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