guidance, flight mechanics and trajectory optimization

Woodrow have shown that t#le Neustad, method of solution cihich changes such
problems to maximum-minimum problems can be used in this case. They
illustrate the convergence properties for three methods of solving the max-
imum-minimum problem. However, regardless of the approach taken in solving
the two-point boundary value problem, the most important feature of the
success or lack of it is the starting point assumed. Without a reasonably
good guess for the first attempt, it has been found that optimum transfer and
rendezvous problems may not converge. Fortunately, there are at least two
schemes for obtaining valid initial estimates for the solutions. One method
uses the solution to the quadratic payoff function problem which can be found
explicitly for rendezvous (Section 2.4.2 below). Another scheme bases the
initial guess for the problem solution of the impulsive solution.
Still another approach has been suggested by McCue (Reference 3.14). In
this reference, the application of quasilinearization to solve the two-point
boundary value problem along with the use of impulsive transfers as the scheme
for generating the first guess of the solution has been shown to be capable of
yielding optimum finite thrust coplanar orbital transfers between arbitrary
elliptic orbits. This particular scheme has not been applied to the linear
rendezvous problem as formulated above to the author's knowledge but it should
be a rapidly converging device for the problem. Another demonstration of the
use of the impulsive case as a starting point is given Handelsman (Ref. 4.15).
Actually, the computations involved in locating optimal rendezvous are
required only in the case of low thrust systems. The pratical solution for
optimal rendezvous to any problem where the thrust capability is of the order
of 1 m/sec2 (3 ft/ sec2) can be obtained from the optimal impulsive solution
by replacing the impulsive thrusts by finite burns as is shown by Robbins
(Ref. 4.24). In this analysis, it was assumed that the thrust was of
sufficiently large magnitude that thrusting arcs would not be excessively
long. This is a very modest requirement because, as Robbins points out, an
error of not more than .2$ in the velocity requirement can be assured if the
central angle traversed during the burn does not exceed .22 radians. As an
example, consider a circular orbit at 300 KM above the earth; this angle
corresponds to a burn of nearly 200 seconds (this time, generally increases
as the altitude increases).
As an example of the application of this material, consider Reference 4.11.
In this reference Paiewonsky and Woodrow analyze the problem of time optimal
rendezvous in three dimension from two separate conditions at a series of
available AV'S. For the case illustrated, the situation is somewhat like
an abort problem in that the two craft are separating at the initial time and
they are to be rejoined. Thus, an immediate first burn is required which must
at least change the sign of the relative velocity. Several tables of error
sensitivities were shown which illustrated the capability of the procedure
and the interesting feature that three burns are optimal in certain cases;
some resIiLtS are reproduced in Tables 4.1 and 4.2. When the amount of fuel
available is large, a single burn tine-optimal trajectory occurs as is ex-
pected. For a moderate amount of fuel (the case of Table 4.1), two burns
occur; while for a reduced amount of fuel near the absolute minimum (the case
of Table 4.2), an intermediate burn also develops. The thrust required during
this intermediate burn is largely out-of-plane and has the effect of
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