guidance, flight mechanics and trajectory optimization

Goldstein et al, (Reference 4.9) utilizes linear equations for the in-
plane motion only and describes a procedure for obtaining the optimum thrusting
and steering program. The problem is formulated as a Mayer problem in the
calculus of variations and the switching function is developed. A sequence
of thrusting programs is then constructed and developed in such a way as to
approach that one which satisfies the maximizing conditions. Computational'
results for a series of cases are given and these will be summarized below.
The analysis to be presented is similar to that of Paiewonsky and Woodrow.
In this case, the full set of equations for the circular reference orbit is
used because the function to be minimized involves all three thr,ust directions,
that is, Equation 4.1 is used. In addition, both the time optimal and fuel
optimal rendezvous problems are developed. Much of the analysis is common
to the two procedures; therefore, these discussions will be carried together
until it finally becomes necessary to distinguish one problem from the other.
These two problems are in fact very similar to the time and fuel optimal or-
bit transfer problems which are described by McIntyre (Ref. 4.12, Section
2.3.4, pages 61-68) as illustrations of the Pontryagin Maximum Principle.
The differences between rendezvous and transfer problems are: first, that
for the three dimensional transfer problem only five variables corresponding
to some set of five distinct orbital elements have to be matched at the end,
whereas for rendezvous six variables are needed so that the final position
on the target orbit matches a given phase situation; and second, that the
transfer problem is extremely non-linear while the differential equations
for the present problem are linear with constant coefficients.
In order to attack either of these two problems in three dimensions, it
is necessary to combine the differential equations of motion and one for a
measure of fuel consumption into a set of seven linear differential equations.
The equations of motion were integrated in section 2.1.4.1 as follows: to
begin with, the differential equations of motion are: (Eq. 1.21).
where the els are distances expressed in units of the radius of the circular
reference orbit; the independent variable, 8, can be taken as the true
anomaly or the mean anomaly in radians (time in units of the time for the
reference particle to move one radian around its orbit); and ui are the
forces/unit mass in units of the acceleration of gravity at the reference
orbit.
The variables (51; 32, r; si 53, s; >= 'rT were transformed
to WT according to Eq. 1.59 and thi sol&on was obtained as (Eq. 1.64) as
4.10
w = F (Q) (w 0 - z) 4.l-l
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