guidance, flight mechanics and trajectory optimization

significantly reducing the out-of-plane velocity when the out-of-plane
distance is small. The body of Tables 4.1 and 4.2 contain measurement error
sensitivity coefficients 2 I ?&.co, where the six variables are ~'(5, $",
5', g, f~,s')
* It is%een that the quantities in the matrix depend
only on i&e &it of time (here it is the second). The results represent the
effects of errors in the control due to the measurement errors of the initial
conditions, the'vehicle being at its nominal position. (They are not the
usually obtained errors due to errors in actual position with nominal control.)
The very high dependence of rendezvous errors on initial velocity errors
indicates the need for updating the relative velocity information and for
making the required alterations in the thrusting program.
As another example, Goldstein et. al. (Reference 4.9) compare numerical
results of four methods of determining the velocity increment to rendezvous
for a series of planar
n. mi = 557 KM altitude.
rendezvous problems for
Two of their figures
a circular target orbit at 300
are shown as Figure 4.3 and 4.4.
These figures show the velocity increment to rendezvous as a function of the
time from rendezvous. The four methods of determining the velocity increment
and the labels for the figures are: (1) the power limited optimal theory of
Section 2.4.2 (a curve labeled Phase I); (2) the fuel optimal (fixed time)
theory above (a few points labeled P or L depending on the initial
Method 1 or 4), (3) t wo impulse computation for a series of times
guess from
(a curve
labeled Two Impulse), and (4) logarithmic or proportional guidance for the
case where the relative separation is initially decreasing (a series of points
labeled with crosses_and the value of K (=&@~/fp/p) ). These transfers
were optimized onr( a/ir)/lj/p)
The thrust magnitudes vary between the
methods used, but range below . b ft/ sec2 = .2m/sec2 for the cases in
Figure 4.3 and below 1.5 ft/sec2 = .5 m/sec2 for the cases in Figure 4.4.
The most important conclusion to be observed from these results is that the
two-impulse rendezvous is an extremely good approximation to the optimum if
the relative phase is such that the rendezvous transfer is nearly an optimal
transfer. Furthermore, as has already been noted, the impulse function for
for optimal two impulse transfers for such orbits in the rendezvous problem
has a very broad minima.
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