In modern industrial applications, large-scale involute cylindrical gears play a critical role in heavy-duty machinery such as mining, marine, metallurgy, and energy systems, where they transmit high power and withstand significant loads. The gear profile deviation is a key indicator of gear operational stability, and controlling it during manufacturing is essential for ensuring product quality and yield. Traditional offline measurement methods, which rely on coordinate measuring machines or dedicated CNC gear measuring centers, suffer from drawbacks like re-clamping errors,基准 misalignment, and inefficiencies due to the large size and weight of these gears. Integrated on-machine measurement technology addresses these issues by embedding precision measurement into the manufacturing process, enabling real-time error detection and compensation in a closed-loop “machining-measurement” mode. This approach enhances accuracy, productivity, and automation, making it ideal for large-scale gear grinding applications where gear profile grinding precision is paramount.
The on-machine measurement system for large-scale gear grinding involves a three-dimensional scanning probe that interacts with the gear tooth surface during motion. In a typical setup, such as the YK73 series CNC form gear grinding machine, the system comprises multiple servo axes: radial feed axis (X), longitudinal wheel feed axis (Z), gear indexing axis (C), helix angle rotation axis (A), and wheel dressing axes (Y and W). The topological relationship between these axes forms two kinematic chains: “probe-Y-axis-A-axis-Z-axis-X-axis-bed” and “bed-C-axis-gear.” During measurement, the probe follows a theoretical profile curve relative to the gear, and any deviations in the actual tooth surface cause the probe to deflect, which is recorded and processed to determine profile errors. This process is crucial for minimizing grinding cracks that can arise from inaccuracies in gear profile grinding.
To achieve high-precision on-machine measurement of involute profiles, various methods were analyzed, including the Cartesian coordinate method, generation method (normal polar coordinate method), meshing line method, and polar coordinate method. Each method was evaluated based on criteria such as minimal involvement of servo axes, least center-of-gravity shift, and shortest length基准. For instance, the generation method requires long, high-precision linear guides, which are difficult to manufacture and maintain in machine tool environments, especially for large gears. The Cartesian coordinate method involves significant重心偏移 due to probe installation offsets, leading to geometric and motion errors. The meshing line method necessitates interpolation of multiple axes (X, Y, C), amplifying error accumulation. In contrast, the polar coordinate method, which uses the C-axis and X-axis for interpolation, avoids long guideways and重心偏移, making it the optimal choice for on-machine measurement in large-scale gear grinding. This method ensures that the probe’s ball center follows the polar trajectory of the involute, reducing the impact of machine errors.
The comprehensive measurement model for the involute helical surface was established based on the engagement relationship between the probe and the tooth surface. Multiple coordinate systems were defined: ΣOb for profile deviation measurement in an end section, ΣOk for the normal plane at measurement point k, ΣOc for the contact point on the probe sphere, ΣOt for the probe sphere center, ΣOtb for the probe section in the measurement plane, and ΣOm for the machine (gear) coordinate system. Using homogeneous coordinate transformations, the position of the probe center Ot in ΣOm was derived as:
$$
\begin{aligned}
\mathbf{T}_m &= \mathbf{W}_{m,b} \cdot \mathbf{R}_{b,t} \cdot \mathbf{W}_{b,t} \\
\mathbf{P}_t^m &= \begin{bmatrix}
\cos\gamma & -\sin\gamma & 0 & 0 \\
\sin\gamma & \cos\gamma & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix} \cdot \begin{bmatrix}
\cos\phi_k & -\sin\phi_k & 0 & 0 \\
\sin\phi_k & \cos\phi_k & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix} \cdot \begin{bmatrix}
1 & 0 & 0 & r_b \cos\phi_k \\
0 & 1 & 0 & r_b \sin\phi_k \\
0 & 0 & 1 & b \\
0 & 0 & 0 & 1
\end{bmatrix} \cdot \begin{bmatrix}
1 & 0 & 0 & r_0 \cos\beta_b \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & r_0 \sin\beta_b \\
0 & 0 & 0 & 1
\end{bmatrix}
\end{aligned}
$$
where φk is the unfolding angle at point k, rb is the base circle radius, r0 is the probe sphere radius, and βb is the base helix angle. For on-machine measurement, the gear rotates via the C-axis, while the probe moves linearly via the X-axis, with their relationship given by:
$$
\theta_k = \arcsin\left(\frac{P_{t,Y}^m}{e}\right), \quad X_k = P_{t,X}^m \cos\theta_k – P_{t,Z}^m \sin\theta_k
$$
Here, θk and Xk are the coordinates of the C-axis and X-axis in the machine coordinate system, and e is the fixed Y-axis position. This model ensures accurate trajectory generation for profile measurement during gear grinding.
Error compensation is vital to counteract geometric and motion errors in the machine tool, which cause deviations between the actual and theoretical probe paths. The compensation principle involves establishing a mapping between machine errors and measurement errors, calculating error values at each measurement point, and applying post-compensation to the raw data. Based on multi-body system theory and homogeneous coordinate transformations, the machine’s accuracy model for the polar coordinate method was derived. The overall spatial error includes servo axis motion errors and perpendicularity errors between axes, as summarized in Table 1.
| Error Type | Characteristic Matrix |
|---|---|
| Servo Axis Motion Error |
$$\Delta \mathbf{U}_i = \begin{bmatrix} 1 & -\Delta \gamma_i & \Delta \beta_i & \Delta x_i \\ \Delta \gamma_i & 1 & -\Delta \alpha_i & \Delta y_i \\ -\Delta \beta_i & \Delta \alpha_i & 1 & \Delta z_i \\ 0 & 0 & 0 & 1 \end{bmatrix}$$ |
| Axis Perpendicularity Error |
$$\Delta \mathbf{W}_{i+1} = \begin{bmatrix} 1 & -\Delta \gamma_{i+1} & \Delta \beta_{i+1} & 0 \\ \Delta \gamma_{i+1} & 1 & -\Delta \alpha_{i+1} & 0 \\ -\Delta \beta_{i+1} & \Delta \alpha_{i+1} & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$ |
The machine accuracy model for the measurement state is expressed as:
$$
\mathbf{D}_{\text{machine}} = \prod_{i=1}^{n} \Delta \mathbf{W}_{i+1} \cdot \mathbf{U}_i \cdot \Delta \mathbf{U}_i
$$
Using this, the actual theoretical profile curve coordinates under spatial errors are computed, and the measurement error at each point is:
$$
\text{Error}(C_k, X_k) = f(C_k, X_k) – f'(C_k’, X_k’)
$$
where f and f’ represent the ideal and actual coordinate functions, respectively. This error is subtracted from the raw measurement data for compensation, improving accuracy without modifying the NC program—a post-compensation approach that avoids secondary errors.
Experiments were conducted on a five-axis CNC form gear grinding machine to validate the on-machine measurement and error compensation method. The gear parameters are listed in Table 2.
| Parameter | Value |
|---|---|
| Module | 10 mm |
| Number of Teeth | 36 |
| Helix Angle | 0° |
| Modification Coefficient | 0 |
| Pressure Angle | 20° |
The measurement involved programming the machine based on calculated C-axis and X-axis coordinates, with a scanning probe recording deviations. Laser interferometry was used to measure axis errors and perpendicularity across the full travel range. Compensation was applied to the raw data, and profile deviations were evaluated according to ISO 1328-1:2013. To illustrate the measurement setup, the following figure shows a typical grinding cracks inspection scenario in gear profile grinding:
Results from a sample tooth flank demonstrated the effectiveness of compensation: before compensation, profile deviation parameters showed significant errors, but after compensation, they closely matched those from a gear measuring instrument. For example, the total profile deviation Fα reduced from 10.6 μm to 4.3 μm, compared to 5.1 μm from the instrument. Table 3 compares the compensated and uncompensated results with instrument measurements.
| Parameter | Before Compensation | After Compensation | Gear Measuring Instrument |
|---|---|---|---|
| fHα | -6.3 | -4.1 | -3.3 |
| Fα | 10.6 | 4.3 | 5.1 |
| ffα | 2.7 | 3.2 | 3.5 |
Repeatability tests on three evenly distributed teeth, with five independent repetitions, confirmed the method’s reliability. For instance, on tooth No. 1, the standard deviations for left flank parameters were as low as 0.2 μm for Fα, indicating high consistency. The error fluctuation ranges were mostly under 1 μm, demonstrating precision in gear profile grinding applications. A sample measurement report showed close alignment with instrument results in deviation magnitude, position, and shape, underscoring the method’s accuracy.
In conclusion, the polar coordinate method for on-machine measurement of large-scale involute profiles in gear grinding machines is effective, leveraging minimal servo axes and avoiding重心偏移. The comprehensive measurement model and error compensation approach, based on multi-body system theory, successfully mitigate machine tool errors, enhancing measurement accuracy. Experimental results validate the method’s high precision, reliability, and suitability for industrial gear grinding, reducing the risk of grinding cracks and improving overall manufacturing quality. Future work could focus on real-time compensation integration and application to other gear types.