In modern industrial applications, internal gears play a critical role in transmission systems, robotics, and aerospace engineering. As an internal gear manufacturer, we constantly seek methods to enhance machining efficiency and precision. Traditional internal gear machining techniques, such as hobbing or grinding, often face limitations in speed and accuracy. Power skiving, or gear turning, has emerged as a high-efficiency alternative, but it suffers from servo system issues like steady-state error and overshoot. In this article, we explore a feedforward control strategy combined with differential feedback to improve the dynamic performance of servo systems in internal gear turning. We analyze the motion trajectory, derive control models, and validate the approach through simulations and experiments, focusing on how internal gears can be manufactured with higher precision.
The gear turning process involves complex interactions between the tool and workpiece. For internal gears, the machining requires precise coordination of multiple axes. We begin by establishing a coordinate system to describe the motion. Let the fixed coordinate system for the tool be denoted as $o – xyz$, and for the workpiece as $o_p – x_p y_p z_p$. The rotating coordinate systems are $o – x_1 y_1 z_1$ for the tool and $o_2 – x_2 y_2 z_2$ for the workpiece, with rotation angles $\phi_1$ and $\phi_2$, respectively. The axis angle is $\Sigma$, the center distance is $a$, and $l_2$ represents the offset due to workpiece feed. The transformation from the tool to workpiece coordinate system is given by:
$$ \mathbf{r}_2 = \mathbf{M}_{2p} \mathbf{M}_{po} \mathbf{M}_{o1} \mathbf{r}_1 $$
where $\mathbf{M}_{o1}$ is the transformation matrix from $o – x_1 y_1 z_1$ to $o – xyz$:
$$ \mathbf{M}_{o1} = \begin{bmatrix} \cos \phi_1 & -\sin \phi_1 & 0 & 0 \\ \sin \phi_1 & \cos \phi_1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$
$\mathbf{M}_{po}$ transforms from $o – xyz$ to $o_p – x_p y_p z_p$:
$$ \mathbf{M}_{po} = \begin{bmatrix} 1 & 0 & 0 & a \\ 0 & \cos \Sigma & -\sin \Sigma & 0 \\ 0 & \sin \Sigma & \cos \Sigma & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$
and $\mathbf{M}_{2p}$ from $o_p – x_p y_p z_p$ to $o_2 – x_2 y_2 z_2$:
$$ \mathbf{M}_{2p} = \begin{bmatrix} \cos \phi_2 & -\sin \phi_2 & 0 & 0 \\ \sin \phi_2 & \cos \phi_2 & 0 & 0 \\ 0 & 0 & 1 & -l_2 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$
This kinematic model helps in understanding the relative motion during internal gear turning. However, the servo system’s performance is crucial for accuracy. Traditional CNC servo systems use a three-loop control structure (position, velocity, and current loops), which can lead to significant following errors. The following error $E$ is defined as:
$$ E = \frac{v}{K_v} $$
where $v$ is the command velocity and $K_v$ is the velocity gain. Reducing $E$ by increasing $K_v$ may cause instability, while decreasing $v$ lowers efficiency. Thus, we propose a feedforward control approach to mitigate this issue.
In servo systems, feedforward control compensates for disturbances in an open-loop manner. We integrate velocity and acceleration feedforward with differential negative feedback to enhance tracking performance. The block diagram of the proposed system includes feedforward functions $F_v(s)$ for velocity and $F_c(s)$ for acceleration, along with a differential feedback loop. The velocity loop transfer function is $G_n(s) = \frac{K_n}{\tau_n s + 1}$, the current loop is $G_c(s) = \frac{K_c}{\tau_c s + 1}$, and the motor transfer function is $G_t(s) = \frac{K_t}{J s^2}$, where $K_n$, $\tau_n$, $K_c$, $\tau_c$, $K_t$, and $J$ are system parameters. The closed-loop transfer function with feedforward is:
$$ \frac{C(s)}{R(s)} = \frac{G_c(s) G_t(s) [F_v(s) G_n(s) + F_c(s) + K_p G_n(s)]}{1 + K_p G_n(s) G_c(s) G_t(s) + G_n(s) G_c(s) G_t(s)} $$
To achieve zero tracking error, we set this equal to 1, yielding the feedforward functions:
$$ F_v(s) = s \quad \text{and} \quad F_c(s) = \frac{J s^2}{K_c K_t} $$
When differential feedback is added, with $G_d = 1 + \tau_d s$, the modified feedforward functions become:
$$ F_v(s) = (1 + K_p \tau_d) s \quad \text{and} \quad F_c(s) = \frac{J}{K_c K_t} s^2 $$
This composite control strategy reduces steady-state error and overshoot, improving dynamic performance for internal gear manufacturer applications.
We simulated the traditional, feedforward, and feedforward-differential composite control systems in MATLAB/Simulink to evaluate their performance. The parameters for the simulation are summarized in the table below, which includes different configurations for comparison.
| System Type | $K_p$ | $\tau_n$ | $J / (K_c K_t)$ |
|---|---|---|---|
| Traditional Servo | 250 | 0 | 0 |
| Feedforward Servo | 250 | 0.0004 | 0.006 |
| Feedforward-Differential Servo | 250 | 0.0004 | 0.006 |
| Feedforward Servo | 150 | 0.0004 | 0.006 |
| Feedforward-Differential Servo | 150 | 0.0004 | 0.006 |
The simulation results show that the feedforward-differential composite control achieves smaller overshoot and steady-state error in step responses. For sinusoidal tracking, the composite system exhibits superior accuracy, as illustrated by the reduced phase lag and amplitude error. This is crucial for internal gears, where tooth profile precision is paramount. The following error comparison in the time domain demonstrates a significant reduction, enhancing the overall machining quality for internal gear manufacturer processes.
To validate the simulation findings, we conducted experiments on a gear turning machine. The setup included monitoring the following errors for the X, Z, A, and C axes under both traditional and feedforward control. The results are summarized in the table below, highlighting the improvement with feedforward control.
| Axis | Following Error without Feedforward (μm or mm) | Following Error with Feedforward (μm or mm) | Reduction Percentage |
|---|---|---|---|
| X-axis | -2.8 μm | -1.0 μm | 64.3% |
| Z-axis | -7.6 μm | -2.5 μm | 67.1% |
| A-axis | -31.59 mm | -3.16 mm | 90.0% |
| C-axis | 10.80 mm | 2.26 mm | 79.0% |
These results confirm that feedforward control drastically reduces following errors, leading to better servo system performance. For internal gears, this translates to higher accuracy in tooth formation. We further assessed the gear quality by measuring tooth profile and pitch deviations using a Gleason 650GMS system. The table below compares the gear errors with and without feedforward-differential control.
| Error Type | Without Feedforward (μm) | With Feedforward-Differential (μm) | Accuracy Grade Improvement |
|---|---|---|---|
| Right Flank Total Cumulative Deviation $F_P$ | 70.9 | 44.3 | From Grade 8 to 7 |
| Left Flank Total Cumulative Deviation $F_P’$ | 56.4 | 47.5 | From Grade 8 to 7 |
| Right Flank Single Pitch Deviation $f_P$ | 10.8 | 8.9 | From Grade 8 to 7 |
| Left Flank Single Pitch Deviation $f_P’$ | 13.7 | 10.6 | From Grade 8 to 7 |
| Total Profile Deviation $F_\alpha$ | 21.1 | 11.8 | From Grade 9 to 6 |
The data shows that the feedforward-differential composite control significantly improves gear accuracy, with reductions in pitch and profile deviations. This is essential for internal gear manufacturer standards, as it ensures reliable performance in high-precision applications. The composite control strategy not only enhances dynamic response but also suppresses overshoot, making it ideal for internal gears production.
In conclusion, the integration of feedforward and differential feedback control in servo systems for internal gear turning offers substantial benefits. By deriving and implementing velocity and acceleration feedforward functions, we reduce following errors and improve tracking precision. The simulations and experiments validate that this approach minimizes steady-state error and overshoot, leading to higher-quality internal gears. For internal gear manufacturer processes, this method enhances efficiency and accuracy, supporting advancements in industries like aerospace and robotics. Future work could focus on adaptive control techniques to further optimize performance for complex internal gears geometries.