guidance, flight mechanics and trajectory optimization
guidance, flight mechanics and trajectory optimization
guidance, flight mechanics and trajectory optimization
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
.<br />
will attain, z , can be calculated by substituting the value of t, from<br />
Equation (2.5) into Equation (2.4). This substitution gives<br />
Thus, assuming l&j L4 Ia,1 (this approximation becomes better as<br />
r-0 as will be seen when motion along the LOS is discussed) the ex-<br />
pression becomes<br />
The overall behavior of 6 can thus be determined from this equation <strong>and</strong><br />
from the one given earlier for a no-thrust condition L Equation (2.3)].<br />
This overall behavior will be a limit cycle between the values f0 <strong>and</strong> x<br />
as characterized in Figure (2.4). The time between applications of normal<br />
thrust ( tc) <strong>and</strong> fl can be<br />
calculated from Equations (2.3)<br />
<strong>and</strong> (2.6). Letting z= -$<br />
(the time until rendezvous<br />
occurs assuming no radial<br />
thrust is applied) quation<br />
(2.3) results in<br />
a'Limit Cycle<br />
Figure 2.2<br />
Finally, substituting for Z$ from Equation (2.6) gives<br />
A certain minimum time will be necessary to take <strong>and</strong> process the data<br />
to determine the current position <strong>and</strong> velocity. Therefore, it will be<br />
necessary to require that the time between corrections, t,, be longer,than<br />
the minimum data taking time. Since p is to be driven to zero <strong>and</strong> F, is<br />
small, it can be seen from Equation,(2.7) that a minimum t, will be maintained<br />
ifaminimum 1 is maintained. In turn, a constant7 ( or more practically<br />
a constant range of values for T ) is obtained by a proper choice of<br />
thrusting periods for the LOS thruster. A discussion of motion along the LOS<br />
is now called for so that the behavior of 2 during periods of LOS thrusting<br />
34<br />
(2.6)<br />
(2.7)