Kinematics for XYZAB mill

Aciera wrote:

... Note that it is not just the Z axis that differs between the diagram and your image in the first post. In the image the B axis rotates with the A axis while in the diagram its the other way around...
 

@rodw: Also note that the machine X and Y and inverted on the diagram. The picture on the machine show that the X axis is front-back, not the usual left-right depicted on the diagram. The whole (half) trunnion then rotate around the A axis, while the rotary table (B axis) rotate around the Y axis.

You then have to swap both X-Y and A-B on the diagram to have the correct representation of the machine.

Jeez guys, thanks for all your support.
I've asked for a heap of data so will await a reply.

I've spent all my time today looking at the kinematics paper Aciera shared.  I still don't understand how they derived  formulas 6 & 7 from formulas 2 & 3 and figure 2. I'd like to be able to understand that so at least I can follow along.

(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 ( 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. 

Thanks, its slowly sinking in. That was very helpful.
For our XYZAB mill,  I think we should be looking for 
WAT = WAB . BAY. AAP. PAT
Does that sound right?
I need to redo Figure 3 for our new model.

And yes, the tedious part is yet to come..

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.