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For controlling the rockets to stay on course, continuous engine control is require which is done by using gimbal actuators.

What type of gimbal actuators have been used on SSME, Rocketlab's Rutherford, Spacex Merlin series.

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For Merlin-1D I have cheked this answer but the type of actuator is not mentioned anywhere.

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SSME & Merlin use hydraulic actuators. For instance,

The engine is gimballed through two planes by hydraulic actuators for vehicle pitch, yaw, and roll control. - Space Transportation System, Space Shuttle Main Engine Orientation, pg 4 (PDF)

and

Engine failure modes are minimized by eliminating separate subsystems where appropriate. For example, the first-stage thrust vector control system pulls from the high-pressure rocket-grade kerosene system, rather than using a separate hydraulic fluid and pressurization system. Using fuel as the hydraulic fluid eliminates potential failures associated with a separate hydraulic system and with the depletion of hydraulic fluid. - Falcon 9 Users Guide, pg 8 (PDF)

The Rutherford engine is generally believed to use linear actuators as gimbal actuators, but I'm having difficulty finding an official source. I'll update if I locate that.

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    $\begingroup$ Looking forward for the reference of Rutherford engine. Linear actuator with what kind of motor (stepper, BLDC, servo)? $\endgroup$ – Amar Nov 23 '18 at 17:46
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    $\begingroup$ I wish I knew, I'll let you know if I do find it. I have found reference to 'brushless DC motors' in company literature, but I think that's specific to the pump motors. It does indicate company experience with BLDC at least. $\endgroup$ – Saiboogu Nov 23 '18 at 17:57
  • $\begingroup$ You could link to this answer for the Rutherford. $\endgroup$ – uhoh Nov 24 '18 at 0:24

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