In the development of robotic technology, the demand for high-performance reducers is critical. Traditional reducers like harmonic and RV types often rely on imports, highlighting the need for domestic alternatives. This paper presents a novel beveloid internal gear plane enveloping external-rotor drum-worm transmission device, designed to offer compact structure, adjustable backlash, and high transmission efficiency. By integrating an internal gear with a drum-shaped worm, this device leverages the advantages of internal meshing to reduce size and enhance performance. The design focuses on achieving drive, transmission, and support integration, making it suitable for robotic joints. Key aspects include the use of symmetrical wedge teeth in the internal gears, which allow for axial adjustments to compensate for wear and backlash. This approach not only improves utilization but also reduces manufacturing costs for internal gears. Throughout this work, the importance of collaboration with internal gear manufacturers is emphasized to ensure precision in production. The theoretical foundation, design parameters, contact analysis, and structural integration are discussed in detail, supported by equations and tables.
The transmission mechanism involves a drum-worm paired with a beveloid internal gear, where the internal gear’s tooth surfaces act as tool surfaces. The internal meshing motion is analyzed using multiple coordinate frames to derive the relationship between rotation angles. Specifically, six frames are established: fixed frames for the worm and internal gear, and moving frames attached to them. An auxiliary frame on the internal gear helps define the working angle, linking the worm’s rotation angle to the internal gear’s motion. The meshing equations for both the A-side and B-side tooth surfaces are derived based on engagement principles. For the A-side, the relative velocity between the worm and internal gear is expressed in the moving frame, leading to the meshing equation. Similarly, the B-side analysis accounts for reverse rotation conditions. These equations are fundamental for determining the contact lines and boundaries, ensuring proper meshing without interference. The involvement of internal gear manufacturers is crucial here, as they can produce gears with the required symmetrical wedge profile to facilitate adjustments.
To illustrate the coordinate systems, consider the A-side frames: a fixed frame Σ₀ for the worm and Σ₀’ for the internal gear, separated by the center distance a. Moving frames σ₁ (attached to the worm) and σ₂, σ₃ (attached to the internal gear) are defined. The rotation angles φ₂ for the internal gear and φ₁ for the worm relate through the transmission ratio i₁₂, given by φ₁ = i₁₂ φ₂. The working angle φ₃ is determined from the auxiliary frame σ₂. The relative velocity vector v^(31) in frame σ₃ is derived as:
$$ \mathbf{v}^{(31)} = v_x^{(31)} \mathbf{i}_3 + v_y^{(31)} \mathbf{j}_3 + v_z^{(31)} \mathbf{k}_3 $$
where the components are:
$$ v_x^{(31)} = \omega_1 u – v \cos \beta \sin \varphi_3 $$
$$ v_y^{(31)} = \omega_1 v \sin \beta – v \cos \beta \cos \varphi_3 – r_b \omega_1 $$
$$ v_z^{(31)} = -v \sin \beta \sin \varphi_3 + u \sin \varphi_3 + a \omega_1 \cos \varphi_3 $$
The meshing equation Φ = v^(31) · n^(3) = 0 yields the A-side engagement condition:
$$ v = \left( r_b + \frac{a}{\sin \varphi_3} – \frac{1}{\sin \varphi_3} \right) \sin \beta + \frac{a \tan \beta}{\sin \varphi_3} \cos \beta $$
For the B-side, similar frames are set up for reverse rotation, with the worm angle φ₁’ related to φ₃ by φ₁’ = i₁₂ φ₃ – (π – θ) i₁₂. The relative velocity v^(21) in frame σ₂ leads to the B-side meshing equation:
$$ v’ = \left( r_b + \frac{a}{\sin \varphi_2} – \frac{1}{\sin \varphi_2} \right) \sin \beta + \frac{a \tan \beta}{\sin \varphi_2} \cos \beta $$
These equations are solved using MATLAB to model the tooth surfaces and boundaries, ensuring the design avoids undercutting and interference. Internal gear manufacturers can use these models to produce accurate internal gears with the desired profiles.
The design parameters are selected based on the U10PLUS KV170 motor specifications, which influence the worm’s internal dimensions. The motor is integrated within the worm, contributing to the compact external-rotor configuration. Key parameters include the center distance, module, number of teeth, and base circle radius. The internal gears are designed with symmetrical wedge teeth, allowing axial adjustment for backlash control. The table below summarizes the design parameters:
| Parameter | Value | Description |
|---|---|---|
| Center Distance (a) | 100 mm | Distance between worm and internal gear axes |
| Module (m) | 4 mm | Gear module |
| Number of Worm Threads (z₁) | 1 | Single-start worm |
| Number of Internal Gear Teeth (z₂) | 50 | High reduction ratio |
| Base Circle Radius (r_b) | 62.5 mm | Base circle for meshing |
| Internal Gear Width (B) | 110 mm | Initial design width |
| Pressure Angle (α) | 22.99° | Calculated from design |
| Transmission Ratio (i₁₂) | 50 | z₂ / z₁ |
The internal gear is modeled with a symmetrical wedge profile, enabling axial displacement for adjustment. The worm is designed as a drum-shaped component enveloped by the internal gear’s plane surfaces. Using CREO, 3D models are generated from MATLAB outputs, ensuring accurate geometry. The assembly allows for a 7 mm axial shift of the internal gear to eliminate interference, showcasing the flexibility offered by the beveloid design. Compared to traditional toroidal worm drives with a 220 mm center distance, this design achieves a more compact 100 mm center distance, highlighting the space-saving benefits of internal gears. Collaboration with internal gear manufacturers is essential to achieve the precise tooth geometry required for this adjustment capability.
Contact analysis is crucial for verifying meshing performance. The contact lines on the A-side and B-side tooth surfaces are determined from the meshing equations. For the A-side, the parameter u ranges from the root to the tip diameter of the internal gear, while v spans the effective width. The contact lines are plotted in MATLAB, showing that the A-side contact distribution in the v-direction is between 4.40 mm and 26.35 mm, and the B-side between -8.58 mm and -1.44 mm. Based on this, the working width of the internal gears is reduced to 75 mm from the initial 110 mm, optimizing material usage and cost. This reduction is feasible due to the concentrated contact areas, and internal gear manufacturers can produce these gears with narrower widths without compromising performance.
The second boundary curves, which envelope the contact lines, are derived from the meshing equation and its derivative. The condition for the second boundary is given by:
$$ \Phi = 0 \quad \text{and} \quad \frac{\partial \Phi}{\partial t} = 0 $$
This leads to the equation for the second boundary curve:
$$ u = \frac{a \tan \beta \cos \varphi_3}{\sin \varphi_3} \quad \text{and} \quad v = \frac{a \tan \beta \sin \varphi_3}{\sin \varphi_3} $$
Plotting these curves alongside the contact lines reveals that the second boundary curves are tangent to the contact lines, indicating proper meshing without edge contact. This ensures that the load is distributed evenly across the tooth surfaces, enhancing durability. Internal gear manufacturers must ensure that the tooth profiles adhere to these boundaries to prevent premature wear.
The first boundary curves, which represent the envelope of meshing lines on the worm surface, are derived to check for undercutting. The equations involve the worm’s tooth root lines and the trajectory of the internal gear’s tip contact points. The first boundary condition is expressed as:
$$ \Psi = \begin{vmatrix}
\mathbf{F}^{(3)} & \mathbf{G}^{(3)} \\
\Phi & 0
\end{vmatrix} = 0 $$
Solving this yields the first boundary curve equations:
$$ u = \frac{2 i_{12} r_b \sin \eta \cos \eta \sin \varphi_3 \cos \varphi_3 + u \cos \beta \sin^3 \varphi_3 + i_{12}^2 r_b \sin \eta \cos \beta + i_{12} a \cos \eta \sin \varphi_3 + 2 i_{12} a \sin \eta \cos \varphi_3 – a \cos^2 \varphi_3 \sin^2 \eta \cos \beta – a \sin^2 \varphi_3 \cos \beta + u \sin \eta \cos \beta \sin \varphi_3 \cos^2 \varphi_3}{2 i_{12} \sin \beta – 3 i_{12} \sin \beta \cos \eta \cos^2 \varphi_3 – \cos \beta \sin^2 \varphi_3 \cos \varphi_3 + i_{12} \sin \beta \cos \eta + 3 i_{12} \sin \eta \cos \beta \cos \varphi_3 – \sin \eta \cos \beta \cos^3 \varphi_3} $$
$$ v = \left( r_b + \frac{a}{\sin \varphi_3} – \frac{1}{\sin \varphi_3} \right) \sin \beta + \frac{a \tan \beta}{\sin \varphi_3} \cos \beta $$
Analysis shows that the first boundary curves lie inside the worm’s root lines, confirming that no undercutting occurs. This validates the meshing theory and ensures smooth operation. Internal gear manufacturers can use this analysis to guarantee that their gears meet the required standards for robotic applications.
The structural design integrates drive, transmission, and support functions. The worm houses the U10PLUS KV170 motor, creating an external-rotor configuration that saves space. The internal gears are mounted on a gear seat, with shims allowing axial adjustment for backlash control. The support system uses shafts and plates instead of a bulky housing, further reducing size. This integrated approach simplifies assembly and maintenance, making it ideal for robotic joints. The table below compares this design with a traditional toroidal worm drive:
| Feature | Drum-Worm Design | Traditional Toroidal Worm |
|---|---|---|
| Center Distance | 100 mm | 220 mm |
| Backlash Adjustment | Axial shift of internal gears | Limited or none |
| Structure | Integrated drive and support | Separate components |
| Manufacturing | Requires precision internal gears | Standard gears |
In conclusion, the beveloid internal gear plane enveloping external-rotor drum-worm transmission device offers a compact, efficient solution for robotic reducers. The symmetrical wedge teeth of the internal gears enable adjustable backlash and wear compensation, extending service life. The meshing analysis confirms no undercutting and optimal contact patterns. By integrating the motor within the worm and using a shaft-based support system, the design achieves a minimalist structure. Future work should focus on developing custom motors to further reduce size. Internal gear manufacturers play a vital role in realizing this design, as their expertise in producing high-precision internal gears is key to performance. This innovation supports the localization of robotic components, reducing dependence on imports and advancing automation technology.