Using RSI format - koasas

REVIEW OF SCIENTIFIC INSTRUMENTS

VOLUME 71, NUMBER 8

AUGUST 2000

Six-degree-of-freedom displacement measurement system using a diffraction grating Jong-Ahn Kim, Kyung-Chan Kim, Eui Won Bae, Soohyun Kim, and Yoon Keun Kwaka) Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, 3731, Kusong-dong, Yusung-gu, Taejon 305-701, Republic of Korea

共Received 25 January 2000; accepted for publication 12 May 2000兲 Six-degree-of-freedom displacement measurement systems are applicable in many fields: precision machine control, precision assembly, vibration analysis, and so on. This article presents a new six-degree-of-freedom displacement measurement system utilizing typical features of a diffraction grating. It is composed of a laser source, three position sensitive detectors, a diffraction grating target, and several optical components. Six-degree-of-freedom displacement is calculated from the coordinates of diffracted rays on the detectors. A forward and an inverse problem were solved to compute the full pose of an object through kinematic analysis. The experimental results show that the measurement system had a maximum error of ⫾10 ␮m for translation and ⫾0.012° for rotation. The repeatability is about 10 ␮m for translation and 0.01° for rotation. © 2000 American Institute of Physics. 关S0034-6748共00兲04708-0兴

ferometry,7 optical triangulation,8 amplitude modulation,9 and phase modulation.10 Since most amplitude and phase modulation methods were focused on robotics applications, they had limitations in resolution and were very sensitive to variations of surface reflectivity. In this article, we present a new method for measuring rigid body motion and derive the kinematic relationship between the coordinates of diffracted rays on detectors and six-DOF displacement of an object. Also, we investigate the validity of the proposed six-DOF sensing system and evaluate the performance through experiments.

I. INTRODUCTION

Fine manipulation systems generating six-degree-offreedom 共DOF兲 motion can increase efficiency of operations and execute complex tasks with dexterity. Such devices require precision six-DOF sensor systems to measure and control the required motion with high accuracy. Six-DOF sensor systems can also provide complete information in motion and vibration analyses. However, six-DOF sensing is a more complicated problem than one would guess by just imaging a sensor system comprised of multiple pieces each of which is sensitive to only one-dimensional motion. The reason for the additional complexity is that crosstalk between sensing channels is almost inevitable. Further, assembling a multidimensional position sensor from one-dimensional sensing components generally results in a large sensor system. Several methods are applicable to six-DOF displacement measurement, but optical sensing systems are superior to others in many aspects: noncontact measurement, resolution of light wavelength, immunity to electromagnetic interference 共EMI兲, and compactness.1 Many studies on six-DOF optical sensing can be classified into several categories. There were some results using a solid specular structure fixed on an object.2,3 Multiple light sources and reflective marks on the surface of an object were utilized to measure six-DOF displacement.4 However, their geometric accuracy directly affected the performance of the measurement systems and therefore they had to be manufactured very precisely. Six-DOF displacement of stage was measured using a complex optic system.5,6 In a small sensing range, displacement could be measured with high accuracy, but this approach had difficulty in the setup and optical alignment. The studies on multi-DOF optical sensing have been executed in various ways: multiple-beam inter-

II. DESIGN OF THE MEASUREMENT SYSTEM

A monochromatic incident ray is diffracted into discrete directions on a diffraction grating. The directions of the diffracted rays are determined by the wavelength of an incident ray, the pitch of a diffraction grating, and an incident angle. This relationship is well known as ‘‘the grating equation’’ when an incidence plane, which contains an incident ray and the normal of a grating, is perpendicular to the direction of the grating pitch. If the incident plane is oblique to this direction, the propagation of diffracted rays must be analyzed in three-dimensional space 共Fig. 1兲. In Fig. 1, A is the unit direction vector of the incident ray and B is one of the diffracted rays. The subscripts of B represent orders of the diffracted rays. Three threedimensional grating equation can be expressed11 as B mx ⫽A x ⫹m

共1兲

B y ⫽A y ,

where A x and A y are the x- and y-axis components of the unit direction vector of the incident ray and similarly B mx and B y are the x- and y-axis components of the m-order diffracted ray. ␭ is the wavelength of the incident ray and d is the pitch

a兲

Electronic mail: [email protected]

0034-6748/2000/71(8)/3214/6/$17.00

␭ , d

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© 2000 American Institute of Physics

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Rev. Sci. Instrum., Vol. 71, No. 8, August 2000

Displacement measurement using a gravity

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FIG. 1. Three-dimensional diffraction of a monochromatic incident ray on a grating. FIG. 3. Coordinates of the six-DOF displacement measurement system.

of the diffraction grating. The change of the grating pose varies A x and A y and also induces the variation of B mx and By . However, the directions of the diffracted rays do not change with translation of the grating, which is parallel to the x-y plane of the grating coordinate. A diverging incident ray and a small diffraction grating target on a nonreflective surface are used to measure this motion. The target has a circular shape and grooves of a conventional diffraction grating. If the target has the in-plane translation inside the diverging incident ray, the position where the diffracted rays originate also changes. The six-DOF displacement measurement system is composed of a laser source, a diffraction grating target, three two-dimensional position sensors, and several optical elements 共Fig. 2兲. It utilizes ⫹1, 0, and ⫺1-order diffracted rays, because they are generated over a wide range of the incident angle and the measurement system is able to have a symmetric structure. A convex lens adjusts the direction of the 0-order diffracted ray and increases the performance of the sensing system. Six-DOF displacement can be calculated from the coordinates of diffracted rays on the detectors. III. KINEMATIC ANALYSIS

The six-DOF motion of an object can be calculated kinematically. The calculation process is composed of two

steps: a forward and an inverse problem solving step. In the forward step, the coordinates of the diffracted rays on the detectors are estimated. In the inverse step, an actual sixDOF displacement is determined. A. Coordinate system

Figure 3 shows the coordinates of the six-DOF displacement measurement system. A grating coordinate 兵G其 is affixed to the diffraction grating target. The x axis of 兵G其 is perpendicular to the grating pitch direction, and the origin of 兵G其 coincides with the geometric center of the grating. 兵R其 stands for the reference coordinate. A source coordinate 兵S其 aligns with the direction of the laser source. 兵 D ⫹1 其 , 兵 D 0 其 , and 兵 D ⫺1 其 are detector coordinates attached to the position sensors which detect ⫹1, 0, and ⫺1-order diffracted rays, respectively. The origin of the detector coordinate is the center of each sensor and the x-y plane is each detecting surface. A lens coordinate 兵L其 is fixed on the surface of the convex lens. The center and the optical axis of the lens are assigned as the origin and the z axis of 兵L其. Three translational parameters (p x ,p y ,p z ) and three rotational parameters ( ␣ , ␤ , ␥ ), representing the Z-Y -X Euler angles, determine the coordinate transform matrix T. In Eq. 共2兲, c and s represent cos and sin for simplicity:

T⫽

冋

c␣c␤

c ␣ s ␤ s ␥ ⫺s ␣ c ␥

c ␣ s ␤ c ␥ ⫹s ␣ s ␥

px

s␣c␤

s ␣ s ␤ s ␥ ⫹c ␣ c ␥

s ␣ s ␤ c ␥ ⫺c ␣ s ␥

py

⫺s ␤

c␤s␥

c␤c␥

pz

0

0

0

1

册

. 共2兲

B. Forward problem

The incident ray can be considered to propagate from the origin of 兵S其 to that of 兵G其 and the unit incident vector ls is obtained as R

FIG. 2. A configuration of the six-DOF displacement measurement system.

ls⫽

Pgo⫺Pso , 兩 Pgo⫺Pso兩

共3兲

where Pgo and Pso are the origin of 兵G其 and 兵S其. the superscript of ls represents the coordinate where it is described. To


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obtained the unit direction vector of the m-order diffracted ray lmg, the unit incident vector ls in 兵G其 is substituted into Eq. 共1兲 and the results are given as follows: G

ls⫽ RG RR ls⫽ 关 l sx

冋册冋

l mgx G lmg⫽ l mgy ⫽ l mgz

l sy l sz 兴 , T

l sx ⫹m

␭ d

l sy

2 2 冑1⫺l mgx ⫺l mgy

册

共 m⫽⫹1, 0,⫺1 兲 ,

共4兲

describes 兵R其 relative to 兵G其. The where a rotation matrix ⫹1 and ⫺1-order diffracted rays directly progress to each detector, but the 0-order diffracted ray is refracted by a concave lens. Therefore, two cases are analyzed separately to compute the coordinates on the detectors. First, in the case of the ⫹1 and ⫺1-order diffracted rays, P Pgo and G lmg are expressed in each detector coordinate system 兵 D m 其 as G RR

D

Dm

Pgo⫽ R m TR Pgo⫽ 关 x mgo y mgo z mgo 1 兴 T ,

Dm

lmg⫽ G m RG lmg⫽ 关 l mgx l mgy l mgz 兴 T 共 m⫽⫹1,⫺1 兲 . 共5兲

TABLE I. The coordinate transform matrices between coordinates of the six-DOF displacement measurement system. Parameter

R ST

R D ⫹1 T

R LT

L D0T

R D ⫺1 T

p x 共mm兲 p y 共mm兲 p z 共mm兲 ␣ 共degrees兲 ␤ 共degrees兲 ␥ 共degrees兲

0.0 0.0 250.0 0.0 0.0 0.0

37.7 0.0 32.3 0.0 50.1 0.0

0.0 0.0 100.0 0.0 0.0 0.0

0.0 0.0 181.3 0.0 0.0 0.0

⫺37.7 0.0 32.3 0.0 ⫺50.1 0.0

冋册冋册冋册冋册 x lo x 0go ⫽M l lx l 0gx

x md ⫽x mgo ⫺

l mgx z , l mgz mgo

y md ⫽y mgo ⫺

l mgy z 共 m⫽⫹1,⫺1 兲 . l mgz mgo

共6兲

Next, in the case of the 0-order diffracted ray, additional procedures are needed to calculate the coordinate on the detector. Since the diffracted ray can be assumed to arrive at shallow angles with respect to the optical axis, the paraxial theorem is applicable and a lens transform matrix M is expressed12 as follows:

M⫽

冋

1⫺

t n

t 2n f

1 t ⫺ ⫹ f 4n f 2

t 1⫺ 2n f

册

共7兲

,

where t is the thickness, f is the focal length, and n is a refractive index of the convex lens. As a first step, R Pgo and G l0g are expressed in 兵L其 by a similar method as in Eq. 共5兲. The coordinate of the 0-order diffracted ray on the lens surface is determined from the line equation of the 0-order diffracted ray and the plane equation of the lens surface. L Plo and L ll are obtained using the lens transform matrix in Eq. 共8兲. L Plo is the coordinate on the lens surface and L ll is the unit direction vector of the 0-order diffracted ray after refraction.

冋册

x lo y lo L Plo⫽ t 1

L

ll ⫽

冋

册

l lx l ly , 冑1⫺l 2lx ⫺l 2ly

共8兲

where x 0go and y 0go are the x- and y-axis coordinates on the lens surface, l 0gx and l 0gy are the x- and y-axis components of the unit direction vector of the 0-order diffracted ray before refraction, and t is the thickness of the lens. D

D0

Plo⫽ L 0 TL Plo⫽ 关 x lo y lo z lo 1 兴 T ,

D0

ll⫽ L 0 RL ll⫽ 关 l lx l ly l lz 兴 T ,

D

D

From the line equations of the diffracted rays and the plane equations of the detecting surfaces, the coordinates on the detectors are determined as

y lo y 0go ⫽M , l ly l 0gy

x 0d ⫽x lo ⫺

l lx z , l lz lo

y 0d ⫽y lo ⫺

l ly z . l lz lo

共9兲

共10兲

In Eq. 共9兲, L Plo and L ll are expressed in 兵 D 0 其 and the coordinate on the detector is obtained from the line equation of the 0-order diffracted ray after refraction and the plane equation of the detecting surface as in Eq. 共10兲. C. Inverse problem R The parameters of the coordinate transform matrix G T can be calculated from the coordinates of the diffracted rays on the detectors to measure the six-DOF displacement. However, it is not easy to compute those values directly unless the equations are linearized in a small selected sensing range, R T are coupled with each other. In because the parameters in G this work, the six-DOF displacement is obtained through a numerical iterative algorithm, Newton’s method13 without linearization.

IV. EXPERIMENTS

Table I shows the coordinate transform matrices between coordinates of the measurement system. These should be determined carefully with an optimal design concept, because they determine the sensing range and performance of the measurement system. In an experimental setup, some were set to zero and others had definite values to obtain optimal performance under dimensional constraints. Figure 4 shows the experimental setup. A diffraction grating target was fixed on a six-DOF stage system. A He–Ne laser was used as a monochromatic light source and three position sensitive detectors 共PSDs兲 were employed as two-dimensional position sensors. A concave lens and a polarization beamsplitter projected an incident ray perpendicularly on the grating about 10 mm in diameter. A quarter-




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TABLE II. Specificatioins of devices in the experimental setup. Device Circular diffraction grating Laser source Concave lens Convex lens Six-DOF stage system External displacement sensor PSD

FIG. 4. Experimental setup for evaluation of the six-DOF displacement measurement system.

wave plate made the 0-order diffracted ray propagate to a concave lens, which refracted the 0-order diffracted ray. Output signals of the PSDs were converted into voltage signals proportional to the coordinates by signal processing circuits. The voltage signals were interfaced to a computer through an analog-to-digital convertor 共A/D兲 board and used in calculation of the six-DOF displacement 共Fig. 5兲. Table II summarizes the specifications of the devices composing the experimental setup. In this article, performance of the six-DOF displacement measurement system was evaluated from the point of view of measurement error and repeatability, because these are the principle indices in evaluation of a sensing system. First, displacement was generated with a constant step in each measurement direction using the six-DOF stage system and the measurement error, which was the difference between measured and actual displacement, was recorded at each position. In translation, the output value of the measurement system was taken at every 0.1 mm interval between ⫾1 mm. In rotation, 21 output values were measured between ⫾0.54° 共rotation about x axis: roll兲, ⫾0.39° 共rotation about y axis: pitch兲, and ⫾1.64° 共rotation about z axis: yaw兲. The experiment was performed six times with the same procedure.

FIG. 5. A composition of data processing system.

Specification Pitch: 1200 grooves/mm Diameter: 1 mm Output power: 2 mW Wavelength: 632.8 nm Focal length: 32 mm Focal length: 65 mm Translational resolution: 0.07 ␮m Rotational resolution: 2 arcsec Resolution: 0.01 ␮m Linearity: ⫾0.05% of F.S. Tetra-lateral type Sensitive are: 4.1 mm⫻4.1 mm

Since sensitivity was not equal in each measurement direction, measurement ranges were set differently so that the diffracted rays did not get out of the limited ranges on the detectors. The actual displacement was obtained from the index of the micrometer and calibrated with an external displacement sensor. Changes of the index could be considered as the actual displacement in translation, but they should be transformed to the actual angle using basic trigonometric in rotation. Next, an identical pose of the stage system was measured 125 times to evaluate the precision of the measurement system with the path taken to reach that pose which varies randomly. V. DISCUSSIONS

Figure 6 shows typical experimental results for the evaluation of measurement errors. The solid lines without symbols represent the actual displacement. Measurement errors occurred in the first experimental results. There existed crosstalk between the sensing channels, and the errors increased linearly as the measured displacement increased. From these facts, we knew that the parameter errors, the differences between the actual and nominal values of the kinematic parameters, caused the crosstalk and the measurement error. These could be eliminated, only if the actual values of the parameters were known. However, it was not easy to find the real values directly from the measurement errors. Therefore, the values of the parameters were calibrated to minimize the measurement errors using Marquardt-based algorithm. There was a little disagreement between the experimental results and the kinematic analysis, because the actual ray was not a line as assumed in the analysis and a laser source had a Gaussian intensity distribution.14 This could be compensated through an optical analysis. Figure 7 shows the measurement error in each direction against the variation of displacement. Data points are the averages of errors obtained at each point in the experiment repeated six times and thin lines denote the standard deviation. The measured crosstalk for x-axis translation and rollaxis rotation are presented in Fig. 8. The calibration of parameter errors reduced the magnitude of systematic errors and crosstalk remarkably. They did not increase with displacement, but fluctuated around zero.


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FIG. 6. Measured displacement against the variation of displacement. 共a兲 Translation in x axis. 共b兲 Rotation about roll axis.

FIG. 8. Measured crosstalk against the variation of displacement. 共a兲 Translation in x axis. 共b兲 Rotation about roll axis.

The standard deviations of crosstalk for the each axial displacement input are summarized in Table III. They had a smaller value than the measurement errors, so they had little influence on the performance of the measurement system. Some sensing channels showed relatively large crosstalk with each input channel: z and pitch-axis sensing channels 共x-axis translational input兲, roll and yaw-axis and yaw-axis sensing channels 共y-axis translational input兲, and so on. There were similar variations of the coordinates of ⫾1-order diffracted rays on the detectors, therefore they represented more considerable crosstalk than other sensing channels. This could be diminished by adjusting the kinematic parameters or employing additional optical elements. In addition, a calibration matrix mapping the actual and the measured displacement is effective in reducing the crosstalk. The measurement error in each sensing direction is shown in Fig. 9. Circles, thin lines, and bold lines represent the average, the range, and the standard deviation, respectively. The error ranges were about ⫾10 ␮m for translation TABLE III. Standard deviations of measured crosstalk for the each directional displacement.

X

FIG. 7. Measurement error against the variation of displacement. 共a兲 Translation in x, y, and z axis. 共b兲 Rotation about roll, pitch, and yaw axis.

X 共␮m兲 Y 共␮m兲 Z 共␮m兲 Roll 共arcsec兲 Pitch 共arcsec兲 Yaw 共arcsec兲

¯ 0.182 3.639 1.752 8.733 1.678

Direction of input displacement Y Z Roll Pitch 0.407 ¯ 0.918 9.773 1.690 9.615

1.760 2.676 ¯ 9.212 7.711 8.204

1.664 3.323 1.730 ¯ 2.073 7.559

0.781 0.208 3.079 1.195 ¯ 1.639


Yaw 0.540 0.870 2.225 3.947 5.678 ¯



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TABLE IV. Results of repeatability evaluation.

X 共mm兲 Y 共mm兲 Z 共mm兲 Roll 共degrees兲 Pitch 共degrees兲 Yaw 共degrees兲

FIG. 9. Measurement error in each sensing direction.

and ⫾0.012° for rotation, but the averages were approximately zero. This means that the calibration of parameter values enabled the measurement error to be distributed almost uniformly over the error range, because this could minimize the norm of error. As remarked previously, the sensitivity of each sensing direction was not equal and the yaw-axis direction was less sensitive than the roll and the pitch axis. Therefore, output signals of the PSDs changed less for the yaw-axis rotation when the same amount of displacement occurred. Conversely, the identical position detecting error of the PSDs caused a larger measurement error in the yaw direction. The repeatability of the measurement system was evaluated about 10 ␮m for translation and 0.01° for rotation referring to six times of the standard deviation 共Table IV兲. There was rather low repeatability in the yaw direction and this could be explained with the same reason as the case of measurement error. To improve accuracy of the measurement system, position detectors with higher position accuracy and more precise calibration of detectors are required. More thorough optical analysis will be necessary to enhance the performance of the sensing system in further works. In this article, optimal design and optical analysis of the six-DOF displacement measurement system were outlined briefly and details of

Mean value

6␴

⫺0.18043 0.12717 0.86232 0.32163 0.04159 ⫺0.27634

0.01050 0.00636 0.00996 0.00612 0.00474 0.01718

these subjects will be discussed in our subsequent articles. This measurement system can be manufactured in a small sensor head module through miniaturization and applied in various fields where optical precision six-DOF sensors are required. ACKNOWLEDGMENT

This work was supported in part by the Brain Korea 21 Project. D. C. Williams, Optical Methods in Engineering Metrology 共Chapman and Hall, London, 1993兲. 2 E. H. Bokelberg, H. S. Sommer III, and M. W. Trethewey, J. Sound Vib. 178, 643 共1994兲. 3 C. S. Vann, U.S. Patent No. 5,883,803 共1999兲. 4 I. J. Busch-Vishniac, A. B. Buckman, W. Wang, D. Qian, and V. Mancevski, U.S. Patent No. 5,367,373 共1994兲. 5 N. K. S. Lee, Y. Cai, and A. Joneja, Opt. Eng. 36, 2287 共1997兲. 6 K. C. Fan, M. J. Chen, and W. M. Huang, Int. J. Mach. Tools Manuf. 38, 155 共1998兲. 7 O. Nakamura and M. Goto, Appl. Opt. 33, 31 共1994兲. 8 H. Aoyama, K. Yamazaki, and M. Sawabe, Trans. ASME, Ser. B: J. Manuf. Sci. Eng. 118, 400 共1996兲. 9 A. Bonen, R. E. Saad, K. C. Smith, and B. Benhabib, IEEE Trans. Rob. Autom. 13, 377 共1997兲. 10 R. Masuda, J. Rob. Syst. 3, 137 共1986兲. 11 D. Post, B. Han, and P. Ifju, High Sensitivity Moire´: Experimental Analysis for Mechanics and Materials 共Springer, New York, 1994兲. 12 E. Hecht, Optics, 2nd ed. 共Addison-Wesley, New York, 1987兲. 13 N. C. Shammas, C/C⫹⫹ Mathematical Algorithms for Scientists and Engineers 共McGraw-Hill, New York, 1996兲. 14 A. Yariv, Introduction to Optical Electronics 共Holt, Rinehart and Winston, New York, 1971兲. 1
