Hi guys, in this page I'll be detailing my design process and research for rev 1 of the cycloidal gearbox. In this page, I'm going to try to define as much terminology as I can so it can also be viewed as a design guide.
Step 1: Cycloidal Disk 1 Profile Generation
To start off this design, we will first need to figure out how to generate the profile of the cycloidal disk. I've highlighted exactly what I am referring to in orange in the image below.
Parametric Equation Variable Definition
Luckily, this is actually a lot more straight forward than it seems. The outer profile of the cycloidal disk is based off the shape of an "ordinary cycloid". I think the best way to describe this shape is through this video, but pretty much it's a curve that theoretically is produced from the rolling of a ball around a fixed diameter. This curve can be modelled using a parametric equation, as described through this SolidWorks blog post, and uses the following variables to generate the curve:
Note 1: Gonna use different terminology from the SolidWorks blog post to reflect the diagram above. Think the consistency of terminology might make things easier to follow.
Note 2: Monash used the same SW blog guide to make their cycloids. Still waiting to hear how Michigan and Cornell did it.
- R (pitch radius of fixed ring pins)
- E (eccentricity offset)
- Rr (radius of ring pins)
- N (number of ring pins)
Once you define these variables, the parametric equation for the X and Y components of the cycloid curve are:
- X = (R*cos(t))-(Rr*cos(t+arctan(sin((1-N)*t)/((R/EN)-cos((1-N)*t)))))-(E*cos(N*t))
- Y = (-R*sin(t))+(Rr*sin(t+arctan(sin((1-N)*t)/((R/EN)-cos((1-N)*t)))))+(E*sin(N*t))
Values that require replacement are listed in bold. It's probably important to note that for the R/EN part of the curve should be interpreted as R/(EN).
For the sake of this initial guide, I will be using these values for the variables above. Justification will be listed in the next section
- R = 50mm
- E = 0.75mm
- Rr = 2.5mm
- N = 40
Defining Values for Your Variables
Here is what you need to consider when defining values for these variables.
- R (pitch radius of fixed ring pins)
- This is pretty much gonna define the size of your gearbox. It roughly refers to this dimension (in red), so housing is gonna be slightly bigger. This is pretty arbitrary unless you have some strict diameter requirements for the gearbox.
- For prototype 1, I've selected a radius of 50mm completely arbitrarily.
- This is pretty much gonna define the size of your gearbox. It roughly refers to this dimension (in red), so housing is gonna be slightly bigger. This is pretty arbitrary unless you have some strict diameter requirements for the gearbox.
- E (eccentricity offset)
- The eccentricity offset is a very important value to select. Just for some background, this page does a great job to explain how eccentricity works but I'll do my best here. The eccentric movement of the cam shaft drives the rotation of the cycloidal disk. If you have 0 eccentricity offset you just have a circle, with no cycloidal gearing profile. The closer the eccentricty offset is to 0, the closer the outer profile will be shaped as a regular circle. However, the larger the offset is, the more you approach an "ordinary cycloid" shape. This is also really bad, as ordinary cycloids are more difficult to work with mechanically. They have larger tangent variation on the outer lobes, which results in an outer profile that is difficult to work with mechanically, you need tighter and tighter tolerances to get everything to work properly.
Essentially, you want a select a good middle ground eccentricity to "smoothe" out the ordinary cycloid profile to a contracted cycloid, but also provide enough of depth on the outer profile for load transmission. From YouTube resources and the page linked above, the general guideline I've found is that E < R/N. - For prototype 1, that means E< 50/40, or E <1.25. So, how did I select 0.75 as the offset? Also arbitrarily (somewhat). I modelled out the options for 1mm, 0.75mm and 0.5mm (left to right below) and landed on 0.75mm as the outer profile looks most consistent to other cycloidal disks I've seen online. I think testing these out would definitely be ideal to select the best offset for our applications though.
- The eccentricity offset is a very important value to select. Just for some background, this page does a great job to explain how eccentricity works but I'll do my best here. The eccentric movement of the cam shaft drives the rotation of the cycloidal disk. If you have 0 eccentricity offset you just have a circle, with no cycloidal gearing profile. The closer the eccentricty offset is to 0, the closer the outer profile will be shaped as a regular circle. However, the larger the offset is, the more you approach an "ordinary cycloid" shape. This is also really bad, as ordinary cycloids are more difficult to work with mechanically. They have larger tangent variation on the outer lobes, which results in an outer profile that is difficult to work with mechanically, you need tighter and tighter tolerances to get everything to work properly.
- Rr (ring pin radius)
- Ring pin radius isn't something that has a strict guideline to adhere to, it mostly depends on how you want your design to end up looking. It seems like you just need to make sure you select a suitable radius in relation to the number of ring pins you have on your outer housing, as if you don't want to have large open gaps between each of the ring pins.
- I selected Rr = 2.5mm, as this comes out to a diameter of 5mm which can easily be adapted to dowel pin selection. More on that later tho
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- I selected Rr = 2.5mm, as this comes out to a diameter of 5mm which can easily be adapted to dowel pin selection. More on that later tho
- Ring pin radius isn't something that has a strict guideline to adhere to, it mostly depends on how you want your design to end up looking. It seems like you just need to make sure you select a suitable radius in relation to the number of ring pins you have on your outer housing, as if you don't want to have large open gaps between each of the ring pins.
- N (number of ring pins)
- This is probably the most important variable to define, as N directly relates to your transmission ratio. The way this equation works is that it will generate a cycloidal disk with one less lobe than the number of ring pins you choose. The transmission ratio is given by: (N-NL)/NL, thus your ratio is going to be 1:(N-1). (NL = number of lobes in cycloidal disk)
- I wanted a ratio of about 1:40, but I just picked flat 40 because it just makes the equation a bit easier to work with. Thus, the ratio of this cycloidal drive is going to be 1:39
Generating Cycloidal Disks in SolidWorks
Using all the variables selected above, the parametric equations for the cycloidal disk profile will end up as:
- X = (50*cos(t))-(2.5*cos(t+arctan(sin((-39)*t)/((5/3)-cos((-39)*t)))))-(0.75*cos(40*t))
- Y = (-50*sin(t))+(2.5*sin(t+arctan(sin((-39)*t)/((5/3)-cos((-39)*t)))))+(0.75*sin(40*t))
Using the equation driven curve tool, you can convert this equation to a sketch. IDK why but SW doesn't let you input the equation from 0→ 2pi, so I just input it from 0 → pi, then again from pi → 2pi
Step 2: Selecting Ring Pins
Ring Pin Configurations
Next up, I had to decide how I wanted the outer ring pins to be configured. Ultimately, there are two main design configurations for the ring pins - it can either be a solid piece of the housing (outlined in red), or it can be a free roller (point to in blue).
I talked to Rocomo (really nice ME robotics designer whos one of Lance's friends), and he said that a free rolling configuration will have lower friction and help with load transmission, and from other research I've heard that it will also result in a drive with lower operation noise. However, going with a solid ring pin design also has some benefits - you have less parts for the gearbox, lower assembly time, cheaper (as bearings are pretty damn expensive), and there are less points of failure for the gearbox. If you go with free rollers, you need to make sure you spec out bearings or bushing that can sustain the wear and load from the cycloidal disk, and you need to make sure that the "shafts" for the bearings are able to sustain the load without failing or deforming. Furthermore, the bearing design also has a tolerance stack up to consider, as you need to make sure your rolling pins, mounting holes and bushings all line up properly with the cycloidal lobe.
The loading scenario on the outer ring pins is described in this paper (~ pg 31), so it is something we can design for if we really need to. I'm still delving through this paper and trying to get a good understanding of it - some of it is pretty complex (especially the eccentricity section) however I think if we REALLY need to design for free rolling ring pins, we should be able to perform failure analysis with the info in the paper:
However, for rev1 prototype I decided to go with solid ring pins as it is much easier to deal with and test for an initial prototype. This is also how all of our competition (Monash, Cornell, Michigan) design their cycloids.
Adding Ring Pins to SolidWorks Schematic
The way I designed this first drive was by creating a sketch schematic as a basis for the key components. In the same part document used to create the cycloidal lobe sketch, I added the PCD of the ring pins, and ring pins themselves offset from the cycloidal disk in a new sketch layer. It's a bit of a cluster fuck, but I've outlined all the variables currently in the sketch.
This is a really good time to take a look at your cycloidal disk profile, as you can see if you set it up properly. There should be one point of contact between all ring pins and the cycloidal disk, and there should not be any interference between the two components. As shown in the picture, it looks like I've set things up correctly so I'm gonna move on the next step.
Step 3: Cycloidal Disk 2 Generation
Next up, we will need to generate the second cycloidal disk. I've discussed this in Actuator and Joint Drive Research, but to cover things again quickly: a singular cycloidal disk generates a lot of vibration due to it's eccentric motion, which can lead to more wear. Adding a second cycloidal disc phase shifted 180 degrees from the first will cancel out the vibrations generated by both disks. Two disks are used on pretty much all cycloidal gearboxes. I've followed this guide to design the second disc, as it isn't mentioned in the SW guide.
That being said, setting up the second one is pretty damn easy. All I'm going to do is mirror the first disk along the centerline of the ring pins. You can check if you did this step correctly if once again, there is point of contact an no interference between the ring pins and the cycloidal disc.
Step 4: Adding Camshaft Holes
This is kinda working backwards, but next up I went back to each individual sketch and added the camshaft holes for each of the cycloidal discs. The diameter of these holes should be large enough to fit a bearing, as the camshaft needs to roll around this cutout and isn't fixed in place. Honestly, I need to do more research into selecting cam shaft diameter as I haven't found too much information into this - but I have a feeling it has to do with the loading of the shaft. I picked a cam shaft diameter of 30mm pretty much arbitrarily for the time being - I looked at other cycloidal drives, noticed the approximate ratio of cycloidal disc PCD to cam shaft hole and picked a round number in the same ballpark. Then, I spec'd out this needle roller bearing for the cam shaft. Needle roller bearings were selected over ball bearings or plain bushings due to their high load capacity. However, we will need to make sure that the cam shaft is hard and smooth enough to match the rating of the bearing. Thus, I cut out a 37.02mm hole on the cycloidal disk for a 37mm OD bearing. I based this off the clearance for a transitional ISO fit, but I haven't really learned ISO limits and fits properly so I may have fucked that one up a bit.
Step 5: Output Ring Holes
Next, we need to add the holes on each of the cycloidal disc for the output ring pins. This brings about a lot of design decisions that need to be made for the actual output ring. First of all, a rolling or sliding element is incredible useful on the output rings to improve load transmission. I've seen people put bearings/plain bushings on the holes on the cycloidal discs, or around the output pins themselves. See below the pictures of the two options (red arrow pointing to rolling output pins, blue arrow pointing to solid pins and bushing on holes). Some people actually use both bushings on output holes and have rotating bearing pins.
However, for prototype 1 I decided to go with only bushings on the output pins. I wanted to reduce the number of machined parts as much as possible, as we don't have too many machining capabilities at the moment. I had trouble finding compatible bearing/bushing sizes given the eccentricity of my shaft. The hole for the output pins on the cycloidal ring (in red) is equal to (Diameter of Output Pin) + 2E. My eccentricity was 0.75 and all standard dowel pins come in integer values. It was pretty difficult to find compatible values.
Increasing the number of output pins will help distribute loads better, but obviously you poke more holes in your disc which can complicate it's force loading. Also, more pins means more components and more assembly. I went with a conservative 12 pins for the time being, although this number can definitely be optimized through force analysis. The paper linked in step 2 describes loading calculations for pins we can use down the road.
For prototype 1, I spec'd out some cheap 5mm diameter dowel pins that will act as the output pins, so the diameter of the holes on the cycloidal disk are (5mm + 2*0.75mm) = 6.5mm. Just as reference, Monash uses 3mm output pins. I chose an output pin centerline diameter of 65mm as it was approximately in the middle of the disc. I applied this cutout to both cycloidal discs.
You will know you did everything correctly if you add the output ring diameter to the middle of the housings, and the 5mm pins line up between the "eye" of both holes.
With that, the sketch schematic is finished! Note: I've found that the equation driven curve is a bit wonky and randomly flips planes while I work. Found that if I save and close the sketch it goes back to normal. Not too sure what the deal with that is.
Step 6: 3D Modelling
Next, it's time to move to 3D modelling. Lets start with the cycloidal discs. Both discs will be extruded to a length of 12mm to match the needle roller bearing I spec'd out. I added some ribbing to reduce weight because a 12mm thick cycloidal disc seemed super chonky. These aren't backed by any load calculations lol, I just offset everything by 3mm. Disk A (bottom disc) is on left, and disc B (top disc) is on right. I gave them different ribbing patterns so you know which ones are which. The rib patterns will actually face each other (this will make more sense when I show the assembly. Both have a material of 7075-O aluminum, but ideally 7075 hardened aluminum should be used for these discs (as that is what other teams use).
After the cycloidal discs, it's pretty easy to design the ring pin housing. The only notable thing here is that I made my ring pins 25mm high despite the cycloidal lobes being 24mm high. That's because I want to have a slight gap to sauce in grease, and will also be putting a gap in the cam to create some distance between disks 1 and 2. More on that later though. Also, it's a good idea to have good clearance for your cam shaft here - make it as big as you can while making sure the cam/bearings are actually supported on the plate. Motor mounting holes are also important to put here, but we don't have any motors picked out so I just randomly put M4 tapped holes inside this part in a circle. 7075 aluminum is also the material to choose here - as explained by Monash. Not ideal compared to steel in terms of wear, but light so it is good for the arm.
For rev 1, I'm also going to arbitrarily add mounting holes to the outer profile for the cycloidal drive lid.
