Schaublin 125-CNC retrofit.

  personally I would have all three under linuxcnc control. That way you can have the speed and torque control you need . Their's a reason
modern automatic car/truck transmissions have so many gears. That said dialing in when to favor which method will take some
 experimenting.

I am thinking that most turning cuts, the gearbox range set first, then set a standard frequency like 50Hz, and use the variator to set the desired RPM.

For facing, set the gearbox range, then the variator slow, and use the VFD to control up to nearly it top frequency, and then do one full variator move, with the VFD compensating to maintain target CSS rpm.

Can that variator be controlled by the PC/LinuxCNC/Mesa ?
I would make it!
Then do some math and some logic or look up table with speeds and stuff to always choose the most torque, but also leave some room for CSS.
-
Mental note:
I should really start working on my Mazak, poor thing already had it's social security expire !

Controlling the variator is only two pins on the 7i84 so the hard ware side is trivial.  This is a good input from the comments:

W Hogg:
"It's an interesting problem, I would look at both mechanism's response time and precision. working under the assumption that the variator is the slower reaction system, I would would control both simultaneously, using the vfd as a tuning adjustment to the variator. At an algorithmic level I would have the variator constantly trying to take the vfd frequency to it's mid point, that way you always have the most overhead in the most responsive system. With the encoder output as feedback, tuning the system would be a fascinating video. You could also mount a sensor on the variator swing arm or the lead screw to get positional feed back, to supplement the two limit switches. About the gearbox, first I would have a software sanity check, for a known variator and vfd setting you should get one of three encoder outputs, gearbox in high, gearbox in low, and powertrain fault. Ideally fixing a switch in or on the gearbox would be ideal. You could also have the machine stop the spindle and force the gearbox to low speed if it ever looses track of it's state, which should pretty much be when the controller loses power. Selecting desired gearbox speed should be done based off looking ahead in the gcode at requested spindle speeds and setting the gearbox for whatever settings coveres the most upcoming speeds, assuming there is a reasonable amount of overlap in all three speed governing systems. Just a few things that sprang to mind, I would be interested to know if I have made some glaring error or totally miss understood one of the systems at play."

My response:
That is an excellent input, only it hits up angainst the one significant limitation of the variator control. It's ratio change follows a fixed schedule (not rate as it is non-linear). To use the variator as primary adjust ment would mean either bang bang control, or replacing the contactors with a second VFD. The gearbox has no feedback, but I can infer gearbox position from the pneumatic control selections. I will probably initialise the machine in gearbox High, to give a known start point. I dont think you can do speed look ahead like that. I would assume I can only react to the next line. Basically the gearbox component will need to stop the spindle if a gearbox ratio change is necessary. That will have to be done based on RPM at the start of the cut, as you cant start a large diamter facing cut in high gear, as that the the speed yo would ideally end in, it would burn up the tool at the start of the cut.

Poster from youtube here, been thinking about this all day.

First, up the variator, i think it depends on whether you want to control it reactively or predictively, reactively you could somewhat abstract it and use a PID controller taking the VFD deviance from 50% and outputting a pulse length to power the contactors. you would need the VFD to be responsive to the change in spindle speed, in order to compensate for the changing ratio.
If you wanted to be solely predictive you could just use a dead reckoning strategy, and just power the contactors for a best-guess approximation of how long it would take to have the varitor move, this could probably be achieved with a lookup table.
I think the first method would be a lot more cool and flashy, and maybe make a more interesting video, but it's also more complicated and will have a slower instantaneous reaction, in exchange for a greater continuous range of motion.
The second is definitely simpler, at least in concept. and would allow for faster speed changes. However less flashy and may be harder to deal with for long continuous speed changes.
Finally, the gearbox, if you had gearbox control exposed as an m command, then it would be pretty simple to write a little bash script to go through a g-code file and look for g96/7 commands and then insert the m code where appropriate. this would be a simple calculation for g97 (fixed spindle speed), and for g96 (constant surface speed) you would need to calculate the minimum and maximum radial move, to find your minimum and maximum rpm "move".

Also, love this series, your methodical and meticulous approach is wonderfully different and enjoyable. Hope my extra thoughts are helpful or at least legible. And I look forward to this machine making some chips, it sounds like a jet engine at full whack.

We are still talking about the variator from that video, right?
Here's a kicker:
Remove old variator control, bolt an encoder to its control arm/shaft, do a position loop in LinuxCNC.
Since it has quite a broad range with the included gearbox, maybe much better that using a VFD as far as torque is concerned.
I have seen it change speed while spinning so even Constant Surface Speed can be done with no VFD to a certain extent.