[Al Skierkiewicz]

Greg,
In looking at the data I am guessing the 250 ms is the delay between the transition of the PWM and the first change in the gyro output centered on 3000ms in the top graph? I think we need a little more info. Are you feeding the gyro directly to the RC or are you using a sub processor to interpret data from the gyro? Are you using someone's code or your own?
I am not a software guy here so I am working from memory on what I think our software team has discussed in the past. Their thought had been to look at several samples from a gyro to determine if the data was in error or that an actual trend was in effect. It was only then that they used the data to effect a change. In reaction to the drive spinning backward while the robot is trying to stop is an indication of fairly high loop gain which also appears in the graph. If you are attempting directional control, I would expect smaller changes in the PWM data. The gyro graph does show some damping but I think it should be even greater which may occur with a lower gain. I would also expect that the PWM values would decrease as the gyro output decreases.
The tougher thing to do is make predictions based on the variables of your drive system, robot weight and wheel friction. Since the wheels are breaking friction with the floor, all bets are off. I would think that speed changes (PWM values) that reduce to near zero (127) would be the most desirable response. In this way you would be getting closer to being on target without stopping forward motion. I would also expect lower magnitude PWM values as you get closer to the target. Does any of this make sense? It was a long day preceded by a short night.
BTW, the view in your photo looks familiar, where in NM was the pic taken?