guidance, flight mechanics and trajectory optimization

2.4.1 Optimal Stepwise Thrusting
Assuming that a spacecraft is propelled by means of a chemical rocket
engine, optimization of space maneuvers on the basis of minimum fuel implies
optimization on the velocity increment provided by the engine. That is
aminimum 4.1
It is assumed that the engine, when it is turned on, operates at a fixed fuel
rate and that this condition implies a fixed acceleration because the total
change in velocity for the rendezvous is expected to be provided by less
than 5% of the total weight in fuel. If an unlimited time is allowed for a
rendezvous or transfer maneuver, the optimal thrust program solution will
revert to the impulsive solution in which many very short duration pulses at
each passage of certain points on the orbit (Breakwell Ref. 4.2) are applied.
Another variation of the rendezvous problem formulation is the time optimal
problem in which a fixed acceleration and an upper limit to the total amount
of fuel (i.e. AV) available are assumed; the solution then seeks the pro-
gram for rendezvous in the shortest time possible. 30th procedures lead to
a series of engine burns and both procedures require the solution of two
point boundary value problem as will be shown.
Most of the results to be presented in this section refer to the linear
problem with a circular reference orbit. The solutions to the equations of
motion are found in section 2.1. ( Equations 1 - 21 and 1 - 48 to 1 -
Before proceeding to the application of the Pontryagin Maximum Prin-
:iAie to these problems
some results obtained by Tschauner and Hempel
(Ref. 1.4) will be indiiated. (Tschauner has recently extended &he results
to the case of targets in elliptical orbits (Ref. 1.5 and 1.6).) These
authors consider the rendezvous maneuver on the basis of linear terms
and a circular reference orbit and separate the motion into the in-plane
and out-of-plane motions so that the quantity being minimized is
where
8
&the acceleration of gravity at the target orbit.
The out-of-plane motion is simple harmonic and is driven to zero by a
series of oppositely directed thrusts applied every half cycle, the last one
of which yields 5 3 = 0, 5 3 = 0. The duration ($) of each pulse depends
on the initial
of the engine.
motion, the number of pulses, desired,
The optimum cost is that of a single
and the maximum thrust
impulse applied at the
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