工具機動力學講義-誤差補償(2008/10/22上傳)

Transcription

OUTLINE
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Introduction of Geometrical Errors for a Machine Tool and Error Compensation Technology
J. J. Junz Wang ME NCKU
ME, NCKU
Induction of machining quality of productions
Error sources of machine tool
Definition and evaluation of errors
Measurements of geometrical errors
Measurements of geometrical errors
Error compensation technology
Variance of analysis
1
2
Conventional Machine Tools
• Among the basic machine tools:
– Lathe, shaper, planer and milling machine, drilling ,
p ,p
g
,
g
and boring, grinders, saws.
Induction of Machining Quality of Induction
of Machining Quality of
Productions
• Lathe
• Oldest and most common type of turning machine, holds and rotates metal or wood while cutting tool
holds and rotates metal or wood while cutting tool shapes material. 3
4
Conventional Machine Tools contd
Conventional Machine Tools‐contd.
• CNC milling machines ( 5 axis)
– The manual programming or command generation p g
g
g
is nearly impossible.
– Thus, they are practically always programmed Thus they are practically always programmed
with CAM.
• Computer‐Aided
Computer‐Aided Manufacturing
Manufacturing is the software used is the software used
to generate the instruction codes for a CNC machine in order for it to cut out a shape designed in a Computer‐
p
g
p
Aided Design (CAD) system.
Cylinder Head manufacture using 5-axis CNC machine
5
6
7
8
Micro‐EDM Process
Si bevel gear
Si S
S-shaped
shaped beam
Which quality of work is the most important?
Microgripper jaw with force sensor
Steel micropropeller
Error Budget in a Machine Tool
Error Budget in a Machine Tool
Error Sources for a Machine Tool
9
Machine Tool Performance Testing
Machine Tool Performance Testing
Source: Ramesh et al., Int. J Mach, Tools Manuf., 40, 1235‐1256
10
Machine Tool Performance Testing contd
Machine Tool Performance Testing‐contd.
• Th
The error motion of spindle has to be measured ti
f i dl h t b
d
dynamically.
• The response of the machine to internal heat sources The response of the machine to internal heat sources
such as spindles and axes drives has to be measured.
• The contouring capability of the machine tool has to be measured using as many axes as possible. (Ex: circle test)
d i
ibl (E i l
)
• The cutting performance (in case of machining centers) has to be measured using specified cutting tests
has to be measured using specified cutting tests.
• The short‐term reliability of the machine must also be measured. • Th
The response of the machine to the installation f h
hi
h i
ll i
environment has to be measured (Vibrations, Temperature Variation Error etc )
Temperature Variation Error etc.)
• Linear positioning errors at key locations have to be measured.
• Angular positioning errors at key locations have to be measured.
• The critical machine alignments have to be measured (i.e., squareness and parallelism)
11
12
Uncertainty of Nominal Differential Expansion (UNDE)
Thermal Error Index
Thermal Error Index
• Th
The thermal error index (TEI) for a machine and its h
l
i d (TEI) f
hi
di
environment has two components
– the uncertainty of nominal differential expansion the uncertainty of nominal differential expansion
(UNDE)
– the temperature variation error (TVE)
the temperature variation error (TVE)
• The UNDE is a measure of the machine to make accurate parts in its environment due to an ‘incorrect’
parts in its environment due to an incorrect mean mean
temperature.
• The TVE is a measure of positional drift due to p
environmental temperature changes.
• The length (or size) of mechanical parts is defined by international law at a temperature of 20 oC (68 oF)
• The thermal expansion coefficients of materials have uncertainties of approximately 1 ppm.
• Correction to the defined temperature therefore co a s u ce a es a us be ecog ed a d
contains uncertainties that must be recognized and quantified.
13
Drift Test
Drift Test
Temperature Variation Error (TVE)
Temperature Variation Error (TVE)
• It
It is also known as the is also known as the “drift”
drift test.
• The structural loop of a The str ct ral loop of a
machining center responds to changes in the ambient
to changes in the ambient temperature and gradients, causing unmeasured changes
causing unmeasured changes in relative position of tool and workpiece.
p
• This test is performed over a p
period of time to measure these errors.
14
Spindle
Test Sphere
Table
LVDT’s/
Capacitance
Gages
15
16
Geometrical Errors in a Machine tool
Geometrical Errors in a Machine tool
Displacement Accuracy
Displacement Accuracy
Z
yyaw
Z straightness of X
roll
pitch
Y
Y straightness of X
X
Three axis machine
Linear error
1 x 3 = 3
g
Straightness
2 x 3 = 6
Angular error
3 x 3 = 9
Squareness
Total
• The displacement accuracy is performed with a laser interferometer corrected for atmospheric conditions.
• The linear displacement accuracy is essential for correct performance of the machine tool.
• The linear positioning accuracy is checked through the center of th
the work zone on machining centers.
k
hi i
t
• The test is usually performed both uni‐directionally and bi‐
directionally.
directionally
= 3
21
Linear Position Accuracy
17
18
Straightness Errors
Straightness Errors
Linear Displacement Accuracy
Linear Displacement Accuracy
• Straightness errors are caused when a linear axis deviates towards the other two axes when moving along a straight line.
19
20
Angular Errors of Linear Axis Motion
Angular Errors of Linear Axis Motion
Squareness Error between the Axes
Error between the Axes
• For a linear axis, squareness is the angular deviation from 90 degrees measured between the best fit lines drawn through two sets of straightness data derived from two orthogonal axes.
• When the linear axes move they also rotate
move they also rotate through small angles. These angular motions g
are known as pitch, yaw and roll.
• Angular errors can lead to large positioning errors due to Abbe’s offsets.
Y
Straightness data
Least squares fit
Least squares fit
θ
X
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22
Definition of Accuracy (By American National Bureau of Standards, NBS)
• Accuracy is the ability to tell the truth or:
Accuracy is the ability to tell the truth or:
– Accuracy can be represented as the difference between the root mean square radius of all bullet holes in a target
the root mean square radius of all bullet holes in a target and the radius of the bull’s eye. – Linear, planar, and volumetric accuracy can all be defined Linear, planar, and volumetric accuracy can all be defined
for a machine.
Definition and Evaluation of Errors
23
24
Definition of Accuracyy‐contd.
Definition of Accuracy‐contd.
Definition of Accuracy
contd
• Repeatability (Precision) is the ability to tell the same p
y(
)
y
story over and over again or:
• Resolution is how detailed your story is
– The error between a number of successive attempts to p
move the machine to the same position.
– Repeatability is often considered to be the most important parameter of a computer controlled machine (or sensor).
– Th
The smallest
ll discernible change in the parameter of di
ibl h
i h
f
interest that can be detected by the instrument, – The smallest
The smallest positional increment that can be commanded positional increment that can be commanded
of a motion control system, – The smallest programmable step, The smallest programmable step
– The smallest mechanical step the machine can make during point to point motion.
during point to point motion.
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26
Accuracy and Repeatability
Accuracy and Repeatability
單目標點單向定位及其平均值(
單目標點單向定位及其平均值(mean)
)
Accuracy is the ability to tell the truth. And, repeatability (Precision) i h bili
is the ability to tell the same story over and over again. As shown in ll h
d
i
h
i
following figures, poor precision may cause poor accuracy to machine tool when the average radius of bullet holes to target is
machine tool when the average radius of bullet holes to target is close to zero.
Poor accuracy
Poor precision
Poor accuracy
Good precision
After
Before
correction

x 為目標位置
將所有點資料加總後除以樣本個數可
得到所有點資料的平均位置 x
Δx 為所有資料的平均偏移量(mean deviation)

Good accuracy
Good precision
After
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28
單點定位的頻率圖 (frequency plot)
(frequency plot)
平均偏移量 (mean deviation)計算
(mean deviation)計算
目標位置
0.1213 (in)
Run
Deviation(μin)
1
40
2
30
3
20
4
10
5
-10
6
-30
7
-40
Δx
3
60
Deviation
(μin ) 40
20
data
0
mean
-20
20
Δx
為所有資料的平均偏移量 (mean deviation)
(
)
Δx
跟準確度(accuracy)有關
-40
-60
0
0.05
0.1
0.15
Poistion (in)
29
30
標準差 (standard deviation) &重現性
( t d d d i ti ) &重現性 (repeatability)
(
t bilit )
X=
• X ′真實位置
∑ X′
3σˆ = 3
N
∑(X ′ − X )
N −1
2
• X 真實位置的平均值
• N 取樣次數
• 3σˆ is dispersion on each side of the mean
of the mean
31
重現性 (repeatability) 計算
(
t bilit ) 計算
目標位置
0.1213 (in)
Run
Deviation(μin)
1
40
2
30
3
20
4
10
5
-10
6
-30
7
-40
Δx
3
3σˆ
91
32
單目標點單向定位的準確度 &重現性
多目標點單向定位及其平均值(
多目標點單向定位及其平均值(mean)
)

xi為目標位置
將所有點資料加總後除以樣本
個數可得到所有點資料的平均位
置 x
i
Δxi
為所有資料的平均偏移量
(mean deviation)
1
33
3.9439
7.5601
22.0005
26.4583
1
40
270
-420
370
400
420
2
30
240
-420
370
370
400
3
20
240
-430
430
380
350
390
4
5
10
240
-450
340
350
400
-10
210
-410
360
350
360
Run
13.3727
34
重現性 (repeatability)
600
400
Deviation
i i
(μin )
6
-30
210
-470
340
330
360
7
-40
190
-480
310
320
350
Δx
3
229
-440
353
353
383
D e v ia tio n
0.1213
2
2
1
平均偏移量 (mean deviation) 計算
目標位
置
2
1
200
0
-200
-400
1
1
-600
0
5
10
15
20
25
2
1
30
1
Poistion (in )
2
2
1
2
2
以所有目標位置所對應的重現性當中最大的值
當作多目標點單向定位的重現性

35
36
機器的定位為單向 (Unidirectional) 或
同時考慮兩個單向亦即雙向 (Bidirectional)
單目標點雙向定位及其平均值(mean)
單向：
• 得到雙向的點資料，該如何處理平均值?
1 當作一大群單筆資料
1.當作一大群單筆資料
2.將正方向及負方向的點資料分兩群處理
雙向：
•
↑ or ↓ ‐
正方向或負方向
37
38
將雙向的點資料當作一筆資料
處理平均值 (mean value)
將雙向的點資料分別處理
平均值 (mean value)
平均偏移量 (mean deviation)
(
d i ti ) 計算
目標位置
50
f
r
Deviation
(μin )
b
0
forward data
-50
forward mean
reverse data
-100
reverse mean
-150
b
f
-200
0
r
0.05
0.1
Poistion

0.15
(in )
x 為目標位置
50
將所有點資料加總後除以樣本
個數可得到所有點資料的平均位
置 xj

Deviat
ation
(μin )
Δx j為所有資料的平均偏移量
((mean deviation))
0
-50
data
x
mean
-100
-150

-200
0
0 05
0.05
01
0.1
Poistion
39
(in )
0 15
0.15
0.1213
Forward
Reverse
Run
Deviation
1
40
-40
2
30
-60
3
20
-80
4
10
-90
5
-10
-130
6
-30
30
-130
130
7
-40
-160
Δx f / Δxr
3
-99
99
Δxb
-48
40
失步 (lost motion)
(l t
ti )
當分別處理兩筆數據的時候，
其平均值會有所差異，因此會
其平均值會有所差異
因此會
有失步產生
產生原因為背隙 (backlash)或
延滯 ((hysteresis)
y
)

f
將雙向的點資料當作一筆資料處理
r
b
f
b
f
f
b
r
xf
r
xr

b
r
單向定位的重現性

雙向定位的重現性
41
42
準確度 (accuracy)
重現性 (repeatability) 計算
(repeatabilit ) 計算
目標位置
Forward
Run
0.1213 (in)
Reverse
Deviation(μin)
1
40
-40
2
30
-60
60
3
20
-80
4
10
-90
5
-10
-130
6
-30
-130
7
-40
40
-160
160
x f / xr
3
-99
準確度 (accuracy) 在單目標點雙向
定位的過程中亦可解釋為不確定度
(uncertainty)
3σˆ f / 3σˆ r
91
129

3σˆ b
將雙向的點資料當作一筆資料處理

因此有所謂的單向準確度及雙向
準確度

其定義為 (mean+3s) & (mean‐3s)之
間的最大寬度
191
43
44
平均值 (mean)
多目標點雙向定位及其平均值(
多目標點雙向定位及其平均值(mean)
)
得到雙向的點資料，該如何
處理平均值？
1.當作一大群單筆資料
2.將正方向及負方向的點資料分
兩群處理

平均值 (mean)
1r
1f
1
1f
2f
2r
2
2f
2r
2
2b
1r
將雙向的點資料當作一筆
資料處理平均值 (mean )
1
2
1f
1f
1f
1r
以所有目標位置所對應
400
mean forward
200
mean forward+3s
0
mean forward-3s
-200
mean reverse
-400
mean reverse+3s
-600
mean reverse-3s
-800
2f
0.121 3.944 7.56 13.37
2r
2r

因此有所謂的單向準確度及雙向準
確度
以所有目標位置所對應
的重現性當中最大的值
將雙向的點資料
當作一筆資料處理
當作多目標點雙向定位
的重現性
其定義為 (mean+3s) & (mean‐3s)之
間的最大寬度

26.46
1500
(μin )
1000
Devia
ation
1r
1r
22
Position (in )
2r
1r
準確度 (accuracy) 在多目標點雙向
定位的過程中亦可解釋為不確定度
(uncertainty)

600
(μin )
的重現性
2f
2f
1f
當作多目標點單向定位
2r
2f
46
重現性 (repeatability) &準確度
(repeatability) &準確度 (accuracy) 計算
(accuracy) 計算
的重現性當中最大的值
2
2b
1b
Deviation

將雙向的點資
料分別處理
1
1b
1
45
500
mean+3s
0
mean-3s
3
mean
-500
-1000
1b
1
2b
2
-1500
0 1213 3.9439
0.1213
3 9439 7.5601
7 5601 13.373
13 373 22.001
22 001 26.458
26 458
1b
1b
2b
1b
2b
Position
2b
47
(in )
48
各國工具機標準名稱
工具機常用的檢驗標準
• NMTBA NMTBA 1968, 1972
1968 1972
• NMTBA
( National Machine Tool Builders Association ) ‐ United States
• VDI/DGQ 3441 VDI/DGQ 3441 1977, 1982
1977 1982
(Verein Deutscher Ingenieure ) ‐ Germany
• ISO 230‐2
ISO 230 2 1988, 1997
1988 1997
( International Organization for Standardization )
• ASME B5.54 ASME B5 54 1992, 2005
1992 2005
( The American Society of Mechanical Engineers ) ‐ United States
Definition and evaluation of accuracyy and repeatability
p
y for numerically controlled machine tools
• ISO 230‐2
Test code for machine tools ‐
d f
hi
l Part 2
Determination of accuracy and repeatability of positioning numerically controlled axes
• ASME B5.54
Methods for performance evaluation of computer numerically controlled machining centers
controlled machining centers
• VDI/DGQ 3441
Statistical testing of the operational and positional accuracy of machine Tools
hi T l
49
50
Lost Motion
Lost Motion

NMTBA
51
The distance between the right‐
i h and left‐
d l f approach h
means at any given target point
target point
52
Lost Motion
Lost Motion
Repeatability
• Dispersion on each side of the mean
Right
Left
1
4.492026
9 0 6
5.79874
5
98
2
4.532614
5.407237
3
4.190211
5.225438
4
3.832765
4.970977
5
3.551299
4.760545
ΔX u 4.119783
5.232587
Lost Motion :
4 119783 - 5.232587
4.119783
5 232587 = 1.1128044
1 1128044
Unidirectional repeatability
53
54
Accuracy
E
Example
l
Right
Left
1
4.492026
5.79874
2
4.532614
5.407237
3
4 190211
4.190211
5 225438
5.225438
4
3.832765
4.970977
5
3 551299
3.551299
4 760545
4.760545
3σˆ u
1.272357066
1.20219773
3σˆ b
Bidirectional repeatability
• Accuracy (A)
Au = ΔX u ± 3σˆ u
Ab = ΔX b ± 3σˆ b
2.111323815
55
56
Accuracy
Evaluation of Machine Accuracy
Evaluation of Machine Accuracy
•
After A has been determined for a determined
for a
number of points, two extremes of A are averaged to
averaged to determine a “zero”
• As is based value of accuracy.
is based value of accuracy
• Bs is the specified distance of As.
• For a longer travel, the For a longer travel the
specification may include an additional allowance Cs per additional allowance C
per
additional unit length.
58
57
Deviation from target
Example :
( μin )
Point NO. Target position
1
2
3
4
5
6
The extremes of the ΔXu ± 3σˆ u data are:
7
1
-14.7993
109
-2
11
92
44
3
-91
2
-9.4491
108
-16
80
7
99
28
65
4
-2.0725
87
-186
89
51
105
-11
102
19
45.6279
-611
-726
-583
-585
-524
-668
-604
20
48.9045
-732
-717
-685
-700
-691
-720
-784
Before zero shift
Point NO.
∑X
3σˆ u
1
166
200
24
-176
176
224
2
371
142
53
-89
195

maximum = 349 (point 4)
4
237
315
34
-281
349

minimum
i i
= ‐819 (point 20)
819 ( i 20)
19
-4301
195
-614 -809
-419
20
-5029
100
-719 -819
-619
ΔX u ΔXu −3σˆu ΔXu + 3σˆ u
maximum = 349 (point4)
minimum = -819
819 (point20)
the zero shift will then be
349 819
349-819
= −235
2
The extremes of the
point NO.
ΔX u ± 3σˆ u data are ：
data are ：
59
∑X
3σˆ u
After zero shift
ΔX u ΔXu −3σˆu ΔXu + 3σˆ u ΔX u ΔX u − 3σˆ u ΔX u + 3σˆ u
1
166
200
24
-176
224
259
59
459
2
371
142
53
-89
195
288
146
430
4
237
315
34
-281
349
269
-46
584
19
-4301
195
-614
-809
-419
-379
-574
-184
20
-5029
100
-719
-819
-619
-484
-584
-384
60
R lt 2：f
Result 2：for additional length
dditi
l l th
Result 1

584

(μin )
The minimum is
The
minimum ΔX u − 3σˆ u is
‐ 584

Accuracy specification were ± 600 (μin )
• Accuracy specification were ± 500 (μin ) in a foot ± 100 (μin )
in each additional foot
•
The maximum ΔX u + 3σˆ u is (μin )
Accuracy specification were
Accuracy specification were ± 600
(μin )
62
61
ISO 230
Test code for machine tools
ISO 230 is standard method of testing the accuracy of machine tool.
•
230‐ 1 Part 1: Geometric accuracy of machines operating under no‐load or finishing conditions.
– 了解兩軸以上同動時的軌跡精度
– 分析進給系統的動態特性
• ISO 230‐6對角線位移測試
ISO 230 6對角線位移測試
• 230‐ 2 Part 2:
– 評估機器的空間性能
– 節省時間及成本
•
•
•
•
• ISO 230‐4 循圓測試
230‐ 3 Part 3: Determination of thermal effects 230‐ 4 Part 4: Circular tests for numerically controlled machine tools 230‐ 5 Part 5: Determination of the noise emission p
g
y
y
230‐ 6 Part 6: Determination of positioning accuracy on body and face diagonals
63
64
Test Procedure
Test Procedure
•
ISO 230 ‐ 2
positioning numerically controlled axes
•
Tests for linear axes up to 2000 mm
– A minimum of five
A minimum of five target positions target positions
per meter
– With an overall minimum of five
target positions
– Each target position shall be attained five times in each direction
Tests for linear axes exceeding 2000 mm
– Target positions selected with an average Target positions selected with an average
interval length of 250 mm
65
66
Positional Deviation
os t o a e at o
Mean unidirectional positional deviation at a position
1 n
x i ↑=
∑ x ij ↑
n j =1
x i ↓=

1
n
n
∑
j =1
x ij ↓
bidirectional
bidirectional x ↑ + xi ↓
xi = i
2
Positive ↑
Positive ↑
Negative ↓
1
4.492026
5.79874
2
4.532614
5.407237
3
4.190211
5.225438
N ti ↓
Negative
4
3.832765
4.970977
ΔX u = (5.79874 + 5.407237
5
3 551299
3.551299
4 760545
4.760545
xi ↑ / xi ↓
4.119783
5.232587

xi
4.6761852
ΔX u = (4.492026 + 4.532614
+ 4.190211 + 3.832765
+ 3.551299)/5 = 4.119783
+ 5.225438 + 4.970977
+ 4.760545)/5 = 5.232587
Bidirectional
ΔX b = (4.119783 + 5.232587)/2
= 5.232587
67
68
Reversal Value
Reversal Value
•
Reversal value at a position
Positive ↑
Bi = x i ↑ − x i ↓
•
Reversal value of an axis
B = max[ Bi
•
]
Mean reversal value of an axis
B=
Negative ↓
1
4.492026
5.79874
2
4.532614
5.407237
3
4.190211
5.225438
4
3.832765
4.970977
5
3 551299
3.551299
4 760545
4.760545
xi ↑ / xi ↓
4.119783
5.232587
Reversal value at a position:
Bi = x i ↑ − x i ↓
= 4.119783 - 5.232587
= −1.1128044
1 m
∑ Bi
m i =1
70
69
Standard Uncertainty
Standard Uncertainty
•
Repeatability
Unidirectional standard uncertainty at a position
Unidirectional repeatability at a position
•
s i ↑=
n
1
(x ij ↑ − x i ↑ )2
∑
n − 1 j =1
s i ↓=
n
2
1
(
x ij ↓ − x i ↓ )
∑
n − 1 j =1
Ri ↑= 4 si ↑ ; Ri ↓= 4 si ↓

Bidirectional repeatability at a position
Ri = max ⎡⎣ 2si ↑ +2si ↓ + Bi ; Ri ↑; Ri ↓ ⎤⎦
Unidirectional repeatability U idi ti
l
t bilit
of an axis
[
R ↑= max Ri ↑

]
[ ]
R ↓= max Ri ↓
of an axis
R = max[Ri ]
71
72
1
Pos ↑
1
Positive ↑
Negative↓
4.492026
5.79874
Unidirectional repeatability at a position
Ri ↑= 4si ↑= 1.69647608

2
4.532614
5.407237
3
4.190211
5.225438
4
3.832765
4.970977
Bidirectional repeatability at a position
5
3 551299
3.551299
4 760545
4.760545
Ri = max 2 si ↑ +2 si ↓ + Bi ; Ri ↑; Ri ↓
si ↑ / si ↓ 0.42411902 0.400732577
Ri ↓= 4si ↓= 1.602930308

2
Neg ↓
Pos ↑
3
Neg ↓
Pos ↑
8.566713 9.81866
Neg ↓
1
4.492026 5.79874
10.32070 11.00806
2
4.532614 5.407237 8.471852 9.619713 10.28639 10.70958
3
4.190211 5.225438 8.425074 9.279222 10.32760 10.57642
4
3.832765 4.970977 7.945062
5
3.551299 4.760545 7.909447 9.056468 9.648938 10.28608
9.18549
9.894999 10.37214
1 696476 1.602930
1 602930 1.246131
1 246131 1.267815
1 267815 1.233934
1 233934 1.147325
1 147325
Ri ↑ / Ri ↓ 1.696476
[
]
2.762507598
Ri
2.385254486
Unidirectional repeatability of an axis
R ↑= max Ri ↑ = 1.696476
1 696476

= max[2.762507598 ; 1.69647608;
1 602930308]
1.60293030
= 2.762507598
1.685365592
of an axis

[ ]
R ↓= max [R ↓] = 1.602930
R = max[Ri ] = 2.76250759
2 76250759 8
i
73
74
1
Systematic Positional Deviation
Systematic Positional Deviation
•
Pos ↑
Unidirectional systematic positional deviation of an axis
E ↑= max x i ↑ − min x i ↑
[ ] [ ]
E ↓= max [x ↓] − min [x ↓]
i

i
Bidirectional systematic positional deviation of an axis
[
]
[

Mean bidirectional systematic
Mean bidirectional systematic positional deviation of an axis
[ ]
[ ]
M = max x i − min x i
]
•
Pos ↑
Neg ↓
Pos ↑
8.566713 9.81866
Neg ↓
1
4.492026 5.79874
4.532614 5.407237 8.471852 9.619713 10.28639 10.70958
3
4.190211 5.225438 8.425074 9.279222 10.32760 10.57642
4
3.832765 4.970977 7.945062
5
3.551299 4.760545 7.909447 9.056468 9.648938 10.28608
xi ↑ / xi ↓
4 119783 5.232587
4.119783
5 232587 8.263629
8 263629 9.391910
9 391910 10.09572
10 09572 10.59046
10 59046
4.6761852
10.32070 11.00806
9.18549
9.894999 10.37214
8.8277701
10.3430959
Unidirectional systematic Bidirectional systematic Mean bidirectional systematic
positional deviation of an positional deviation of an positional deviation of an axis
axis E ↑=
axis
↑ max [x i ↑ ]− min [x i ↑ ]
= 10.09572 - 4.119783
= 5.975937
[ ]
75
Neg ↓
3
2
xi
E = max x i ↑; x i ↓ − min x i ↑; x i ↓
2
[ ]
E ↓=
↓ max x i ↓ − min x i ↓
= 10.59046 - 5.232587
= 5.357873
[
]
[
E = max x i ↑; x i ↓ − min x i ↑; x i ↓
]
[ ]
[ ]
M = max x i − min x i
= 10.59046 - 4.119783
= 10.3430959 - 4.6761852
= 6.47067
= 5.6669107
76
1
Accuracy
Pos ↑
Unidirectional accuracy of an axis
•
[
] [
]
A↓
↓= max[x ↓ +2s ↓]− min[x ↓ −2s ↓]
A ↑= max x i ↑ +2si ↑ − min x i ↑ −2si ↑
i

i
i
i
i
i
i
8.566713 9.81866
Pos ↑
Neg ↓
10.32070
11.00806
2
4.532614 5.407237 8.471852 9.619713 10.28639
10.70958
3
4.190211 5.225438 8.425074 9.279222 10.32760
10.57642
4
3.832765 4.970977 7.945062
9.894999
10.37214
5
3.551299 4.760545 7.909447 9.056468 9.648938
10.28608
x i ↑ +2si ↑ /
4 968021 6.034052
4.968021
6 034052 8.886695
8 886695 10.02581
10 02581 10.71269
10 71269
11 16412639
11.16412639
3.271544 4.431122 7.640563 8.758002 9.478760
10.01680081
9.18549
x i ↓ −2si ↓
•
Unidirectional accuracy of an axis Bidirectional accuracy of an axis
[
]
[
]
= 10. 71269 - 3.271544
= 7.4411
A ↓= max x i ↓ +2si ↓ − min x i ↓ −2si ↓
[
]
[
= 11.16412639 - 4.431122
ASME B5.54 ‐ 2005
Neg ↓
4.492026 5.79874
A ↑=
↑ max
ma x i ↑ +2si ↑ − min x i ↑ −2si ↑
77
Pos ↑
1
x i ↑ −2si ↑ /
A = max x i ↑ +2si ↑; x i ↓ +2si ↓
i
Neg ↓
3
x i ↓ +2si ↓
Bidirectional accuracy of an axis
[
]
− min[x ↑ −2s ↑; x ↓ −2s ↓]
2
= 6.733
[
[
]
]
A = max x i ↑ +2si ↑; x i ↓ +2si ↓
]
− min x i ↑ −2si ↑; x i ↓ −2si ↓
= 11.16412639 - 3.271544
= 7.8925
78
VDI/DGQ 3441
ASME B5.54 is almost the same as ASME
B5.54 is almost the same as
ISO 230 ‐ 2 standard
79
80
Reversal Error
Reversal Error
•
Reversal error at a position
U j = x j ↓ −x j ↑
Positive ↑
Negative ↓
1
4.492026
5.79874
2
4.532614
5.407237
3
4.190211
5.225438
4
3.832765
4.970977
5
3 551299
3.551299
4 760545
4.760545
xj ↑ / xj ↓
4.119783
5.232587
Reversal error at position:
U j = x j ↑ −x j ↓
= 4.119783 - 5.232587
= 1.1128044
82
81
Positional Scatter
Positional Scatter
•
Mean standard deviation at a position
sj =

Negative ↓
s j ↑ +s j ↓
1
4.492026
5.79874
2
2
4.532614
5.407237
3
4.190211
5.225438
P iti
Positional scatter at a position
l
tt
t
iti
Ps j = 6 ⋅ s j

Positive ↑
Maximum positional scatter
Maximum positional scatter Ps max = Ps j max
4
3.832765
4.970977
5
3 551299
3.551299
4 760545
4.760545
s j ↑ / s j ↓ 0.42411902 0.400732577
83

Mean standard deviation at a position
sj =
s j ↑ +s j ↓
2
0.42411902 + 0.400732577
=
2
= 0.412425799

P iti
Positional scatter at a position
l
tt
t
iti
Ps j = 6 ⋅ s j = 2 .474554796
84
1
•
Pos ↑
2
Neg ↓
1
4.492026 5.79874
Pos ↑
3
Neg ↓
8.566713 9.81866
Pos ↑
Neg ↓
10.32070 11.00806
2
4.532614 5.407237 8.471852 9.619713 10.28639 10.70958
Positional deviation
3
4.190211 5.225438 8.425074 9.279222 10.32760 10.57642
Pa = x jmax − x jmin
4
3.832765 4.970977 7.945062
5
3.551299 4.760545 7.909447 9.056468 9.648938 10.28608
9.18549
9.894999 10.37214
4 119783 5.232587
5 232587 8.263629
8 263629 9.391910
9 391910 10.09572
10 09572 10.59046
10 59046
x j ↑ / x j ↓ 4.119783
4.6761852
xj
8.8277701
10.3430959

Pa = x jmax − x jmin
= 10.3430959 - 4.6761852
= 5.6669107
85
86
1
Positional Uncertainty
Positional Uncertainty
•
Pos ↑
Positional uncertainty
2
Neg ↓
Pos ↑
3
Neg ↓
8.566713 9.81866
Pos ↑
Neg ↓
1
4.492026 5.79874
2
4.532614 5.407237 8.471852 9.619713 10.28639 10.70958
10.32070 11.00806
3
4.190211 5.225438 8.425074 9.279222 10.32760 10.57642
(
)⎤⎥
4
3.832765 4.970977 7.945062
5
3.551299 4.760545 7.909447 9.056468 9.648938 10.28608
(
)
1
x j + U j + Ps j
2
1
x j − U j + Ps j
2
1
⎡
P = ⎢ x j + U j + Ps j
2
⎣
1
⎡
− ⎢ x j − U j + Ps j
2
⎣
⎦ max
(
(
⎤
⎥
⎦ min
•
)
)
9.18549
9.894999 10.37214
6 469864798
6.469864798
10 33464072
10.33464072
11 48343624
11.48343624
2.882505602
7.320899485
9.202755556
(
)
(
)
1
1
⎡
⎤
⎡
⎤
P = ⎢ x j + U j + Ps j ⎥ − ⎢ x j − U j + Ps j ⎥
2
2
⎦ max ⎣
⎦ min
⎣
= 11.48343624 - 2.882505602
= 8.6009
87
88
東台精機立式銑床機械精度檢測 (TMV‐1500A)
不同標準間準確度 & 重現性的比較
89
NMTBA Bidirectional
NMTBA Bidirectional
90
ISO 230 2
ISO 230‐2
B : reversal value
R : repeatability
M, E : positional d i ti
deviation
A : accuracy
91
92
VDI/DGQ 3441
VDI/DGQ 3441
A
Accuracies
i
Ps : positional scatter
Pa : positional deviation
NMTBA VDI ISO definition gguidelines standard
p
y
P : positional uncertainty
Accuracy
±7 282245
±7.282245
14 77652965
14.77652965
14 08510733
14.08510733
ASME
B5.54
standard
14 08510733
14.08510733
94
93
綜合比較(1)
綜合比較(2)
NMTBA
ISO 230‐2
VDI/DGQ 3441
Lost motion
Lost
motion
Standard deviation
Repeatability

Reversal value
Reversal
value
Standard uncertainty
Repeatability
Accuracy

Accuracy
• The number of target points
• The number of runs
• The runs approach from one direction or both directions
Reversal error
Reversal
error
Standard deviation
Positional scatter

ASME B5 54
ASME B5.54
Reversal deviation
Standard uncertainty
Repeatability
Systematic deviation
Accuracy

95
96
Machine tool accuracy standards compared
Term or other
T
th
info
ISO standard
t d d
Target position Position to which the
VDI
guidelines
NMTBA
definition
JIS
standard
Same as ISO
Same as ISO
Same as ISO
Term or other
T
th
into
moving part is
programmed
Actual position Measured position
Same as ISO
Same as ISO
Same as ISO
Unidirectional
/
Bidirectional
reached by the moving
part
Number of
runs to the
target position
Minimum of five for
each direction
Depends on
length of axis.
Minimum of
fi
five
Minimum of
seven
One run in each
direction for
positional
accuracy and
d
seven for
repeatability
N b off
Number
target position
required
Five per meter up to 2
meters.
More for longer
Ten per meter
for each
direction
Unspecified ;
example
shown is 20
Depends on
length. Every
50mm up to
1000mm;then
every 100mm
ISO standard
t d d
Unidirectional
involves a series of
measurements in
which the approach to
target is always form
same direction.
Bidirectional refers to
movement in both
directions.
directions
Bidirectional
recommended
VDI
guidelines
NMTBA
definition
JIS
standard
Same as ISO.
Recommends
bidirectional
Same as ISO.
Recommends
unidirectional
Same as ISO.
R
Recommends
d
bidirectional
97
Term or other
T
th
into
ISO standard
t d d
Positional
deviation
Mean
positional
deviation
98
NMTBA
definition
JIS
standard
Term or other
T
th
into
ISO standard
t d d
Differences between Different form ISO
actual position
Maximum
reached and target
difference of the
mean values of the
actual positions
versus the
individual target
positions along an
axis
Agrees with
ISO definition
however noted
as ‘deviation
from target’
Term not noted
since only one
run is made to
a target two for
bidirectional
Reversal error
Value of the difference
Same as ISO
between the mean positional
deviation at a position for the
two direction of approach
Algebraic mean of
the positional
deviations at a
target position
Agrees with
ISO noted as
‘mean’
VDI guidelines
id li
Agrees with ISO
however noted as
‘mean value’
Not considered
Standard
deviation
(
1 n
∑ xij − x i
n − 1 j =1
VDI
guidelines
)
2
Same as ISO
NMTBA JIS
definition standard
Same as ISO. Not
Referred to
considered
as ‘lost
motion’
Same as ISO.
Not
considered
Where:
n=number of runs;
j
j=any
one run;
xij=positional deviation for
any one run
=mean positional deviation
xi
99
100
M hi tooll accuracy standards
Machine
d d comparedd
Term or
other into
ISO standard
VDI guidelines
Positional
accuracy
Maximum difference
between extreme values of
x+3sigma and x-3sigma
regardless of the position
or direction
di ti off motion.
ti
Applies to unidirectional as
well as bidirectional.
Due to reversal error the
spread will be greater and
the position accuracy less
for bidirectional
No specific
p
term for
Comparable
p
to
accuracy, however the
ISO. Referred
term ‘position uncertainty’ to as accuracy
as described is
comparable
bl tto th
the ISO
definition for ‘position
accuracy’ although the
calculations differ
NMTBA
definition
JIS standard
Term or other
i
into
Differs
considerably
from any one of
the other three.
P iti i
Positioning
accuracy is
measured as the
g variation
largest
of any actual
position from a
target position
Spread at the target
Repeatability
R
t bilit
position having the
(unidirectional largest spread
or bidirectional)
ISO standard
VDI
guidelines
id li
NMTBA
d fi i i
definitio
n
JIS
standard
d d
Comparable to the
ISO standard
Comparable
to the ISO
standard
Differs
considerably
from the other
three and is
expressed as a
value based on
dividing the
read at the
target position
α maximum
spread by two
101
102
Resolution of Measurement
Resolution of Measurement
Measurements of Geometrical Errors
103
104
Laser Vector Measurement Technique Laser Vector Measurement Technique‐contd.
(C. Wang, 2000)
• Machine accuracy can be improved by indentifying all geometric y
p
y
y g g
error and then compensating for these errors. The key is how to measure these error accurately and quickly.
• ASME standard is noted that linear displacement measurement along four diagonals can used to check volumetric positioning accuracy quickly.
i kl
• First mount the laser beam in one of the body diagonal directions the same as in the body diagonal measurement
directions, the same as in the body diagonal measurement. Instead of moving x, y, and z continuously to the next increment R, stop and take a measurement. • Move the x axis to X, stop and take a measurement, th
then move the y axis to Y, stop and take a measurement, th
i t Y t
dt k
t
then move the z axis to Z, stop and take a measurement.
• R= (X2+Y2+Z2)0.5 is the increment in the diagonal direction, and X, Y, and Z are the increments in the x, y, and z directions, respectively.
39
106/39
Laser Vector Measurement Technique‐contd
Laser Vector Measurement Technique
contd.
Laser Vector Measurement Technique‐contd.
Laser Vector Measurement Technique
contd
• Compared to the conventional body diagonal measurement where only one data point is collected at each increment R, h
l
d
i i
ll
d
hi
the vector measurement collects three data points, one at X, one at Y and one at Z Hence three times more data are
one at Y, and one at Z. Hence three times more data are collected.
Central limit theory,
• Central limit theory,
• For conventional body diagonal measurement, the displacement is a straight line along the body diagonal; hence a laser interferometer can be used to do the measurement.
• The displacements are along the x axis, then along the y axis, and then along the z axis. The trajectory of the target or the retroreflector is not parallel to the diagonal direction.
is not parallel to the diagonal direction
σx =
σ
n
σ x : STEDV of sample mean
σ : STEDV of population
n: smapling number
107
108
Assumptions
Volumetric Displacement Errors
To simplify the analysis, the following assumptions are made.
To
simplify the analysis the following assumptions are made
• The motion is repeatable to within certain uncertainty. The y
p
y
accuracy of the method is limited to the repeatability of the motion.
• The position errors can be superpositioned, i.e., the position error is much smaller than the travel distance.
• The angular errors are small compared to the other errors.
• Rigid body motion.
• Consider a rigid body motion from PA to PB. The motion can be described by 6 degrees of freedom (1‐linear, 2‐straightness, 3‐
angular errors)
angular errors).
• Now move PA such that it is at the origin of coordinate system.
• The point P
The point PB would be at the origin if there was no error.
would be at the origin if there was no error
109
Measurement along the Body Diagonal
Measurement along the Body Diagonal
• Else, P
El PB = Xu
X x + E(x) [from vector diagram]
E( ) [f
di
]
• The error vector E(x) can be expressed as
• The diagonal measurement R is achieved by increments in X, Y and Z directions. The unit vector R is given by
E( x ) = E x ( x )u x + E y ( x )u y + E z ( x )u z
• Similarly,
E( y ) = E x ( y )u x + E y ( y )u y + E z ( y )u z
110
R=
x
y
z
u x + uy + uz
R
R
R
PB
PA
• The error along the diagonal direction is given by dR
A
B
E( z ) = E x ( z )u x + E y ( z )u y + E z ( z )u z
111
x
y
z
+ Ey ( x ) + Ez ( x )
R
R
R
x
y
z
dR( ) = E x (y ) + E y ( y ) + E z ( y )
dR(y)
R
R
R
x
y
z
dR(z) = E x (z ) + E y ( z ) + E z ( z )
R
R
R
dR(x) = E x ( x )
112
Measurement along the Four Body Diagonals
Measurement along the Four Body Diagonals‐contd.
contd
• There
There are four diagonals, namely, those from (0, 0, 0) to (nX, nY, are four diagonals namely those from (0 0 0) to (nX nY
nZ), denoted by ppp.
( , , ) ( , , ),
y pp
• (nX, 0, 0) to (0, nY, nZ), denoted by npp.
• (0, nY, 0) to (nX, 0, nZ), denoted by pnp.
(0, 0, nZ) to (nX, nY, 0), denoted by ppn.
• (0, 0, nZ) to (nX, nY, 0), denoted by ppn.
dR ( x )NPP
(nX, nY, nZ)
(0, nY, nZ)
Z
dR ( x )PNP
PPP
NPP
(nX, 0, nZ)
(0, 0, nZ)
PNP
PPN
dR ( x )PPN
Y
(nX, nY, 0)
(0, nY, 0)
(0, 0, 0)
X
(nX, 0, 0)
114
113
Measurement along the Four Body Diagonals‐contd.
contd
R
E x ( x ) = [dR ( x )PPP − dR ( x )NPP ]
2X
R
E y ( x ) = [dR ( x )PPP − dR ( x )PNP ]
2Y
R
E z ( x ) = [dR ( x )PPP − dR ( x )PPN ]
2Z
x
y
z
+ Ey ( x ) + Ez ( x )
R
R
R
x
y
z
= −E x ( x ) + E y ( x ) + E z ( x )
R
R
R
x
y
z
= E x ( x ) − Ey ( x ) + Ez ( x )
R
R
R
x
y
z
= E x ( x ) + Ey ( x ) − Ez ( x )
R
R
R
dR ( x )PPP = E x ( x )
R
E x ( z ) = [dR ( z )PPP − dR ( z )NPP ]
2X
R
E y ( z ) = [dR ( z )PPP − dR ( z )PNP ]
2Y
R
E z ( z ) = [dR ( z )PPP − dR ( z )PPN ]
2Z
Squareness Errors
When the angles between xy, yz, and xz
g
y, y ,
are not exactlyy
90°, then the diagonal distances can be expressed as
dRNPP
dRPNP
R
2X
R
E y ( y ) = [dR ( y )PPP − dR ( y )PNP ]
2Y
R
E z ( y ) = [dR ( y )PPP − dR ( y )PPN ]
2Z
E x ( y ) = [dR ( y )PPP − dR ( y )NPP ]
dRPPP
115
XY
YZ
ZX
+ θ yz
+ θ zx
R
R
R
XY
YZ
ZX
= −θ xy
+ θ yz
− θ zx
R
R
R
XY
YZ
ZX
= −θ xy
− θ yz
+ θ zx
R
R
R
XY
YZ
ZX
= θ xy
− θ yz
− θ zx
R
R
R
dRPPP = θ xy
R
2 XY
R
+ dRPNP )
2ZX
R
+ dRNPP )
2YZ
θ xy = (dRPPP + dRPPN )
θ zx = (dRPPP
θ yz = (dRPPP
116
Relation Between the Measured Volumetric Errors
and the Conventional 21 Errors
Relation Between the Measured Volumetric Errors
and the Conventional 21 Errors‐contd.
• For a machine type FXYZ, the position errors can be expressed as
Ex ( X ) = δ x ( X )
ΔX = δ x ( X ) + δ x (Y ) + δ x (Z ) − Y ε z ( X ) + Z (ε y ( X ) + ε y (Y )) + Zθ ZX − Y θ XY
E x (Z ) = δ x (Z ) + Zθ zx
ΔY = δ y (Y ) + δ y ( X ) + δ y (Z ) − Z (ε x ( X ) + ε X (Y )) − ZθYZ
Ey ( X ) = δ y ( X )
E x (Y ) = δ x (Y ) − Y θ xy
E y (Y ) = δ y (Y )
ΔZ = δ z (Z ) + δ z ( X ) + δ z (Y ) + Y ⋅ ε X ( X )
Remember, only for FXYZ!
E y (Z ) = δ y (Z ) − Zθ yz
• Since
Ez ( X ) = δ z ( X )
Ez (Y ) = δ z (Y )
ΔX = E x ( x ) + E x ( y ) + E x ( z )
Ez (Z ) = δ z (Z )
ΔY = E y ( x ) + E y ( y ) + E y ( z )
ΔZ = Ez ( x ) + Ez ( y ) + Ez ( z )
117
118
Measurement Errors
•
•
•
•
•
•
Projection error
Laser beam alignment error
b
l
Flat‐mirror alignment error
Error due to machine angular motion
symmetry
The sensitivity of squareness results to machine aspect ratio
Error Compensation Technology
119
120
Flow of Error Compensation in “Precimatics”
In this case, some new issues is essential. Such as how to
Such as how to 補償
get 21 geometric error?
get correction?
generate a new correct NC code?
121
dx
X axis values
X‐axis values
Flow of Volumetric Error Flow
of Volumetric Error
Simulation
Y‐axis values
Z‐axis values
Squareness values
dy
dz
ex
ey
ez
122
Flow of Volumetric Error Simulation
0 0 ‐0.0030 ‐0.0026 0 0.0002 ‐0.0013
1.0000 0.0026 0.0014 ‐0.0024 0 0.0001 0.0021
2.0000 ‐0.0014 ‐0.0012 ‐0.0019 0 ‐0.0023 0.0002
3.0000 ‐0.0016 ‐0.0008 0.0033 0 0.0001 ‐0.0001
4.0000 0.0022 0.0019 0.0039 0 0.0004 0.0001
5.0000 ‐0.0006 0.0002 0.0058 0 0 0.0023
6 0000 0 0062 0 0058 0 0022
6.0000 0.0062 0.0058 0.0022 0 0.0011 0.0019
0 0 0011 0 0019
7.0000 ‐0.0026 ‐0.0025 ‐0.0006 0 0.0002 ‐0.0011
8.0000 ‐0.0024 ‐0.0034 0.0062 0 ‐0.0001 0
9 0000 0 0019 0 0025 0 0027
9.0000 ‐0.0019 ‐0.0025 0.0027 0 0.0035 ‐0.0002
0 0 0035 0 0002
10.0000 0.0033 0.0027 ‐0.0010 0 ‐0.0001 0.0001
0 0 0 0 0 0 0
1.0000 0.0026 0.0044 0.0002 0 ‐0.0001 0.0034
2.0000 ‐0.0014 0.0018 0.0007 0 ‐0.0025 0.0015
3.0000 ‐0.0016 0.0022 0.0059 0 ‐0.0001 0.0012
4.0000 0.0022 0.0049 0.0065 0 0.0002 0.0014
Initialize
5.0000 ‐0.0006 0.0032 0.0084 0 ‐0.0002 0.0036
6.0000 0.0062 0.0088 0.0048 0 0.0009 0.0032
7.0000 ‐0.0026 0.0005 0.0020 0 0 0.0002
8.0000 ‐0.0024 ‐0.0004 0.0088 0 ‐0.0003 0.0013
9.0000 ‐0.0019 0.0005 0.0053 0 0.0033 0.0011
10.0000 0.0033 0.0057 0.0016 0 ‐0.0003 0.0014
124
0 0 0 0 0 0 0
1.0000 0.0026 0.0044 0.0002 0 ‐0.0001 0.0034
2.0000 ‐0.0014 0.0018 0.0007 0 ‐0.0025 0.0015
3.0000 ‐0.0016 0.0022 0.0059 0 ‐0.0001 0.0012
4 0000 0 0022 0 0049 0 0065
4.0000 0.0022 0.0049 0.0065 0 0.0002 0.0014
0 0 0002 0 0014
5.0000 ‐0.0006 0.0032 0.0084 0 ‐0.0002 0.0036
6.0000 0.0062 0.0088 0.0048 0 0.0009 0.0032
7 0000 0 0026 0 0005 0 0020
7.0000 ‐0.0026 0.0005 0.0020 0 0 0.0002
0
0 0 0002
8.0000 ‐0.0024 ‐0.0004 0.0088 0 ‐0.0003 0.0013
9.0000 ‐0.0019 0.0005 0.0053 0 0.0033 0.0011
x 10
0
10 0000 0 0033 0 0057 0 0016
10.0000 0.0033 0.0057 0.0016 0 ‐0.0003 0.0014
0 0 0003
0 0014
0 0 0 0 0 0 0
1.0000 0.0026 0.0044 0.0002 0 ‐0.0001 0.0034
2.0000 ‐0.0014 0.0018 0.0007 0 ‐0.0025 0.0015
3.0000 ‐0.0016 0.0022 0.0059 0 ‐0.0001 0.0012
4.0000 0.0022 0.0049 0.0065 0 0.0002 0.0014
5.0000 ‐0.0006 0.0032 0.0084 0 ‐0.0002 0.0036
6.0000 0.0062 0.0088 0.0048 0 0.0009 0.0032
7.0000 ‐0.0026 0.0005 0.0020 0 0 0.0002
8.0000 ‐0.0024 ‐0.0004 0.0088 0 ‐0.0003 0.0013
9.0000 ‐0.0019 0.0005 0.0053 0 0.0033 0.0011
10.0000 0.0033 0.0057 0.0016 0 ‐0.0003 0.0014
-3
Nominal Coordinates
Actual Coordinates
Compensated Coordinates
8
-3
x 10
0
Distance along Y-a
axis (mm)
-2
XFYZ
-3
FXYZ
-4
-5
-6
Nominal Coordinates
Actual Coordinates
Distance along Y-axis (mm)
7
-1
6
5
4
3
2
1
-7
0
-8
125
0
1
2
3
4
5
6
Distance along X-axis (mm)
7
8
9
10
Nominal Coordinates
Actual Coordinates
5
4.8
8
Distance
e along Z-axis (m
mm)
4.6
4.4
4.2
Nominal Coordinates
4
Actual Coordinates
3.8
6
4
5
2
3.6
3.4
0
3
3.2
3
3
3.2
3.4
3.6
3.8
4
4.2
4.4
4.6
4.8
5
4
3.5
4
4.5
5
3
127
0
1
2
3
4
5
6
7
8
9
10
126