This paper presents a machine vision-based method for online measurement of helical gear helix angles using curve fitting algorithms. The proposed system integrates end-face measurement modules with side-view analysis to achieve rapid parameter detection while maintaining industrial compatibility.
1. Measurement System Architecture
The vision system combines dual imaging modules for comprehensive gear analysis:
| Component | Specification | Function |
|---|---|---|
| Double telecentric lens | 2×10⁶ pixels, 18mm DoF | Side-view image acquisition |
| CCD Camera | Global shutter, 4fps | Motion capture |
| LED Array | 6500K, 1200lm | Multi-angle illumination |
2. Mathematical Modeling
The spatial helix curve of helical gears can be expressed as:
$$
\begin{cases}
x = r\cos\left(\theta + \frac{2(n-1)\pi}{c} + \theta_0\right) \\
y = r\sin\left(\theta + \frac{2(n-1)\pi}{c} + \theta_0\right) \\
z = \pm r\theta\cot\beta
\end{cases}
$$
Where β represents the helix angle, c denotes tooth count, and r is the reference circle radius. The projection on x-z plane satisfies:
$$
x = r\cos\left(\frac{z\tan\beta}{r} + \frac{2n\pi}{c} + \theta_0\right)
$$
3. Image Processing Pipeline
The edge detection workflow combines homomorphic filtering with morphological operations:
| Processing Stage | Algorithm | Parameters |
|---|---|---|
| Illumination Correction | Homomorphic Filter | γ_L=0.5, γ_H=2.0 |
| Edge Detection | Canny Operator | σ=1.5, T_low=0.1 |
| Morphological Processing | Hit-Miss Transform | SE=[0 1;1 0] |
The directional filtering using structured elements effectively identifies tooth traces:
$$
N_{+45^\circ} = \sum_{i,j} (E_{i,j} \otimes SE_{+45^\circ}) \\
N_{-45^\circ} = \sum_{i,j} (E_{i,j} \otimes SE_{-45^\circ})
$$
Where SE denotes structural elements. Helix direction is determined by:
$$
\text{Handedness} = \begin{cases}
\text{Left} & N_{+45^\circ} > 2N_{-45^\circ} \\
\text{Right} & \text{otherwise}
\end{cases}
$$
4. Curve Fitting Optimization
The nonlinear least squares method optimizes helix angle estimation:
$$
\min_{\beta} \sum_{k=1}^m \left[x_k – r\cos\left(\frac{z_k\tan\beta}{r} + \phi_n\right)\right]^2
$$
Where φ_n represents the phase difference between teeth. The iterative optimization process achieves sub-pixel accuracy through:
| Parameter | Initial Value | Convergence Threshold |
|---|---|---|
| β | 20° (estimated) | 0.001° |
| r | End-face measurement | ±0.01mm |
5. Experimental Validation
Testing results demonstrate the method’s performance on various helical gears:
| Gear Specification | Nominal β | Measured β | Error |
|---|---|---|---|
| 1.5m 26z Left | 45.00° | 45.11° | +0.24% |
| 1.25m 25z Left | 19.52° | 19.45° | -0.36% |
| 1.5m 26z Right | 45.00° | 44.87° | -0.29% |
The system achieves measurement speeds under 1.2s per gear with repeatability:
$$
\sigma_\beta = \sqrt{\frac{1}{N-1}\sum_{i=1}^N (\beta_i – \bar{\beta})^2} < 0.08^\circ
$$
6. Error Analysis
Major error sources include:
$$
\delta\beta = \sqrt{\left(\frac{\partial\beta}{\partial r}\delta r\right)^2 + \left(\frac{\partial\beta}{\partial d_a}\delta d_a\right)^2 + \left(\frac{\partial\beta}{\partial\theta}\delta\theta\right)^2}
$$
Where d_a represents tip diameter. Calibration experiments show measurement uncertainty:
| Error Source | Magnitude | Contribution |
|---|---|---|
| Image Scaling | ±0.024mm/pixel | 42% |
| Edge Detection | ±0.5 pixel | 33% |
| Mechanical Vibration | ±0.1° | 25% |
This machine vision solution enables efficient online inspection of helical gears while maintaining compatibility with industrial production environments. The integration of optical correction and advanced image processing algorithms successfully addresses challenges in tooth trace identification and parameter calculation.