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What is the relationship between gyroscopes and accelerometers in the orientation algorithm?

An accelerometer, when held vertically and stationary at sea level, will yield an output that corresponds to +1g. When horizontal on its sensing axis, its output is ideally zero and when vertical in the opposite rotation, the output represents -1g. That relationship makes it usable as a tilt sensor. But it also senses lateral acceleration, like a flat straight-line speed increase in an automobile, that will falsely appear to be tilt in that application, and mass-produced devices vary somewhat in sensitivity.  

A rate gyro yields an output that corresponds to rotation rate, like 0.015v/degree/second, and yields an ideal zero output when it is not rotating on its sensing axis. Mass-produced devices, though, both vary in rate sensitivity, are relatively noisy, and exhibit a DC offset, the bias, at zero rate.  

The combination of a rate gyro and an accelerometer, via a Kalman filter, can produce the current attitude by using one sensor to correct or compensate for the other sensor's weaknesses.  

If the rate gyro bias were zero, simply integrating its output (summing successive samples at a constant rate) should produce the current attitude - which should match the tilted accelerometer ideal output. Similarly, the ideal accelerometer output should be matched by the integrated rate gyro data. In practice neither is correct at all times, but the smart math of the well-tuned Kalman is able to infer and use the errors of each to produce a correct output. The DC bias of the rate gyro is, in fact, one of the values that many Kalman implementations yield even while the sensors are in constant motion. 

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