Originally Posted by
philba
Any G-Code based controller (including GRBL) that can do arcs can trace those lines with a high degree of accuracy. Also, different gearing for the axes is not a problem - each axis can be controlled independently You tell the controller, for each axis, how many steps to go 1 mm (or inch). In your G-Code you specify how far to go and the controller sends out the right number of steps.. I'm not sure about fabric stretch but that could probably be compensated in G-Code.
I wouldn't try to write a controller from scratch - even an Arduino Nano with GRBL ($3) could run a machine like that. I would spend my time figuring out how to create the G-Code to run those patterns.