A new kinematic theorem for rotational motion

The Goodman-Robinson theorem (ASME Jour. of App. Mech., vol.25, p, 210-213, 1968), used to explain kinematic drift in strapdown attitude algorithms due to coning motion, can be thought of as an integral form of the rotation vector differential equation. This theorem states that, in the absence of instrument errors, the delta-theta count of a rate-integrating-type gyro is equal to the time integral of the angular velocity component along the gyro sensitive axis, plus the area that the sensitive axis traces out on a sphere of unit radius. This paper utilizes the Darboux frame from differential geometry to obtain an expression for the area term in the Goodman-Robinson formula. It turns out that this term is equal to the time integral of the component along the gyro sensitive axis of the angular velocity of the angular velocity of the sensitive axis, plus exterior angle terms. The results of this paper provide a geometric explanation of how movement of the direction of the angular velocity vector contributes to kinematic drift.