Kinematics for XYZAB mill

... but then the current Y axis of the machine (left-right) would be 'somehow' the Z axis for a lathe BUT at an angle determined by the current A axis of the machine. This thing can get real complicated very fast I think.

(and what about the current X axis that would now be a Y axis as a lathe ???????????) image missing /media/kunena/emoticons/sideways.png

JPL wrote:

... but then the current Y axis of the machine (left-right) would be 'somehow' the Z axis for a lathe BUT at an angle determined by the current A axis of the machine. This thing can get real complicated very fast I think.No I don't think its super hard. We are already using custom kinematics so the additional mode would be based on something like the attached: 

we just need to add the offsets b & c to the Z abd Y positions If my maths is right!
we'd need a pin to set the distance from the chuck to the pivot point of the B axis.
 
 

I forgot to mention that the millturn example shows how to swap joints around and change axis length etc.

Yes, I'm pretty sure that you can indeed solve the Z axis at an angle but then I'm still very much lost about what to do with the X axis becoming a Y axis since I've never seen a lathe where you can control the 'height' of the tool.

Oh, milturn, I'm not very familiar with that (lathe guy here) but yes, that maybe the solution image missing /media/kunena/emoticons/wink.png
 

Thats actually handled by remapping the joints in 2.9 when the switchkins mode changes
as shown here in millturn.comp
github.com/LinuxCNC/linuxcnc/blob/master...s/millturn.comp#L156
This is a standard sim in 2.9

I see your point and there are millturn machines (hence the sim you mentioned) but to justify the term 'turn' you need a rotary axis that can position as well as rotate at high speeds. If it's just rotating while the mill bit is engaged in the material then I would call that milling. I don't expect the particular machine you need the kinematics for has the ability to run the B axis as a spindle, but I don't know.
Usually a 5/6 axis millturn machine has a horizontal A axis that also works as the lathe spindle and a tilting head as the B axis. This is also preferable since bar stock can be loaded in the spindle/A-axis and the milling head can also be used on work held in the tailstock axis (ie 6th axis). On a millturn machine non rotating lathe tools can also be used instead of a rotating millbit.