A quick update on this project. I have built all three encoders now, and mounted them on a make-shift frame, and I connected the belt ends to the router. I wrote a small Python script that sends gcode to Marlin, and reads the encoders via a separate Teensy microcontroller.
At first I thought I would only need the three distances between the encoders, but in addition to that I also need all three belt lengths to one specific point, for example the home location. Measuring those things with a ruler or tape measure only gets within a millimeter or so, and I would like (much) higher accuracy than that.
So I started thinking about a way to automatically calibrate it by having the router probe something. My plan is to string a bunch of thin wires in various orientations, and have the router probe those. Each line gives rise to some constraints. Given a model of the system with some parameters, those constraints can be used to estimate these parameters. I think, though I haven’t done it yet, that it will be possible to get very good calibration this way, without needing any known size of something (!).
I then got side tracked with this little project to calibrate the router itself. The approach in that thread does not work (yet), but I’m still working on it and it looks very promising.