Line Laser 3D Measurement Method and Experiments of Gears

Abstract

Line laser measurement, as a typical method of laser triangulation, makes the acquisition of 3D tooth-surface data more accurate, efficient, and informative. Thus, a line laser 3D measurement model of gears is established, and a specialized polyhedral artifact with specific geometric features is invented to determine the pose parameters of the line laser sensor in measuring space. Based on this, a single-spindle gear-measuring instrument is developed and a series of experimental studies are conducted for gears with different module and flank directions in this instrument, including profile deviation, helix deviation, pitch deviation, topological deviation, etc. A comparative experiment with traditional contact measurement methods validates the correctness of the methods mentioned in this paper for the accurate evaluation of tested gears. In further research, the mining and utilization of big data obtained from the line laser 3D measurement of gears will be an important topic.

1. Introduction

In 1765, the involute was introduced into the gear profile by mathematician L. Euler. In the late 19th century, gear generative machining methods which realized the efficient machining of involute gears were invented by Reinecker, Fellows, Pfauter, and others. In 1923, an involute tester which realized the precise measurement of involute gears was invented by Zeiss in Germany [1,2,3]. In the early 20th century, a large demand for gear transmission emerged with the development of the automobile industry, and the involute gear could meet this demand with high-cost performance. Since then, the involute gear has occupied the leading position in the field of gear transmission [4,5].

It has been 100 years since the emergence of gear precision measurement, and gear measurement technologies have experienced three generations of development from pure mechanization and electrification to CNC [6]. At present, gear measurement is in a critical stage of the transition from the third generation to the next generation, whose main feature is full information [7,8].

Recently, optical non-contact measurement methods have been the main highlights within holistic gear deviation measurement, such as laser triangulation [9,10], laser holography [11], industrial CT [12], and so on. As a typical method of laser triangulation, line laser measurement has become a main method for capturing holistic gear deviations due to its high measurement efficiency [13,14]. Matthias developed an online inspection tool for large wind power gears to detect fractures and provide quantitative damage assessment capabilities [15]. The method enables full-tooth-surface scanning of individual gear teeth with the ability to simultaneously measure 1280 points in a single line scan in under 10 milliseconds. Goch employed an optical CMM (HN3030, Nikon Corporation, Tokyo, Japan) for gear inspection using line laser scanning to capture comprehensive tooth-surface data [16]. Approximately 480,000 raw measurement points were acquired per tooth surface, covering the root, tip, and tip platform regions. Of these, roughly 25% fell within the evaluation zone and were suitable for surface area assessment. Shi developed a 3D gear measurement system based on a high-precision turntable and integrated it into a gear measurement cloud computing platform [9]. The work introduced a novel method for calculating 3D tooth-surface deviations and established a comprehensive gear lifecycle data exchange standard. Yu introduced a combined laser–vision detection method for online gear measurement, effectively addressing the limitations, namely, inefficiency and a dependency on gear mounting references, of contact measurement techniques [17,18]. This method is capable of fulfilling the requirement for online inspection of small modulus gears, thus offering a robust and reliable solution for precision gear inspection.

Since 2015, similar pieces of equipment based on the line laser 3D measurement method of gears have been introduced one after another [9]. Recently, depending on the application scenario and the device being installed, line laser 3D measurement schemes of gears have been proposed, especially based on a gear measurement center (GMC) [19], coordinate measuring machine (CMM) [20], precision turntable [21], comprehensive measuring instrument [22], machine tool [23], articulated arm CMM [24], and so on. The schemes based on a precision turntable, comprehensive measuring instrument, and machine tools are suitable for gear production sites. The first two types are specialized pieces of equipment that are suitable for the large-scale, online, full inspection of gears. The third type is universal equipment that is compatible with various models of DMGMORI machining machines to achieve the on-machine full inspection of gears. The scheme based on a GMC has high measurement accuracy, but it is still necessary to use contact probes for eccentricity correction and initial positioning in order to ensure high measurement efficiency. The scheme based on a CMM is limited in its application scenarios due to the influence of precision turntable components. The scheme based on an articulated arm CMM eliminates the need for gear clamping and repeated sensor calibration, but it is difficult for its measurement accuracy to meet the requirements of high-precision gear measurement.

A solution based on the precision turntable for line laser 3D measurement of gears is introduced in this paper. It is ideal for gear production lines, enabling 100% online inspection of mass-produced gears. Experimental research is conducted on gears with different module and flank directions, which can achieve the accurate evaluation of tested gears, including profile deviation, helix deviation, pitch deviation, topological deviation, etc.

2. Line Laser 3D Measurement Model of Gears

The tested gear is installed on the spindle of the instrument and rotates with it. The installation of the tested gear has either high-precision chuck fixtures, or double tips through a core shaft that matches the tested gear. The rotation signal triggers the line laser sensor to sample the geometric information of the tooth surface in real time. As the gear rotates at a certain speed, the measuring light scans across all the teeth, so full tooth-surface information is captured in the circumference of the gear.

In Figure 1, taking the installation of the tested gear with the high-precision chuck fixtures as an example, four coordinate systems, namely, the machine coordinate system ${\sigma }_0:(O_0-x_0,y_0,z_0)$, the sensor coordinate system ${\sigma }_s:(O_s-x_s,y_s,z_s)$, the measuring light coordinate system ${\sigma }_l:(O_l-x_l,y_l,z_l)$, and the gear coordinate system ${\sigma }_g:(O_g-x_g,y_g,z_g)$, are set up. Among them, ${\sigma }_0$, ${\sigma }_s$ and ${\sigma }_l$ are fixed. ${\sigma }_g$ is attached to the tested gear and rotates around its axis.

The line laser measurement model of the tested gear is

$$D_g=M_0^g\cdot M_s^0\cdot M_l^s\cdot D_l$$

(1)

$D_l$ is the coordinate value of the measurement point cloud in ${\sigma }_l$. $D_g$ is the coordinate value of the measurement point cloud in ${\sigma }_g$.

$$D_g={\left(x_g,y_g,z_g,1\right)}^{\mathrm{T}},D_l={\left(x_l,y_l,z_l,1\right)}^{\mathrm{T}}$$

(2)

$M_l^s$ is the transformation matrix from ${\sigma }_l$ to ${\sigma }_g$.

$$M_l^s=\left(\begin{array}{cccc}c\_s& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& -h_0\\ 0& 0& 0& 1\end{array}\right)$$

(3)

where $c\_s$ represents the installation direction of the line laser sensor, and its value is +1 or −1. $h_0$ is the nominal distance of the line laser sensor, which is determined by the type of selected sensor.

$M_s^0$ is the transformation matrix from ${\sigma }_s$ to ${\sigma }_0$.

$$M_s^0=\left(\begin{array}{cccc}{C}_{\gamma }{C}_{\beta }-S_{\gamma }{S}_{\alpha }{S}_{\beta }& -S_{\gamma }{C}_{\alpha }& C_{\gamma }{S}_{\beta }+S_{\gamma }{S}_{\alpha }{C}_{\beta }& a_0\\ S_{\gamma }{C}_{\beta }+C_{\gamma }{S}_{\alpha }{S}_{\beta }& C_{\gamma }{C}_{\alpha }& S_{\gamma }{S}_{\beta }-C_{\gamma }{S}_{\alpha }{C}_{\beta }& b_0\\ -C_{\alpha }{S}_{\beta }& S_{\alpha }& C_{\alpha }{C}_{\beta }& c_0\\ 0& 0& 0& 1\end{array}\right)$$

(4)

where $C$ and $S$ represent cos() and sin(), respectively. The subscripts α, β, and γ denote the angles ${\omega }_{\alpha }$, ${\omega }_{\beta }$ and ${\omega }_{\gamma }$, respectively. For example, $C_{\alpha }=\cos ({\omega }_{\alpha })$. ${\omega }_{\alpha }$, ${\omega }_{\beta }$, ${\omega }_{\gamma }$, $a_0$, $b_0$, and $c_0$ are the three Euler angles and three coordinate values of the line laser sensor in ${\sigma }_0$, respectively. These six parameters are used to determine the spatial position and attitude of the line laser sensor in the 3D measurement of gears.

$M_0^g$ is the transformation matrix from ${\sigma }_0$ to ${\sigma }_g$.

$$M_0^g=\left(\begin{array}{cccc}\cos \phi & \sin \phi & 0& 0\\ -\sin \phi & \cos \phi & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$$

(5)

where $\phi$ is the rotation angle of the tested gear.

So far, the actual three-dimensional geometry information of the tested tooth surfaces can be obtained by combining Equations (1)–(5).

Due to the evaluation of tooth surface characteristics requiring measurement and calculation in the direction of the meshing line of the gear pair (i.e., the normal direction of the tooth surface), the relationship between the actual tooth surface measurement point P_actu of the tested gear and the corresponding theoretical tooth surface measurement point P can be expressed in the form of a vector sum [8], as shown in Figure 2.

$$p_{actu}=p+e_{norm}=p+\left|e_{norm}\right|\cdot n$$

(6)

where $p_{actu}$ is the vector of the actual tooth surface measurement point, $p$ is the vector of the corresponding theoretical tooth surface measurement point, $e_{norm}$ is the deviation vector from the theoretical measurement point along the normal direction of the tooth surface to the actual measurement point, and $n$ is the unit normal vector at the measurement point on the tooth surface.

According to the tooth surface information $D_g$ of the tested gear obtained using Equation (1), the normal error $\left|e_{norm}\right|$ at any point on the tooth surface is

$$\left|e_{norm}\right|={\displaystyle \frac{r_b}{\sqrt{1+{\tan }^2{\beta }_b}}}\left[\sqrt{{\displaystyle \frac{x_g^2+y_g^2}{r_b^2}}-1}\right.\left.-\arctan \left(\sqrt{{\displaystyle \frac{x_g^2+y_g^2}{r_b^2}}-1}\right)-\arctan \left({\displaystyle \frac{y_{\mathrm{g}}}{x_g}}\right)+z_g\cdot {\displaystyle \frac{\tan {\beta }_b}{r_b}}\right]$$

(7)

where $r_b$, ${\beta }_b$ are the basic radius and the basic helix angle of the tested gear, respectively.

According to the criterion that the directions of plus and minus materials are positive and negative, respectively, the normal error e at any point on the tooth surface can be expressed as follows:

$$e=\mathrm{type}\cdot \left|e_{norm}\right| \begin{array}{c}\left\{\begin{array}{l}\mathrm{type}=1 : \mathrm{plus} \mathrm{material}\\ \mathrm{type}=-1 : \mathrm{minus} \mathrm{material}\end{array}\right.\end{array}$$

(8)

3. Calibration of Pose Parameters for Line Laser Sensors

In line laser 3D measurement of gears, it is crucial to determine the pose parameters of the line laser sensor in measuring space, which determines the accuracy and reliability of the tooth surface data collected by the sensor [25,26]. That is to say, we determine the six degrees of freedom parameters of the line laser sensor in the machine coordinate system of the line laser 3D measurement model of gears, including three Euler angle parameters (${\omega }_{\alpha }$, ${\omega }_{\beta }$, ${\omega }_{\gamma }$) and three position parameters ($a_0$, $b_0$, $c_0$). For this purpose, a specialized polyhedral artefact with specific geometric features is invented to accurately calibrate the pose parameters of line laser sensors in measuring space, as shown in Figure 3.

The polyhedral artefact is a central rotating structure with an inner cylindrical surface in the middle. There are various specific geometric features on the side, including an outer cylindrical surface, planes, and V-grooves. Three planes divide the outer cylindrical surface into three parts, forming phase angles of 0°, 90°, and 180°. In order to ensure the consistency of the installation positioning between the polyhedral artefact and the tested gear, the size of the artefact matches that of the tested gear. The inner cylindrical diameter of the artefact is the same as the inner hole diameter of the tested gear, the outer cylindrical diameter of the artefact is the same as the tooth tip diameter of the tested gear, and the distance between the two planes of the artefact is the same as the tooth root diameter of the tested gear. Among them, the roundness and cylindricity of the cylindrical surface, the coaxiality of the inner and outer cylindrical surfaces, the flatness of the plane, and the perpendicularity between the planes are all less than 1 μm. The actual measurement verification was carried out using a coordinate measuring machine (PMM-C 12.10.7/456 Ultra, Leitz, Germany), as shown in Figure 4, and the results showed that the design requirements are met.

A calibration model for pose parameters of line laser sensors in measuring space is established, as shown in Figure 5. Among them, the six free parameters of the line laser sensor in the measurement space are demonstrated in the figure, which are the pose parameters that need to be determined through calibration. The line laser sensor scans the artefact surface with a certain pose to obtain point cloud data ${}^{\delta }{D}_l$ of geometric features in the line coordinate system, which can be expressed as

$${}^{\delta }D_l=\left({}^{\delta }D_l^{(1)},{}^{\delta }D_l^{(2)},\dots ,{}^{\delta }D_l^{(\mathrm{i})},\dots ,{}^{\delta }D_l^{(n_{\delta })}\right), \begin{array}{c}\delta =C_o,P_1,P_2\end{array},P_3,V_g$$

(9)

where $n_{\delta }$ is the total number of feature points collected by the line laser sensor. $C_o$, $P$, $V_g$ are the outer cylindrical surface feature, planar features and V-groove feature of the polyhedral artefact, respectively.

Referring to Equations (1)–(5), the point cloud data ${}^{\delta }D_c$ of geometric features in the polyhedral artefact coordinate system is obtained.

$$\begin{array}{c}{}^{\delta }D_c=\left({}^{\delta }D_c^{(1)},{}^{\delta }D_c^{(2)},\dots ,{}^{\delta }D_c^{(\mathrm{i})},\dots ,{}^{\delta }D_c^{(n_{\delta })}\right), \begin{array}{c}\delta =C_o,P_1,P_2\end{array},P_3,V_g\\ {}^{\delta }D_c^{(\mathrm{i})}={\left(x_{\delta }^i,y_{\delta }^i,z_{\delta }^i,1\right)}^{\mathrm{T}}\end{array}$$

(10)

Because the measurement range of the line laser sensor covers the entire tooth width, $c_0$ is taken as a relative measurement result and does not affect other parameters. Therefore, the coordinate transformation matrix contains five unknown parameters (${\omega }_{\alpha }$, ${\omega }_{\beta }$, ${\omega }_{\gamma }$, $a_0$, $b_0$), and the calibration of the pose parameters of the line laser sensor is equivalent to the estimation problem of the five unknown parameters.

The parameter estimation problem can be solved by transforming it into a nonlinear optimization problem. Geometric features containing unknown parameters are obtained thought a calibration model of pose parameters for line laser sensors, including ${}^{C_o}D_c$, ${}^{P_1}D_c$, ${}^{P_2}D_c$, ${}^{P_3}D_c$. Among them, the planar features P1, P2, and P3 are used to determine the Euler angle parameters (${\omega }_{\alpha }$, ${\omega }_{\beta }$, ${\omega }_{\gamma }$), and the radius of the cylindrical feature C_o is used to determine the position parameters ($a_0$, $b_0$).

The algebraic error of the cylinder radius at each measurement point in ${}^{C_o}D_c$ is defined as $f_1\left(\stackrel{\wedge }{\mathrm{W}}\right)$.

$$f_1\left(\stackrel{\wedge }{\mathrm{W}}\right)={\left(x_{C_o}^i\right)}^2+{\left(y_{C_o}^i\right)}^2-{\left(R_{C_o}\right)}^2,\stackrel{\wedge }{\mathrm{W}}={\left({\omega }_{\alpha },{\omega }_{\beta },{\omega }_{\gamma },a_0,b_0\right)}^{\mathrm{T}}$$

(11)

where $R_{C_o}$ is the design value of the outer cylindrical radius of the polyhedral artefact, used as a reference value.

The algebraic distance error between each measurement point in ${}^{P}D_c$ and the plane is defined as $f_2\left(\stackrel{\wedge }{\mathrm{W}}\right)$.

$$f_2\left(\stackrel{\wedge }{\mathrm{W}}\right)={\displaystyle \frac{A_k{x}_{S_k}^i+B_k{y}_{S_k}^i+C_k{z}_{S_k}^i+D_k}{\sqrt{{A_k}^2+{B_k}^2+{C_k}^2}}}, k=1,2,3$$

(12)

where $A_k$, $B_k$, $C_k$, and $D_k$ are the coefficients of the plane equation $A_{k}x+B_{k}y+C_{k}z+D_k=0$.

Algebraic errors between planar and cylindrical features are used as constraint equations for the least squares method to solve unknown parameters. According to Equations (11) and (12), the objective function is established as shown in Equation (13).

$$\min F\left(\stackrel{\wedge }{\mathrm{W}}\right)\to \min \left({\displaystyle \frac{1}{2}}\left({\displaystyle \sum _{i=1}^{n_{C_o}}{f_1}^2\left(\stackrel{\wedge }{\mathrm{W}}\right)}+{\displaystyle \sum _{k=1}^3{\displaystyle \sum _{i=1}^{n_{P_k}}{f_2}^2\left(\stackrel{\wedge }{\mathrm{W}}\right)}}\right)\right)$$

(13)

The second-order Taylor series expansion at the $\stackrel{\wedge }{\mathrm{W}}$ for $F\left(\stackrel{\wedge }{\mathrm{W}}\right)$ is performed, and then $F\left(\stackrel{\wedge }{\mathrm{W}}\right)$ can be approximately linearly expressed as Equation (14).

$$F\left(\stackrel{\wedge }{\mathrm{W}}+\Delta \stackrel{\wedge }{\mathrm{W}}\right)=F\left(\stackrel{\wedge }{\mathrm{W}}\right)+\Delta {\stackrel{\wedge }{\mathrm{W}}}^{\mathrm{T}}\nabla F\left(\stackrel{\wedge }{\mathrm{W}}\right)+{\displaystyle \frac{1}{2}}\Delta {\stackrel{\wedge }{\mathrm{W}}}^{\mathrm{T}}{\nabla }^{2}F\left(\stackrel{\wedge }{\mathrm{W}}\right)\Delta \stackrel{\wedge }{\mathrm{W}}+O{‖\Delta \stackrel{\wedge }{\mathrm{W}}‖}^3$$

(14)

If the high-order infinitesimal term represented by $O{‖\Delta \stackrel{\wedge }{\mathrm{W}}‖}^3$ is omitted, there is Equation (15).

$$\epsilon \left(\Delta \stackrel{\wedge }{\mathrm{W}}\right)=F\left(\stackrel{\wedge }{\mathrm{W}}+\Delta \stackrel{\wedge }{\mathrm{W}}\right)-F\left(\stackrel{\wedge }{\mathrm{W}}\right)=\Delta {\stackrel{\wedge }{\mathrm{W}}}^{\mathrm{T}}\nabla F\left(\stackrel{\wedge }{\mathrm{W}}\right)+{\displaystyle \frac{1}{2}}\Delta {\stackrel{\wedge }{\mathrm{W}}}^{\mathrm{T}}{\nabla }^{2}F\left(\stackrel{\wedge }{\mathrm{W}}\right)\Delta \stackrel{\wedge }{\mathrm{W}}$$

(15)

By taking the first derivative of Equation (15) and making the derivative 0, Equation (16) can be obtained.

$$\Delta \stackrel{\wedge }{\mathrm{W}}=-{\left({\nabla }^{2}F\left(\stackrel{\wedge }{\mathrm{W}}\right)\right)}^{-1}\nabla F\left(\stackrel{\wedge }{\mathrm{W}}\right)$$

(16)

The corresponding coordinates are updated by minimizing the sum of squared errors. When $\epsilon \left(\Delta \stackrel{\wedge }{\mathrm{W}}\right)$ reaches its minimum value, $\stackrel{\wedge }{\mathrm{W}}$ is the result of parameter optimization, which is

$${\stackrel{\wedge }{\mathrm{W}}}^{\left(j+1\right)}={\stackrel{\wedge }{\mathrm{W}}}^{\left(j\right)}-{\left({\nabla }^{2}F\left({\stackrel{\wedge }{\mathrm{W}}}^{\left(j\right)}\right)\right)}^{-1}\nabla F\left({\stackrel{\wedge }{\mathrm{W}}}^{\left(j\right)}\right)$$

(17)

where j is the jth iteration.

At this point, the five unknown parameters (${\omega }_{\alpha }$, ${\omega }_{\beta }$, ${\omega }_{\gamma }$, $a_0$, $b_0$) in the coordinate transformation matrix can be obtained through nonlinear least squares iterative optimization.

4. Line Laser 3D Measurement Instrument of Gears

A line laser 3D measurement instrument of gears, which was developed by the authors, as well as its specifications, are shown in Figure 6. This is a single-spindle instrument that consists of main spindle, circular grating, servo control system, line laser sensors, data processing unit and gear measuring software. The main spindle adopts a precision dense ball bearing with circular grating sensor. Two line laser sensors are symmetrically distributed around the spindle and triggered by the grating signal to capture the geometric information of the tooth surface. According to the intrinsic attribute of line laser gear measurement, measuring software is developed, including determination of the position and attitude of the line laser sensors, evaluation of the gear accuracy, utilization of the holistic gear deviations and other functions. The 3D point cloud of a gear is checked and illustrated by this instrument. The technical index and performance parameters of this instrument are shown in

Table 1. Technical index and performance parameters of the instrument.

Technical Index	Parameters
Tested gear module	1~5 mm
Maximum addendum diameter of tested gears	230 mm
Flank direction of tested gears	Spur gears and helix gears
Accuracy evaluation class	grades 5 (ISO 1328-1:2013 [27])
Measurement repeatability	0.002 mm
Minimum resolution of rotating shaft system	0.0001°
Measurement items	Profile deviation, helix deviation, pitch deviation, etc.

.

5. Experiment and Discussion

5.1. Tested Gears and Line Laser Sensors

To verify the correctness of the measurement model and the measurement instrument, four types of gears with different parameters, as shown in Figure 7, are selected for measurement experiments. Their specific parameters are shown in

Table 2. Parameters of the tested gear.

Parameters	Symbols/Units	Gear 1	Gear 2	Gear 3	Gear 4
Tooth number	Z	84	38	24	48
Module	mn/mm	1	4	5	4
Pressure angle	α/°	20	20	24	20
Helix angle	β/°	18.75	15	—	30
Face width	b/mm	20	45	53	40
Flank direction	—	Left-handed	Left-handed	—	Right-handed
Modification	—	None	None	Crowned	Twist
Tip diameter	da/mm	90.85 ± 0.02	${163.75}_{-0.10}^{-0.05}$	${132.7}_{-0.20}^0$	229.70
Root diameter	df/mm	85.94 ± 0.1	145.817	${109.27}_{-0.54}^0$	211.70
Accuracy class	—	5	7	6	6

.

The KEYENCE’s ultra-high-speed contour measuring instrument is used as the ex-perimental line laser sensor, with specific parameters shown in

Table 3. Parameters of the line laser sensor.

Parameters	Units	Value
Measuring mode	—	data diffuse reflection
Reference distance	mm	60
Measuring range: Z-axis	mm	±8
Measuring range: X-axis (Near)	mm	13.5
Measuring range: X-axis (Far)	mm	15
Measuring accuracy: Z-axis	μm	0.4
Measuring accuracy: X-axis	μm	5
Linearity (Z-axis)	—	±0.1% of F.S.
Measuring point interval	μm	20
Sampling period (trigger interval)	μs	16 (high-speed), 32 (efficient)

. It adopts dual polarization technology, HSE3-CMOS photosensitive device, Sham’s Law and 2D Ernos-tar objective lens. The sensor has good stability, relatively high accuracy and good envi-ronmental adaptability.

5.2. Sensor Calibration and Measurement Process

In order to meet the demand for rapid and 100% full inspection of gears online, a series of specialized polyhedral artefacts matching different batches of gears are designed based on their basic parameters. As shown in Figure 8, based on the line laser 3D measurement principle and model of gears, different sensor positioning mechanisms are designed for different types of gear measurement. The pose parameters of the line laser sensor are accurately calibrated with the polyhedral artefacts. Furthermore, the measurement of the tested gear is completed for the acquisition of 3D data of the gear.

It is worth noting that the clamping mechanism of the gear adopts a ball bushing cage positioning method, which improves the centering between the positioning core shaft and the tested gear, and facilitates the automation of online detection of the tested gear.

5.3. Discussion on Experimental Results

5.3.1. Raw Data of 3D Tooth Surface

According to the experimental conditions of Section 5.1 and Section 5.2, the raw data of the 3D tooth surface point cloud (single measurement) obtained from gear 1 to gear 4 are shown in Figure 9.

5.3.2. Measurement Repeatability Analysis

Taking gear 2 as an example, repeatability analysis is conducted after repeating the measurement five times, as shown in

Table 4. Measurement repeatability of raw data.

Measuring Point	1st Time	2nd Time	3rd Time	4th Time	5th Time	Standard Deviation	Range
1	2.2315	2.2318	2.2308	2.2319	2.2316	0.000387	0.0011
2	2.229	2.229	2.229	2.2295	2.2299	0.000366	0.0009
3	2.2233	2.2231	2.2235	2.2236	2.2242	0.000372	0.0011
4	2.2222	2.2226	2.2226	2.2233	2.2234	0.000458	0.0012
5	2.22	2.2197	2.2195	2.2197	2.2199	0.000174	0.0005
6	2.2189	2.219	2.2188	2.2193	2.2193	0.000206	0.0005
7	2.2144	2.2147	2.2148	2.2151	2.215	0.000245	0.0007
8	2.2128	2.2129	2.2129	2.2132	2.2132	0.000167	0.0004
9	2.2013	2.201	2.2016	2.2016	2.2026	0.000538	0.0016
10	2.1742	2.1753	2.1746	2.176	2.1754	0.000632	0.0018
11	2.1605	2.1597	2.1607	2.1598	2.1605	0.000408	0.001
12	2.094	2.0954	2.0951	2.0953	2.0944	0.000546	0.0014
13	2.0717	2.0725	2.0722	2.0731	2.0727	0.000472	0.0014
14	2.0385	2.0382	2.0392	2.039	2.0393	0.000422	0.0011
15	1.982	1.9805	1.9809	1.9808	1.9809	0.000511	0.0015
16	1.6776	1.6784	1.6771	1.6775	1.6778	0.000426	0.0013
17	1.6357	1.6371	1.6365	1.6363	1.6353	0.000627	0.0018
18	1.5746	1.5758	1.5751	1.5752	1.576	0.000504	0.0014
19	1.3636	1.3652	1.3636	1.3638	1.3638	0.000607	0.0016
20	1.3184	1.3188	1.3186	1.3194	1.3186	0.000344	0.001
21	1.3013	1.3024	1.3028	1.3017	1.3029	0.000624	0.0016
22	1.2175	1.2167	1.2176	1.2185	1.2170	0.000615	0.0018
23	1.0682	1.0680	1.0692	1.0689	1.0681	0.000479	0.0012
24	1.0435	1.0441	1.0431	1.0446	1.0440	0.000516	0.0015
25	0.9629	0.9638	0.9633	0.9634	0.9630	0.000319	0.0009
					maximum	0.000632	0.0018

. The first 25 measurement points are selected for repeatability analysis, with a maximum standard deviation of 0.632 μm and a maximum range of 1.8 μm.

5.3.3. Accuracy Inspection of 3D Tooth Surface

• Profile Deviation.

Taking gear 2 as an example, the middle part of the tooth width is selected for tooth surface scanning, and one of the cross-sectional data parallel to the end section is taken for analysis of the involute profile deviation on the gear tooth surface. Figure 10 shows the complete tooth profile data and local magnification of the right and left tooth surfaces of gear 2, respectively.

According to the evaluation criteria for gears in ISO 1328-1:2013, four evenly distributed teeth of the tested gear are selected for analysis and calculation of involute profile deviation. The evaluation curve and results are shown in Figure 11a. Among them, the left profile deviations of the 29th gear tooth of the tested gear 2 are selected for detailed analysis, as shown in Figure 12. The evaluation results of profile deviations are as follows. f_Hα = −10.54 μm, F_α = 10.21 μm, f_fα = 5.62 μm.

Based on the evaluation results of gear profile deviation mentioned above, a comparative analysis is conducted with the evaluation results of traditional gear measurement centers (issued by the National Institute of Metrology in China; see Figure 11b). The comparison results show that the tooth profile evaluation results of optical measurement based on line laser sensors are slightly larger in numerical value than traditional contact measurement results but comply with the 3σ principles, and the evaluation levels fully correspond, which allows for the accurate identification of the quality level of gears. The reason for the differences is that the cross-sections where the tooth profiles used for evaluation are not completely the same, and there are differences in data processing methods, such as the selection of measurement points, filtering algorithms, etc.

• Helix Deviation.

By multiple scans of different cross-sections, the full tooth width information of the tested tooth surface is obtained, capturing the effective tooth height measurement area. Figure 13 shows the left tooth surface and the helix on the tooth surface of the first tooth of gear 2. The reference cylinder of the tested gear intersects with the gear tooth surface one by one to extract the helix curve of each tooth surface, and then we analyze the helix deviation on the gear tooth surface.

Similarly, according to the evaluation criteria for gears in ISO 1328-1:2013 [27], four evenly distributed teeth of the tested gear were selected for analysis and calculation of helix deviation. The evaluation curve and results are shown in Figure 14a. Based on the evaluation results of the gear helix deviation mentioned above, a comparative analysis is conducted with the evaluation results of the traditional gear measurement center (issued by the National Institute of Metrology in China; see Figure 14b. The same conclusion can be drawn as in the evaluation of tooth profile deviation.

• Pitch Deviation.

Based on the tooth profile curve obtained from scanning the tested gear, the evaluation area is cropped and retained. Then pitch deviation of the tested gear is measured and evaluated at the position of the reference circle. As shown in Figure 15a, for gear 2, the individual pitch deviation on the left tooth surface is 3.75 μm, and that on the right tooth surface is 3.67 μm. The cumulative pitch deviation on the left tooth surface is 12.54 μm, and that on the right tooth surface is 14.31 μm. Figure 15b shows the results issued by the National Institute of Metrology in China.

There is a certain degree of error in the inspection results of the two methods, because it is difficult for them to achieve the same measuring point on the pitch measurement circle. But this level of error is acceptable and does not affect the discrimination of gear accuracy level.

• Topological Deviation.

The topological deviation is shown in Figure 16 for the right tooth surface of the 29th tooth of gear 2, which can clearly identify the morphology of the entire tooth surface.

The distribution of normal deviation at different positions on the tooth surface, especially the shape and amount of tooth surface modification, can be intuitively represented from the topological deviation graph. In addition, the position with the highest normal deviation of the tooth surface, as well as the trend and rate of change of the normal deviation, can be further highlighted from the figure. At the same time, some new evaluation indicators can be discovered to better characterize the entire tooth surface, such as the flank twist, etc.

6. Conclusions

A measurement method using line laser sensors is proposed in this paper to obtain 3D tooth surface information and holistic deviations of tested gears. For the calibration issues of the pose parameters of the line laser sensor in measuring space, a specialized polyhedral artefact with specific geometric features is invented to determine the six degrees of freedom parameters of the line laser sensor in the machine coordinate system. Based on this, a single-spindle gear measuring instrument is developed and a series of experimental studies are conducted with this instrument. For gears with different module and flank directions, conventional indicators specified in international standards such as tooth profile, helix, and pitch are calculated to evaluate gear accuracy, and then compared with traditional contact measurement methods. This validates the correctness of the methods mentioned in this paper. In the future, how to fully utilize and mine big data obtained from the line laser 3D measurement of gears is a topic worth discussing and exploring. In particular, gear machining error tracing and performance prediction should be carried out based on big data of the entire tooth surface, such as the establishment of machining error maps, wave analysis, NVH analysis, etc. This can not only guide gear machining but also predict service performance. Furthermore, data processing and accuracy improvement of tooth surface information are also important links, which will affect the further expansion of the application direction of line laser measurement.