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