L
Lx =■ /,«, = LsmO sin^ L2~ /2u2= Lsin0cosij> 1^=* I3u3=> LcosS
We can choose 9 to lie between 0 deg and 180 deg, so Eq. (1680c) determines 9 completely, with <o3 given by Eq. (1660c) or (1669c). Note that with these conventions, 9 is the nutation angle introduced in Section 15.1. Then, Eqs. (1680a) and (1680b) determine if> completely, including the quadrant, with to, given by Eq. (1659a) or (1669a) and <o2 by Eq (1659c) or (1669b). We cannot determine $ in this fashion, so we use Eq. (1612b), which in the notation of this section is d<t>
Using Eq. (1680) yields the equivalent, and more useful, form d<2> /,«?+/,«2
In the asymmetric case, /, I2 and Eqs. (1669a) and (1669b) can be substituted into Eq. (1682). Integration results in a closedform expression for </>, which involves an incomplete elliptic integral of the third land [MilneThomson, 1965; Byrd and Friedman, 1971; Morton, et ai, 1974]. In the axial symmetry case, on the other hand, d$/df is a constant, and we have and
where the inertial nutation rate, co;> was introduced in Section 15.1 and Eq. (1667a). The initial value of <fi in Eq. (1684), is arbitrary because the definition of the inertial reference system only specifies the location of the inertial three axis.
Because the kinematic equations of motion for the 313 Euler angles have now been solved, the direction cosine matrix can Ik found from Eq. (1220). Any other set of kinematic parameters can then be evaluated by the techniques of Section 12.1, e.g., the Euler symmetric parameters from Eq. (1214), the Gibbs vector from Eq. (1218), or the 312 Euler angles from Eq. (1223). The resulting parameters specify the orientation of the spacecraft body principal axes relative to an inertial frame in which the angular momentum vector is along the inertial three axis. It is frequently more convenient to specify the orientation of the spacecraft relative to some other inertial frame, such as the celestial coordinate frame. This is especially important if the resulting closedform solution is to be used as the starting point for a variationofparameters analysis of the motion in the presence of torques, as described Mow, because the angular momentum vector is not fixed in inertial space when the torque does not vanish. Changing this reference system is straightforward if there is a convenient rule for the parameters representing the product of two successive orthogonal transformations. The most convenient product rule is Eq. (1215) for the Euler symmetric parameters, so we will write the closedform solution for this kinematic representation. This solution, in the axial symmetry case, is

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