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On a typical pot you have three wires, a +5 volts input into the pot (out from the controller), a ground wire, and an analog wire that carries a signal 0-5 volts back into controller. At one extreme on a pot, the voltage will be 0 volts, and on the other extreme the voltage will be +5 volts, depending on the shaft location. The robot controller takes this analog voltage signal and runs it through a analog to digital converter (ADC), and expresses this voltage as an integer somewhere between 0 and 1024.
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Matt's right on the mark. The other application in FRC is using pots to build custom operator interfaces. If you haven't opened a joystick before, it has one or two (depending on whether its a one or two axis stick) pots connected to the joystick's gimbal. As the joystick is moved it changes the analog signal. After its converted, the value is used to set the PWM value (0-255) being sent to the programmed Victor speed controller or being used to match the value to the feedback pot on the robot arm. Teams have built small arms on their OI which mimic the location of the arm on their robot. The pots are used at the joints to tell the bot's arm what position to go to.
Just one other bit on pots. FRC usually uses rotary pots, but there are also linear pots (think of sound mixing board slider pots). Rotaries also come in different resistance values (10k, 100k, 500k) and single turn or multi-turn versions. Single turn pots give you full range 0-5 volts in about 270 degrees of motion. This is good for arm feedback where the arm doesn't exceed the 270 degrees of motion. Multi-turn pots require several complete revolutions (usually 10 turns) to complete the full feedback range. These might be better on gearboxes or elevator feedback where the drive system is making several full rotations to complete its range of motion.