In modern manufacturing, gear hobbing is a highly efficient process for producing both conventional and non-circular gears. The gear hobbing machine plays a critical role in achieving precision in gear profiles. However, factors such as guideway straightness errors, ball screw backlash, tool wear, and thermal deformation can introduce significant errors in gear contour accuracy. Non-circular gears, with their unique advantages in motion transmission, require high-precision machining, but their non-circular pitch curves lead to frequent acceleration and deceleration in multi-axis motion, causing vibrations and sudden changes in cutting forces. This exacerbates the degradation of machining accuracy. As the operational speeds of gear hobbing machines increase, motion control errors become a dominant factor affecting the quality of non-circular gear surfaces. In this study, we focus on improving the tracking accuracy of the workpiece axis (C-axis) and radial feed axis (X-axis) in a CNC gear hobbing machine through advanced control strategies.
The gear hobbing process for non-circular gears involves complex kinematic relationships. For spur non-circular gears, the machining motions include generating motion, radial feed motion, and axial feed motion. The generating motion ensures the workpiece (C-axis) follows the hob (B-axis) rotation according to the meshing relationship, while the radial feed motion (X-axis) adjusts the center distance dynamically due to the varying polar radius of the workpiece. The axial feed motion (Z-axis) controls the tool movement along the workpiece axis. The mathematical model for the motion relationship between the hob and workpiece is given by:
$$ \omega_c = \frac{k m}{2} \cdot \frac{r^2 + (dr/d\theta)^2}{r} \cdot \omega_b $$
$$ v_x = \frac{k m}{2} \cdot \frac{dr}{d\theta} \cdot \omega_b $$
where $\omega_b$ is the hob rotational speed, $\omega_c$ is the workpiece rotational speed, $v_x$ is the radial feed velocity, $r$ is the polar radius of the pitch curve, $\theta$ is the polar angle, $k$ is the number of hob starts, and $m$ is the module. In each interpolation cycle, the Electronic Gear Box (EGB) module calculates the incremental movements $\Delta \theta_c$ for the C-axis and $\Delta x$ for the X-axis based on the feedback from the B-axis encoder and the multi-axis control model, enabling master-slave synchronization.
Servo control systems in gear hobbing machines typically employ Proportional-Integral-Derivative (PID) controllers. While PID feedback control reduces the impact of external disturbances and parameter variations, it often results in poor tracking performance in nonlinear contouring applications, leading to significant tracking errors. To enhance servo tracking performance, we incorporate feedforward control, which anticipates future command signals and broadens the system bandwidth. The combined feedforward and feedback control structure reduces the phase lag inherent in feedback systems and compensates for deviations in real-time. The error transfer function for a single-axis system is:
$$ E(s) = \frac{1 – F(s) P(s)}{1 + G(s) P(s)} R(s) $$
where $R(s)$ is the input signal, $Y(s)$ is the output, $E(s)$ is the error, $P(s)$ is the plant transfer function, $G(s)$ is the PID controller transfer function, and $F(s)$ is the feedforward transfer function. To achieve zero error ($E(s) = 0$), the feedforward transfer function must satisfy $F(s) = 1/P(s)$. Expanding $F(s)$ as a series in $s$ yields:
$$ F(s) = a_1 s + a_2 s^2 + a_3 s^3 + \cdots + a_n s^n $$
In practice, higher-order derivatives are challenging to implement due to nonlinearities and noise sensitivity. Therefore, we limit the feedforward to velocity (first derivative) and acceleration (second derivative) terms. The velocity feedforward compensates for command position changes by adding a gain-scaled first derivative to the velocity command, improving response speed. The acceleration feedforward adds a gain-scaled second derivative to the acceleration command, suppressing overshoot from velocity feedforward. For the C-axis and X-axis of the gear hobbing machine, we adopt a “velocity/acceleration feedforward + PID/PI” servo control model. The control law for each axis can be expressed as:
$$ u(t) = K_p e(t) + K_i \int e(t) dt + K_d \frac{de(t)}{dt} + K_{fv} \frac{dr(t)}{dt} + K_{fa} \frac{d^2 r(t)}{dt^2} $$
where $u(t)$ is the control signal, $e(t)$ is the tracking error, $r(t)$ is the reference signal, $K_p$, $K_i$, $K_d$ are the PID gains, and $K_{fv}$, $K_{fa}$ are the velocity and acceleration feedforward gains, respectively.
To optimize the controller parameters, we employ the Particle Swarm Optimization (PSO) algorithm. PSO is a global optimization technique inspired by bird flocking behavior, where each particle represents a potential solution (i.e., a set of controller parameters). Each particle has a position and velocity, and it updates its state based on individual and collective experiences. The update equations for the $i$-th particle are:
$$ x_i^{t+1} = x_i^t + v_i^{t+1} $$
$$ v_i^{t+1} = \omega v_i^t + c_1 r_1 (p_i^t – x_i^t) + c_2 r_2 (g^t – x_i^t) $$
where $x_i^t$ and $v_i^t$ are the position and velocity at iteration $t$, $\omega$ is the inertia weight, $c_1$ and $c_2$ are acceleration constants, $r_1$ and $r_2$ are random numbers in [0,1], $p_i^t$ is the particle’s best position, and $g^t$ is the global best position. The fitness function for optimization is the Integrated Time Absolute Error (ITAE), defined as:
$$ J_{\text{ITAE}} = \int_0^\infty t |e(t)| dt $$
ITAE penalizes errors that persist over time, leading to improved dynamic and static performance. The optimization process involves initializing a swarm of particles within specified bounds for the controller parameters, simulating the system in MATLAB/Simulink to compute ITAE, and iteratively updating particle positions until convergence.
For simulation, we model the servo drive system of the gear hobbing machine’s C-axis and X-axis. The plant transfer function for each axis considers the motor torque constant, inertia, and damping. The parameters for the C-axis and X-axis are listed in the table below:
| Parameter | C-axis | X-axis |
|---|---|---|
| KaKt (Nm/V) | 0.6 | 0.6 |
| J (kg·m²) | 0.95×10⁻³ | 0.95×10⁻³ |
| B (kg·m²/s) | 0.63×10⁻³ | 0.63×10⁻³ |
| Transmission ratio rg | 57.296 °/rad | 0.796 mm/rad |
The initial controller parameters for the C-axis and X-axis are set as follows:
| Parameter | C-axis | X-axis |
|---|---|---|
| Kp | 30 | 180 |
| Ki | 0.5 | 5 |
| Kd | 5 | 5 |
| Kfv | 0.0174 | 1.256 |
| Kpv | 30 | 50 |
| Kiv | 0.3 | 5 |
| Kfa | 0.000028 | 0.002 |
We configure the PSO algorithm with an inertia weight of 0.9, acceleration constants of 2, a swarm size of 100, and a maximum of 200 iterations. The search bounds for the C-axis parameters are [0,50] for Kp, [0,0.2] for Ki, [0,5] for Kd, [0,0.02] for Kfv, [0,50] for Kpv, [0,15] for Kiv, and [0,0.02] for Kfa. For the X-axis, the bounds are [0,300] for Kp, [0,5] for Ki, [0,5] for Kd, [0,0.02] for Kfv, [0,200] for Kpv, [0,15] for Kiv, and [0,0.02] for Kfa. A nonlinear friction value Tc = 0.016 Nm is included based on prior identification.
After optimization, the PSO-derived parameters for the C-axis and X-axis are:
| Parameter | C-axis | X-axis |
|---|---|---|
| Kp | 44.27310 | 296.13690 |
| Ki | 0.03940 | 3.32350 |
| Kd | 4.48520 | 2.26350 |
| Kfv | 0.00031 | 0.00667 |
| Kpv | 48.77500 | 54.53080 |
| Kiv | 9.83560 | 11.16490 |
| Kfa | 0.00026 | 0.01084 |
The fitness iteration curves for the C-axis and X-axis show convergence within 200 iterations, indicating effective optimization. To evaluate performance, we simulate the system with a sinusoidal input $x(t) = \sin(t)$ over 6.28 seconds. The tracking errors for the C-axis and X-axis are compared before and after optimization. For the C-axis, the maximum tracking error decreases from $5.85 \times 10^{-4}$ rad to $3.96 \times 10^{-4}$ rad. For the X-axis, the error reduces from 7.2 μm to 4.36 μm. The optimized feedforward PID control significantly reduces tracking errors, enhancing the gear hobbing machine’s motion accuracy.
The improved tracking performance directly translates to higher precision in non-circular gear hobbing. By minimizing errors in the C-axis and X-axis, the gear hobbing machine can better follow the complex trajectories required for non-circular pitch curves, reducing profile deviations and improving overall gear quality. This approach demonstrates the effectiveness of combining advanced control strategies with optimization algorithms in industrial applications like gear hobbing.
In conclusion, this study presents a comprehensive method for optimizing PID control parameters in a gear hobbing machine using particle swarm optimization. The integration of velocity and acceleration feedforward with PID feedback control, coupled with PSO-based tuning, effectively reduces tracking errors in the workpiece and radial feed axes. The simulation results validate the proposed approach, showing significant improvements in tracking accuracy and non-circular gear machining precision. Future work could explore real-time adaptation of controller parameters under varying machining conditions in gear hobbing processes.