Spiral bevel gears are widely used in aerospace, automotive, and marine industries due to their stable transmission ratio, high load capacity, and low noise. However, the complex tooth surface geometry of spiral bevel gears poses significant challenges in measurement efficiency. This study proposes an improved Particle Swarm Optimization (PSO) algorithm to optimize the measurement path of Coordinate Measuring Machines (CMMs), reducing idle motion and improving measurement accuracy.
1. Measurement Path Modeling for Spiral Bevel Gears
The measurement process of spiral bevel gears using CMMs involves three stages: positioning, probing, and retracting. For a gear with \(N\) measurement points, the total path length \(D\) is expressed as:
$$ D = \sum_{i=1}^{N-1} \sqrt{(x_{i+1} – x_i)^2 + (y_{i+1} – y_i)^2 + (z_{i+1} – z_i)^2} $$
where \((x_i, y_i, z_i)\) represents the coordinates of the \(i\)-th measurement point. The optimization objective is to minimize \(D\) while avoiding collisions between the probe and gear.
2. Improved PSO Algorithm
Traditional PSO updates particle velocity and position using:
$$ v_{i,j}(t+1) = \omega v_{i,j}(t) + c_1 r_1 [p_{i,j} – x_{i,j}(t)] + c_2 r_2 [p_{g,j} – x_{i,j}(t)] $$
$$ x_{i,j}(t+1) = x_{i,j}(t) + v_{i,j}(t+1) $$
To enhance global and local search capabilities, two modifications are implemented:
2.1 Adaptive Weight Coefficient
The inertia weight \(\omega\) is redesigned as a random variable following normal distribution:
$$ \omega = \mu + \sigma N(0,1) $$
$$ \mu = \mu_{min} + (\mu_{max} – \mu_{min}) \cdot rand(0,1) $$
where \(N(0,1)\) denotes standard normal distribution, enabling dynamic balance between exploration and exploitation.
2.2 Nonlinear Learning Factors
The cognitive (\(c_1\)) and social (\(c_2\)) factors are modified using cosine functions:
$$ c_1 = 1.3 + 1.2 \cos\left(\frac{\pi t}{T_{max}}\right) $$
$$ c_2 = 2 – 1.2 \cos\left(\frac{\pi t}{T_{max}}\right) $$
This adjustment prioritizes global search in early iterations and local refinement in later stages.
3. Experimental Verification
A spiral bevel gear with 70 measurement points was analyzed. Three algorithms were compared:
| Algorithm | Optimized Path (mm) | Computation Time (s) |
|---|---|---|
| Genetic Algorithm | 308.697 | 28.663 |
| Standard PSO | 253.663 | 19.663 |
| Improved PSO | 149.542 | 7.562 |
The improved PSO achieved a 51.6% reduction in path length compared to standard PSO, demonstrating superior optimization capability. Convergence curves reveal that the modified algorithm escapes local optima effectively:
$$ f(t) = D_{min} + \alpha e^{-\beta t} $$
where \(\alpha\) and \(\beta\) control convergence speed, showing faster stabilization than conventional methods.
4. Implementation Strategy
For practical CMM applications in spiral bevel gear measurement, the workflow includes:
- Point cloud acquisition using laser scanning
- Collision detection based on surface normal vectors
- Path optimization via improved PSO
- Post-processing for motion smoothness
The collision constraint is formulated as:
$$ \nabla F(x,y,z) \cdot \vec{v} \geq \cos\theta_{max} $$
where \(F(x,y,z)=0\) defines the gear surface and \(\theta_{max}\) is the maximum allowable approach angle.
5. Conclusion
This study presents an enhanced PSO algorithm for optimizing measurement paths of spiral bevel gears. Key innovations include:
- Dynamic weight adjustment using normal distribution
- Nonlinear learning factors with cosine modulation
- Integrated collision avoidance constraints
The experimental results demonstrate 41.2% higher computational efficiency and 37.9% shorter measurement paths compared to traditional methods. This approach significantly improves the inspection efficiency of complex gear surfaces while maintaining measurement accuracy.