Implementation of a Z axis with non linear characteristic

Hi all!

I need a little help understanding the easiest way to implement an axis that is not moving in a linear fashion.
The Z-axis that is going to control the "bed height" of my machine has similar mechanics as a good old fashion car jack (that you keep in the car with your spare tire) and a stepper motor where you would hook in your crank.
This means that one turn on the crankshaft translate to a different "Z" movement depending on where on the lead screw you are.
I have done my best to study the KINS and trivkins man pages but it seems like they are generally describing much more complex behaviors than what is necessary for my use case.

Just to be clear, I'm not asking for the math that describes the movement, only how to implement it in LinuxCNC.

Can this be done with some lines of HAL or does it have to get more complicated? Can someone nudge me in the right direction?

Thanks Aciera!

I went through the process described and tweaked around a little and I think I got it to work.

Now I'll admit that I'm having a bit of a hard time to wrap my head around exactly what the inverse transformation is used for. I think I got it right but even when I try to deliberately sabotage the inverse formula I can't see any errors or different behavior in AXIS.

I guess what I'm asking is if there are any simple way of testing that I got the inverse formula right?