Kinematic model for a 5axis mill with universal (nutating) head

3rd UPDATE :
The zip folder below now contains two configs:
1. Table rotary C - spindle rotary B - spindle rotating (nutating 0..90°) A
2. Table rotary B - spindle rotary C - spindle rotating (nutating 0..90°) A

The spindle setup is defined in a [TWP] section in the ini file.

 

I have expanded the original config and documentation to a generic configuration with variable nutation angles 0-90°.
Also the Tilted Work Plane (TWP) section in the documentation has been expanded.
Note that that the TWP remap is still fairly experimental!

 

 

ORIGINAL POST:

This is from work I've been doing to derive a kinematic model for a milling machine with a 45° nutating head like this one:




Notes:
1. TCP kinematic works
2. TOOL kinematics work also but there are some quirks that I have not been able to figure out. Like an odd behavior when issuing an incremental move immediately after switching to tool kinematics.
3. The attempted implementation of 'Tilted Work Plane' GCode is still rather experimental and would need a fair bit more work.
4. The config requires a custom vismach.py file for visualisation.
5. This kinematics requires version 2.10pre (ie master)
6. Documentation is included in the config folder as an html document.
 

 
Maybe somebody finds this useful

For a very quick demonstration of TCP/ TOOL kinematics:
github.com/Sigma1912/linuxcnc/assets/460...d6-b125-af95fb13b218

Here a quick demo of the Tilted Work Plane feature.

github.com/Sigma1912/linuxcnc/assets/460...38-8f23-50dc38e17c57

That would be the transpose of the rotation matrix for the table rotary C.
Since you have a fixed table and an AC spindle rotary assembly you would omit the C table rotation in my writeup and replace the B spindle rotation with a C spindle rotation (ie you would have 'Rc' instead of 'Rb' on the spindle side).
Also there is a typo in my write up: both of the lower two formulas are for inverse transformation (P_A_Q) and not just the last one.

I can't read the numbers in the data sheet but if your nutation axis is not 45° then you would need to change the rotation matrix for A using the formula in my documentation.
I would suggest to start with the A axis (C = 0°) and work backwards. You will also need to think about which geometrical offsets are going to be relevant in your particular assembly.

[edit]
This section is where you will need to adjust for angles other than 45°.
 
Note that in your case you may not be able to use a simple algebraic expression like 1 over the square root of two.