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where B is the transformation from the body frame to the reference frame defined by Eq. (18-46). The matrix A transforms any vector, V, from orbital coordinates (V0) to body coordinates (Va); that is, A\0 in Eq. (18-48) is irrelevant because infinitesimal rotations commute.

The angular velocity vector in the body frame is (see Section 16.1) approximately

where «*=/*©/R3 is the orbital angular velocity of a spacecraft in a circular orbit of radius R, and jute = GMm is the Earth's gravitational constant The zenith vector in body coordinates is

and hence the gravity-gradient torque (see Section 17.2) is

Ncc = 3^ X(/rfi) = 3^(^(7, - Iy ),$,(/, - ),0)T (18-51)

where the moment of inertia tensor, /, is assumed diagonal with components Ix, Iy, and It along the body axes. Note that to first order there is no gravity-gradient torque along the yaw axis. (There is, however, a yaw-restoring torque which results from gyroscopic roll/yaw coupling, as described in the next subsection.)

With the previous definitions, Euler's equations in body coordinates for a spacecraft with internal angular momentum, hx, hy, and ht along the body X, Y, and Z axes are (see Section 16.2)

j-L+mXL1 at

where N£, Nc, and Ncc are the environmental, external control, and gravity-gradient torques, respectively. Writing Eq. (18-52) in component form gives

* For orbits of nonzero eccentricity, the velocity vector is replaced by the cross product of the negative orbit normal and the nadir vector.

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