I have an issue which I would love any guidance at all with.
The Background:
I have been working on an experimental CNC router which looks like this:
The X and Y position is controlled by adjusting the lengths of the two chains.
We have been working hard to both reduce the complexity of the math that goes into computing the position from the chain lengths and making sure the math corrects for all of the important factors like the radius of the sprocket, however we are still seeing some distortion in our parts when we cut them.
The distortion amounts to only about 5mm over a 2000mm part, which is small enough that even a slight error in any measurement could be causing it. It is large enough to matter for large parts which need to fit together.
The Real Question:
My question relates to how to measure and correct for this distortion.
If I draw a grid of 100mm squares on a sheet of plywood we have a known coordinate system with no distortion.
Moving the router to each intersection point and recording both the position on the grid and the position reported by the machine can give us a measurement of the distortion. For each intersection point we now have $(X,Y)$ in the true grid coordinate system and $(X', Y')$ in the machine's distorted coordinate system.
How do we take this information and define equations which can relate to an arbitrary $(X,Y)$ point to the amount of correction which needs to be applied at that point?
Thank you for any help!