Remote control a lever

Hello,
I need some help.
I would like to remote control a push/pull lever with a joystick. I have attached an image to be more clear.


The null point of the lever is in the central position. I need a sort of actuator to pull and push the lever for about 5cm (10cm total).
The push pull force needed is about 5kg.
I know there are different type of linear actuators, but those I found out are to slow. I can't measure the speed I need, but I can make an example. Let think that you are driving uphill and you have to low down the gear because you are loosing speed and power. Ok the movement that you make with your hand when you downshift from 3rd to 2nd gear should represent the speed I need. Sorry, I know it's a stupid example :roll:

Another problem that I found using linear actuators is that you can't measure the exact position of the actuator, without external position control devices.
In fact I need a resolution of minimum 50 steps (200 or more would be perfect), so I can calculate the exact position of the lever.
Using a step motor, perhaps a nema motor with a screw, would be perfect but they are pricey and I don't know if they are enough fast.
I have excluded the servo motors because they don't have so much power.
As said, I would like to remote control the lever with a joystick and for the software part, I will use probably arduino.
If someone have any idea on how to solve this problem or links to useful stuff it would be very helpful.

Best regard,
Martin

so....

so much movement of the joy-stick translates into X movement of the lever?

if full movement of the joystick is 90 degrees, then moving the joystick 30 degrees means the lever should move 3.3cm?

Hmm, I did not think about this because this issue is related on which actuator I will use. I figure that 90 deg joystick movement is translated to 10 cm X travel of the actuator, so yes you are correct.
If I use a step motor attached to a worm, I have to use an arduino to control it. If I use arduino I can manage and map the joystick input value as I wish and I can vary the relation between deg movement and x movement.
If I use the joystick just like a potentiometer linked somehow to an actuator without a software in the middle then things goes a little different.