DIY lathe conversions for machines of that size sometimes use stepper motors to drive the axis positions. Stepper motors are typically run "open loop", meaning that there is no feedback to the controller to indicate the axis position, other than home and limit switches. The controller starts from a known position and then counts steps in either direction to determine position along the axis (taking into account the motor's steps per revolution rating, the amount of microstepping used by the controller, if any, and the pitch of the leadscrew).
If the conversion uses a servomotor instead of a stepper, then typically an encoder attached to the motor shaft is used to keep track of the shaft rotation (along with switches and the other factors mentioned previously).
Some conversions that I have seen use belt drive to change the motor orientation (to save space). Others may use a belt drive to modify the speed or resolution from the motor's original characteristics.
Some people use Oldham couplers to compensate for possible axial misalignment between the motor shaft and the leadscrew, without affecting the backlash.


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