VEX Planetary Guide

[9935E]

VEX Planetary Guide

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Introduction

By using 2 inputs we get 1 output which can vary in speed and torque. This is a differential that is setup in a way to achieve 2 different output speeds depending on which way the motors are spinning. I also know that the swerve drives and spur differentials should be covered but I don’t have the time to figure out how that works and I need to start getting my robot built.

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The Most Common Design and It’s Flaws

The most common design uses chain as the ring gear but has back driving problems when in high torque mode and will end up slipping and losing all torque. While it can still achieve higher torque than in speed mode it can’t use the full force of both motors.

Why it slips

Since it doesn’t use a spur gear and instead uses chain on sprockets it changes the radius depending on sprocket location.

This is a major problem and causes most people to avoid using planetary gear sets since it adds another layer of complexity without any major benefits.

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Different design that solves this problem

By using an internal ring gear we can solve the problem of back driving. In a configuration like this. Aka having 2 sun gears and 2 planets. You might be thinking that you could have same size planets and sun gears but this ends up making the carrier useless.

Reason for this

This means that the b of the system would be 1 and that means the carrier is essential dividing by 1 which does nothing. (The gear ratio between the 2 sun gears with the carrier fixed)

Imagine if you kept an sun as an input stationary and rotate the carrier then the output sun wouldn’t move. Meaning that the carrier makes no affect on the output of the system and the only thing we are spinning is the planet gears.

But if we have a system like this where the planet gears are different sizes then the carrier is a functioning input. Imagine we hold the input sun gear and spin the carrier the 36 tooth planet will spin around the 12 tooth sun which will cause it to spin and since the 12 tooth planet and 36 tooth sun have a different gear ratio it will cause the output sun to spin, instead of rotating around it.

So this means as long as the gear ratio between the 2 suns is not 1 then this system will work.

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Math to solve for the output

Math

Lets assume that Green Sun and the Carrier are the inputs and Orange Sun is the output.
Imagine we have 2 suns one shaft that each mesh with one sun gear each which are both at the center of the assembly (Free spinning from each other).

Variable Names and whatnot

N = number of teeth
p1 = Big Planet
p2= Small Planet
s1 = Green Sun (Input)
s2 = Orange Sun (Output)
c = carrier
b = basic ratio = faster sun gear/speed of slower sun gear
(With the carrier held fixed)
ω= Angular velocity (Aka Rpm)

First we need to find b which is the basic ratio
Let’s say the green sun is spins at 100 rpm many fast would orange sun spin?

Solving for speed of bigger planet gear
ωp1 = (-Ns1 / Np1) ωs1 = -24/24 (100 rev) = -100 rpm

Since they’re on the same axle they experience the same rpm
ωp2 = ωp1 = -100

Now the orange sun rotates
ωs2 = (-Np2 / Ns2) * ω = -32/16 (-100 rpm) = 200 rpm

So this means the b = 200 = faster sun gear/speed of slower sun gear

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This time lets allow the carrier to rotate. Define ϕs1 as the number of rotations the green sun rotates relative to the carrier.
ϕs1 = ωs1 - ωc

Lets also define ϕs2 as the number of rotations the orange sun rotates relative to the carrier.
ϕs1 = ωs2 - ωc

Since the b of the 2 sun gears must stay the same even if the carrier rotates, we can say

b = ϕs1 / ϕs2 = (ωs1-ωc) / (ωs2-ωc)

Then move b to the other side
ωs1 - ωc - b (ωs2 - wc) = 0

Get rid of parenthesis
ωs1 - ωc - bωs2 + bωc = 0

Move ωs2 to other side since it’s our output
ωs1 - ωc + bωc = bωs2

Then divide by b to isolate ωs2
1/b (ωs1 - ωc) +

b=9, ωs2 = ωc * (|b-1| / b) + (ωs1 / b)
This also works,
b=9, ωs2 = 1/b[ωs1 + ωc (b-1)]
(1st equation from Kyle, 2nd equation from the website)

Here is some math in desmos so you can mess around with the sliders

Planetary math

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Some Designs that utilize this

Pictures of the Robots

Here are some of mine

4 motor tank drive with planetary- 26 holes long 6 holes for wheels and gears and side bracing (This is the size from end to end of the wheels)
Ignore the Missing gear

6 motor drive with planetary and strafing capabilities- 26 holes long 6 holes for wheels and gears and side bracing (This is the size from end to end of the wheels)

Here are some of kyle’s from 81818X

4 motor tank drive with planetary- 31 holes long 7 holes for wheels and gears and side bracing (This is the size from end to end of the wheels)

6 motor drive with planetary and strafing capabilities- 32 holes long 7 holes for wheels and gears and side bracing (This is the size from end to end of the wheels)

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Also if you have any additional questions you can either DM me and I might be able to answer it. I also highly recommenced going Analysis of Planetary Gearsets since it has animations and a more in depth look at the math.

Citation and references

Constans, Eric. 2013. Analysis of Planetary Gearsets

users.rowan.edu - Expired Page - Public Knowledge - Rowan University - Rowan...

Miller, Kyle. 2020 June 18. Reply to “Where would I locate strafing wheel”

Where would I locate a strafing wheel

Hi, I’m going to assume you are using epicyclic gearing for your transmission. An alternate would be using the relative directions of your motors as a shifting, but that would be pretty terrible. Epicyclic gearing also has the nice advantage that it creates an infinitely variable transmission, which benefits acceleration time. The first epicyclic transmission I’ve seen in vex was all the way back in 2013, by Stanley Shi - so its certainly nothing new. [Vex Planetary 2 Speed Transmission…

Also I got some of the info from DM with Kyle

Disclaimer- First saw this from Kyle1 so I’m going to assume he made it first in VEX. This Design may have been designed a long time ago but all the older images and videos I can’t access. Correct me if I’m wrong on math or anything else.

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Edit: Resized some images and Added a new one
Edit: Forgot about giving credit to where I got the equations from (In math section)

[Xenon27]

images are really small/low quality, hard to discern much from them tbh. Good stuff though, looks like you could gain knowledge from thoroughly reading this whole thing.

I don’t know how practical a planetary transmission is in vex, because friction losses and complexity are real downsides, and 4m worth of power is more than enough for both the drive and for the scoring mechanisms of a robot (especially this year)

[9935E]

I can’t do much about bad image quality since Jpeg compression but I resized the renders but left most of my examples small so it was easier to read. If you think there is anything else I should change just tell me.

[Sylvie]

I won’t have time to read through all of this until tomorrow, but it looks decent from what I can see.

[Railgunawesome]

@Xenon27

I don’t know how practical a planetary transmission is in vex,

Any form of transmission is just to boost the coolness factor, but doesnt really serve a useful purpose. VEX parts are just too bulky and limiting to make a truly effective transmission.

[7686B_Ian]

I 100% agree. It is really hard to make a useful transmission in Vex because of friction loss of the whole system. The only way for you to make it practical is to implement motor sharing, otherwise it can be a waste of energy. I am not saying that all transmissions are bad, I am just saying that it is hard to make a good one.

[technik3k]

@9935E

has back driving problems when in high torque mode and will end up slipping and losing all torque. While it can still achieve higher torque than in speed mode it can’t use the full force of both motors.

That is a very important observation that is applicable to any kind of non-locking dual-input differential based transmission.

This was discussed multiple times before.

Lots of math here: Automatic Transmission (non-shifting, multi-speed) - #9 by technik3k
Additional explanations here: Planetary CVT (Continuously Variable Transmission) Help! - #11 by technik3k

So what is the math that describes torques for the @Kyle1’s differential design?

@Kyle1

Here is it’s simplified half-diagram:

Input power from the first motor (τ₁ω₁) is applied to carrier gear and from the second motor (τ₂ω₂) to the input sun gear.

Output sun gear delivers output power (τ₀ω₀)

Gear sizes are, obviously, related to the number of teeth on the gear through π.

If you lock the output sun gear (ω₀=0) you want both inputs to have matched torques to avoid one of the inputs from overpowering and backdriving another:

Assuming both input motors generate same power, this tells you how much inputs should be geared up or down relative to each other.

Another way to look at it would be construct a static force diagram around the connected planetary gears or the point where they contact output sun gear:

Here is the table for cases where you use (36T:12T)=(3:1) or (60T:12T)=(5:1) in the differential stages:

If the input torques are matched using formula (2), you could verify that the output power will indeed be equal to the power of two individual motors.

Otherwise, if the torques don’t match, you will not be able to utilize full power of one of the motors, because at full power it will be backdriving another motor.

In the (3:1,3:1) case you will need to gear up sun input x8 relative to the carrier gear and in (3:3,5:1) case you will need sun input to run at x4 of the carrier.

Also, please, note that this type of differential has input velocities and torques both adding up to the output with the positive coefficients, which means that you achieve maximum output velocity and maximum (added) torque at a time when both motors spin in the same direction.

This means that you cannot use it to build dual speed transmission, where you want to have either high speed or high torque mode, depending on the relative motor spin direction.

However, you can use this type of differential for continuous power sharing between two outputs, where relative direction of the input motors determines which output is spinning. But you have to be cautious about outputs that could be pushing back into the system when they are not active, for example, a lift arm.

[Ryan_4253B]

Always wondered how far people can push vex components. iirc this bot uses 6 motors and have a 2 speed transmission + power transfer to lift using a combination of planetary gear, differential and pneumatics

[9935E]

1. I’ve finished my change up bot and have started getting more interested in differentials again. Correct me if I’m wrong but isn’t kyle’s design a 1:1 ratio so it wouldn’t spin the carriers (At Least effectively)? Unless he has the motors on different cartridges what I just said would be true.
   Any clarification would help. Either if I still don’t understand differentials or he has different cartridges.

2. With the “2 speed drive” (One has best torque and speed) where does the energy go for the weak output (worse torque and speed output) go?

3. Would you be able to somehow measure the torque or speed and then change the motor velocity or torque to stop the system from powering the wrong output? Pretty sure if you could measure torque you could do this but I’m unsure about speed.

[9935E]

Since I’m unable to edit my most recent post-

1. Think I’m right. Also could the motors still be back powered in kyle’s power sharing diffrential?
2. It backpowers into the the motors in high torque. It’s almost like techniks3k’s whole post was about that.
3. You could probably measure the acceleration but no EDR sensors that can detect torque.

[Ethan5956F]

@9935E

You could probably measure the acceleration but no EDR sensors that can detect torque.

technically speaking inertia sensors measure acceleration, which would have the equivalent vector quanity as the torque output. you probably wouldn’t want to use an inertia sensor this way though lol

[technik3k]

I agree that differentials could be very confusing - if I don’t think about them for a few months then, in my mind, they start to blend into a melted blob of gears.

Writing down equations for the forces (torques) acting on the gears is what helps me to detangle what’s going on. And while it is not obvious how motor outputs are connected from the image of @Kyle1’s design, this is my best guess:

@technik3k

Input power from the first motor (τ1ω1) is applied to carrier gear and from the second motor (τ2ω2) to the input sun gear.

After you go through all the math and gear sizes, the equivalent simplified force (torque) diagram will look something like this:

(It is not the actual gear ratio from @Kyle’s model - I choose 3:1 to make this example easier to read).

If the gear ratio between the motors is 3:1 then, for everything to be balanced, one motor needs to output (10) units of torque and another motor needs to output (30) units of torque. Together they could push the output that generates reactive torque of (40) units.

However, if the torques out of the motors are not balanced then, when reactive torque (of whatever is connected to the differential output) gets too large and stalls the output, the motor that is geared for more torque will backdrive the other motor that is geared for less (or everything will stall if the motor control system could detect backdriving or other overpower condition and shuts down the power).

@9935E

Correct me if I’m wrong but isn’t kyle’s design a 1:1 ratio?

According to my math a few posts above - motor torque at the carrier needs to be 8x the torque driving the input sun. Otherwise you are creating unbalanced torque situation and motors need to be shut down before one of them reaches 100% of max torque to prevent potential backdriving of the other (that is already at 100%). This prevents you from utilizing both motors at 100%.

@9935E

With the “2 speed drive” (One has best torque and speed) where does the energy go for the weak output (worse torque and speed output) go?

In the high torque mode both motors push in the same direction, adding their forces.
Imagine you and your friend both standing on the ground lifting a heavy box together. You try to lift it up with the same speed and then your efforts (forces) add up.

In the high speed mode, one of the motors runs forward, another runs in the opposite direction, and through the clever gearing adds extra output speed by bouncing off the first motor.

Imagine that now you lift your friend and your friend lifts the box. The box velocity is now the sum of how fast your friend lifts the box and how fast you lift your friend, but if neither of you is strong enough to lift the heavy box - you can only lift the final load that is lighter than half of the max load in the first case (high torque mode).

In actual designs (for example, this worm gear based differential) the max load in high speed mode is 1/3 of the max load in high torque mode. However, the velocity is 3x, so you end up with 100% motor utilization in both modes, minus the friction to run differential gear train.

@9935E

Would you be able to somehow measure the torque or speed and then change the motor velocity or torque to stop the system from powering the wrong output? Pretty sure if you could measure torque you could do this but I’m unsure about speed.

If you know the current of V5 motor then you can estimate the torque it produces (minus the friction and electrical resistance losses, for which you could derive coefficients with experiments). Knowing torque out of both motors and differential gear ratios you could determine output torque and detect any potentially dangerous condition that would warrant power shutdown.

@9935E

could the motors still be back powered in kyle’s power sharing diffrential?

Yes, if you use stock motor cartridges and don’t do external gearing to get 8x input torque difference as described above, then you can end up backdriving one of the motors and don’t get any benefits of differential (i.e. extended torque and velocity ranges).

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To summarize, two speed differentials only make sense when the game calls for two very distinctive operating modes, where one calls for something like 3x torque at 1/3 speed, and the other calls for 3x speed at 1/3 the torque. Then the benefits could justify increased complexity and friction losses in the longer drivetrain. Otherwise, it’s just an interesting concept without practical use.

On the other hand, if you want to use differential to share motors between two actions that don’t have to happen at the same time, like the driving and lifting heavy goals during ITZ season, then you could have more practical success, given that you are careful about the build quality and keep friction losses from exploding.