Not that I know anything about animatronics, but it's got to be very complicated.
I imagine movement is done with a servo when the limb is lightweight. There are light duty servos, and heavy duty type. Servos require a specific pulse length in order to position them.
Or a motor with gearing. Probably switches positioned at the extremes of the range of motion, to turn off power to the motor.
For a heavy limb, a counterweight can reduce the amount of electrical power needed to move it.
See if you can find a historical account about Disney animatronics, and how they do it. Computers make programming easier, I'm sure.
Or various tv programs (Nova episodes, etc.) about robotics, and how to make machines act more human. As to their walk, facial expressions, etc.
These might be available for you to view on the internet.