guidance, flight mechanics and trajectory optimization

with the rocket motor which is to supply the braking thrust. Data taking
and processing requirements could also place a particular requirement on .
For a generalization to finite thrusts, see below Section 2.3.1 and the work
of Tschauner and Hempel (Reference 1.k).
2.2.4.3 Extension to Non-Circular Orbits
The state transition matrix used in the derivation of the required
velocity in the last section was limited to circular orbit . If the
transition .matrix of Equation (1.58) is replaced by the symbolic notation
--pi $1 ,
the technique of the previous section can be'applied to any orbit. In the
case of a circular orbit, the transition matrix has been given (Equation 1.58
or Figure 1.58). For other orbits, the matrix is available in an analytic
form (Reference 1.1, 1.5, 1.6, 1.9). References 1.1 and 1.9 gives ex-
'pressions for transition matrices for elliptic and hyperbolic orbits. For
extension to finite thrust, see the work of Tschanner (Reference 1.5 and 1.6)
Very little use appears to have been made of the results for elliptic
target orbits and it is suggested here that they should be of significant use.
For example, possibly the use of the long set of approximate equations for
orbits of small eccentricity developed by Anthony and Sasaki (Reference 1.2
and Section 2.2.4.5 below) could be replaced entirely and with no limitation
on eccentricity.
2.2.4.4. Multiple Impulse Rendezvous
A modification of the previous scheme developed by Shapiro (Reference 2.11)
employs several velocity corrections and drives the range rate to zero by the
successive application of velocity pulses rather than by a single impulse at
zero range. This technique achieves a degree of flexibility over the previous
scheme in that it is also applicable to targets in an elliptical orbit.
The first velocity correction is computed from Equation (2.11) based on
a fictitious time to rendezvous r . Subsequent velocity corrections are
computed on the basis of a time to rendezvous equal to ( f - kt) where t is
the time elapsed since the initiation of the .rendezvous'maneuver and k is a
constant. The desired velocity magnitude decreases at each computation
period because of the dilated
see that this is so, consider
the approximations &ir-g~!)
are valid and Equation (2.2.1)
time scale and approaches zero as t --c 74 .
the solution (2.11) as t -G T/J . Inn $$-$y,
c fi(-r-dt) and Eaon (-r-At) =
reduces to
2
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