In my research on advanced gear transmissions for robotic applications, I have focused on developing a compact, high-precision worm gear reducer that can replace traditional harmonic or RV reducers. The core innovation is a beveloid internal gear plane enveloping external-rotor drum worm gear transmission device. This worm gear system integrates the motor inside the worm, uses a symmetrically wedge-shaped internal gear for backlash adjustment, and eliminates the need for a bulky housing through a support shaft design. In this article, I present the complete theoretical foundation, parametric design, contact analysis, and a practical device architecture for this novel worm gear.
Coordinate Frames and Meshing Equations
To derive the tooth surface equations of the drum worm gear, I established six coordinate frames for both the drive side (A-side) and the reverse side (B-side) of the worm gear pair. The fixed frames \( \sigma_0 (O_0, x_0, y_0, z_0) \) and \( \sigma_4 (O_4, x_4, y_4, z_4) \) represent the initial positions of the worm and the internal gear, separated by the center distance \( a \). The moving frames \( \sigma_1 (O_1, x_1, y_1, z_1) \) attach to the worm, while \( \sigma_2 (O_2, x_2, y_2, z_2) \) and \( \sigma_3 (O_3, x_3, y_3, z_3) \) attach to the internal gear. The working angle \( \varphi_3 \) is related to the worm rotation angle \( \varphi_1 \) and the internal gear rotation angle \( \varphi_2 \) by:
\[
\varphi_3 = \frac{r_b}{r_2} \varphi_1, \quad \varphi_1 = i_{12} \varphi_2, \quad i_{12} = \frac{z_2}{z_1}
\]
where \( r_b \) is the base circle radius, \( r_2 \) is the pitch circle radius of the internal gear, \( z_2 \) is the number of internal gear teeth, and \( z_1 \) is the number of worm starts. The relative velocity vector \( \mathbf{v}^{(31)} \) at the meshing point expressed in frame \( \sigma_3 \) is:
\[
\begin{aligned}
v_x^{(31)} &= 2 f_{21} u – v \cos \beta \sin \varphi_3, \\
v_y^{(31)} &= f_{21} v \sin \beta – v \cos \beta \cos \varphi_3 – r_b f_{21}, \\
v_z^{(31)} &= -v \sin \beta \sin \varphi_3 + u \sin \varphi_3 + a \cos \varphi_3 – H
\end{aligned}
\]
The meshing condition for the A-side plane tooth surface is \( \Phi = \mathbf{v}^{(31)} \cdot \mathbf{n}^{(3)} = 0 \), which yields the A-side meshing equation:
\[
v = \left( r_b + \frac{a}{\sin \varphi_3} – \frac{H}{\sin \varphi_3} \right) \sin \beta – \frac{a \cos \beta}{\tan \varphi_3}
\]
For the B-side, the reverse rotation involves a different relationship between worm rotation and internal gear rotation:
\[
\varphi_1′ = i_{12} \varphi_3 – (\pi – \theta) i_{12}
\]
The B-side meshing equation is derived similarly:
\[
v’ = \left( r_b + \frac{a}{\sin \varphi_2} – \frac{H}{\sin \varphi_2} \right) \sin \beta – \frac{a \cos \beta}{\tan \varphi_2}
\]
These equations define the tooth surfaces of the drum worm gear, which are generated by the plane surfaces of the beveloid internal gear.
Design Parameters of the Worm Gear Pair
Based on the selected U10PLUS KV170 motor (parameters shown below), I determined the geometric parameters of the worm gear transmission. The motor sits inside the worm, so its outer diameter constrains the worm’s internal cavity.
| Parameter | Value |
|---|---|
| Motor type | U10PLUS KV170 |
| Outer diameter (mm) | 85 |
| Length (mm) | 100 |
| Rated speed (rpm) | 3000 |
| Rated torque (Nm) | 3.5 |
| Parameter | Symbol | Value |
|---|---|---|
| Center distance | a (mm) | 100 |
| Base circle radius of internal gear | rb (mm) | 62.5 |
| Number of internal gear teeth | z2 | 62 |
| Number of worm starts | z1 | 1 |
| Plane inclination angle | β (°) | 1 |
| Internal gear width (designed) | B (mm) | 110 |
| Working width | B’ (mm) | 75 |
The worm gear’s pressure angle α is calculated as:
\[
\alpha = \arcsin\left(\frac{2r_b}{d_2}\right) = \arcsin\left(\frac{2 \times 62.5}{320}\right) = 22.99^\circ
\]
The working semi-angle of the worm φw is given by:
\[
\varphi_w = \frac{(z_7 + 0.45)\pi}{i_{12}}, \quad z_7 = 7
\]
where z7 is the number of worm teeth in engagement. The initial working angle φ0 = α − φw. The worm works within the range (φ0, φ0 + 2φw).
Geometric Design of the Worm Gear Pair
Internal Gear with Beveloid Teeth
The internal gear of this worm gear system uses symmetrical wedge-shaped teeth. The two flanks (A-side and B-side) are planar surfaces inclined at ±β relative to the base cone generatrix. This wedge geometry allows axial adjustment of the internal gear relative to the worm to eliminate backlash and compensate for wear.
The internal gear’s working width B’ = 75 mm is determined from the contact line distribution analysis. The contact lines on the A-side extend in the v-direction from 4.40 mm to 26.35 mm, while on the B-side they range from −8.58 mm to −1.44 mm. The designed width of 110 mm is trimmed to 75 mm to reduce material cost without sacrificing meshing capability.
Drum Worm Geometry
The drum worm has a contoured profile that matches the internal gear teeth. Using the meshing equations and a MATLAB R2013b script, I generated the spiral lines of the worm tooth surfaces. The output .ibl files were imported into Creo 2.0 to build the 3D solid model. The worm accommodates the motor inside its hollow shaft, acting as the external rotor.
Assembly and Backlash Adjustment
During assembly in the Creo virtual environment, I adjusted the axial position of the internal gear by 7 mm to achieve an interference-free fit with the worm. This simple axial shift (using shims) demonstrates the key advantage of the beveloid internal gear worm gear: backlash between the worm gear teeth can be nullified without complex mechanisms. Compared to a toroidal worm gear with the same design parameters (which required a center distance of 220 mm), my drum worm gear achieves a center distance of only 100 mm, resulting in a much more compact transmission.
Contact Analysis
Contact Lines
For the A-side, the meshing region of each tooth corresponds to a specific working angle φ3(i) = φ0 + (i−1)θ, where θ = 2π/z2. The contact lines on the worm gear tooth surface are curves in the (u, v) parameter space. The limits on u and v are:
\[
u \in \left[0, \sqrt{\frac{d_{a2}^2}{4} – c^2}\right], \quad v \in \left[-\frac{B’}{2\cos\beta}, \frac{B’}{2\cos\beta}\right]
\]
where da2 is the internal gear tip diameter and c is the clearance. The contact lines for the A- and B-sides are shown in the analysis results; their v‑range justifies the selected working width B’.
Second Limit Curve
The second limit curve is the envelope of contact lines on the internal gear tooth. It satisfies the condition Ψ = 0:
\[
\Psi = \begin{vmatrix}
F^{(3)} & G^{(3)} \\
\frac{\partial \Phi}{\partial u} & \frac{\partial \Phi}{\partial v}
\end{vmatrix} = 0
\]
Solving this together with the meshing equation yields:
\[
u = \frac{a \tan \beta \cos \varphi_3}{\sqrt{\sin^2 \varphi_3 + (a \tan \beta \sin \varphi_3)^2}}, \quad v = \frac{a \tan \beta \sin \varphi_3}{\sqrt{\sin^2 \varphi_3 + (a \tan \beta \sin \varphi_3)^2}}
\]
In my worm gear design, the second limit curve is tangent to each contact line, confirming that the internal gear tooth surfaces are properly formed.
First Limit Curve (Undercut Check)
The first limit curve is the envelope of meshing lines on the worm gear tooth. The condition Γ = 0 is:
\[
\Gamma = \begin{vmatrix}
E^{(1)} & F^{(1)} & G^{(1)} \\
\frac{\partial \Phi}{\partial u} & \frac{\partial \Phi}{\partial v} & 0 \\
\Phi_u & \Phi_v & 0
\end{vmatrix} = 0
\]
After expansion, the first limit curve equations become:
\[
u = \frac{2i_{21} r_b \sin \beta \cos \beta \sin \varphi_3 \cos \varphi_3 + \cdots}{2i_{21} \sin \beta – 3i_{21} \sin \beta \cos \beta \cos^2 \varphi_3 – \cos \beta \sin^2 \varphi_3 \cos \varphi_3 + \cdots}
\]
\[
v = \left( r_b + \frac{a}{\sin \varphi_3} – \frac{H}{\sin \varphi_3} \right) \sin \beta + \frac{a \cos \beta}{\sin \varphi_3}
\]
By plotting the first limit curve together with the worm tooth root line and the trajectory of the internal gear tip contact points, I verified that the first limit curve lies entirely inside the worm tooth root boundary. No undercutting occurs in this worm gear transmission.
Structure of the Worm Gear Transmission Device
I designed the complete worm gear transmission device as an integrated unit combining drive, transmission, and support functions:
- Drive: The U10PLUS KV170 motor is installed inside the hollow worm shaft. This “external-rotor” arrangement makes the worm gear unit extremely compact compared to conventional external-motor configurations.
- Transmission: The beveloid internal gear is fixed between an upper gear seat and a lower gear seat using set screws. Axial adjustment of the internal gear (via shims between the cover and gear seat) enables backlash elimination and wear compensation of the worm gear.
- Support: A support shaft and fixed shaft hold all components, eliminating the need for a separate housing. The internal gear rotates together with the gear seats, which are supported by bearings on the fixed shaft.
This design results in a highly compact robot joint actuator. The 3D model created in Creo shows all components assembled without interference, ready for prototyping.
Conclusion
In this research, I have successfully designed a beveloid internal gear plane enveloping external-rotor drum worm gear transmission device. The key findings are:
- The symmetrical wedge teeth of the internal gear allow simple axial adjustment to eliminate backlash and compensate for wear in the worm gear pair.
- Compared to a toroidal worm gear, the drum worm gear reduces the center distance from 220 mm to 100 mm, making the transmission much more compact.
- By analyzing the contact line distribution, the internal gear working width is reduced from 110 mm to 75 mm, lowering manufacturing cost.
- The first limit curve lies inside the worm tooth root, confirming no undercut occurs.
- The integrated drive, transmission, and support structure eliminates the need for an external motor and housing, providing a clean solution for robot joints.
Future work will focus on manufacturing and testing a prototype to verify the predicted performance of this novel worm gear transmission. The use of grinding to finish both the worm and the internal gear surfaces will ensure high precision and efficiency, making this worm gear a promising candidate for next-generation robot reducers.