Generally, you'll choose from AC or DC type motors. The former includes stepper, PM and induction types; the latter, usually a regular (wound rotor + commutator) type, but the rotor is twisted just so, and probably something about the magnets, and the cut of the rotor itself, to minimize cogging: a good DC servo motor has very smooth motion.
Steppers tend not to move smoothly (as the name suggests), but despite their cogging motion, can be controlled smoothly by microstepping. This is usually digital control anyway, so you just need to decide on an acceptable step size and go from there.
PMAC motors control the same way, but with huge steps, so usually use something a bit more specific like field-oriented control and a full three-phase inverter or something like that. Whereas steppers are generally current controlled and just chopped with whatever to maintain that current.
Induction motors aren't often used this way AFAIK, primarily because the torque at low speeds is pitiful (aside from stopping, which can be done easily with DC bias); they can be precisely controlled at speed.
Other than electrical, there are hybrid mechanical types; cameras often use piezo motors for their low mechanical noise. These use a revolving wave of deformation of a piezo material, to cause motion in rolling contact with the rotor. These aren't very fast, so may be augmented by a faster but louder electromagnetic motor.
There are other oddball and special purpose things, mostly that are slower but very precise (down to nanometers if you like). Or extremely fast but narrow range (piezo stacks). Or like the actuators on JWST, apparently are a combo leadscrew + flexture mechanism that offers wide range at the cost of slow speed and nonlinear transfer function (the leadscrew is driven by a pin on a disc, maximizing backlash, while the flexture is driven by a cam on the gearmotor shaft, providing fine control inbetween).
Tim
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