As an engineer specializing in gear transmission systems, I have always been fascinated by the potential of screw gear mechanisms in high-precision applications, particularly in robotics. The demand for compact, efficient, and adjustable speed reducers in robotic joints has driven me to explore innovative designs that leverage the advantages of planar enveloping techniques. In this article, I will present a comprehensive study on the design of a beveloid internal gear planar enveloping external-rotor drum-type screw gear transmission device. This system integrates driving, transmission, and support functions into a unified structure, aiming to achieve high transmission ratios, precise motion control, and adaptability through backlash adjustment. The core of this design lies in the use of a screw gear—specifically a drum-shaped worm—that meshes with a beveloid internal gear, enabling unique kinematic and contact characteristics. Throughout this discussion, I will emphasize the role of the screw gear in enhancing performance, utilizing mathematical models, tables, and simulations to validate the design. The goal is to contribute to the localization of robotic reducers by offering a viable alternative to traditional harmonic and RV reducers.
The fundamental principle behind this screw gear transmission is based on internal meshing between a planar tooth surface and a generated screw gear surface. Unlike conventional worm gears, this design employs a beveloid internal gear with symmetrical wedge-shaped teeth, which allows for axial adjustment to control backlash and compensate for wear. The screw gear, in this context, refers to the drum-shaped worm that envelops the internal gear teeth, creating multiple contact points for load distribution. To analyze the motion, I establish coordinate frames for both the driving and driven sides. Let us consider the A-side (primary meshing side) and B-side (secondary meshing side) of the transmission, each with its own set of frames to describe the relative movement between the screw gear and the internal gear.
For the A-side, I define a fixed frame \( \Sigma_0 (O_0, x_0, y_0, z_0) \) attached to the screw gear’s initial position and another fixed frame \( \Sigma_f (O_f, x_f, y_f, z_f) \) for the internal gear, separated by the center distance \( a \). Moving frames \( \Sigma_1 (O_1, x_1, y_1, z_1) \) and \( \Sigma_2 (O_2, x_2, y_2, z_2) \) are fixed to the screw gear and internal gear, respectively, with an auxiliary frame \( \Sigma_3 (O_3, x_3, y_3, z_3) \) used to determine the working angle. The rotation angles are related by the transmission ratio \( i_{12} = z_2 / z_1 \), where \( z_1 \) is the number of starts on the screw gear and \( z_2 \) is the number of teeth on the internal gear. The working angle \( \phi_3 \) on the A-side satisfies:
$$ \phi_1 = i_{12} \phi_2, $$
where \( \phi_1 \) is the screw gear rotation angle and \( \phi_2 \) is the internal gear rotation angle. On the A-side, the meshing surface is a plane tangent to the base cone of the internal gear, inclined at an angle \( \beta \). The relative velocity \( \mathbf{v}^{(31)} \) between the screw gear and internal gear at a contact point \( P \) is derived through coordinate transformations. In frame \( \Sigma_3 \), the coordinates of \( P \) are \( (u, v, 0) \), and the relative velocity components are:
$$ v_x^{(31)} = \Omega_2 u – v \cos \beta \sin \phi_3, $$
$$ v_y^{(31)} = \Omega_1 v \sin \beta – v \cos \beta \cos \phi_3 – r_b \Omega_1, $$
$$ v_z^{(31)} = -v \sin \beta \sin \phi_3 + u \sin \phi_3 + a \cos \phi_3 – \dot{a}, $$
where \( \Omega_1 \) and \( \Omega_2 \) are angular velocities, \( r_b \) is the base radius, and \( \dot{a} \) is the rate of change of center distance (assumed zero for static analysis). The meshing equation \( \Phi = \mathbf{v}^{(31)} \cdot \mathbf{n}^{(3)} = 0 \) yields the condition for contact on the A-side:
$$ v = \left( r_b + \frac{a}{\sin \phi_3} – \frac{u}{\tan \phi_3} \right) \sin \beta – \frac{a \cos \beta}{\sin \phi_3}. $$
This equation defines the relationship between parameters for effective screw gear meshing. Similarly, for the B-side, I establish frames and derive the meshing equation. The B-side corresponds to reverse rotation, where the screw gear angle \( \phi_1′ \) relates to the working angle \( \phi_3′ \) as:
$$ \phi_1′ = i_{12} \phi_3′ – (\pi – \theta) i_{12}, $$
with \( \theta \) being the pressure angle. The B-side meshing equation in frame \( \Sigma_2 \) is:
$$ v’ = \left( r_b + \frac{a}{\sin \phi_2} – \frac{u’}{\tan \phi_2} \right) \sin \beta – \frac{a \cos \beta}{\sin \phi_2}. $$
These equations form the basis for analyzing the screw gear transmission’s kinematic behavior. To ensure proper design, I also consider the limit curves that define the boundaries of meshing. The second limit curve, which envelopes the contact lines on the gear tooth surface, is derived from the condition \( \Phi = 0 \) and \( \frac{\partial \Phi}{\partial u} = 0 \). For the A-side, this gives:
$$ u = \frac{a \tan \beta \cos \phi_3}{\sqrt{\sin^2 \phi_3 + \cos^2 \phi_3 \sin^2 \beta}}, \quad v = \left( r_b + \frac{a}{\sin \phi_3} – \frac{u}{\tan \phi_3} \right) \sin \beta – \frac{a \cos \beta}{\sin \phi_3}. $$
The first limit curve, which envelopes the lines of contact on the screw gear surface, is obtained from \( \Phi = 0 \) and the Jacobian condition \( \frac{\partial (F, G)}{\partial (u, v)} = 0 \), where \( F \) and \( G \) represent surface equations. This results in a complex expression that ensures no undercutting occurs on the screw gear teeth. By evaluating these curves, I can verify the integrity of the screw gear transmission design.
To translate theory into practice, I define key design parameters based on a typical robotic joint application. The screw gear transmission must be compact, so I select a center distance of 100 mm, which is significantly smaller than traditional toroidal worm gears with similar specs. The internal gear has a beveloid tooth profile with variable thickness, allowing axial adjustment for backlash control. I choose a screw gear with a single start (\( z_1 = 1 \)) to achieve a high reduction ratio, and the internal gear has \( z_2 = 100 \) teeth for a transmission ratio of 100:1. The base radius \( r_b \) is set to 62.5 mm, and the plane inclination angle \( \beta \) is 1 degree to facilitate meshing. The motor integrated into the screw gear is a brushless DC type with an external rotor, fitting inside the drum-shaped worm. Below is a table summarizing the design parameters for this screw gear system:
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Center distance | \( a \) | 100 | mm |
| Base radius | \( r_b \) | 62.5 | mm |
| Internal gear width (design) | \( B \) | 110 | mm |
| Internal gear width (working) | \( B’ \) | 75 | mm |
| Screw gear starts | \( z_1 \) | 1 | – |
| Internal gear teeth | \( z_2 \) | 100 | – |
| Transmission ratio | \( i_{12} \) | 100 | – |
| Plane inclination angle | \( \beta \) | 1 | degree |
| Pressure angle | \( \alpha \) | 22.99 | degree |
| Screw gear enveloping teeth | \( z_7 \) | 7 | – |
Using these parameters, I model the screw gear and internal gear in CAD software. The internal gear features symmetrical wedge teeth, which enable axial displacement of up to 7 mm for interference-free assembly and backlash adjustment. The screw gear is a drum-shaped worm generated by enveloping the planar tooth surfaces. Its working angle range is \( \phi_3 \in (\phi_0, \phi_0 + 2\phi_w) \), where \( \phi_0 = \alpha – \phi_w \) and \( \phi_w = (z_7 + 0.45) i_{12} \pi / 180 \). This ensures multiple teeth are in contact simultaneously, enhancing load capacity. The compactness of this screw gear transmission is evident when compared to a toroidal worm gear with the same parameters but a center distance of 220 mm; our design reduces size by over 50%, making it ideal for space-constrained robotic joints.
Contact analysis is crucial for ensuring smooth operation and longevity of the screw gear transmission. I evaluate the contact lines on both the A-side and B-side tooth surfaces. For the A-side, the contact line distribution along the \( v \)-axis ranges from 4.40 mm to 26.35 mm, while for the B-side, it ranges from -8.58 mm to -1.44 mm. These ranges determine the effective working width of the internal gear. Based on this, I reduce the design width from 110 mm to 75 mm, optimizing material usage and cost without compromising performance. The contact lines are plotted using computational tools, and they show a consistent pattern across multiple tooth pairs. At any given moment, up to 8 teeth are in contact, thanks to the enveloping action of the screw gear. This multi-tooth contact distributes loads evenly, reducing stress and wear on individual teeth.
Next, I examine the limit curves to prevent common gear issues. The second limit curve, derived earlier, is tangent to the contact lines on the gear tooth surface. Plotting these curves confirms that the contact lines are properly enveloped within the tooth boundaries, ensuring full utilization of the tooth surface. The first limit curve, which defines the boundary for undercutting on the screw gear teeth, is compared with the screw gear’s root line. Analysis shows that the first limit curve lies inside the root line, indicating no undercutting occurs. This is vital for maintaining the strength and durability of the screw gear. The mathematical verification involves solving the limit curve equations and checking spatial relationships. For instance, the first limit curve equation for the A-side is given by:
$$ u = \frac{2 i_{12} r_b \sin \beta \cos \beta \sin \phi_3 \cos \phi_3 + u \cos \beta \sin^3 \phi_3 + i_{12}^2 r_b \sin \beta \cos \beta + i_{12} a \cos \beta \sin \phi_3 + 2 i_{12} a \sin \beta \cos \phi_3 – a \cos^2 \phi_3 \sin^2 \beta \cos \beta – a \sin^2 \phi_3 \cos \beta + u \sin \beta \cos \beta \sin \phi_3 \cos^2 \phi_3}{2 i_{12} \sin \beta – 3 i_{12} \sin \beta \cos \beta \cos^2 \phi_3 – \cos \beta \sin^2 \phi_3 \cos \phi_3 + i_{12} \sin \beta \cos \beta + 3 i_{12} \sin \beta \cos \beta \cos \phi_3 – \sin \beta \cos \beta \cos^3 \phi_3}, $$
$$ v = \left( r_b + \frac{a}{\sin \phi_3} – \frac{u}{\tan \phi_3} \right) \sin \beta – \frac{a \cos \beta}{\sin \phi_3}. $$
By substituting design parameters, I confirm that the curve values remain within safe limits. This rigorous analysis underscores the reliability of the screw gear transmission design.
The overall device design integrates driving, transmission, and support into a single compact unit. The screw gear serves as the core transmission element, directly coupled to an external-rotor motor housed inside its drum-shaped body. This integration eliminates the need for additional couplings, reducing inertia and alignment issues. The internal gear is mounted on a support shaft via a gear seat, with spacers allowing axial adjustment for backlash control. Tightening screws secure the assembly, ensuring precise positioning. The support structure consists of a frame and plates, eliminating the need for a bulky gearbox. This streamlined approach not only saves space but also enhances rigidity and reduces weight—key advantages for robotic joints. The screw gear transmission system thus embodies a holistic design philosophy where every component serves multiple functions.
In terms of performance, this screw gear transmission offers several benefits. The adjustable backlash feature, enabled by the beveloid internal gear, allows for compensation of manufacturing tolerances and wear over time. This extends the service life and maintains accuracy, which is critical in robotics. The high contact ratio due to planar enveloping ensures smooth motion and high torque capacity. Efficiency is improved through optimized tooth profiles and reduced sliding friction. Experimental validation, though not detailed here, would involve testing prototype units under load conditions to measure transmission error, efficiency, and durability. Based on simulations, I anticipate efficiency levels above 90% for certain operating ranges, comparable to premium reducers. The screw gear’s ability to handle misalignment and shock loads further makes it suitable for dynamic robotic applications.
To further illustrate the design process, I summarize key equations and parameters in another table. This table encapsulates the kinematic relationships and geometric constraints that govern the screw gear transmission:
| Aspect | Equation or Value | Description |
|---|---|---|
| Transmission ratio | \( i_{12} = z_2 / z_1 = 100 \) | Ratio of internal gear teeth to screw gear starts |
| A-side meshing equation | \( v = \left( r_b + \frac{a}{\sin \phi_3} – \frac{u}{\tan \phi_3} \right) \sin \beta – \frac{a \cos \beta}{\sin \phi_3} \) | Defines contact condition on primary side |
| B-side meshing equation | \( v’ = \left( r_b + \frac{a}{\sin \phi_2} – \frac{u’}{\tan \phi_2} \right) \sin \beta – \frac{a \cos \beta}{\sin \phi_2} \) | Defines contact condition on secondary side |
| Second limit curve (A-side) | \( u = \frac{a \tan \beta \cos \phi_3}{\sqrt{\sin^2 \phi_3 + \cos^2 \phi_3 \sin^2 \beta}} \) | Envelope of contact lines on gear tooth |
| Working angle range | \( \phi_3 \in (\alpha – \phi_w, \alpha + \phi_w) \) with \( \phi_w = (z_7 + 0.45) i_{12} \pi / 180 \) | Angular range for effective meshing |
| Contact line span (A-side) | \( v \in [4.40, 26.35] \) mm | Distribution along tooth width |
| Contact line span (B-side) | \( v’ \in [-8.58, -1.44] \) mm | Distribution along tooth width |
| Undercutting check | First limit curve inside screw gear root line | Ensures no root interference |
The screw gear transmission system also presents opportunities for further optimization. For instance, the plane inclination angle \( \beta \) can be fine-tuned to balance contact pressure and efficiency. Material selection—such as using hardened steel for the screw gear and polymer composites for the internal gear—could reduce weight and noise. Additionally, advanced manufacturing techniques like grinding can improve the accuracy of the screw gear teeth, enhancing transmission precision. The integration of sensors for real-time backlash monitoring could enable adaptive control in robotic systems. These innovations would build upon the core advantages of the screw gear design, pushing the boundaries of what is possible in compact power transmission.
In conclusion, the planar enveloping screw gear transmission system I have described offers a compelling solution for robotic joint reducers. By leveraging a beveloid internal gear and a drum-shaped screw gear, it achieves compactness, adjustability, and high performance. The mathematical models for meshing and limit curves ensure reliable operation without undercutting. The integrated design combines drive, transmission, and support, simplifying assembly and reducing footprint. This screw gear approach addresses key challenges in robotics, such as the need for precise, durable, and space-efficient speed reduction. Future work will focus on prototyping and experimental validation to confirm theoretical predictions. As robotics continues to evolve, innovative screw gear transmissions like this will play a pivotal role in advancing automation technology.