renishaw rotary measuring system RX10 is compared with it detected by this method. Table 10 shows the measurement deviations for the two methods at some measuring points. From Table 10, the identiﬁcation results of two methods are close to each other, so the multi-station and timesharing measurement method is feasible. Only distance is involved in the measurement, and the length measurement accuracy of laser tracker has greater effect on the measurement results. The ETALON laser tracker is used in the experiment. This laser tracker use a high accuracy sphere as the optical reference for the interferometric measurement, which is different from other types of laser trackers. This laser tracker can achieve the very accurate length measurement. Meanwhile, with the

Fig. 5. The error detection of the turntable.

Table 9 The calibration results of part measuring points.

Fig. 6. The part error curves of turntable.

Measuring point

x

y

z

A1 A2 A3 A4 A5 A6

365.02221 268.95333 164.71107 55.46788 55.46497 164.71160

521.30662 576.77101 614.71171 633.97329 633.97437 614.71132

149.99771 149.99891 150.00298 150.00252 150.00159 150.00224

Table 10 The comparison of two measurement method lrad. Measuring points

A5

A10

A15

A20

A25

Deviation

1.97422

2.65901

3.70245

2.87374

3.16986

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J.D. Wang et al. / Measurement 45 (2012) 1713–1722

environmental compensation for temperature, pressure and humidity, the length measurement uncertainty of the laser tracker is U = 0.2 lm + 0.3 lm/m. So, the length measurement uncertainty of the laser tracker has smaller effect on the measurement. 7. Conclusion (1) A laser tracker is used to adopt the multi-station and time-sharing measurement principle to detect the rotary axis of NC machine in the paper. Only distance is involved in the measurement, so this method has higher measurement accuracy. Meanwhile, this method has higher measurement efﬁciency. (2) The algorithms of rotary axis measurement and error separation are deduced, and the simulations verify the feasibility of the algorithms. To avoid the singular matrix in calculation, the positions of base station should not be arranged in a plane. (3) The experiment veriﬁes the feasibility and accuracy of this method. In order to further improve the measurement accuracy of rotary axis, the errors of linear axis can be ﬁrstly measured and compensated. (4) Further research is needed to improve the measurement accuracy and efﬁciency of this method. Meanwhile, the laser tracker is considered to be placed on the machine tool bed, and the cat eye is installed on the turntable. In this measurement layout, the measurement results of turntable will be not affected by the error of linear axis.

[2]

[3]

[4]

[5]

[6]

[7]

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[10]

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[13]

[14]

Acknowledgement [15]

This research has been supported by China Technology Major Project (No. 2011ZX04016-101) and National Natural Science Foundation of China (No. 51175415). References [1] J.S. Chen, C.C. Ling, Improving the machine accuracy through machine tool metrology and error correction, The International

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