Hello, I'm afraid I've got no idea what your connection table actually indicates, do you have motors A and a or A and B or a and b? Not really clear what your notation means, what you have already tried (beyond the single wiring scheme that you know doesn't work) and what you tried from the results of googling "polargraph motor wiring" that didn't work. You're leaving us quite a lot of assumptions to make.
If you mean that you are connecting one motor to M1 and M2, and the other motor to M3 and M4, then that is correct. On each of your motors, four wires are connected to two circuits, two coils. So there are two pairs of wires that you're interested in. You have already worked out which these are, I guess. Each pair goes to one motorshield port. So
MotorA.coil1 goes to M1
MotorA.coil2 goes to M2
and
MotorB.coil2 goes to M3
MotorB.coil1 goes to M4
Above is just a restating of what you'll find in your google search, so I assume you know it already, BUT, if it's not working, I'd suggest working through some of the adafruit motorshield examples (https://learn.adafruit.com/adafruit-motor-shield) to make sure your motors actually work one at a time, you've got your power supply rigged correctly, that kind of thing.
good luck,
sn
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