You hold the machine like a normal router. But you don’t have to be super accurate in your movements. You would have to watch the entire video to understand.
A microcontroller or maybe an FPGA does all the super fine left/right control.
Question: do you think that it is uses a stepper motor for the super fine left/right movement of the motor?
You’d need to ask the manufacturer to be sure, but I can’t think of anything other than a stepper motor for something like this. A stepper would have the precision needed and if you gear them down and microstep them, they are very smooth.
The could be using some sort of linear induction motor, but steppers would be much less expensive.
