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. When vertical in the opposite orientation, the output ideally shows -1g. This relationship makes it usable as a tilt sensor. But it also senses lateral acceleration, like a linear speed increase in an automobile, which may falsely appear as tilt. Furthermore, mass-produced devices vary somewhat in sensitivity.  

A rate gyro yields an output that corresponds to rotation rate, like 0.015 degree/second, and yields an ideal zero output when it is not rotating on its sensing axis. Mass-produced devices, though, vary in rate sensitivity, are relatively noisy, and exhibit a DC offset (or sensor 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 should produce the current attitude - which should match the tilted accelerometer's 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 filter is able to infer and use the errors of each to produce a correct output. The sensor bias of the rate gyro is, in fact, one of the values that many Kalman filter implementations yield even while the sensors are in constant motion. 

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