A frame with a stepper motor at each corner has a pulley on which winds up a cable. The cable only winds up in a single layer so there's no diameter change to figure.
The four cables all connect to the same point and that point can be moved around the interior of the frame via differential rotation of all four motors. All four motors will have to rotate for all moves except for arcs that happen to center on one motor. The cables have to maintain tension at all times so their connection point doesn't move vertically.
It's like one of those flying skycams but working only in a 2D plane.
Now the tricky mathy part. How would X-Y coordinates be translated to step and direction commands to the motors to move the connection point of the four cables? Or translate directly from G-code to positioning the connecting point using LinuxCNC.
I've seen this done -slowly- on vertical wall plotters using two motors. The problem with such an arrangement is it's vertical and control authority and accuracy is very poor outside a small central area due to jitter and cable sag. Four cables would fix that and allow the thing to work in any orientation. That should also enable much higher speeds, though not the 10+ G acceleration of the IPAnema robot
Here's an overhead image of the design. http://www.partsbyemc.com/pub/pattern-torch-jpg
The application for this? Adding inexpensive CNC to an old pattern torch similar to this one
. The motor torque required should be quite small. I can easily move the end of the arm around with very light pressure on a little finger tip.
There are kits with four NEMA17 motors, power supply and integrated driver board available for around $200. The shop with the torch already has plenty of steel to build the frame and the wire cable will be easy to get and inexpensive. With an older desktop PC, free software, various other bits of hardware, should be able to put it together for around $500. Far less expensive than an X-Y gantry table as large as the area this arm can reach. But the sticky bit is the coding to make it move.
What's wrong with just using the torch as-is? The patterns. How it works is there's a motor and gearbox which rotates a 1/4" diameter, knurled, magnetic shaft directly above the torch. To adjust for the drive magnet diameter, the patterns have to be 1/8" undersize all around. It doesn't handle sharp inside corners, the magnet can get stuck if it's tangent to two sides. It doesn't handle sharp outside corners too well, the magnet can come loose and the torch goes sailing off randomly. Sharp corners on the pattern translate to slightly rounded on the metal being cut. Within its limitations, it's OK for when a pattern will be used several times but for jobs that require only a few or one of a part it's a time consuming job to make the pattern which will end up in the scrap bin. Need one complex shaped part in 1/2" or thicker steel *and* the edges have to be cut straight and clean? Not happening freehand. Draw out the pattern then rough cut out of 1/8" plate then grind, file etc by hand until it's right. Still faster than working directly on the thick plate and goofs on a pattern are less expensive than goofs on the thick metal.
Drawing up a pattern on a PC, to the exact size, then cutting it out and being able to weld it directly onto the machine being rebuilt would be soo much easier and faster.
The guy who wants this pattern torch retrofitted with CNC rebuilds large pellet mills and often spends hours cutting and fitting a single piece of steel plate.