guidance, flight mechanics and trajectory optimization

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