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I would reccomend using an electronic compass from Digi_Key, I know they have them (though they may not be cheap).
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I have checked these compasses out, they are too expensive to fit the < 100 rule. They are in excess of $600.
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Also, a note on the system you are trying to build: They are very mechanically complex and use many motors.
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I have designed this system in such a way that it only uses the two drill motors for driving, but still giving us 4 wheel drive. The system allows for Stafing as well as 0 degree turning. I think it will be worth it.
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I use something exactly like you are thinking in my RC Helicopter, called a heading hold gyro which is a combination gyro and electronic compass that keeps the tail rotor locked into to a direction unless you change it. Most Gyros (for RC) have a gain that if too low won't help you, too high and causes an oscillation so the heading hold helps out pretty nicely.
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So then are you saying that this would be a good thing for us to use. Where can I find this part?