Relative Quaternion State Transition Relation

The attitude of a maneuvering spacecraft relative to a desired noninertial reference is compactly represented in the quaternion format by the relative quaternion. The popular technique for bootstrapping the relative quaternion relies on the state transition matrix for the quaternion strapdown equations of motion wherein the rates are estimates of spacecraft rates relative to the desired reference written in body coordinates. Even with a perfect three-axis gyro pack, whose signals are noiseless and always proportional to spacecraft inertial rates, the mere fact that the transformation from reference to body coordinates is not exact causes the relative quaternion estimate by the popular technique to diverge from the truth. It is shown that the rate of divergence from the truth is a function of the post-update attitude error, the maneuver rate, and the gyro sample frequency. An alternate form of the state transition matrix is derived which is invariant under all transformations from reference to body coordinates. With perfect gyros and for a spacecraft spinning at a constant rate, the error in the relative quaternion estimate, using the invariant state transition matrix, remains bounded to the postupdate attitude error.