This paper was contributed to the Journal of the Japan Society for Precision Engineering, Vol.59, No.2, 1993 (in Japanese).
Spatial Coordinate Measurement Using Structured Laser Light
and a Spherical Mirror
Takaaki OIWA* , Minoru SUZUKI** and Akira KYUSOJIN***
*Faculty of Engineering, Shizuoka University, Hamamatsu, 432 Japan
**NTN, Osaka, 550 Japan
***Faculty of Engineering, Nagaoka University of Technology, Nagaoka, 940-21 Japan
A coordinate measurement method using a spherical mirror as a target is developed. A slit-ray laser scanner and a photo-detector are employed to detect the direction of the reflection from the target mirror. This method is suitable for robot performance measurements at which the attitude of the target changes, because only curvature center of the mirror can be detected. Moreover, the use of the spherical mirror allows precise calibration and easy setting. To demonstrate this measurement method, coordinate values in one-dimension were measured at various distances. In this paper, the results from this one-dimensional test are presented. The proposed measurement method was shown to be feasible through this experiment. The small beam size and the small divergent angle of the light source were indispensable to high accuracy, while the detector size was found to have little effect on the accuracy. The standard deviation in the measured value was evaluated to be 1 arc-second at a distance of 750 mm.
Key words: coordinate measurement, spherical mirror, laser scanning, structured projection
1. Introduction
Noncontact coordinate measurement methods for a spatially moving object have been strongly required in various fields of industries. In particular, because there is tendency to automate manufacturing systems with industrial robots in recent years, various methods have been presented to measure the robot arm performance1-8. A target or a sensor, e.g. a photo-detector, a cat's eye and a cube corner retroreflector, attached to the robot arm, are scanned or tracked generally by the laser light in these methods.3-8. These taget and sensor must be non-directional, because the attitude of the target will change frequently in actual robot. Nevertheless, the incidence angles on above targets are basically limited to about less than 90 degree. Moreover, the sensitivity of the sensor e.g. a photodiode is dependent on the incidence direction or the incidence angle.
The method presented in this paper is based on optical triangulation and uses a spherical mirror to remove the effect of changes in the attitude. Since this method detects the center of curvature of the mirror only, it is suitable for the robot metrology. The spherical mirror, moreover, can be used in the incidence angle much larger than 90 degree. In addition, the high accuracy of the spherical mirror allows precise calibration and easy setting. Because the out-of-roundness and the center position of the spherical surface can be measured easily. In this paper, to demonstrate this measurement method, coordinate values in one-dimension were measured.
2. Principle
2.1 Reflection of a spherical mirror
When rays are incident on the spherical mirror as shown in Fig.1, they reflect in various directions depending on incidence angles at the mirror surface. However, provided an incident light axis coincides with the curvature center of the spherical mirror, the incidence angle becomes 0 owing to optical properties of the spherical surface. Thus, the ray is reflected toward the light source. In other words, when a reflected light is detected on the incident axis, the curvature center of the spherical mirror will be on the extended line of the incident axis. Therefore, the position of the mirror center can be found by detecting the reflected light from the spherical mirror independently of the target attitude, when the spherical mirror is scanned or tracked by the ray throughout workzone.
A spot laser light necessitates a number of scans to catch the target mirror. The laser tracking system, moreover, often needs complicated and expensive instruments. The use of a structured line laser light, however, reduces the number of scans per one dimension to one. Hence, the measurement intervals can be shortened8. In this paper, the structured line light is produced by a cylindrical lens as shown in Fig.2. Sensitive direction or scanning direction is parallel to the lens axis which is perpendicular to the laser projection plane.
For three-dimensional measurement, the target mirror should be scanned by three structured lights projected from three rotary-type scanners as shown in Fig 3. The position of the target center can be computed by triangulation from the scanner information8. This 3-D system is now under construction and will be reported in a later paper. In this paper, we performed the one-dimensional measurement in the parallel scan and the rotational scan.
2.2 Reflected light in parallel scanning
Fig.4 shows a laser beam being incident to the horizontal direction and its reflection from a spherical mirror having a curvature radius r. Now, the spherical target mirror is moving perpendicularly. The laser beam having a width b and the photo-detector having a size d are set to be coaxial. The distance L between the mirror and the laser is extremely larger than the radius r. Because the laser beam has a half divergent angle q, typically between 0.2- 1 mrad, the target travel B within which the detector is certainly receiving the reflection exists as shown in Fig.4. In other words, only reflection from the target being within the travel B strikes the detector. In general, the position detecting accuracy depends on the travel B. Thus, the travel B must be small for accurate measurements. Noting the angles b and q are very small, the expressions for B and the angle b shown in the figure are
Substituting equation (2) into equation (1) and rearranging with and r<
Eq(3) suggests that a large divergent angle q and a width b of the laser beam deteriorate the measuring accuracy under a long distance. Moreover, a photo-detector having a large receiving screen can be employed, because the detector size d has little effect on B due to the large denominator in the third term in equation (3).
2.3 Reflected light in rotational scanning
Secondly, we discuss the time width of the detector output signal when the target is scanned with a rotational plane mirror. Since the signal width has an influence on the angle detection accuracy, it should be small like the travel B about the parallel scan in section 2.2. As shown in Fig.5(a), a very small rotation angle of the scanning mirror, a, causes a displacement of the reflected light center, s, along the photo-receiving plane as shown in Fig.5. Relations between s, a and an angle b in the figure are
where L is the distance between the target mirror and the receiving plane.
Eliminating the angle b from equations (4) and (5), and noting that 2r<
The peripheral velocity of the reflected light at the photo-receiving plane, v, is as follows.
where w is the angular velocity of the scanning mirror (rad/s).
When the laser beam has a half divergent angle q, a reflected light width at the receiving plane, B, is shown in Fig. 5(b).
It is expected easily that the width B will be remarkably larger than the laser size b or the detector size d. The time t during which the reflected light passes over a photo-detector having a size d is obtained by considering rb=b/2+Lq, rq<
Eq(9) shows that the increment of the distance L decreases an influence of the distance L on the time t or the signal width. The time t gradually will finally approach the fixed value, q/w. Moreover, the radius of the spherical mirror has little effect on the time t. Actually, the third term in Eq(9) is approximately zero. Because the signal width t influences the angle detecting accuracy da, considering Eq(9) we can expect that the relationship between da and the distance L is
However, the position detecting accuracy Lda becomes
Consequently, the accuracy is estimated to be the same as Eq(3) in section 2.2. To inhibit the drop in the accuracy needs a small beam size and a small divergent angle of laser for scanning the spherical mirror. The detector size has little effect on the accuracy, so that we can use a large detector or a condenser lens to improve the sensitivity of the detector.
3. Experiments
3.1 Parallel scanning
The fundamental accuracy of this position detection method has to be evaluated independently from the angle measuring accuracy of the rotary-type scanner. Accordingly, we observed how the detector signal changes when the spherical mirror is moving in the sensitive direction without the rotary-type scanner. A schematic of the experimental setup is shown in Fig.6. The cylindrical lens produces the structured line projection from the laser light. The reflection light from the spherical mirror is transmitted to the detector through the beam splitter. A He-Ne laser was used as the light source. The photo-detector was composed of a photodiode and a condenser lens having a diameter of 15mm. Spherical mirrors having curvature radii of 80 and 30 mm were primarily used. The probing head of a commercial coordinate-measuring-machine was used as the moving mechanism for the target mirror. It also simultaneously measured the target position. The moving speed was adjusted to about 50 mm/s.
Fig.7 shows the variations of the detector signal with the target travel. Since an irradiance distribution of the laser was Gaussian, the signal profile was similar to a Gaussian distribution. To evaluate this signal width, we calculated the travel width within which the signal voltage exceeds its maximum value multiplied by 1/e2. The results is shown in Fig.8. The straight line in the figure refers a regression based on b+2Lq in Eq(3). This figure shows that the beam size of the laser is 0.73 mm and the half divergent angle q is 0.31 mrad. Moreover, the mirror radius r and the distance L have little effect on the signal width.
In this method, a technique for detecting the peak-timing becomes the important problem. Because the information of the scan angle at which the detector signal is maximum decides the target coordinate value. In general, the peak timing can be detected by the differentiator and the zero-cross detector as shown in Fig.9(a). However, we found that a noise adjacent to the signal peak decreases the detection accuracy in this technique. Because an inclination of the signal near the peak is small, any noise strongly affects the differentiated signal. Thus, in this paper, the technique shown in Fig.9(b) was introduced. A pulse signal was generated from the detector signal using a comparator. The threshold level of the comparator was determined in steep slope of the signal. This determination of the level which is approximately half of the peak level, was not a great concern because the level was relatively insensitive to the peak-timing. The mean of the two edges of the pulse signal refers the peak timing. Incidentally, an adjustment of the amplifier gain becomes simple because the edges can be detected also from a signal saturated near the peak.
The peak-timing obtained by the above technique decides the coordinate value. Fig.10 shows the influence of the distance to the target on the variation of the measured value. The position detection accuracy was evaluated to be better than 10 mm. Moreover, no increase in the variation with the distance was observed in our experiment.
3.2 Rotational scanning
Fig.11 shows a diagram of the one-dimensional coordinate measuring system using a rotational plane mirror. We added a rotational scanner to the system shown in Fig.6. The rotary encoder having a resolution of 18 arc-second and the interpolating circuit measured the angle of the scanning mirror with a resolution of 0.2 arc-second. The rotational speed of the scanning mirror is controlled to 60 rpm by a PLL circuit with the encoder pulses.
Fig.12 shows measurement results when the target travels 100 mm with steps of 10 mm. The target was measured ten times at each position. The vertical axis represents the measured value as an angle of the scanning mirror. The curvature radius of the target mirror was 80 mm. When the distance to the target is 600 mm, a travel of 100 mm corresponds to one pulse of the rotary encoder. The measured value, therefore, contains no instrumental error of the encoder. Fig.12 demonstrates that the resolution measured using the interpolation was less than that of one pulse. However, the variation in measured values of 2 arc-second existed.
The Fig.13 shows the deviation of the measured value throughout a long travel of the target. Since the encoder used here has a instrumental error of approximately 20 arc second, the variation in the measured value has an amplitude of approximately 40 mm. This error, being a systematic error, can be eliminated by using a higher precision encoder and compensating of the encoder. Variation in the measured values was 10 to 20 mm, viz. 2 to 4 arc-second.
Fig.14 shows results using a target mirror having a radius of curvature of 30 mm. The specifications about the surface accuracy and coatings of the mirror were identical to the 80 mm mirror. However, the variation in the measured values increased to 15-30mm. The amplifier gain had to be made 10 times higher approximately, because the intensity of the reflected light decreased considerably when using 30 mm mirror. Thus, frequency response and S-N ratio of the detector amplifier, which became worse, increased the variation in the measured value.
The influence of the distance to the target on the variation in the measured values is shown in Fig.15. The vertical axis in the figure represents the standard deviation of the measured values. The target was measured ten times at each position, and their standard deviations calculated were plotted 10 or 15 times. The targets having curvature radii of 30mm and 80mm were able to measure throughout distances of 550mm and 750mm, respectively. The standard deviation of the measured value using 80mm mirror is less than 1 arc-second. Moreover, no increment of the angle measurement error with the distance was observed in our experiments. This conforms well to the da shown in the equation (10) in chapter 2.3. Consequently, these results suggest that improvements in the laser power and the detector sensitivity will enable us to expand the measuring zone while maintaining the accuracy.
4. Discussion
In general, the detecting-accuracy is dependent on the intensity of the reflected light, because an increment of the gain of the detector amplifier make the frequency response and S-N ratio worse. The intensity of the reflected light, furthermore, depends on the laser power, the curvature of the mirror and the distance to the target. To improve the accuracy, therefore, an increase of the laser power, condensation of the reflected light and higher sensitive detector will be required. However, there is a limitation in detection accuracy basically because the deflection of the laser light through the air has a strong effect on the accuracy.
To increase the sampling rate, it will be necessary to employ polygonal mirrors instead of the plane mirror and to increase the rotation speed of the mirror. In three-dimensional measurement, because the reflection light undesirably strikes other detectors when using three scanners, each phases of the scanner angles must be different.
5. Conclusions
In spatial coordinate measurement using the structured laser light and the spherical mirror as a target, the following conclusions were obtained from one-dimensional measurements.
(1) The curvature center position of the spherical mirror can be found independently of the target attitude by detecting the reflection light.
(2) A small beam size, a small divergent angle and a high power are required of the light source to keep the accuracy high. The detector size was found to have little effect on the accuracy.
(3) In parallel scanning, the variation in the measured values was less than 10mm regardless of the distance to the target.
(4) In rotational scanning, the standard deviation of the measured values was less than 1 arc second, and was independent of the distance.
We are sure that improvements of the laser power, the detection sensitivity and the scannnig speed will expand the workzone and will achieve an accurate and dynamic measurement. Furthermore, installation of two scanners to the 1-D measurement system proven in this paper will introduces the 3-D measurement.
References
1 H.Van Brussel, Evaluating and Testing of Robots, Annals of the CIIR, 1990,39(2), 657-662.
2 K.Lau and R.J.Hocken, A Survey of Current Robot Metrology Methods, Annals of the CIIR. 1984, 33(2), 485-488.
3 J.H.Gilby and G.A.Parker, Laser tracking system to measure robot arm performance, Sensor Review, 1982, 10, 180-184.
4 K.Lau, R.J.Hocken and W.C.Haight, Automatic laser tracking interferometer system for robot metrology, Precision Engineering, 1986, 8(1), 3-8.
5 K.Lau, R.Hocken and L.Haynes, Robot performance measurements using automatic laser tracking techniques, Robotics & Computer-Integrated Manufacturing, 1985, 2(3/4), 227-236.
6 Tilo Pfeifer und Albrecht Hof, Selbstkalibrierendes rŠumliches Wegme§system zur Koordinatenbestimmung im Raum, VDI-Z, 1985, 127(12), 441-444.
7 Nakamura, O., Goto, M., Tanimura, Y. and Kurosawa, T.Coordinate measuring system by laser tracking, J Japan Soc Precision Engng, 1991, 57(5), 831-836. (in Japanese)
8 Kanamori, Y et al. Proceedings of '90 Japan Soc Precision Engng conference, 1990, 269-270. (in Japanese)