# Geometrical Model of Laser Triangulation System Based on Synchronized Scanners

Authors: Dawei Tu; Pan Jin; Xu Zhang
Journal: Mathematical Problems in Engineering
 (2019)

---

## Abstract

In traditional synchronized scanning system, the trajectory of the 3D scanning point is modeled as a circle when the thickness of rotated mirror is assumed to be zero. In this paper, a novel method to model the geometric measurement of synchronized scanning triangulation in the unfolded light path is proposed. Unlike most existing recent works, the 3D coordinate of the target is reasoned from the geometrical model, which includes all 14 system parameters. Further, the performance of the system precision can be analyzed and the importance of the thickness of rotated mirror is confirmed. In the experiment, a synchronized scanning system is developed. The experimental result demonstrates that the value of measurement uncertainty at a distance of 0.5m is 0.75mm and at a distance of 5m is 6.68mm. The standard deviations from the measurement point to the fitting plane at a distance of 0.5 m and 5 m are 1.10 mm and 19.73 mm, respectively.


---

## Body

## 1. Introduction

Laser triangulation technology is a method of non-contact measurement, overcoming the limitation of traditional measurement technology, which has the qualities of fast scanning speed, strong real-time, high precision, and intelligence. It has been widely used in production manufacturing [1–3], biomedical [4, 5], and cultural relics [6, 7].The active source of laser triangulation technology [8, 9] is one of the most common methods for 3D measurement, which generally includes point laser [10] and line laser [11, 12]. There are two kinds of methods to obtain points cloud, such as probe movement and laser swing. The method of probe movement is to fix the laser triangulation probe [13–15] in moving parts and scan the object by the spatial motion of moving parts. The laser swing method realizes the two-dimensional scanning by laser movement [16, 17], vibrating mirror rotation [18], and other methods. However, when a large-scale scene is measured, the probe needs to fix in large moving parts; thus the precision of moving parts affects the final measurement precision; and if the laser oscillation method is used, the horizontal and vertical scanning areas are mutually constrained, which will greatly influence the scanning range.In order to solve the scanning constraint of horizontal and vertical of traditional triangulation, M. Rioux et al. [19, 20] propose a synchronized scanning measurement technique; the receiving light and the transmitting light are synchronous by installing a scanner to both optical paths. When the scanner rotates an angle, the receiving light and the transmitting light have the same angle variation. Because of this mechanism, the depth of field and the measuring range are greatly expanded. Following this technique, the literatures [21, 22] further studied the characteristics of the laser synchronized scanning triangulation system and theoretically derived the general meaning of the trajectory circle equation, and the system parameters of the maximum and minimum range, distance resolution, and other system parameters. C. Samson et al. [23] studied the accuracy of the synchronized scanning triangulation system and the error caused by the deviation of system parameters, such as mirror angle, are obtained. English Chad et al. [24] designed a laser triangulation system based on synchronized scanners with lidar, which makes full use of advantages of triangulation and ladar technologies. Triangulation can be highly precise at close range, while pulsed ladar technology is capable of very long range measurements. Haibo Zhang et al. [25] proposed a novel method using prism-based optical structure to correct the nonlinear problem of laser triangulation displacement measurement and the results showed the nonlinear problem was significantly improved. What is more, Xiong Shengjun et al. [26] designed a line structure-light three-dimensional shape measurement system based on laser triangulation with auto-synchronous scanners designed, which reduce the system weight and size. The above study establishes a trajectory circle model of three-dimensional scanning, which is of great value for describing the measuring mechanism of synchronized scanning triangulation.In this paper, we proposed a synchronized scanning trigonometric geometric parameter model based on unfolded light path. The position and pose of different parts are accurately described by 14 system parameters, and then the relationship between three-dimensional space points and system parameters is expressed. In the simulation, the system’s error could achieve 400mm, when the thickness of double-faced reflector is 2mm. This analysis results confirm the importance of the thickness of the rotated mirror. Then, the parameters are ranked in the aspect of influence on measurement precision, which is instructive for design the system. Finally the experiment results show that the system can scan the object and display its point cloud.


## 2. Model of Synchronized Scanning Triangulation

### 2.1. Theorem of Laser Triangulation System Based on Synchronized Scanners

As is shown in Figure1, the collimated laser beam emitted by the laser travels the upper reflector of double-faced reflector M3, reflector M1, and reflector M4, arriving at Q~ and diffusing reflection. Part of diffuse light goes through reflector M4, reflector M2, the lower reflector of double-faced reflector M3, lens and achieving the camera. According to the principle of triangulation, the coordinates of Q~ are determined by the position of converged imaging spot on the camera. M3 and M4 are driven by the motor to implement plane-scanning. When M3 oscillates around the axis, light spot Q~ scans the surface of the measured object in the X-axis; likewise, light spot Q~ scans in the Y-axis direction with M4 oscillates around the axis. The surface of measured object can be scanned when those two motors are synchronously driven.Figure 1
Principle of synchronized scanning triangulation.


### 2.2. Unfolded Optical Path of Synchronized Scanning Triangulation

The parameters of the synchronous scanning triangulation system is shown in Figure2. In the unfold light path of system (see Figure 3), the dashed line indicates initial position. And after the M3 mirror is rotated by a certain angle, light path is represented by the solid line. For any point Q~[x,y,z] in space, the corresponding point imaged on the CCD is point p, and the distance between point p relative and the pixel position of the CCD main point (pccd) is q. q is a vector, which is positive in the same direction of CCD rotation angle, and vice versa.Figure 2
System parameters.Figure 3
Unfolded light path.
(a)
X-Y plane (b)
Y-Z plane


#### 2.2.1. System Initial Position

As the angle between M3 and X-axis isθ3, the direction of incident laser is dirE′P′=2θ1-2θ3-(π/2) and the direction of optical axis is dirP′H′=2θ2-2θ3-(π/2). The coordinates of O′,O′′ are as follows:(1)O′=PM1-O′PM1=-d13-d13cos⁡2θ1-π-d13-cos⁡2θ1-π0TO′′=PM2+O′′PM1=d32+d32cos⁡2θ2-πd32-cos⁡2θ2-π0TAccording to formula (1), the positions of O′ and O′′ are independent of the rotation of M3. Then set the distance from origin to upper reflector of M3 is ThM3up. The coordinates of E′ and H′ can be expressed as follows:(2)E′=O′+O′E′=-d13-d13cos⁡2θ1-π+ThM3upcos⁡θ3cos⁡2θ1-π2-d13cos⁡2θ1-π+ThM3upcos⁡θ3sin⁡2θ1-π20(3)H′=O′+O′H′=d32+d32cos⁡2θ2-π+ThM3up-ThM3cos⁡θ3cos⁡2θ2-π2d32cos⁡2θ2-π+ThM3up-ThM3cos⁡θ3sin⁡2θ2-π20Due to the thickness of M3, a slight change in the optical path travel occurs. The stroke of lens center to pointH is as follows:(4)s′=cH=s+ThM3up-ThM3cos⁡θ3And the coordinate ofc′ corresponding to optical center point c is as follows:(5)c′=H′+H′c′=d32+d32cos⁡2θ2-π+ThM3up-ThM3cos⁡θ3cos⁡2θ2-π2+s′cos⁡2θ2-2θ3-π2d32cos⁡2θ2-π+ThM3up-ThM3cos⁡θ3sin⁡2θ2-π2+s′sin⁡2θ2-2θ3-π20


#### 2.2.2. CCD Direction and Position

As is shown in Figure4, the points on line DG are ideally imaged on CCD when Scheimpflug principle [27] is satisfied. D is intersection point of c′D and DG, and c′D direction is dirc′D=dirPH′-π/2=2θ2+π/2. The coordinate of point D can be expressed:(6)D=Ex′-Ey′-cy′cot⁡2θ1-π-cx′tan⁡2θ2+π/2cot⁡2θ1-π1-tan⁡2θ2+π/2cot⁡2θ1-πDx-cx′tan⁡2θ2+π2+cy′0Figure 4
Illustration of calculating the CCD’s angle with Scheimpflug principle.In the triangle△c′DG, we know ∠c′DG=2θ1-2θ2-π/2 and Dc′=(Dy-cy′)2+(Dx-cx′)2. According to the triangle geometry, the distance D and G is as follows:(7)DG=fsin⁡∠cDG=fsin⁡2θ1-2θ2-π/2According to sine theorem, the following is given:(8)DGsin⁡-β=Dc′sin⁡∠DGc′SubstitutingDG and Dc′ into (8), the angle β can be expressed as follows:(9)β=arccot⁡Dc′f-tan⁡2θ1-2θ2Since△Dpccdc′ is a right triangle and ∠Dc′pccd=π/2, the expression of f0 and Dc′ is as follows:(10)f0=Dc′tan⁡-β


#### 2.2.3. Imaging Light Direction

In△c′pccdp (see Figure 5), we know ∠c′pccdp=π/2-β, pccdc′=f0, pccdp=q. The expression of imaging light direction is as follows:(11)dirQp=dircp=2θ2-2θ3-π2+αα=arctan⁡q·cos⁡βf0-q·sin⁡βFigure 5
Relationship between the CCD imaging point and imaging of light.


#### 2.2.4. The Spatial Coordinates of Points

On the X-Y unfolding plane, the spatial pointQ~ corresponds to point Q. We know that point Q is both in the laser emission direction and on the imaging ray, which can be expressed as follows:(12)cot⁡2θ1-2θ3-π2=Qx-Ex′Qy-Ey′cot⁡2θ2-2θ3-π2+α=cx′-Qxcy′-QyAccording to expression (12), the coordinates of point Q on the X-Y unfolding plane are as follows:(13)Q=Qy-Ey′cot⁡2θ1-2θ3-π2+Ex′cy′cot⁡2θ2-2θ3-π/2+α-Ey′cot⁡2θ1-2θ3-π/2+Ex′-cx′cot⁡2θ2-2θ3-π/2+α-cot⁡2θ1-2θ3-π/20The coordinate ofQ~ can be obtained:(14)Q~=Qx~Qy~Qz~=QxQy-hcos⁡2θ4+hQy-hsin⁡2θ4


## 2.1. Theorem of Laser Triangulation System Based on Synchronized Scanners

As is shown in Figure1, the collimated laser beam emitted by the laser travels the upper reflector of double-faced reflector M3, reflector M1, and reflector M4, arriving at Q~ and diffusing reflection. Part of diffuse light goes through reflector M4, reflector M2, the lower reflector of double-faced reflector M3, lens and achieving the camera. According to the principle of triangulation, the coordinates of Q~ are determined by the position of converged imaging spot on the camera. M3 and M4 are driven by the motor to implement plane-scanning. When M3 oscillates around the axis, light spot Q~ scans the surface of the measured object in the X-axis; likewise, light spot Q~ scans in the Y-axis direction with M4 oscillates around the axis. The surface of measured object can be scanned when those two motors are synchronously driven.Figure 1
Principle of synchronized scanning triangulation.


## 2.2. Unfolded Optical Path of Synchronized Scanning Triangulation

The parameters of the synchronous scanning triangulation system is shown in Figure2. In the unfold light path of system (see Figure 3), the dashed line indicates initial position. And after the M3 mirror is rotated by a certain angle, light path is represented by the solid line. For any point Q~[x,y,z] in space, the corresponding point imaged on the CCD is point p, and the distance between point p relative and the pixel position of the CCD main point (pccd) is q. q is a vector, which is positive in the same direction of CCD rotation angle, and vice versa.Figure 2
System parameters.Figure 3
Unfolded light path.
(a)
X-Y plane (b)
Y-Z plane


### 2.2.1. System Initial Position

As the angle between M3 and X-axis isθ3, the direction of incident laser is dirE′P′=2θ1-2θ3-(π/2) and the direction of optical axis is dirP′H′=2θ2-2θ3-(π/2). The coordinates of O′,O′′ are as follows:(1)O′=PM1-O′PM1=-d13-d13cos⁡2θ1-π-d13-cos⁡2θ1-π0TO′′=PM2+O′′PM1=d32+d32cos⁡2θ2-πd32-cos⁡2θ2-π0TAccording to formula (1), the positions of O′ and O′′ are independent of the rotation of M3. Then set the distance from origin to upper reflector of M3 is ThM3up. The coordinates of E′ and H′ can be expressed as follows:(2)E′=O′+O′E′=-d13-d13cos⁡2θ1-π+ThM3upcos⁡θ3cos⁡2θ1-π2-d13cos⁡2θ1-π+ThM3upcos⁡θ3sin⁡2θ1-π20(3)H′=O′+O′H′=d32+d32cos⁡2θ2-π+ThM3up-ThM3cos⁡θ3cos⁡2θ2-π2d32cos⁡2θ2-π+ThM3up-ThM3cos⁡θ3sin⁡2θ2-π20Due to the thickness of M3, a slight change in the optical path travel occurs. The stroke of lens center to pointH is as follows:(4)s′=cH=s+ThM3up-ThM3cos⁡θ3And the coordinate ofc′ corresponding to optical center point c is as follows:(5)c′=H′+H′c′=d32+d32cos⁡2θ2-π+ThM3up-ThM3cos⁡θ3cos⁡2θ2-π2+s′cos⁡2θ2-2θ3-π2d32cos⁡2θ2-π+ThM3up-ThM3cos⁡θ3sin⁡2θ2-π2+s′sin⁡2θ2-2θ3-π20


### 2.2.2. CCD Direction and Position

As is shown in Figure4, the points on line DG are ideally imaged on CCD when Scheimpflug principle [27] is satisfied. D is intersection point of c′D and DG, and c′D direction is dirc′D=dirPH′-π/2=2θ2+π/2. The coordinate of point D can be expressed:(6)D=Ex′-Ey′-cy′cot⁡2θ1-π-cx′tan⁡2θ2+π/2cot⁡2θ1-π1-tan⁡2θ2+π/2cot⁡2θ1-πDx-cx′tan⁡2θ2+π2+cy′0Figure 4
Illustration of calculating the CCD’s angle with Scheimpflug principle.In the triangle△c′DG, we know ∠c′DG=2θ1-2θ2-π/2 and Dc′=(Dy-cy′)2+(Dx-cx′)2. According to the triangle geometry, the distance D and G is as follows:(7)DG=fsin⁡∠cDG=fsin⁡2θ1-2θ2-π/2According to sine theorem, the following is given:(8)DGsin⁡-β=Dc′sin⁡∠DGc′SubstitutingDG and Dc′ into (8), the angle β can be expressed as follows:(9)β=arccot⁡Dc′f-tan⁡2θ1-2θ2Since△Dpccdc′ is a right triangle and ∠Dc′pccd=π/2, the expression of f0 and Dc′ is as follows:(10)f0=Dc′tan⁡-β


### 2.2.3. Imaging Light Direction

In△c′pccdp (see Figure 5), we know ∠c′pccdp=π/2-β, pccdc′=f0, pccdp=q. The expression of imaging light direction is as follows:(11)dirQp=dircp=2θ2-2θ3-π2+αα=arctan⁡q·cos⁡βf0-q·sin⁡βFigure 5
Relationship between the CCD imaging point and imaging of light.


### 2.2.4. The Spatial Coordinates of Points

On the X-Y unfolding plane, the spatial pointQ~ corresponds to point Q. We know that point Q is both in the laser emission direction and on the imaging ray, which can be expressed as follows:(12)cot⁡2θ1-2θ3-π2=Qx-Ex′Qy-Ey′cot⁡2θ2-2θ3-π2+α=cx′-Qxcy′-QyAccording to expression (12), the coordinates of point Q on the X-Y unfolding plane are as follows:(13)Q=Qy-Ey′cot⁡2θ1-2θ3-π2+Ex′cy′cot⁡2θ2-2θ3-π/2+α-Ey′cot⁡2θ1-2θ3-π/2+Ex′-cx′cot⁡2θ2-2θ3-π/2+α-cot⁡2θ1-2θ3-π/20The coordinate ofQ~ can be obtained:(14)Q~=Qx~Qy~Qz~=QxQy-hcos⁡2θ4+hQy-hsin⁡2θ4


## 2.2.1. System Initial Position

As the angle between M3 and X-axis isθ3, the direction of incident laser is dirE′P′=2θ1-2θ3-(π/2) and the direction of optical axis is dirP′H′=2θ2-2θ3-(π/2). The coordinates of O′,O′′ are as follows:(1)O′=PM1-O′PM1=-d13-d13cos⁡2θ1-π-d13-cos⁡2θ1-π0TO′′=PM2+O′′PM1=d32+d32cos⁡2θ2-πd32-cos⁡2θ2-π0TAccording to formula (1), the positions of O′ and O′′ are independent of the rotation of M3. Then set the distance from origin to upper reflector of M3 is ThM3up. The coordinates of E′ and H′ can be expressed as follows:(2)E′=O′+O′E′=-d13-d13cos⁡2θ1-π+ThM3upcos⁡θ3cos⁡2θ1-π2-d13cos⁡2θ1-π+ThM3upcos⁡θ3sin⁡2θ1-π20(3)H′=O′+O′H′=d32+d32cos⁡2θ2-π+ThM3up-ThM3cos⁡θ3cos⁡2θ2-π2d32cos⁡2θ2-π+ThM3up-ThM3cos⁡θ3sin⁡2θ2-π20Due to the thickness of M3, a slight change in the optical path travel occurs. The stroke of lens center to pointH is as follows:(4)s′=cH=s+ThM3up-ThM3cos⁡θ3And the coordinate ofc′ corresponding to optical center point c is as follows:(5)c′=H′+H′c′=d32+d32cos⁡2θ2-π+ThM3up-ThM3cos⁡θ3cos⁡2θ2-π2+s′cos⁡2θ2-2θ3-π2d32cos⁡2θ2-π+ThM3up-ThM3cos⁡θ3sin⁡2θ2-π2+s′sin⁡2θ2-2θ3-π20


## 2.2.2. CCD Direction and Position

As is shown in Figure4, the points on line DG are ideally imaged on CCD when Scheimpflug principle [27] is satisfied. D is intersection point of c′D and DG, and c′D direction is dirc′D=dirPH′-π/2=2θ2+π/2. The coordinate of point D can be expressed:(6)D=Ex′-Ey′-cy′cot⁡2θ1-π-cx′tan⁡2θ2+π/2cot⁡2θ1-π1-tan⁡2θ2+π/2cot⁡2θ1-πDx-cx′tan⁡2θ2+π2+cy′0Figure 4
Illustration of calculating the CCD’s angle with Scheimpflug principle.In the triangle△c′DG, we know ∠c′DG=2θ1-2θ2-π/2 and Dc′=(Dy-cy′)2+(Dx-cx′)2. According to the triangle geometry, the distance D and G is as follows:(7)DG=fsin⁡∠cDG=fsin⁡2θ1-2θ2-π/2According to sine theorem, the following is given:(8)DGsin⁡-β=Dc′sin⁡∠DGc′SubstitutingDG and Dc′ into (8), the angle β can be expressed as follows:(9)β=arccot⁡Dc′f-tan⁡2θ1-2θ2Since△Dpccdc′ is a right triangle and ∠Dc′pccd=π/2, the expression of f0 and Dc′ is as follows:(10)f0=Dc′tan⁡-β


## 2.2.3. Imaging Light Direction

In△c′pccdp (see Figure 5), we know ∠c′pccdp=π/2-β, pccdc′=f0, pccdp=q. The expression of imaging light direction is as follows:(11)dirQp=dircp=2θ2-2θ3-π2+αα=arctan⁡q·cos⁡βf0-q·sin⁡βFigure 5
Relationship between the CCD imaging point and imaging of light.


## 2.2.4. The Spatial Coordinates of Points

On the X-Y unfolding plane, the spatial pointQ~ corresponds to point Q. We know that point Q is both in the laser emission direction and on the imaging ray, which can be expressed as follows:(12)cot⁡2θ1-2θ3-π2=Qx-Ex′Qy-Ey′cot⁡2θ2-2θ3-π2+α=cx′-Qxcy′-QyAccording to expression (12), the coordinates of point Q on the X-Y unfolding plane are as follows:(13)Q=Qy-Ey′cot⁡2θ1-2θ3-π2+Ex′cy′cot⁡2θ2-2θ3-π/2+α-Ey′cot⁡2θ1-2θ3-π/2+Ex′-cx′cot⁡2θ2-2θ3-π/2+α-cot⁡2θ1-2θ3-π/20The coordinate ofQ~ can be obtained:(14)Q~=Qx~Qy~Qz~=QxQy-hcos⁡2θ4+hQy-hsin⁡2θ4


## 3. Simulation Analysis

### 3.1. Influence of M3 on Measurement Precision

In this section, we focus on analyzing the influence of the M3 thickness parameterThM3 and ThM3up. As shown in Figure 6, receiving and transmitting path will produce deviation because of the thickness of M3, which is equivalent to the horizontal displacement of M1 and M2, namely, the errors on d13 and d32. There is a significant correlation between measurement precision and the thickness of M3. The system parameters in this model are consistent with the traditional model as shown in the Table 1, and we take M3 thickness into consideration.Table 1
Simulation parameters of the system.
Parameter Value Parameter Value d/mm 100 ThM3/mm 1 d32/mm 50 T h M 3 u p / m m 0.5 s/mm 50 θ 1 / ∘ 45 h/mm 105 A M 1 / m m 65×40 f/mm 100 θ 2 / ∘ 45 f0/mm 100 A M 2 / m m 114×40 β/∘ 45 θ 2 / ∘ 45, Scan range (37.5 52.5) pccd/pixel -800 A M 3 / m m 58×40 D/mm 30 θ 2 / ∘ 45, Scan range (37.5 52.5) q/mm 023.072 A M 4 / m m 250×40Figure 6
Influence of thickness of M3 on other parameters.For the purpose of facilitating to analyze the influence of the thickness parameter on the measurement error, the following indicators are used:(1) the maximum measurement error in the whole area; (2) the maximum measurement error at a distance of 10m; (3) the maximum measurement error at 0.6m. Calculate the error of the emission path when Thm3up is 0, 0.5, 1, 1.5, and 2 mm, and the error of the emission path when ThM3-ThM3up is 0, 0.5, 1, 1.5, and 2 mm. It can be seen from the simulation results (Figure 7) that there is about 400 mm at a distance of 10 m when thickness of M3 is 2mm. Therefore, when calculating 3D coordinates of spatial points, the thickness of M3 must be paid enough attention.Figure 7
Influence of M3’s thickness on system errors.
(a)
Light path of launching (b)
Light path of receiving


### 3.2. Influence of M3 on Measurement Precision

Simulation experiments are carried out according to the parameters given in Table1. The measurement precision of any point Q~ is correlated with the parameters (θ1,θ2,θ3,θ4,d,f0,s,h,β,q). Differentiate these parameters:(15)△Q~=J△θ1△θ2△θ3△θ4△d△f0△s△h△β△qJ=∂Qx~,Qy~,Qz~∂θ1,θ2,θ3,θ4,d,f0,s,h,β,qAccording to actual processing characteristics, we take the position error±0.005mm, the angle error ±0.02∘, and the thickness error ±1mm. The influence of each error source on the measurement accuracy is shown in Table 2 (length: mm, angle: degree).Table 2
Influence of errors on measuring accuracy.
Order Error source Error value Max error in the whole rang Max error at 10m Max error at 0.6m D Position error at x of M1 ±0.0005 0.145 0.128 0.009 D Position error at x of M2 ±0.0005 0.149 0.132 0.008 E Position error at x of M3 ±0.0005 0.065 0.057 0.005 E Position error at x of M4 ±0.0005 0.001 0.001 0.001 A Angle error of M1 ±0.02 2097.335 1380.223 6.691 A Angle error of M2 ±0.02 2097.341 1875.158 7.173 C Rotating angle error of M3 ±0.02 26.711 23.664 1.680 C Rotating angle error of M4 ±0.02 16.894 14.087 0.838 D Position error at x of lens ±0.0005 0.175 0.150 0.011 D Position error at x of CCD ±0.0005 0.174 0.149 0.010 C Angle error of CCD ±0.02 6.057 5.207 0.425 C CCD image detect error ±0.0007 24.267 18.429 0.075 B Position error of M3’s up surface ±1 409.086 356.012 24.300 B Position error of M3’s down surface ±1 419.260 364.652 19.223According to the results in Table2, system parameters are ranked in the aspect of significance.(a) The angles of the M1 mirror and the M2 mirror are very important and have the greatest impact on the measurement accuracy of the three-dimensional point, reaching the level of several meters, and must be given A-level attention.(b) The thickness of the M3 mirror has an accuracy effect on three-dimensional points up to hundreds millimeters in the laser projection and receiving circuit, which should be given enough attention and belong to the B-level.(c) M3 mirror rotation error, M4 mirror rotation error, CCD angle error, and CCD image detection error have an accuracy effect to tens millimeters, giving C-level.(d) The position error of the M1 mirror, the position error of the M2 mirror, the lens position error, and the CCD position error have little effect on the measurement accuracy, giving appropriate attention, D-level.(e) The position error of the M3 mirror on the x-axis and the position error of the M4 mirror have the least influence on the measurement accuracy of the three-dimensional measurement point, giving general attention and belong to the E-level.


## 3.1. Influence of M3 on Measurement Precision

In this section, we focus on analyzing the influence of the M3 thickness parameterThM3 and ThM3up. As shown in Figure 6, receiving and transmitting path will produce deviation because of the thickness of M3, which is equivalent to the horizontal displacement of M1 and M2, namely, the errors on d13 and d32. There is a significant correlation between measurement precision and the thickness of M3. The system parameters in this model are consistent with the traditional model as shown in the Table 1, and we take M3 thickness into consideration.Table 1
Simulation parameters of the system.
Parameter Value Parameter Value d/mm 100 ThM3/mm 1 d32/mm 50 T h M 3 u p / m m 0.5 s/mm 50 θ 1 / ∘ 45 h/mm 105 A M 1 / m m 65×40 f/mm 100 θ 2 / ∘ 45 f0/mm 100 A M 2 / m m 114×40 β/∘ 45 θ 2 / ∘ 45, Scan range (37.5 52.5) pccd/pixel -800 A M 3 / m m 58×40 D/mm 30 θ 2 / ∘ 45, Scan range (37.5 52.5) q/mm 023.072 A M 4 / m m 250×40Figure 6
Influence of thickness of M3 on other parameters.For the purpose of facilitating to analyze the influence of the thickness parameter on the measurement error, the following indicators are used:(1) the maximum measurement error in the whole area; (2) the maximum measurement error at a distance of 10m; (3) the maximum measurement error at 0.6m. Calculate the error of the emission path when Thm3up is 0, 0.5, 1, 1.5, and 2 mm, and the error of the emission path when ThM3-ThM3up is 0, 0.5, 1, 1.5, and 2 mm. It can be seen from the simulation results (Figure 7) that there is about 400 mm at a distance of 10 m when thickness of M3 is 2mm. Therefore, when calculating 3D coordinates of spatial points, the thickness of M3 must be paid enough attention.Figure 7
Influence of M3’s thickness on system errors.
(a)
Light path of launching (b)
Light path of receiving


## 3.2. Influence of M3 on Measurement Precision

Simulation experiments are carried out according to the parameters given in Table1. The measurement precision of any point Q~ is correlated with the parameters (θ1,θ2,θ3,θ4,d,f0,s,h,β,q). Differentiate these parameters:(15)△Q~=J△θ1△θ2△θ3△θ4△d△f0△s△h△β△qJ=∂Qx~,Qy~,Qz~∂θ1,θ2,θ3,θ4,d,f0,s,h,β,qAccording to actual processing characteristics, we take the position error±0.005mm, the angle error ±0.02∘, and the thickness error ±1mm. The influence of each error source on the measurement accuracy is shown in Table 2 (length: mm, angle: degree).Table 2
Influence of errors on measuring accuracy.
Order Error source Error value Max error in the whole rang Max error at 10m Max error at 0.6m D Position error at x of M1 ±0.0005 0.145 0.128 0.009 D Position error at x of M2 ±0.0005 0.149 0.132 0.008 E Position error at x of M3 ±0.0005 0.065 0.057 0.005 E Position error at x of M4 ±0.0005 0.001 0.001 0.001 A Angle error of M1 ±0.02 2097.335 1380.223 6.691 A Angle error of M2 ±0.02 2097.341 1875.158 7.173 C Rotating angle error of M3 ±0.02 26.711 23.664 1.680 C Rotating angle error of M4 ±0.02 16.894 14.087 0.838 D Position error at x of lens ±0.0005 0.175 0.150 0.011 D Position error at x of CCD ±0.0005 0.174 0.149 0.010 C Angle error of CCD ±0.02 6.057 5.207 0.425 C CCD image detect error ±0.0007 24.267 18.429 0.075 B Position error of M3’s up surface ±1 409.086 356.012 24.300 B Position error of M3’s down surface ±1 419.260 364.652 19.223According to the results in Table2, system parameters are ranked in the aspect of significance.(a) The angles of the M1 mirror and the M2 mirror are very important and have the greatest impact on the measurement accuracy of the three-dimensional point, reaching the level of several meters, and must be given A-level attention.(b) The thickness of the M3 mirror has an accuracy effect on three-dimensional points up to hundreds millimeters in the laser projection and receiving circuit, which should be given enough attention and belong to the B-level.(c) M3 mirror rotation error, M4 mirror rotation error, CCD angle error, and CCD image detection error have an accuracy effect to tens millimeters, giving C-level.(d) The position error of the M1 mirror, the position error of the M2 mirror, the lens position error, and the CCD position error have little effect on the measurement accuracy, giving appropriate attention, D-level.(e) The position error of the M3 mirror on the x-axis and the position error of the M4 mirror have the least influence on the measurement accuracy of the three-dimensional measurement point, giving general attention and belong to the E-level.


## 4. Experiments and Discussions

### 4.1. Experiments

The synchronized scanning triangulation ranging system prototype is designed and manufactured (shown in Figure8(a)). We first carried out the functional experiment. The motor starts scanning and the camera starts to collect after obtaining pulse command from upper computer program. This 3D synchronous scanning system can scan the object and display its point cloud. The system performs scanning capability test at short and long distance, respectively. As is shown in Figure 8, not only can point cloud be displayed correctly, but also the surface characteristics of the object can be clearly displayed.Figure 8
Measurement device and experiment.
(a)
D100 system prototype (b)
The first measurement experiment (c)
The second measurement experimentNext, the experiment measurement uncertainty is carried out. We divide the plane into three linear positions of different heights, as shown in Figure9, and the coordinates of P1, P2, and P3 can be calculated by the trigonometric method. According to the standard deviation of each coordinate, the value of measurement uncertainty can be evaluated. The results are shown in Table 3. It can be seen from the above experimental analysis that the value of measurement uncertainty of the system at a distance of 0.5m is 0.75mm and at a distance of 5m is 6.68mm.Table 3
The value of measurement uncertainty.
Parameter Accuracy in 0.5m Accuracy in 5m p1 x 0.035 0.133 0.069 1.001 0.727 1.298 y 0.034 0.129 0.067 1.600 1.162 2.074 z 0.125 0.481 0.250 3.377 2.452 4.378 p2 x 0.123 0.094 0.093 1.906 1.382 1.517 y 0.256 0.196 0.194 2.453 1.778 1.953 z 0.379 0.290 0.287 6.676 4.839 5.314 p3 x 0.086 0.210 0.046 1.057 0.773 0.969 y 0.301 0.734 0.161 1.021 0.746 0.936 z 0.111 0.271 0.059 3.810 2.785 3.494Figure 9
Reorientation precision experiment.Figures10 and 11 show the flatness of the system at a distance of 0.5m and 5m. Import the scanned point cloud into the geomagic, fitting plane, and then calculate the distance from the measurement point to the fitting plane. The result shows that the standard deviations at a distance of 0.5 m and 5 m are 1.10 mm and 19.73 mm, respectively.Figure 10
Plane scanning experiment at a distance of 0.5m.
(a)
The point cloud of the plane (b)
The fitting result of the planeFigure 11
Plane scanning experiment at a distance of 5m.
(a)
The point cloud of the plane (b)
The fitting result of the plane


### 4.2. Discussions

In this optical measurement system, the measurement accuracy is directly related to the mechanical design and processing precision of the optical system. Therefore, we must pay attention to the adjustability of the optical system when we optimize mechanical structure in the future, so as to compensate for the deficiencies caused by machining. In addition, the principle prototype is not calibrated, and the system must be calibrated to improve measurement accuracy.And in terms of photodetector, CCD is used as a photodetector in this synchronous scanning triangulation system, in which some shortcomings exist. When measuring a point, it takes many sampling periods to collect data and process the collected data. Secondly, the driving circuit of CCD is more complicated. The driving circuit is one of the key problems in practical application, and the CCD driving signals of different manufacturers and different models are not identical, which will bring inconvenience to the design of the system. Because of these shortcomings, PSD will be taken into consideration as a photodetector in the future.


## 4.1. Experiments

The synchronized scanning triangulation ranging system prototype is designed and manufactured (shown in Figure8(a)). We first carried out the functional experiment. The motor starts scanning and the camera starts to collect after obtaining pulse command from upper computer program. This 3D synchronous scanning system can scan the object and display its point cloud. The system performs scanning capability test at short and long distance, respectively. As is shown in Figure 8, not only can point cloud be displayed correctly, but also the surface characteristics of the object can be clearly displayed.Figure 8
Measurement device and experiment.
(a)
D100 system prototype (b)
The first measurement experiment (c)
The second measurement experimentNext, the experiment measurement uncertainty is carried out. We divide the plane into three linear positions of different heights, as shown in Figure9, and the coordinates of P1, P2, and P3 can be calculated by the trigonometric method. According to the standard deviation of each coordinate, the value of measurement uncertainty can be evaluated. The results are shown in Table 3. It can be seen from the above experimental analysis that the value of measurement uncertainty of the system at a distance of 0.5m is 0.75mm and at a distance of 5m is 6.68mm.Table 3
The value of measurement uncertainty.
Parameter Accuracy in 0.5m Accuracy in 5m p1 x 0.035 0.133 0.069 1.001 0.727 1.298 y 0.034 0.129 0.067 1.600 1.162 2.074 z 0.125 0.481 0.250 3.377 2.452 4.378 p2 x 0.123 0.094 0.093 1.906 1.382 1.517 y 0.256 0.196 0.194 2.453 1.778 1.953 z 0.379 0.290 0.287 6.676 4.839 5.314 p3 x 0.086 0.210 0.046 1.057 0.773 0.969 y 0.301 0.734 0.161 1.021 0.746 0.936 z 0.111 0.271 0.059 3.810 2.785 3.494Figure 9
Reorientation precision experiment.Figures10 and 11 show the flatness of the system at a distance of 0.5m and 5m. Import the scanned point cloud into the geomagic, fitting plane, and then calculate the distance from the measurement point to the fitting plane. The result shows that the standard deviations at a distance of 0.5 m and 5 m are 1.10 mm and 19.73 mm, respectively.Figure 10
Plane scanning experiment at a distance of 0.5m.
(a)
The point cloud of the plane (b)
The fitting result of the planeFigure 11
Plane scanning experiment at a distance of 5m.
(a)
The point cloud of the plane (b)
The fitting result of the plane


## 4.2. Discussions

In this optical measurement system, the measurement accuracy is directly related to the mechanical design and processing precision of the optical system. Therefore, we must pay attention to the adjustability of the optical system when we optimize mechanical structure in the future, so as to compensate for the deficiencies caused by machining. In addition, the principle prototype is not calibrated, and the system must be calibrated to improve measurement accuracy.And in terms of photodetector, CCD is used as a photodetector in this synchronous scanning triangulation system, in which some shortcomings exist. When measuring a point, it takes many sampling periods to collect data and process the collected data. Secondly, the driving circuit of CCD is more complicated. The driving circuit is one of the key problems in practical application, and the CCD driving signals of different manufacturers and different models are not identical, which will bring inconvenience to the design of the system. Because of these shortcomings, PSD will be taken into consideration as a photodetector in the future.


## 5. Conclusions

A synchronous scanning triangulation system based on unfold light path has been described. This study focuses on the accurate calculation model of the synchronous scanning triangulation imaging system in the unfolded optical path and analyzed the influence of M3 thickness on the system according to the parameters of the traditional model. Next, the influence of each error source on the measurement accuracy is obtained according to the actual processing characteristics. The feasibility of the system is verified theoretically. The experimental results of this study indicate that the system can scan the object in three dimensions and obtain the surface contour information of the object.



---

*Source: 3503192-2019-03-25.xml*