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Simple three-dimensional laser angle sensor for three-dimensional small-angle measurement Chien-Hung Liu, Wen-Yuh Jywe, and S. C. Tzeng

A simple three-dimensional 共3D兲 laser angle sensor for 3D measurement of small angles based on the diffraction theorem and on ray optics analysis is presented. The possibility of using position-sensitive detectors and a reflective diffraction grating to develop a 3D angle sensor was investigated and a prototype 3D laser angle sensor was designed and built. The system is composed of a laser diode, two position-sensitive detectors, and a reflective diffraction grating. The diffraction grating, mounted upon the rotational center of a 3D rotational stage, divides an incident laser beam into several diffracted rays, and two position-sensitive detectors are set up for detecting the positions of ⫾1st-order diffracted rays. According to the optical path relationship between the three angular motions and the output coordinates of the two position-sensitive detectors, the 3D angles can be obtained through kinematic analysis. The experimental results show the feasibility of the proposed 3D laser angular sensor. Use of this system as an instrument for high-resolution measurement of small-angle rotation is proposed. © 2004 Optical Society of America OCIS codes: 120.0120, 120.4640, 120.4640, 050.0050, 050.1950.

1. Introduction

Several optical methods for measurement of angles of rotation have been developed in the past decades. They include those based on interferometry, speckle interferometry, and electronic speckle pattern interferometry,1–5 a moire and a holographic moire method,6 –9 internal reflection of optical elements, integration of optical elements and gratings, and triangulation.10 –15 Measurement of rotation angles is important for control systems in industry and now is commonly done in microsystems. However, the best-developed systems can measure only onedimensional angles of moving targets. Fang and Cao14 developed a two-dimensional 共2D兲 laser angle sensor that comprises a cube-corner mirror, a tetrahedral prism, a cube-corner prism, and an orthogonal holographic grating and uses a distribution-sensing technique. The 2D laser angle sensor can be used to measure the 2D angle of a moving target. Our main C.-H. Liu and S. C. Tzeng are with the Chien Kuo University of Technology, Changua 500, Taiwan, C. H. Liu with the Institute of Mechtronoptic Systems and S. C. Teng with the Department of Mechanical Engineering. W.-Y. Jywe 共[email protected]兲 is with the Department of Automation Engineering, National Huwei Institute of Technology, Huwei, Yunlin 632, Taiwan. Received 14 August 2003; revised manuscript received 9 February 2004; accepted 18 February 2004. 0003-6935兾04兾142840-06$15.00兾0 © 2004 Optical Society of America 2840

APPLIED OPTICS 兾 Vol. 43, No. 14 兾 10 May 2004

objective in the research reported in this paper has been to develop a simple system for measuring threedimensional 共3D兲 angles of a rotational object by using a diffraction grating. Traditionally, a reflective plane mirror can give only 2D angle signals. We have used a onedimensional reflective diffraction grating to replace the reflective plane mirror. With the diffraction grating the proposed method can measure 3D angles 共Euler angles兲 of a rotational object individually or simultaneously. Utilizing the diffractive theorem and ray optics analysis, we mount a diffraction grating upon the rotational center of the object to divide an incident laser beam into several diffracted rays, and two position-sensitive detectors are set up for detecting directional changes in the ⫾1st-order diffractive beams based on the rotation of the diffraction grating. Based on the outputs of the two positionsensitive detectors, a simple algebraic calculation is used to determine the angles of the rotating object. 2. Optical Setup of the Proposed Three-Dimensional Laser Angle Sensor System

Figure 1 shows the simple optical layout and experimental setup of the system. The system is composed of a laser source, two 2D position-sensitive detectors 共PSDs兲, and a diffraction grating. The laser source and the two 2D PSDs are integrated into one component, namely, a 3D laser angle sensor. The 3D laser angle sensor is fixed on a holder. The

Fig. 2. Spatial relationship of 3D diffraction rays on a grating and two PSDs.

3. Analysis of the Optical System

Fig. 1. Simple optical layout and experimental setup of the system.

diffraction grating is a target mounted on the rotating object. The Z axis of rotation is perpendicular to the diffraction grating. The laser source in the 3D laser angle sensor directs the laser beam into the diffraction grating and reflects it as several diffractive rays. In this research, ⫾1st-order diffracted rays, which have large intensity, are used. When the object rotates, the ⫾1st-order diffracted beams change direction according to the change in angle of the grating. Two PSDs in the 3D laser angle sensor are carefully set up to detect the locations where changes in the ⫾1st-order diffracted beams occur and also the positions of the spots. All three angular motions can affect the ⫾1st-order dif-

Rc

The measurement utilizes directional change in the ⫾1st-order diffraction rays and the two detecting positions of the spots on two PSDs to produce three angular motion errors. When a laser source emitted an incident ray onto a diffraction grating, it was diffracted in several directions. The directions of the diffracted rays are a function of the wavelength of the incident ray, the grating pitch, and the angle between the incident ray and the normal direction of the grating surface. Figure 2 shows the spatial relationship between the 3D diffracted rays on a grating and the two PSDs. Grating coordinate 兵Gc其 is fixed to the center of the diffraction grating. 兵Rc其 is a reference coordinate. The x axis is perpendicular to the direction of the grating’s pitch. Let unit vector RcI of an incident ray in the reference coordinate system 兵Rc其 be represented as Rc

冋册

Ix I ⫽ Iy . Iz

The laser diode directs the laser beam onto the diffraction grating 共the rotational center of the object兲. When the object rotates, the rotation matrix between the reference coordinate and the new grating coordinate that is due to the angular motion of the rigid body can be represented as

冋

册

cos ␣ cos ␤ cos ␣ sin ␤ sin ␥ ⫺ sin ␣ cos ␥ sin ␣ sin ␥ ⫹ sin ␣ sin ␤ cos ␥ RGc ⫽ sin ␣ cos ␤ cos ␣ cos ␥ ⫹ sin ␣ sin ␤ sin ␥ sin ␣ sin ␤ cos ␥ ⫺ cos ␣ sin ␥ , ⫺sin ␤ cos ␤ sin ␥ cos ␤ cos ␥

fracted beams’ directional change. Thus the relationship between the three angular motions of a rotating object and the two detecting positions of the spots on two PSDs can be obtained through kinematic analysis. Based on the diffraction theorem and on optical path analysis, a simple measurement system is implemented.

(1)

where RcRGc describes coordinate system 兵Gc其 relative to coordinate system 兵Rc其. Here three rotational parameters 共␣, ␤, ␥兲 are represented as the z, y, x Euler angles. Thus the change in direction of the incident ray related to the grating can be represented as Gc

I⬘ ⫽ GcRRcRcI ⫽ 关I x⬘, I y⬘, I z⬘兴 T,

10 May 2004 兾 Vol. 43, No. 14 兾 APPLIED OPTICS

(2) 2841

where ray:

Rc

I is represented as the incident unit vector

Rc

I⫽

冋册

0 0 . ⫺1

Pm

From the 3D grating equation,15 the diffraction vector in the 兵Gc其 coordinate system can be formulated as Gc

冋册

b mx B ⫽ by . b mz

Here b mx ⫽ I x⬘ ⫹ m共␭兾d兲,

m ⫽ 1, ⫺1,

b mz ⫽ 共1 ⫺ b mx2 ⫺ b y2兲 1兾2,

冤

冥

0 sin ␪ m a m 1 0 0 , 0 cos ␪ m c m 0 0 1

(3)

where ␪m is the angle displacement and am and cm are the translation displacements between the 兵Pm其 detector coordinate system and the 兵Rc其 reference coordinate system. The directional changes are affected by angular motions ␣, ␤, and ␥ of a diffraction grating. Let the rotational center of the rotational object on the diffraction grating be RcP. That is,

冤冥

We can express the coordinate value RcP and the vector of the mth-order diffracted ray in the 兵Pm其 detector coordinate system by using a coordinate transform matrix. Point PmP related to the 兵Pm其 coordinate system can be obtained from the following coordinate transformation:

冋册


b mpy P mz ⫽ f 共␣, ␤, ␥, a m, c m兲, b mpz

m ⫽ 1, ⫺1. A.

(8)

Method for Calculating Angular Motion ␣

In the proposed measurement system, each of the PSDs can have two coordinated values 共x and y coordinate values兲. For measurement of small angles ␣ the more-sensitive directions are the y directions of the two PSDs. Thus angular motion ␣ can be obtained from the following two equations: y ⫹1p ⫽ P ⫹1y ⫺

b ⫹1py P ⫹1z ⫽ f 共␣, 0, 0, a ⫹1, c ⫹1兲, b ⫹1pz

(9)

y ⫺1p ⫽ P ⫺1y ⫺

b ⫺1py P ⫺1z ⫽ f 共␣, 0, 0, a ⫺1, c ⫺1兲. b ⫺1pz

(10)

Averaging the values from Eqs. 共9兲 and 共10兲 can produce a better measurement result. That is, 共 y ⫹1p ⫹ y ⫺1p兲 1 ⫽ 关 f 共␣, 0, 0, a ⫹1, c ⫹1兲 2 2 ⫹ f 共␣, 0, 0, a ⫺1, c ⫺1兲兴. B.

(11)

Method for Calculating Angular Motion ␤

x ⫹1p ⫽ P ⫹1x ⫺

b ⫹1px P ⫹1z ⫽ f 共0, ␤, 0, a ⫹1, c ⫹1兲, b ⫹1pz

(12)

x ⫺1p ⫽ P ⫺1x ⫺

b ⫺1px P ⫺1z ⫽ f 共0, ␤, 0, a ⫺1, c ⫺1兲. b ⫺1pz

(13)

Also, averaging the values from Eqs. 共12兲 and 共13兲 can produce a better measurement result. That is, (5)

where PmTRc ⫽ RcTPm⫺1. The vector of the mth-order diffracted ray in the 兵Gc其 coordinate system can be 2842

y mp ⫽ P my ⫺

(7)

For measurement of small angles ␤, the more sensitive directions are the x directions of the two PSDs. Thus angular motion ␤ can be obtained from the following two equations:

0 0 Rc . P⫽ 0 1

P mx Pm P ⫽ PmTRcRcP ⫽ P my , P mz

b mpx P mz ⫽ f 共␣, ␤, ␥, a m, c m兲, b mpz

m ⫽ 1, ⫺1,

(4)

(6)

Combining the line equations of the mth-order diffracted ray after reflection and the plane equation of the PSDs yields the spot coordinate values of the PSDs in the 兵Pm其 coordinate system:

b y ⫽ I y⬘,

m ⫽ 1, ⫺1,

冋册

b mpx B ⫽ PmRGcGcB ⫽ b mpy . b mpz

x mp ⫽ P mx ⫺

where Ix⬘ and Iy⬘ are the components of the incident vector and bmx, by, and bmx are the components of the mth-order diffraction vector. ␭ is the wavelength of the incident ray, and d is the pitch of the diffraction grating. As is also shown in Fig. 2, 兵Pm其 is the mth detector coordinate attached to the PSDs that detects the mthorder diffraction ray. Each of the original points in the 兵Pm其 coordinate system attached to a PSD is represented as Pm. Let the coordinate transform matrix between the mth detector coordinate system and the 兵Rc其 reference coordinate system be cos ␪ m 0 Rc TPm ⫽ ⫺sin ␪ m 0

transformed into the 兵Pm其 coordinate system by means of the following equations:

共 x ⫹1p ⫹ x ⫺1p兲 1 ⫽ 关 f 共0, ␤, 0, a ⫹1, c ⫹1兲 2 2 ⫹ f 共0, ␤, 0, a ⫺1, c ⫺1兲兴.

(14)

Fig. 3. Setup for PSD calibration tests with two HP interferometers: A兾D, analog to digital.

C.

Method for Calculating Angular Motion ␥

For measurement of small angles ␥, the more sensitive directions are the y directions of two PSDs. Thus angular motion ␥ can be obtained from the following two equations: y ⫹1p ⫽ P ⫹1y ⫺ y ⫺1p ⫽ P ⫺1y ⫺

b ⫹1py P ⫹1z ⫽ f 共0, 0, ␥, a ⫹1, c ⫹1兲, b ⫹1pz

(15)

b ⫺1py P ⫺1z ⫽ f 共0, 0, ␥, a ⫺1, c ⫺1兲. b ⫺1pz

(16)

Fig. 5. Results of the repeatability tests of a PSD.

Again, averaging the values from Eqs. 共15兲 and 共16兲 can produce a better measurement result. That is, 共 y ⫹1p ⫹ y ⫺1p兲 1 ⫽ 关 f 共0, 0, ␥, a ⫹1, c ⫹1兲 2 2 ⫹ f 共0, 0, ␥, a ⫺1, c ⫺1兲兴.

(17)

4. Performance of the Three-Dimensional Laser Angle Sensor A.

Calibration Test of a Position-Sensitive Detector

Calibration tests for two positioning errors of a PSD were carried out with two HP 5529A interferometer measurement systems. The resolution of the HP interferometer system is 10 nm for displacement. The setup for the calibration tests of a PSD is shown in Fig. 3. Each of the two HP interferometer systems was aligned in the X–Y direction of the PSD. The calibration test results are shown in Fig. 4. The results show that the calibration curve of the X- and Y-axis positions of the PSD is good agreement with linearity within the range ⫾5 mm. The resolution of the PSD is 0.1 ␮m. Also, the results of repeatability analyses are shown in Fig. 5. The overall repeatability is less than 0.8 ␮m in the X-axis position and is less than 0.9 ␮m in the Y-axis position. B. Sensitivity of the Three-Dimensional Laser Angle Sensor

The resolution of an analog-type sensor is defined from the signal-to-noise ratio. However, in the 3D laser angle sensor the design parameters of the relative poTable 1. Resolution and Maximum Measuring Range of the 3D Laser Angle Sensor

Fig. 4. Calibration test results for a PSD: A兾D, analog to digital.

Rotation Angle

Resolution 共min兲

Maximum Measuring Range 共deg兲

␣ ␤ ␥

0.2 0.1 0.2

⫾2 ⫾1 ⫾1


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Fig. 6. Photograph of the prototype 3D laser angle sensor.

sitions of the optical elements and the calculation method determine the change in each detector’s output relative to the input angular displacement and also affect the resolution. The sensitivity of the 3D laser angle sensor can be analyzed from the resolution and the noise of the PSD and by differ-

Fig. 8. 共a兲 Results of measurement of the angle of rotation about the Y axis based on variation of the rotation angle for three runs, 共b兲 standard deviation based on variation of the rotation angle, 共c兲 mean measuring error based on variation of rotation angle.

entiating Eqs. 共9兲–共17兲. A reflective diffraction grating with 1200 grooves is used as the rotational target. The relative positions of the elements in the 3D laser angle sensor were calibrated and are as follows: a⫹1, 129.9 mm; c⫹1, 111.3 mm, a⫺1, ⫺ 129.9 mm; c⫺1, 111.3 mm. The maximum measuring range of a PSD is ⫾5 mm. The resolution and the maximum measuring range of the 3D laser angle sensor were calculated and are listed in Table 1. 5. Experimental Test

Fig. 7. 共a兲 Results of measurement of the angle of rotation about the X axis based on variation of the rotation angle for three runs, 共b兲 standard deviation based on variation of the rotation angle, 共c兲 mean measuring error based on the variation of the rotation angle. 2844


The simple 3D laser angle sensor was used to measure the rotational error of a 3D rotation stage. The prototype 3D laser angle sensor is shown in Fig. 6. The 3D laser angle sensor was carefully adjusted to direct the laser beam onto the diffraction grating 共the rotational center of the rotation stage兲 to make the diffracted rays focus onto the centers of two PSDs in the initial setup. The resolution of the rotational stage was 0.1°. The measurement errors and the repeatability of the rotational stage were evaluated; Figs. 7–9 show the measurements and the calculated results. As is shown in Fig. 7, the maximum positive angular error about the X axis is 0.05477°, the

motion of a rotational object. A laser diode, two position-sensitive detectors, and a diffraction grating were successfully integrated with an angular measurement system. The proposed sensor has a simple structure and high resolution. Also, one can further improve the resolution, accuracy, and measuring range by optimizing the design parameters of the 3D laser angle sensor and using PSDs with higher accuracy and larger working areas to enhance the performance of the 3D laser angle sensor. This system is proposed for use as an instrument for high-resolution measurement of 3D smallangle rotation. Also, it can be manufactured in a small sensor read-head module through miniaturization for applications in microsystems. This research was supported by the National Science Council of Taiwan 共grant NSC-91-2212-E-270003兲. References

Fig. 9. 共a兲 Results of measurement of the angle of rotation about the Z axis based on variation of the rotation angle for three runs, 共b兲 standard deviation based on variation of the rotation angle, 共c兲 mean measuring error based on variation of rotation angle.

minimum negative angular error is ⬃⫺0.02541°, and the overall repeatability of the test stage about the X-axis is less than 0.00057°. As is shown in Fig. 8, the maximum positive angular error about the Y axis is ⬃0.03799°, the minimum negative angular error is ⬃⫺0.06525°, and the overall repeatability of the test stage about the Y axis is less than 0.00091°. As is shown in Fig. 9, the maximum positive angular error about the Z axis is ⬃0.23215°, the minimum negative angular error is ⬃⫺0.11880°, and the repeatability of the test stage is less than 0.00212°. These measurement results show the feasibility of the proposed 3D laser angular sensor. One can further improve the resolution and the measuring range by optimizing the design parameters of the 3D laser angle sensor. Although the system is not fully optimized, our proposed system has already demonstrated some promising results. 6. Conclusions

In this paper a 3D laser angle sensor has been described that is able to measure the 3D angular

1. A. K. Nassim, L. Joannes, and A. Cornet, “In-plane rotation analysis by two-wavelength electronic speckle interferometry,” Appl. Opt. 38, 2467–2470 共1999兲. 2. R. Jabconaki, “Angle interference measurement,” presented at the First International Symposium on Metrology for Quality Control in Production, Tokyo, July 10 –13, 1984兲. 3. K. M. Abedin, M. Wahadoszamen, and A. F. M. Y. Haider, “Measurement of in-plane motions and rotations using a simple electronic speckle pattern interferometer,” Opt. Laser Technol. 34, 293–298 共2002兲. 4. R. Jones and C. Wykes, Holographic and Speckle Interferometry 共Cambridge U. Press, Cambridge, UK, 1989兲. 5. A. E. Ennos, “Speckle interferometry,” Prog. Opt. 16, 233–288 共1978兲. 6. A. E. Ennos, “Measurement of in-plane surface strain by hologram interferometry,” J. Sci. Instrum. 1, 713–717 共1968兲. 7. C. A. Sciammarella and J. A. Gilbert, “A holographic-moire´ technique to obtain separate patterns for components of displacement,” Exp. Mech. 16, 215–219 共1976兲. 8. L. Pirodda, “Conjugate wave holograph interferometry for the measurement of in-plane deformations,” Appl. Opt. 28, 1842– 1844 共1989兲. 9. Y. L. Lay and W. Y. Chen, “Rotation measurement using a circular moire´ grating,” Opt. Laser Technol. 30, 539 –544 共1998兲. 10. W. D. Zhou, “Interferometer for small-angle measurement based on total internal reflection,” Appl. Opt. 37, 5957–5963 共1998兲. 11. P. S. Huang, S. Kiyono, and O. Kamada, “Angle measurement based on the internal-reflection effect: a new method,” Appl. Opt. 31, 6047– 6045 共1992兲. 12. X. Dai, “High accuracy, wide range, rotation angle measurement by the use of two parallel interference patterns,” Appl. Opt. 36, 6190 – 6195 共1997兲. 13. P. S. Huang and Y. Li, “Small-angle measurement by use of a single prism,” Appl. Opt. 37, 6636 – 6642 共1998兲. 14. X. Y. Fang and M. S. Cao, “Theoretical analysis of 2D laser angle sensor and several design parameters,” Opt. Laser Technol. 34, 225–229 共2004兲. 15. E. W. Bae, J. A. Kim, and S. H. Kim, “Multi-degree-of-freedom displacement measurement system for milli-structures,” Meas. Sci. Technol. 12, 1495–1502 共2001兲.


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