As a leading internal gear manufacturer, we constantly strive to enhance measurement accuracy in the production of internal gears. The laser scanning measurement system represents a breakthrough technology for non-contact inspection of complex gear profiles, enabling comprehensive error assessment in a single operation. This article presents a novel parameter optimization approach for calibrating such systems, which is fundamental for maintaining the stringent quality standards required in internal gear manufacturing. The methodology leverages standard ring gauges and advanced optimization algorithms to achieve unprecedented precision in system parameter estimation.
The internal gear laser scanning measurement system comprises several key components: measurement adjustment mechanisms, work tables, laser displacement sensors, encoders, air-bearing rotary tables, drive motors with controllers, and upper computer processing systems. These elements work in concert to perform circumferential scans of gear profiles. The system’s core innovation lies in its ability to measure both internal and external gears with high precision, making it indispensable for modern internal gear manufacturers. The measurement model transforms sensor readings into spatial coordinates through a carefully derived mathematical relationship.
In the coordinate system definition, the rotary axis serves as the Z-axis, with its projection on the measurement plane as the origin O. The reverse direction of the measurement beam at zero encoder angle defines the positive Y-axis, forming a right-handed coordinate system. For a rotation angle φ, the coordinates of point P are given by:
$$(x, y, z)^T = O_{bv}(\phi, \rho, z_0, r_0, e) = \begin{pmatrix} r_0 + \rho & e & 0 \\ e & -(r_0 + \rho) & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} \sin \phi \\ \cos \phi \\ z_0 \end{pmatrix}$$
Here, ρ represents the displacement sensor reading, z₀ denotes the Z-axis position of the scanning plane, r₀ is the measurement arm length, and e is the distance from the measurement beam to the rotation center. The calibration process focuses on estimating parameters r₀ and e, which are critical for accurate measurements of internal gears. This static model parameter estimation problem forms the foundation of our optimization approach.
Our parameter estimation algorithm utilizes standard ring gauges as calibration artifacts. When the system scans a standard ring with diameter Φ, the ring’s axis projects onto the measurement plane at point O_S(x_S, y_S). For rotation angles φ_{i,i1} and φ_{i,i2}, the corresponding sensor readings are (φ_{i,i1}, ρ_{i,i1}) and (φ_{i,i2}, ρ_{i,i2}). The geometric constraint |P_{i,i1}O_S| = |P_{i,i2}O_S|, combined with the coordinate transformation, yields a nonlinear equation:
$$x^T A_i x + b_i^T x + c_i = 0$$
Where x = [x₁, x₂, x₃, x₄]^T = [r₀, e, x_S, y_S]^T, and the matrices A_i, vectors b_i, and scalars c_i are derived from the measurement data. By selecting multiple point pairs (i=1,2,3,4), we obtain a system of quadratic equations that can be solved through rotational and translational transformations to eliminate linear terms.
To address measurement noise and improve estimation accuracy, we formulate a nonlinear optimization problem based on geometric distance minimization. For each measured point P_i with coordinates (x_i, y_i)^T = O_{bv}(φ_i, ρ_i, r₀, e), the approximate geometric distance to the standard ring is:
$$d_i = \left| \sqrt{(x_i – x_S)^2 + (y_i – y_S)^2} – \frac{\Phi}{2} \right|$$
The nonlinear objective function becomes:
$$F(x) = \sum_{i=1}^{m} d_i^2$$
Where m represents the number of scanning points per revolution. For computational efficiency, we define:
$$f_i(x) = \|P_i – O_S\|^2 – \left(\frac{\Phi}{2}\right)^2 = (x_i – x_S)^2 + (y_i – y_S)^2 – \left(\frac{\Phi}{2}\right)^2$$
Thus, the calibration problem transforms into a nonlinear least squares problem (NLSM):
$$x^* = \arg \min_x \{F(x)\} \quad \text{s.t.} \quad x = [x_m, x_M]$$
Where F(x) = ½‖f(x)‖² = ½f(x)^T f(x), with f: R⁴ → R^m (m > 4) being a vector function, and [x_m, x_M] defines the feasible region for x.
We employ the Levenberg-Marquardt (LM) algorithm to solve this NLSM problem. The approach iteratively approximates the solution through a series of linear least squares problems. At each iteration k, we linearize f_i(x) around the current estimate x^(k) and solve for the next estimate x^(k+1). The damping parameter μ is adaptively controlled using the gain ratio:
$$r = \frac{2[F(x^{(k)}) – F(x^{(k+1)})]}{h_{lm}^T (\mu h_{lm} – g)}$$
Where h_lm represents the LM descent direction, and g = (J(x^(k)))^T f(x), with J(x^(k)) being the Jacobian matrix of F(x). The parameter μ is updated as μ = μ × max{⅓, 1 – (2r – 1)³} when r > 0, and μ = μ × ν (where ν is the gain step size) when r < 0. This adaptive control ensures stable convergence toward the optimal solution.
Experimental validation was conducted using a Bowers ring gauge with nominal diameter Φ_nominal = 180.005 mm. Precise measurement with a PLM600 length measuring instrument established the actual diameter at Φ = 180.0109 mm for the specified cross-section. Our internal gear laser scanning measurement system prototype incorporated a KEYENCE LK-H080 laser displacement sensor and a RENISHAW SiGNUM™ RESM rotary encoder system. We performed six independent calibration experiments to evaluate the algorithm’s precision and consistency, with results summarized in the following table:
| Ring Diameter Φ (mm) | Calibrated Parameter r₀ (mm) | Calibrated Parameter e (mm) | Computed Diameter Φ̂ (mm) | Absolute Error |Φ – Φ̂| (mm) |
|---|---|---|---|---|
| 180.0109 | 91.3673 | 0.2015 | 180.00629 | 0.00461 |
| 180.0109 | 91.3674 | 0.2015 | 180.00629 | 0.00461 |
| 180.0109 | 91.3677 | 0.1973 | 180.00630 | 0.00460 |
| 180.0109 | 91.3676 | 0.2015 | 180.00629 | 0.00461 |
| 180.0109 | 91.3677 | 0.2014 | 180.00629 | 0.00461 |
| 180.0109 | 91.3690 | 0.1973 | 180.00630 | 0.00460 |
The calibration results demonstrate exceptional performance, with diameter errors remaining below 4.7 μm across all experiments. The consistency in parameter estimates confirms the algorithm’s reliability for internal gear measurement applications. For internal gear manufacturers, this level of precision ensures that produced internal gears meet the most demanding tolerance requirements. The repeatability of results across multiple independent trials highlights the robustness of our approach.
The mathematical foundation of our method can be extended to various measurement scenarios. The Jacobian matrix J(x) plays a crucial role in the optimization process, with elements derived from partial derivatives of the objective function. For parameter vector x = [r₀, e, x_S, y_S]^T, the components of J(x) are calculated as follows:
$$\frac{\partial f_i}{\partial r_0} = 2(x_i – x_S) \frac{\partial x_i}{\partial r_0} + 2(y_i – y_S) \frac{\partial y_i}{\partial r_0}$$
$$\frac{\partial f_i}{\partial e} = 2(x_i – x_S) \frac{\partial x_i}{\partial e} + 2(y_i – y_S) \frac{\partial y_i}{\partial e}$$
$$\frac{\partial f_i}{\partial x_S} = -2(x_i – x_S)$$
$$\frac{\partial f_i}{\partial y_S} = -2(y_i – y_S)$$
Where the partial derivatives of coordinates with respect to parameters are obtained from the measurement model:
$$\frac{\partial x_i}{\partial r_0} = \sin \phi_i, \quad \frac{\partial x_i}{\partial e} = \cos \phi_i$$
$$\frac{\partial y_i}{\partial r_0} = \cos \phi_i, \quad \frac{\partial y_i}{\partial e} = -\sin \phi_i$$
These derivatives enable efficient computation of the LM update steps, ensuring rapid convergence to the optimal parameters. The algorithm typically converges within 10-15 iterations for practical applications in internal gear measurement systems.
For internal gear manufacturers, the implications of this research are significant. The ability to accurately calibrate measurement systems directly impacts product quality and manufacturing efficiency. Traditional calibration methods often require specialized artifacts and complex procedures, whereas our approach utilizes standard ring gauges readily available in most metrology laboratories. The table below compares key aspects of our method with conventional approaches:
| Feature | Proposed Method | Conventional Methods |
|---|---|---|
| Calibration Artifact | Standard Ring Gauge | Specialized Masters |
| Computational Complexity | Moderate (NLSM) | Varies (Often Simpler) |
| Accuracy (Diameter Error) | < 5 μm | Typically 10-20 μm |
| Repeatability | High (Consistent Results) | Moderate to Low |
| Implementation Cost | Low (Standard Equipment) | High (Special Tools) |
The optimization process incorporates several important considerations for practical implementation. Initial parameter estimates can be obtained through simple geometric measurements or preliminary calculations. The feasible region [x_m, x_M] constrains the parameter space based on physical system limitations. For typical internal gear measurement systems, reasonable bounds might be r₀ ∈ [90, 95] mm and e ∈ [0.15, 0.25] mm, though these values depend on specific system configurations.
Measurement noise presents a significant challenge in practical applications. Our approach accounts for this through the statistical framework of nonlinear least squares estimation. The residual vector f(x*) at the optimal solution provides insights into measurement quality, with the covariance matrix of parameter estimates given by:
$$\text{Cov}(x^*) = \sigma^2 (J(x^*)^T J(x^*))^{-1}$$
Where σ² represents the variance of measurement errors, estimated from the residuals. This statistical analysis enables uncertainty quantification for the calibrated parameters, which is crucial for quality assurance in internal gear manufacturing.
The versatility of our calibration method extends beyond internal gears to various cylindrical components. However, the specific application to internal gears requires additional considerations due to their complex tooth profiles. The measurement system must maintain calibration accuracy across the entire measurement volume, particularly when scanning different sections of internal gears through vertical adjustment of the sensor position.
Future work will focus on several enhancements to the current methodology. First, we plan to investigate the integration of temperature compensation to address thermal effects on measurement accuracy. Second, we will explore automated initial parameter estimation to reduce dependency on manual measurements. Third, we aim to extend the approach to simultaneous multi-parameter calibration, including additional factors such as sensor alignment errors and rotary axis wobble.
In conclusion, our parameter optimization estimation method for internal gear laser scanning measurement systems demonstrates exceptional accuracy and reliability. The combination of standard ring gauges with advanced nonlinear optimization algorithms provides a practical solution for system calibration. For internal gear manufacturers, this approach enables higher quality control standards and more efficient production processes. The consistent sub-5μm accuracy achieved in our experiments meets the most demanding requirements for precision internal gears, establishing a new benchmark in gear metrology.