The Axiom computer is a main computer that controls the Axiom itself. Even though at first it seems that the Axiom computer controls the robots, it doesn't seem so because the captain could not issue a voice command to the computer to stop AUTO.
That may imply that the computer just didn't have the authorization to affect the autopilot but that does not mean that the computer was not responsible for waking up the other robots on the starliner when they were needed.
When the computer's inability to affect AUTO was noticed, it may at first seem that the computer was only a reference database and had no actual control over the starliner, but some details clearly show that that is not the case. So it is unknown if the computer could truly affect robots.
When AUTO was turned off, it was the computer that made the Axiom's hyper jump to Earth. When the captain pushed the green button for the plant alert, it was the computer that closed the doors of the bridge and presented the message from Shelby Forthright about the Operation Recolonize. It is also clear that it was the computer who opened EVEs cavity to receive the plant after the captain "allowed" that to happen. The computer also seemed to have the control over the escape pods and the probe ship.
These points clearly indicate that it was really the computer who ran most of the functions of the Axiom and that AUTO was responsible for piloting the starliner. It seems that he had override over the computer since he was able to disconnect it from the captain's quarters, which was why the captain was unable to order the computer to let him out of the room.
The computer is even more computer-like than AUTO, but has a more human-like voice. The computer only acted in specific, pre-programmed situations.