So 18 joysticks then.
You will not find an Arduino with that many inputs so you will have to use multiplexing and port extenders to get what you want.
Also 25 servos is a lot, not so much for the control but the current requirement will be very big and that will require a big battery.
So look at a Mega and a multiplex shield. However I would guess that this project is a bit beyond your present skill level.start small and build up.