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