Kinematics for XYZAB mill

(2) and (3) describe the four different matrices we have in our toolbox to model the kinematic chain of our machine. We can use translations T and rotations R around X,Y and Z respectively.

(5) shows the transformations that describe the XYZAC-TRT configuration of Figure 3 using some of the above four matrices in our toolbox :
First WAC being the translation from the workpiece to the pivot point (intersection of rotational axis of C and A)
Second CAA is the rotation around Z ie the C axis (note here that the minus sign of the Sc (sine(C) is switched because we are rotating the workpiece rather than the tool)
Third AAP is the rotation around X ie the A axis (note here that the minus sign of the Sc (sine(C) is switched because we are rotating the workpiece rather than the tool)
Fourth PAt is the translation from the pivot point to the tooltip ie the X,Y and Z axes

(6) and (7) show the individual matrices used in (5) and are not a modified 'formula' of (2) and (3)

[edit]
Damn editor just ditched all my formatting. I hope it makes at least somewhat sense.

[edit2]
Maybe I'm overdoing it now but here is an explanation why we use 4x4 matrices:
www.euclideanspace.com/maths/geometry/af.../matrix4x4/index.htm
Just noticed, this information is also relevant to how the 'Inverse Transformation' is derived in the document.
 

It's the matrix multiplication that gets us from (5) to (8) that is the tedious part to do by hand.
Although it involves only sums and products it's easy to make a small mistake in the process.

[edit]
Also here you may notice why it is important to start off with the correct axis configuration of the machine at hand. The transformation matrix in (5) is the result of a series of matrix multiplications. Multiplication of matrices is not commutative so unlike what we are used to when multiplying say 4 and 6 (4*6=24 and 6*4=24) with matrices A*B is usually not equal to B*A.
Hence the sequence is important eg. whether the A axis is mounted on the B axis or the other way around. 

For the image in the opening post I would say :
WAT = WAB * BAA * AAP * PAT
(which is probably what you meant)

for the diagram you posted later:

WAT = WAA * AAB * BAP * PAT



This is under the presumption that the rotary axes A and B intersect.

As for starting out with vismach, I suggest having a look at this:
/home/user/linuxcnc-dev/configs/sim/axis/vismach/3axis-tutorial

Also maybe the xyzac-trt-gui.py is helpful
/home/user/linuxcnc-dev/src/hal/user_comps/vismach


Generally what threw me off the most is that one has to start assembling the moving parts in what seemed to me a backward way. So you don't start assembling the moving parts from the base but rather from the table that would hold the work piece or in the case of my robot arm from the hand backwards to the base.

Grotius, looks very interesting.
Just a quick question, as I don't want to high jack Rods thread,
Is your work intended to be something that can be integrated in LinuxCNC or is this likely to remain your own personal fork that is going to evolve further and further away from the main source code?

Just found this Vismach tutorial pdf:
sa-cnc.com/wp-content/uploads/2021/07/CNCCLubVismachA.zip

There is more to explore here:
sa-cnc.com/linuxcnc-vismach/
And in the blog:
sa-cnc.com/?page_id=5375