where q\ = Uj cosasin/?+u2 sin a sin/3 q2 = cosck sin/?— if| sin a sin/3 q'3 = u3 cos a sin/? +sin a cos/? qi = cos a cos /? - w3sin a sin /? a — 2upt u=Lo/|Lo| = [ii„ii2>ii3]T Lo = [ 1.01,^02, LM ]T

(16-87a) (16-87b) (16-87c) (16-87d) (16-87e) (16-87f) (16-87g) (16-87h)

In this solution, all the constants of the motion have been reexpressed in terms of initial values of the Euler symmetric parameters and Lg, the angular momentum vector in body principal coordinates. These initial values are arbitrary (except that the sum of squares of the Euler parameters must be unity) because the inertial reference frame can be chosen arbitrarily.

A geometrical construction, due to Poinsot, and presented in many texts (e.g-,

Goldstein [1950]; Synge and Griffith [1959]; MacMillan [1936]; Thomson [1963]; and Kaplan [1976]) pictures the rotational motion of a rigid body as the rolling of the inertia ellipsoid on an "invariable plane" normal to the angular momentum vector. In the axial symmetry case, Poinsot's construction is equivalent to the discussion in Section 15.1 of the space and body cones. In the general case, the geometrical construction is not easy to visualize, and the analytic solutions are more useful for spacecraft applications. The results for the asymmetric case are described by Morton, et al., [1974].

The solutions of the attitude dynamics equations in the torque-free case have been obtained above. The variation-of-parameters formulation of attitude dynamics is a method of exploiting the torque-free solutions when torques are present [Fitzpatrick, 1970; Kraige and Junkins, 1976]. Our approach follows that of Kraige and Junkins.

To introduce the basic ideas of the variation-of-parameters approach, we first consider a simple example, the- translational motion of a point mass in one dimension. The equations of motion in this case are

where the dependence of the force on x, o, and t is arbitrary. The solution of these equations when F=0 is x(t)=x0+v0t (16-89a)

This is called the forward solution because it expresses the position and velocity, x and v, of the mass at time t in terms of its position and velocity, jc0 and ©q, at the prior time, / = 0. We can also write the backward solution:

which expresses x0 and ©0 in terms of x(t) and v(t).

The central idea of the variation-of-parameters approach is to use Eqs. (16-89) to represent the motion of the mass even when a force is applied. This is possible if ■x0 and o0 are allowed to be time varying, as shown in Fig. 16-4. At each point on the trajectory of the particle, the position and velocity are the same as those of the force-free motion represented by the tangent line with intercept x<£t) and slope Oo((), i.e., with initial position and velocity x^t) and o^/). In this case, x0 and ©0 are the varying parameters that would be constant in the force-free case. (It is possible to express the motion in terms of other parameters, such as the kinetic energy, but we will only consider initial conditions as the varying parameters.)

To obtain the equations of motion in the variation-of-parameters form we

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