PID translation

I am curious. Do PID values stay the same across all different versions of firmware? Would PIDs be the same in FW that was on older APM board and on newer Pixhawk assuming the frame is the same? If there is a difference, is there a way to recalculate old values into new ones without running tuning?

No, no and no. It all changed. Start with a fresh install of latest stable, set up voltage and current monitoring then press Alt A in mission planner.
Read through the tuning guide.

I sorta assumed first two “no”. But surely there is a formula of sorts that can describe the changes? Wishful thinking…

how FW affects the values of the PID values? is there any doc explaining it?

There’s not actually a doc that I know of, but in each version of the source code there’s a section for parameters that get converted from earlier versions. It could take some time to go through that, and I doubt it will give much value since much has changed (so I gather) in regards to control loops for attitude and motor control in Copter.
Better to start fresh, and we can help where possible.

i don’t think of any physical reason for the values to change if the board is replaced.
i understand that there will be changes if the firmware changes, due to architecture change.
neglecting of course the changes in inertia and gravity center.