1.

1)

2) 5)

6) 8)

9)10)

11) 13)

1

2 a 2

2 3

b 2 2

3

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1 a b 1

c

1

2

14)

1

1

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2

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2 kC2

1 a 3

1 2 3 (0, 0)

(bx2, 0) (bx3, by3)

2 *

** *** Chanin Sinlapeecheewa*** † ‡

Artifact Calibration of Coordinate Measuring Machine (2nd Report)

– Self Calibration of Redundant Coordinate Measuring Machine –

Kiyoshi TAKAMASU, Osamu SATO, Chanin SINLAPEECHEEWA, Ken SHIMOJIMA and Ryoshu FURUTANI

Traceable calibration methods for 3D mechanisms are necessary to use the mechanisms as coordinate measuring

machines. The calibration method of coordinate measuring machine using artifacts, artifact calibration method, is proposed

in taking account of traceability of the mechanism. Coordinate measuring machine which has multiple forward kinematics

for one measuring point call as “redundant coordinate measuring machine”. In this article, the self calibration method for

the two dimensional laser tracker system as the example of the redundant coordinate measuring machine is formulated

base on least squares method. Firstly, the calculation system which takes out the values of kinematic parameters using least

squares method is formulated. Secondly, the estimation value of uncertainty of measuring machine is calculated using

error propagation method. The self calibration method for the redundant coordinate measuring machine is applicable to

various types of 3D redundant mechanisms.

Key words: geometric calibration, kinematic calibration, coordinate measuring machine, laser tracker, redandunt

coordinate measuring machine, self caalibration

* 15 9 18

**

7-3-1

***

†

‡ 2-2

精密工学会誌 Vol.70,No.5,2004 711

11 l2 l3 d1 d2 d3

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(a) 3 laser trackers measure coordinate of cat’s eye

Y

X

camera 1

(bx2, by2, u2)

t1

camera 2

(bx3, by3, u3)

t2

t3

(x, y)

laser

(0, 0, 0)

profile

(b) Laser scanner and 2 cameras measure profile

Y

X manipulator (0, 0)

t1 + d1

l1one point(x, y)

l2 l3

t2 + d2

t3 + d3

(c) 3DOF articulating measuring machine measure one point artifact

Fig. 1 Examples of coordinate measuring with redundancy in two

dimensional space

座標測定機のアーティファクト校正（第2 報）

712 精密工学会誌 Vol.70,No.5,2004

617)18)

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座標測定機のアーティファクト校正（第2 報）

精密工学会誌 Vol.70,No.5,2004 713

qmmq

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9

(a) Total error (b) from parameters (c) from laser trackers

Fig. 2 Error ellipse (size multiplied 2000) of position errors by total errors, from parameters and from laser trackers: no. of calibration points = 722

Table 1 Mean and max of standard deviations of parameters by no. of

calibration points 41 to 1058

pitch of points 10 mm 5 mm 2.5 mm 2 mm

no. of point 41 181 722 1058

mean ( m) 9.1 4.0 2.0 1.7

max ( m) 12.5 5.5 2.7 2.3

Table 2 Standard deviations ( m) and coefficient of correlations of 6

parameters; calibration points = 722

bx2 bx3 by3 d1 d2 d3

bx2

bx3

by3

d1

d2

d3

Fig. 3 Position errors after calibration: no. of calibration points = 722

10 m

m

座標測定機のアーティファクト校正（第2 報）

714 精密工学会誌 Vol.70,No.5,2004

6.

1

2

3

1)

1

69 6 2003 851

2)

5 4 1987 306

3)

15 2 1997 164

4) S. Hayati, K. Tso and G. Roston: Robot geometry calibration, Proc. of

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66 4 2000 578

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11) Guide to the expression of uncertainty in measurement, ISO, (1992).

12)

63 11 1997 1594

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3 1

3

69 6 2003 841

14) D.J. Bennett and J.M. Hollerbach: Autonomous Calibration of

Single-Loop Closed Kinematic Chains Formed by Manipulators with

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15) M.C. Lee and P.M. Ferreira: Auto-Triangulation and Auto-

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17) (1987)

18)

(1982)

(a) with variance and covariance (b) select small error from 3 points (c) simple average

Fig. 4 Comparisons of 3 averaging methods: no. of calibration points = 181

m

座標測定機のアーティファクト校正（第2 報）

精密工学会誌 Vol.70,No.5,2004 715