This is the way to fix an analog stick that stops working while using it, usually causing you to move the stick around and back to where it was to register it.
This issue usually manifests when clicking the stick to run, and suddenly you stop running or you stop moving all together.
This fix is specific to the stick randomly cutting out in a certain position.
NOTE: You will need a soldering iron for this fix, but you won't need soldiering skills.
Step 1: How the Analog Stick Works, and What the Problem Must Be.
The analog stick works by turning one X and one Y potentiometer. A potentiometer provides different electric resistance depending on how far or how little it is turned.
Your analog stick in the resting position keeps the potentiometers right in the middle or their range. This means that if you are pushing the stick forward, the potentiometer is at a high or low limit.
Because of how the stick is assembled, it can't "slip" to the middle position when it's in the forward position without the stick being completely broken. If the stick is forward, the potentiometer is turned.
The problem we're having here, is that the stick suddenly registers away from the forward position, then works when moved around again. The only way this makes sense from a troubleshooting perspective is by the controller losing contact with the potentiometer.
This actually indicates a rather common problem in electronics of a loose solder connection.
This possibility seems even more likely when you consider that you are probably pushing down pretty hard on the stick while you play.
Lucky for us, this problem is easy to fix.