Celestial Rate Sensing

r T 1 H E ANGULAR velocity of a space vehicle can be imJportant to the operations of attitude control and data compensation. Components of angular velocity relative to inertial space resolved along body-fixed axes can be measured by rate gyroscopes fixed in the vehicle with sensitive axes along the prescribed body axes. However, exactly the same components can be determined from the measured velocity of drift of the star field across the fields of view of two or three body-mounted telescopes. This may be called "celestial rate sensing/' and may be an attractive alternative to mechanical rate gyros on mechanization grounds. The purpose of this note is to describe the principle of operation of celestial rate sensors of a particular type, and to derive the relationship between the vehicle's angular velocity in inertial space and the output of such an instrument. Mechanization details are not included, largely because these are proprietary features from the viewpoints of the several companies concerned with such developments. For this analysis the telescope and its field are analogous to a pinhole camera and its field. The camera's photographic plate or photocathode ("reference surface") and the image of a star produced on it are shown schematically in Fig. 1. A vector directed perpendicularly from the pinhole to the reference surface is denoted by f0. The position of a star image on the plate at time t is given by the vector h. The position of the image, a small time increment At, later is given by f2. The star field is rotating at an angular velocity -co with respect to the camera pinhole and plate, where co is the vehicle's angular velocity in inertial space. The vector velocity of the star image at h is given by (-co X fi). The projection of this velocity onto the plate produces v, the velocity of the star image in the plane of the plate. Note that the projection to the plate is not a perpendicular one but is directed from the pinhole along fx and r2. A base vector reference system eh e2, e3 is presented in the figure; £3 is along fo while ei and e2 lie in the plane of the reference surface. Polar coordinates p and 6 may be used to represent the position of the star image in the plane. The first step is to determine v in terms of p, 6, rQ and co. From the figure it is clear that (f2 — fo)-r0 = 0 and that [f\ + (-co X fi)Af] X f2 = 0. Expanding the several vectors in the base system and substituting them into these relations, and using the fact that vAt = r2 — h} it can be shown that in the limit as At -> 0