guidance, flight mechanics and trajectory optimization

where & is the required velocity computed from linearized equations such
as Equation (2.ll). Substitution of Equation (2.15) in Equation (2.11)
yields equations for the +Lz. A second technique which yields essentially
the same result as that of Anthony and Sasaki method was developed by Bonomo
and Schlegel LReference (2.12)_7. This second method seems to be more
formalized and compact for on-board implementation and is, therefore, chosen
for detailed discussion here. The method consists of first determining the
velocity required for rendezvous by the use of linear equations; next the
miss resulting from the use of this velocity is calculated using Equation
(2.14); and finally, the transition matrix is applied to this miss to de-
termine a new velocity correction.
CURRENT Calculate required
POSITION velocity from
-linear equations
V’ -II,
(Equation 2.2.1)
Predict miss using
this velocity from
Apply transition
4~~7) matrix to miss
_ second order
equations
--vector to
determine new
(Equation 2.3.1) velocity correction
I - --..-. -1
Block Diagram of Second Order Velocity Correction
Figure 2.8
If &(+)denotes the miss distance to the second order resulting from the use
of the first order velocity correction at t = 0, then the relation between
the miss and the required correction to the linear velocity increment is
given in terms of the transition matrix GIT,oI as
The transition matrix can be partitioned as
G(?;Ol =
Therefore, the desired relation between the second order miss and the re-
quired velocity correction is given by (since it is assumed that SE (01 = 0)
or
49 (7) = G,, a_v (0)
47