guidance, flight mechanics and trajectory optimization

Manipulation of the equations involving d' results in
From this equation, it is seen that the condition
K2 + K -1 =. 0
must be met if $ is to remain finite as p approaches zero.
2.4 OPTIMIZATION OF THE RENDEZVOUS MANEUVER
The cost of transporting fuel to space is so great that it is essential
in the design of the maneuvers to determine those which require the least
fuel and to choose the guidance mechanizations which approximate the optimum.
Eventhough the considerations of this monograph deal with small orbital
changes, the problem can be studied from the point of view of orbital
transfers. Thus, it is known from the work of several authors (Reference 3.1
through 3.5) that impulsive transfer of either one or two impulses will be the
optimal transfer (minimum A V requirement) between orbits such as are
involved here. Further, a phasing technique similar to that discussed by
Strahly (3.6) which involves splitting one of the two impulses into two
portions which are used as an integral number of revolutions apart can yield
rendezvous with the impulse of optimum two-impulse transfer. Some
possibilities of this technique were demonstrated by Bender (3.7). Con-
sequently, there exists a determinable lower bound to the velocity increment
to rendezvous for any given case. This bound can be used to evaluate the
effectiveness of any chosen scheme for its nearness to being optimal.
There is a feature of optimal two-impulse orbital transfer which
partially removes the need to optimize. This is the fact that for orbit
pairs which do not intersect deeply and which are fairly near to one another
all the way around, the effect of varying the departure point around the
first orbit is not very significant (see Figures 5 and 6 of Reference 3.8).
In addition to this feature of this class of optimum, impulsive transfers is
the well known result due to Lawden (3.1) which states that optimal space
maneuvers employing a rocket motor are constructed of zero and maximum thrust
segments. Thus, rendezvous schemes universally utilize this principle in
their design.
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