where / is the inertia dyadic and may only have the three principal terms Ix, Iy, and Iz, if it is projected onto the principal axes of the body. R is the magnitude of the Earth vector, and eR is the Earth unit vector from the Earth's center to the satellite as shown in Fig. 1. /x is the gravitational constant for Earth which is GM®. The first term on the right hand side of Eq. 1 is the gravity gradient torque and the second term is the gyroscopic torque with angular velocity w. Equation 1 written in matrix form is where (") is the skew-symmetric matrix representing the matrix counterpart of the vector ( ') for vector production. For simplicity, let the orbit be coplanar with the ecliptical plane. In the context of solar pointing, select the inertial frame such that the Z, axis is pointing toward the Sun, the T/ axis is perpendicular to the orbit plane, and the X, axis is the first coordinate of the right-hand coordinate system, as illustrated in Fig. 1. Let the body frame be selected such that the XB, YB, and ZB axes coincide with the Xh Yh and Z, axes when the body is at its solar pointing attitude (Po in Fig. 1). It is assumed that the principal axes and the body axes coincide. Let a be the angle between R and the X, axis. The angle a - is then the angle between R and the XB axis, where 0u is the angle between XB and X,. Because this paper is only interested in planar motion, Eq. 2 becomes Eq. 3 by substituting and since i±/R3 = wg, where <o0 is the orbital rate, the Earth rotation rate, Eq. 3 becomes, in this case, QUASI-INERTIA ATTITUDE TRAJECTORY ANALYSIS
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