Kappa goniostat

I have an old kappa-goniostat (crystallography) that should be disposed and it's breaking my heart. It has 3 rotational axis for the sample in middle of the goniometer and an additional rotation+translation axis where originally detector was mounted.
I had this crazy idea that one could use it to work on spheric shapes in the wood (think about Escher's "Angels and Devils").

Two main problems:
1) I'm newbie to linuxCNC
2) goniometer has a strange geometry. I've attached image how it works.
The object rotates in center of system with three axis omega, kappa and phi. The kappa-angle is 50° to base (or 130°).
I tried to read examples about 5-axis geometry but all of them describe systems with orthogonal axis.

Any suggestions on how I should start this project? I have no idea how to tackle kinematics.
And before I spend hours reverse engineering encoders on DC motors, I need to know whether it is actually possible with LinuxCNC.

all best, Kris

I see your point. I thought to use Autodesk CAM (Inventor or Fusion).

Just a question ? Do you have those softwares at hand?
You might end up with five axis requirements and then looking at a hefty monthly suscription

Just get a new detector and use it, for what it was for . But... did it come with some sort of x-ray light source?
mmmh. Any lead west in the box?

How about a 3d scanner? Woud at least be more close to original function.

Shame on me -- I'm working at university -- non commercial use is free for us. At least it was free last time I checked.

The problem lies at the source. Spare parts can no longer be purchased from the manufacturer. Actually detector (alas slow) is still working. But the machine is at least 20-30 times slower than the new one. Cooling water, electricity and liquid nitrogen consumption are too high on this old equipment.

yepp, 4 rotary axis around center of the universe and one translation.