guidance, flight mechanics and trajectory optimization

where 2((t) is given by the first equation of (4.38). This miss is to be
reduced to zero by an initial coast period of to seconds with
An earlier paper by Kidd and Soule (Reference 4.18) also discusses
optimum rendezvous in a gravity free space. The basis of the optimization
is that the optimum initial conditions for a rendezvous maneuver have the
relative velocity vector oriented along the line of sight. (This conclusion
can be verified by examination of Figure 4.2 which shows that for constant
values of the other parameters, a minimum tf occurs when yo=O.) If it is
assumed that midcourse corrections have placed the rendezvous vehicle on a
trajectory such that the relative velocity vector is nearly parallel to the
line of sight, then the thrust vector will make only small angles with the LOS.
In this paper, the cost function was selected to be the amount of energy
expended normal to the line of sight,(i.e., the integral of the impulse due
to the non-zero angle between the LOS and the thrust vector) and the
optimization was performed by a variational technique. The range of initial
.conditions over which the results are valid are limited by the same
assumption as the first method of this section with the additional re-
striction that the initial relative velocity vector must be nearly parallel
to the LOS. This method seems to have no advantage over Gunckel's method
and, therefore, will not be discussed further here.
2.4.3.2 Linear Gravity Model
The analysis in Section 2.4.1 is limited as an on-board mechanization
because of the necessity of using iterative methods to achieve a solution.
However, a method discussed in Reference (4.18) approximates an optimum
policy for the terminal maneuver; although an iterative solution is still
required, the convergence was found, by the authors of (L+.18), to be rapid
for a wide range of initial conditions. The basis of the approximation is
the observation that for a range of trajectories, there are only small
variations in the thrust angle ! d" I. This observation suggests an ex-
pansion for $ of the form
where $ >7 (a t& tcftr))
(4.41)
The procedure assumes that this is the form of the optimal steering angle,
then inserts this form in the equations of motion and finally determines the
constants so that rendezvous actually occurs. Assuming a circular target
orbit,(the equations of relative motion are derived in Section 2.1.3.2) and
neglecting the out of plane motion, these equations are
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