Development of a Laser Triangulation Displacement Probe with Laser Beam Pointing Control

Abstract: Directional dithering of a laser beam potentially limits the detection accuracy of a laser triangulation displacement probe. A theoretical analysis indicates that the measurement accuracy will linearly decrease as the laser dithering angle increases. To suppress laser dithering, a laser triangulation displacement probe with laser beam pointing control, which consists of a collimated red laser, a laser beam pointing control setup, a receiver lens, and a charge-coupled device, is proposed in this paper. The laser beam pointing control setup is inserted into the source laser beam and the measured object and can separate the source laser beam into two symmetrical laser beams. Hence, at the angle at which the source laser beam dithers, the positional averages of the two laser spots are equal and opposite. Moreover, a laser dithering compensation algorithm is used to maintain a stable average of the positions of the two spots on the imaging side. Experimental results indicate that with laser beam pointing control, the standard variance of the fitting error decreases from 0.3531 to 0.0100, the repeatability accuracy can be decreased from ±7 mm to ±5 μm, and the nonlinear error can be reduced from ±6 %FS to ±0.16 %FS.


Introduction
Laser triangulation displacement probes (LTDPs) have been widely used for industrial detection because of their noncontact and high-precision properties.The principles of an LTDP are illustrated in Figure 1.A collimated laser beam projects a laser dot onto the measured object.Then, the diffused laser light is collected by a receiver lens, and a dot is imaged on a charge-coupled device (CCD).When the object dot moves in a direction perpendicular to the optical axis of the laser, a corresponding displacement will occur for the image dot on the CCD.
As shown in Figure 1, ε is defined as the observed angle between the source laser beam and the optical axis of the receiver lens, and β is defined as the image angle between the CCD and the optical axis.ε and β must satisfy the Scheimpflug condition [1].s is the distance over which the object moves, and UL is the corresponding image distance.If s is upward along the optical axis, the sign is "+"; otherwise, it is "-."The relationship between s and UL is The measurement accuracy of the LTDP is affected by the speckle, the color of the measured object, the surface texture, the ambient light, the variation in the laser beam intensity, the distortion of the receiver lens, and the dithering of the source laser beam [2][3][4][5].Shen et al. [6] introduced a digital correlation method for suppressing the speckle noise.The results showed that the measurement range reached 1 µm, and the experimental errors were reduced below 2 %.Oh et al. [7] improved the hardware structure by inserting a diffraction grating between the receiver lens and the CCD.The diffraction grating simultaneously generated −1and 0-order light intensity distributions on the CCD.This method can reduce the measurement time by averaging the results of the two orders.et al. [8] and Shen et al. [6] interpreted an adaptive control technique to maintain a stable beam intensity.Keyence [9] proposed a real peak detection algorithm that aimed to detect the true peak position value rather than the traditional centroid value to avoid the effect of the oversized diameter.Zbontar et al. [10] introduced a double curve fitting algorithm to compensate the skewed distribution.In addition, they [10] used an ultraviolet (UV) laser to improve the required signal quality.However, the UV laser will induce photochemical effects, which might lead to material degeneration; thus, this method is only used to detect certain materials such as high-end lenses or hot metals.
In this paper, an LTDP that uses a laser beam pointing control setup (LPC) is proposed to decrease the effect of directional dithering of a laser beam.This probe simultaneously generates two symmetrical laser intensity distributions.Since the averages of the two positions on the detected surface are constant, the influence of laser dithering can be avoided.Moreover, the speckle noise related to the measured surface roughness and stray light can be reduced because the two measurement results are averaged.
The remainder of the paper is arranged as follows.We analyze the effect of laser dithering on the measurement accuracy of an LTDP in Section 2. Section 3 describes the LTDP with the LPC and introduces the laser dithering compensation algorithm.Experiments for verifying the performance of the probe are presented in Section 4. Finally, the conclusions are summarized in Section 5.The relationship between the dithering angle and the imaging error is deduced as follows.As shown in Figure 1, a stable pointing laser source emits the light spot Q on an object.The diffused light is collected by a receiver lens, and the light spot G is imaged on the CCD.When the source laser beam is dithered by the angle α, the light spot R is projected onto the object.Then, the diffused light is collected, and the light point G is imaged on the CCD.The angle between the rays O l Q and O l R is defined as α, and the angle between the rays RO r and QO r is defined as ω.Moreover, the auxiliary line GV, which is perpendicular to the ray QG, and the auxiliary line QV , which is perpendicular to the ray QG, are added separately.

Effect of the Laser Beam Directivity on the Measurement Accuracy
V RQ = 90 1, by the law of sines,

QR sin(90
Since V QO r ∼ GVO r , QV is expressed as follows: where l is the length of QO r , and l is the length of O r G. Using Equation 2, In In O l RQ, QR = r • tan α; thus, ω is expressed as follows: where Combining Equation 4and Equation 5, GV is expressed as follows: where T = l/l .Considering the inclination angle β of the CCD, the relationship between the dithering angle α and the dithering error x over the distance GG on the image side can be written as follows: The relationship between x and α is shown in Figure 3 for l = 59 mm, l = 68.5 mm, r = 60 mm, ε = 0.349, β = 0.5916, and α ∈ (−1  As shown in Figure 3, when the source laser beam is dithered by an angle of −1 • to 1 • , the measurement accuracy will decrease linearly as the dithering angle increases.

Basic Layout
To avoid the effect of laser dithering, several effective methods have been proposed.In general, these methods utilize feedback control achieved by using the error or reference signal [11][12][13][14][15].However, these methods are inappropriate for an LTDP because the structures mentioned in these studies have a large volume and there is a dependence on the error or reference signal.Therefore, an LTDP based on laser beam pointing control is designed in this study, which consists of a collimated red laser, an LPC, a receiver lens, and a CCD, as shown in Figure 4.The LPC consists of a right-angle prism, a beam splitter, a pentaprism, a half-pentaprism, and two rhombic prisms.The angle between the beam splitter and the optic axis of the collimated red laser is 22.5 two laser beams by the beam splitter.One laser beam is reflected four times by the pentaprism and rhombic prism, and the laser beam P 1 is formed.Another laser beam is reflected four times by the half-pentaprism and another rhombic prism, and the laser beam P 2 is formed.The two rhombic prisms can shorten the distance traveled by the two laser beams, which are located at positions perpendicular to the optical axes of P 1 and P 2 .
With respect to the LPC, the closer point, zero point and farther point were set at distances of 60 mm, 65 mm and 70 mm with an error of 0.2 mm.
(i, j, k) is defined as the unit vector of the source laser beam P, and (i , j , k ) is the unit vector of P 1 .According to the prism turning theorem, the interaction matrix B 1 of P 1 is Furthermore, (m , n , t ) is the unit vector of P 2 , and the interaction matrix B 2 of P 2 is Comparing Equation 8and Equation 9, the basis vector j of B 1 and n of B 2 are opposite, which means that the directions of P 1 and P 2 are opposite.
To ensure that the positional variations in O 1 and O 2 are equal, the positional relationship among the right-angle prism, pentaprism, half-pentaprism, and rhombic prisms must satisfy ] 1/2 , (x 1 , 1 ) are the coordinates of point B, (x 2 , y 2 ) are the coordinates of point D, (x 3 , y 3 ) are the coordinates of point C, 2h is the thickness of the beam splitter, l 1 is the length of the short side at 45 • in the half-pentaprism, l 2 is the length of the side at 90 • in the pentaprism, x E is the x coordinate of point E, and x F is the x coordinate of point F. In summary, if the positional relationship of the prisms satisfy Equation 10, the average value of O 1 and O 2 remains invariant.

Laser Dithering Compensation Algorithm
As shown in Figure 5, a coordinate system (Oxy) is constructed, where O is the medial point of the receiver lens, and the x axis is coincident with the plane of the receiver lens.The linear function of the CCD is defined as y = k 1 x + b 1 , where k 1 = tan ϕ 1 .Point A is the position at which the pixel value on the CCD is zero.
The laser dithering compensation algorithm is described as follows.First, the peak value y P 1 and the corresponding pixel value x P 1 of the point C 1 as well as the peak value y P 2 and the corresponding pixel value x P 2 of the point C 2 are found.Then, the centroids C d where w is the data width, i is the pixel value before x P 1 , j is the pixel value after x P 1 , p is the pixel value before x P 2 , and q is the pixel value before x P 2 .
The x coordinates of C 1 and C 2 in the (Oxy) system are then expressed as follows: where x A is the x coordinate of point A, and t is the resolution of the CCD.Since the CCD is not parallel to the plane of the object but forms an angle ϕ 1 , the medial position of x C 1 and x C 2 is not appropriate for calibration.Here, we construct a virtual calibrated line y = k 2 x + b 2 , which is parallel to the plane of the object, as shown in Figure 5, where k 2 = tan ϕ 2 .
The x coordinate x 1 of the point H 1 and the x coordinate x 2 of the point H 2 are As a result, the medial position AVG of x 1 and x 2 is Since the positional variations in O 1 and O 2 are equal and opposite, the value of AVG remains constant.An LTDP with an LPC is shown in Figure 6, where the LPC is inserted between the collimated red laser and the detected object.The main devices used in this system are shown in Table 1.

Preprints
According to the design parameters of the LTDP, the linear function of the CCD is y = 0.6009x + 46.09, and the virtual calibrated line is y = 0.6745x + 46.09.

Verification of the Laser Dithering Compensation Algorithm
Figure 7 shows the values of x 1 and x 2 when the ceramic gauge block is located at one fixed position, and the source laser beam P is rotated within ±1.1 • with an increment of 0.2 • .As shown in Figure 7(a), the variations in x 1 and x 2 are equal and opposite.As shown in Figure 7(b), the extreme error of AVG is within ±4 µm.

Calibration
The LTDP with the LPC was calibrated with a RENISHOW XL-80 laser interferometer.The linear resolution of the interferometer is 1 nm.The calibration setup is shown in Figure 8. P 2 is chosen for the comparison of the results.The relative positions of the LTDP with the LPC were calibrated.The criteria ceramic gauge block was moved point-by-point along the optical axis of the source laser beam in an increment of 0.2 mm within 10 mm.At each point, the collimated red laser was rotated with an increment of 0.2 • within ±1.1 • .Here, AVG is used as the calibration criterion for the LTDP with the LPC, and the pixel value of P 2 is used for the calibration criterion of the LTDP without the LPC.The calibration test results are shown in Figure 9.      for three runs.The error between the LTDP and the XL-80 interferometer is shown in Figure 12.As the results show, the nonlinearity with the LPC is within ±0.16 %FS.In comparison, the nonlinearity without the LPC is within ±6 %FS.

Conclusions
Laser beam dithering is considered one of the major error sources for LTDP measurements.A theoretical analysis shows that the measurement error will increase linearly as the drift angle increases.An LTDP with laser beam pointing control has been developed to decrease the influence of laser beam dithering.This probe consists of a collimated red laser, an LPC, a receiver lens, and a CCD.The collimated red laser beam is split into two symmetrical laser beams by the LPC.Therefore, at the angle at which the laser is dithered, the positional average of two laser spots on the measured object remains constant.The experimental tests were effectively verified with a dual-beam laser interferometer within the measurement range of 10 mm.With laser beam pointing control, the repeatability of the measurement displacement is better than ±5 µm, and the nonlinearity is better than ±0.16 %FS.In comparison, without laser beam pointing control, the repeatability of the measurement displacement is ±7 mm, and the nonlinearity is ±6 %FS.

Figure 1 .
Figure 1.Structure of the optical path when the laser beam is dithered by an angle α.

Figure 2 (
Figure 2(a) and Figure 2(b) show two different occurrences of laser dithering produced by two different collimated lasers manufactured by Xi'an Minghui Optoelectronic Technology Co., Ltd.The experimental results show that the laser dithering angle is usually within (−1 • , 1 • ).The relationship between the dithering angle and the imaging error is deduced as follows.As shown in Figure1, a stable pointing laser source emits the light spot Q on an object.The diffused light is collected by a receiver lens, and the light spot G is imaged on the CCD.When the source laser beam is dithered by the angle α, the light spot R is projected onto the object.Then, the diffused light is collected, and the light point G is imaged on the CCD.The angle between the rays O l Q and O l R is defined as α, and the angle between the rays RO r and QO r is defined as ω.Moreover, the auxiliary line GV, which is perpendicular to the ray QG, and the auxiliary line QV , which is perpendicular to the ray QG, are added separately.V RQ = 90 • − (ω + QO r O t ) = 90 • − ω − ε; considering RV Q in Figure1, by the law of sines,

Figure 4 .
Figure 4. Optical structure of the laser triangulation displacement probe with laser beam pointing control.

2 Figure 5 .
Figure 5. Working principle of the laser dithering compensation algorithm.

Figure 6 .
Figure 6.Laser triangulation displacement probe with laser beam pointing control.

Figure 9 .
Figure 9. Calibration results (a) with and (b) without laser beam pointing control.

Figure 10 .
Figure 10.Fitting errors (a) with and (b) without laser beam pointing control.

Figure 12 .
Figure 12.Nonlinearity (a) with and (b) without laser beam pointing control.
1All lengths have units of millimeters.