guidance, flight mechanics and trajectory optimization

2.1.4 Solutions to the Equations of Motion
As has been mentioned in the discussion of the state transition matrices
of the State Determination Monograph (Reference 1.11, this matrix represents
the solutions to the homogeneous parts of various sets of the equations. However,
rather than refer to them directly, solutions will be developed making
use of matrix methods in this section; especially since it is desired to
include the effects of the thrusting acceleration, a3. The sets of equations
will be sp&&mto - \- the out-of-plane motion and the in-plane motion. Further-
more, the techniques of the matrix method will be illustrated in the solution
for the out-of-plane motion; since this motion is seen to be simple harmonic
motion with a forcing function (except when the reference orbit is elliptical
and the independent variable is eccentric anomaly). Matrix methods and results
are given by Leach (Reference 1.31, Tschauner and Hempel (Reference 1.4) for
circular reference orbits and by Tschauner and Hampel (Reference 1.5) and
Tschauner (Reference 1.6) for elliptic reference orbits.
2.1.4.1 The Out-of-Plane Motion
The out-of-plane motion can be represented by the differential equation
for all cases except the set Equation 1.18, which will be considered later.
The matrix methods require that the equations be expressed as linear first-
order equations. This is accomplished by de.IYni.ng the two vectors 5 as
Thus, Equation 1.28 becomes
where
t=(;j = (2)
and 8 =
To proceed, the fundamental matrix (F) for A must be found; that is, a set
of independent solutions to
1.29
1.30