## High-reduction Cycloidal Actuator for Robotics

This page documents my efforts to create a fully-3D printable, high-reduction cycloidal drive. The initial parts of this article describe the construction of a 24:1 drive as a. The latter parts discuss the modifications of this original design to increase its torque output and make it manufacturable.

Precise robotic actuation demands the use of gear reductions for the amplification of torque and accurate translation of control input to mechanical output. For this application, a variety of methods exist including offset spur or helical gears, worm drives, planetary gears, and strain wave (harmonic) drives. Though widespread, many of these designs suffer from backlash, larger form factors, or complexity which diminish their value for use in robotic applications. This actuator design uses a cycloidal drive mechanism to overcome some of these issues.

Cycloidal drives are known for high reduction ratios and compact forms that minimize backlash. Drives consist of cycloidal disks with equation-driven profiles that allow for precise meshing within the mechanism. Cycloidal curves are generated parametrically from following a fixed point on a circle as it rotates about a profile without slipping. Various methods like add-ins, extensions, and equations can be used to generate these profiles. For this project, Otvinta was initially used to extract parametric equations for the drive discs.

Cycloidal drives consist of single or multiple drive discs mounted about an eccentric shaft or bearing. The perimeter of the gearbox commonly contains pins that are fixed, causing the drive gears to exhibit net rotation about the eccentric shaft. Subsequently, output rollers are used to translate the slower rotation of the drive disc linearly based on tangent relations with cutouts in the drive discs. Reduction rate is a function of the number of pins in the gearbox housing (P) and the lobes on the drive discs (N). This equation describes the relation between P and N with R being the reduction ratio. To illustrate this, here is a proof-of-concept design for a 7:1 reduction gearbox:

The above design contains 2 discs with a 180-degree phase offset to minimize vibrations and force imbalance at high RPM. This is the essence of a ‘harmonic’ drive which uses offset gearing to cancel out the effects of vibration. Furthermore, even with 1 disk, and depending on manufacturing methods, reduction ratios of up to 7,569:1 are possible in a smaller length compared to planetary drives. Furthermore, in comparison to other traditional spur and helical gear meshes, cycloidal gears share the output of loads across 1/3 of their profile which leads to a reduction in the forces experienced by each tooth.

Having tested the waters, the following design criteria were established. The mechanism would be able to utilize a standard brushless motor with mounting designed for easy maintenance as well as airflow over the motor bell. A new design strategy modeled after Paul Gould’s work was chosen instead of the more simple 1-sided output ring design shown above. This allowed the output shaft to be the pin casing around the drive disks themselves which removed asymmetric loading conditions and increased the strength of the design. As such, the mechanism would be designed to incorporate a rotary housing, not just an output shaft as is standard with most gearboxes. Alignment between the drive disks would be calculated such that fixed pins can extend through their geometries, thus constraining their rotation and only allowing the outer pin casing to rotate due to tangential meshing. Additionally, an offset cam would be designed to allow the cycloidal gears to eccentrically oscillate. This shaft would need to withstand rotational forces, interlock with the motor shaft, and remain relatively thin as to avoid the need for expensive or large bearings.

The first reduction ratio that I attempted to model was 24:1. To achieve this, I used the Otvinta parameters: D as the ring diameter of 80 mm, d as the pin diameter of 2.5 mm, N is the number of pins (25), and e is the eccentricity offset which was 1 mm for the eccentric cam. These created a set of parametric equations for an epitrochid curve which was then inset to create the desired profile. Later on, I realized that these equations were cumbersome and inhibitory for higher gear ratios but I’ll postpone that discussion for now until I get into the post-semester improvements below. For now though, the equations were used to generate the gears and surrounding output profile shown here with detailed descriptions available in the mentioned semester report:

After this, the components were assembled and a motion study performed to assess the reduction of the mechanism. In order to further reduce vibrations and increase contact area, 3 of the above cycloidal gears were used in the design. 2 of these are identical with one thicker gear (with the same profile) in the center to further distribute loading. Coupled with the surrounding output gear, the resulting design also eliminated backlash!

At this point, I had to research and experiment a fair amount to get the gears to properly mesh. In order for the 180 phase offset to occur, alignment depends on whether the number of lobes on the driving gears is odd or even. Designing for this alignment is crucial in order to generate the proper mating conditions to study motion and, of course, make the gearbox actually work. The following pictures show the differences in alignment and design between such gear types.

Another key consideration is the placement of the internal fixed pins which act as the mechanical constraint for the gears. These pins must be tangential to the inner surfaces of the gears in order to prevent their rotation. To ensure this, the radius of the circle that contains the fixed pins must be equal between the cycloidal gear itself and the housing in which the pins are connected to. The only distinction is that the holes along the circumference of that radius on the gear must be large enough in diameter to have the fixed pins themselves be tangent to their surfaces. In addition to the FEA done in the semester report, a final motion study was performed to determine whether or not the desired 24:1 reduction was achieved. It generated the following angular velocity versus time plots: While the complexity of the gearbox geometries as well as the limitations of my computer yielded some inconsistencies in angular acceleration, comparing the overall average speeds gives an acceptable result. After outputting the data for each plot in discrete intervals and averaging magnitudes, I proved that the output averaged ~600 deg/s while the input averaged ~25 deg/sec: an approximate 24:1 reduction. At this point in the project, the semester had concluded and I finished the first phase of the project by FDM printing the gears.

They ended up working great! This was very exciting but also the beginning of a ‘phase 2’ which was motivated by their application in a parallel project: my autonomous rover. With this new purpose in mind, I identified several improvements that would need to be made in order for their use to make sense in place of commercially available motors like these. Here are the goals that I sought to achieve in the second part of this project:

• Decrease vibration: As shown in the above videos, the large size of the lobes and poor tolerance lead to a fair amount of slop, vibration, and noise at middle-high RPMs (The drive is quite noisy if you un-mute the video)
• Upgrade the case and mounting fixtures to be manufacturable: This was mainly to shift the original focus of the mount (being for a multi-DOF arm) to be mounted to the rover’s chassis
• Make the output gear power a drive shaft: Instead of rotating an arm, the output would have to translate rotation to a parallel axle
• Increase the reduction: Using 3s or 4s LiPo batteries as a power source, the actuator failed to reduce speed enough to maintain torque output at low speeds. Such a range of operation would be crucial to make the rover operate efficiently for long periods of time
The latter of these was definitely easier said than done as it required an entirely new set of equations and methods to build the cycloidal profiles. For this, I defined the following parameters: Rr as the radius of the fixed pins, R as the radius of the circle on which the output pins are situated, N as the number of output pins, and E as the eccentricity or offset of the camshaft. While the radius of the arc on which the fixed pins are located is crucial, its exact value can be arbitrary as long as it is shared between the gears and where the fixed pins are mounted to. As a note of clarification, the fixed pins are the pins that extend through the internal parts of the cycloidal gears and restrain their rotation while the output pins are the lobes in the output gear that mesh with the cycloidal profiles. The parametric equations generate curves based on these variables (might require editing depending on CAD software):

$x(t)=(R*cos(t))-(R_r*cos(t+atan(sin((1-N)*t)/((R/(E*N))-cos((1-N)*t)))))-(E*cos(N*t))$
(R*cos(t))-(R_r*cos(t+atan(sin((1-N)*t)/((R/(E*N))-cos((1-N)*t)))))-(E*cos(N*t))


$y(t)=(-R*sin(t))+(R_r*sin(t+atan(sin((1-N)*t)/((R/(E*N))-cos((1-N)*t)))))+(E*sin(N*t))$
(-R*sin(t))+(R_r*sin(t+atan(sin((1-N)*t)/((R/(E*N))-cos((1-N)*t)))))+(E*sin(N*t))



Before these equations could be applied to the output gear however, its geometry would have to be modified to power a parallel drive shaft instead of an arm joint. I chose to do this with double helical (herringbone) gears. Here, I modified the output gear with a fully-equation driven, involute spur gear according to module, number of teeth, and pressure angle. I also added a helix angle parameter which drove the angle between mirrored helical gear profiles. With this modification, power could be transmitted laterally between drive shafts while minimizing inward forces between the gears in this direction. Using the 24:1 reduction design, I performed a motion study on the experimental helical gears to confirm meshing:

After confirming the validity of the helical gears, I applied the new equations to the cycloid gears to give a new actuator with a reduction ratio of 44:1. Embedded Ball Bearing in the Wall of the Gearbox

In making these improvements, I faced a bunch of challenges. Aside from convoluted motion studies causing my machine to crash, I found that a few of SolidWorks’ built-in features, such as the surface offset tool, break with complex sketches. For example, experiments with equations to add reductions to the cycloid gears usually succeeded in generating curves but could either not mirror, trim, or offset sketches/surfaces. After some experimentation using Desmos, I found that curves pushing the limits of self-intersection (but not actually intersecting) were the most problematic. So, I used trial and error by varying the number of output pins, their diameter, and eccentric offset until I achieved the maximum reduction that I could find. Referencing the above equations, the parameters that gave a functional 44:1 drive were Rr = 2.5 mm, R = 40 mm, N =45, and E = 0.75 mm with a 27.92 mm fixed pin mounting radius.

Further improvements involved designing a slim ball bearing into the walls of the gearbox to reduce friction between the helical output gears and its surfaces. For the balls, I used 6 mm smoothed airsoft BB’s due to their low cost and because I already had a ton of them. These worked surprisingly well and removed enough friction to justify not choosing more expensive nylon or stainless steel bearings. For these and the tangential interfaces of the gears, I chose a silicon-based lubricant to avoid any degradation to printed components as a result of petroleum-based compounds. Lastly, the final version of the gearbox utilized and improved motor housing with a CNC’d motor plate to avoid plastic melt and facilitate air cooling.