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And yea im guessing your turret is just a fraction the weight of my original tanks turret (The one in the video clip)
I have good news. In my troubleshooting I discovered two issues: The first one is that the platform the velocity pivot benefits from being rigid. If it wiggles around too much the stabilizer overcorrects like I described. Small attachments come in handy in this regard.
The second, and best thing I did was lowering the integral gain on the PID. I took it from 0.1 all the way down to 0.0008. If this seems drastic it is because the turret I'm using is quite small, so it is very sensitive. The other two values, proportional and derivative gain, were left at their default values.
During some extreme maneuvering the oscillations return. Toggling the stabilizer off then on has corrected this issue every time. Thank you!