Theoretical Analysis on the Degree of Observability of Attitude and Accelerometer Bias under Static and In-Flight Alignment

This paper presents a theoretical analysis on the degree of observability of attitude and accelerometer bias under static and in-flight alignment by solving the error covariance matrix explicitly. By following the Kalman filter covariance update process, the elements of this matrix can be resolved under simple assumptions, and provide a good conformity to the simulation results. The solution consists of the magnitude of gravity, the given acceleration, and the initial covariance. By using this formula, the covariance of the estimation error, which represents the degree of observability, can be examined accurately before the estimation process. For the initial attitude covariance, set by the coarse alignment, it is shown that the error covariance of the roll, pitch, and accelerometer bias in the x and y axes decrease by half of their respective initial values under static alignment. Moreover, the estimation error variance of the yaw is shown to decrease significantly under in-flight alignment.