guidance, flight mechanics and trajectory optimization

1.0 STATEMENT OF THE PROBLEM
SATELLITE RENDEZVOUS
This monograph is directed to that portion of the rendezvous problem in
which the relative distance and velocity between an active spacecraft and a
passive target satellite must be reduced from moderate values (say 50 km and
5 km/set) to small values (say less than 5.0 meters and 1.5 meters/set).
&-board relative position and velocity sensing are assumed for the purpose of
allowing precise manual or automatic steering. These observed quantities are
to be utilized to drive the state of the system to zero in a reasonable time
with as little fuel as possible. Neither the gross orbital changes which have
been brought about previous to this closing (rendezvous) maneuver, nor the
final phase, known as docking, will be considered. The target satellite will
be assumed to be in a closed orbit; and since perturbative influences (such as
a non central gravity field) will be nearly the same on both vehicles with the
result that their effect will be very small, the orbit will be assumed to be a
two-body orbit, Since this monograph represents an attempt to survey the known
information regarding the rendezvous problem, it will be analytical in nature
and will not refer to any particular spacecraft or its capabilities.
The problem of station keeping is similar to that of rendezvous in that
it is assumed that a satellite is to be maintained in a specified orbit with a
specified phase within tolerances similar to those mentioned for rendezvous.
Thus, in a sense, the target is a point which moves along a desired path (this
path may not correspond to the motion in the actual gravitational field). On
the other hand, the chase vehicle moves along a path relative to this desired
path which is defined by the perturbative influences acting on the vehicle and
the differences in the positions and velocities. Accordingly, the position
coordinates of the chase vehicle may consequently deviate from those of the
target. After such a deviation has accumulated for a period of time the prob-
lem of returning the active craft to the nominal path in a substantially
shorter time is the rendezvous problem as presented. Of course, it is assumed
that the active craft possesses a mechanism by which the deviations from its
nominal trajectory are determined as they may be needed.
The discussions begin with the presentation of the field free case; i.e.,
the case in which the same gravity acts on both satellites. This problem is
of little physical importance; however, it serves to provide valuable insight
into a more rigorously formulated system. It might be surmised, at first
thought, that in this case no guidance technique would be necessary since an
astronaut could effect the rendezvous by line-of-sight thrusting. This would
approach would, however, cause the motion of the active craft to be one of
constant angular momentum about the target. That is, if an angular momentum
caused by an initial small velocity (v ) perpendicular to the line of sight
exists at the distance ro, then, if thg'distance is reduced to 10B3 r. and
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