How to calculate wrist angle given the angle of the arm through a 4 bar and chain ratio?

Hello everyone, this could very well be simple to solve, and I could just be very tired, but I need some help calculating the angle of our wrist mathematically given the angle of the arm.

I’ll try to be descriptive as possible, We have both our wrist motor and arm motor mounted within the carriage of our elevator, the arm motor, straightforwardly takes the output of the gearbox, and runs it through a sprocket that is bolted to the side of the arm, therefore rotating it rotates the arm, simple enough. The wrist motor however, is more difficult, it runs through a chain to a sprocket on the same axle that the arm rotates around, but is affixed to the axle not the arm (obviously), then there is a second sprocket, attached to the same axle, which runs a chain across the arm, over one passive sprocket, to the wrist axle, with a sprocket attached to it. Unfortunately the second sprocket on the arm axle and the sprocket are not the same tooth count (22T vs 12T) meaning there is a ratio. This also causes the problem that when we rotate our arm, because the arm axle stays stationary (due to being attached to the wrist gearbox via a chain) it causes the wrist to act as a 4 bar, but because there is a ratio between the two sprockets driving it, it rotates (as apparent from the floor) as well.

Im sorry I don’t have great pictures, please excuse any janky-ness in the photos it will be fixed eventually. The left side of the chains / sprockets controls the arm, and the right / middle controls the wrist.

What I need is to be able to calculate the rotation of the wrist that is caused by some rotation in the arm. Say the arm rotates 15 degrees, I need to figure out that the wrist, as seen from the floor, or arm, either works, rotates either 5 degrees or 15 + 5 degrees.

Thank you so much for any help you can provide, if you need measurements, tooth counts or anything to help me figure this out let me know and I can get them for you.

Oh and bonus points if you can explain it like I only had half of a geometry course due to covid

My team had something very very similar and we figured this out experimentally. Hold the arm in a specific position, and sweep the entire wrist range of motion all while watching and recording the wrist encoder. Then hold the wrist at a specific angle relative to the arm, and sweep the entire arm range of motion while watching the arm encoder. You should be able to determine the relationship between the arm:wrist and the range of the wrist this way.

If you want a better description or you can’t do it this way and want to do the math(and save the world) let me know!

For problems like this I always covert to what units the transmission natively moves. What I mean by this is:

Shared shafts transfer degrees of rotation. If I move one element on the shaft d degrees, every element attached to the shaft will move the same d degrees, no matter if it is a 12T gear, a 6" lever, or a 19.435mm v-pulley. So a shaft natively transfers degrees.

Chains on the other hand natively transfer movement in teeth. If I turn a 12T sprocket through a full revolution, every sprocket will move 12 teeth, no matter if it is a 9T sprocket, a 47T sprocket, or any other sprocket. Every sprocket on the shared chain will move the same number of teeth.

For belts it’s cm (or inches).

So for a system like this, if I raise the arm 90degrees, then I move the wrist sprocket at the shoulder 90degrees because they are effectively a shared shafts. Now I’m move to the chain from shoulder to wrist, so I convert the 90degrees to teeth (22T/4 = 5.5 teeth). That 5.5 teeth is transfered through the chain and moves the wrist 5.5 teeth. (The idler sprocket also moves 5.5 teeth, but we don’t care because it’s just idling on a shared chain). The wrist sprocket is attached to a shared shaft with the hand, so sine shafts move degrees, convert the teeth to degrees (5.5 teeth/12T = ~.45 revolutions = ~162 degrees). Now I know that the wrist will move 162 degrees when the arm moves 90 degrees.

This breaks almost any transmission into steps that can be intuitively analyzed one at a time.

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Thank you so much, all of that is so intuitive but breaking it down to that level makes the answer much easier to see, thank you!

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