guidance, flight mechanics and trajectory optimization

This matrix is simply written as G(8) (i.e., this notation is used to avoid
writing)
4-3 cos 8 0 Sin 6 2(1 - cos e)
6(Sin 0 - '3) 1 ~(COS 8 -1) 4 Sin e - 3
G(8) = 3 Sin 0 0 cos 0 2 Sin 8
6(Cos 8 - 1) 0 -2 Sin 6 4 cos 0 - 3
This matrix, when combined with that of Equation 1-38 and both expressed in
terms of 8 = nT (and n added as needed to give dimensions correctly), are seen
to be exactly that of page 145 of SID 65-1200-5 (Reference 1.1).
In the case of the representation of state transition matrix for locally
level inertial systems (Table 2.4.2, page 147 of Reference l.l), the coordinate
transformation required is only that between inertial and rotating systems at
the moment they are aligned. Reverting to 6 = nT, the rotation rate is one of
the angular velocity, n, about the z or third axis and the transforma-
tion matrix, T, for 6 =TX is
with
T = [
‘&
---d- I ' 0
1
0 -n!
n o!I
The state transition matrix for inertial locally level (at both times) may thus
be obtained from G(nT) and it is Q!
'1
= T'lG(nT)T =
:
2-Cos nT sin nT l/n Sin nT 2/n(l-Cos nT)
2 Sin nT-3nT 2 Cos nT-1 2/n(Cos nT-1) l/n(4 Sin
n(3nT Sin nT) n(l-Cos nT) 2-Cos nT 3nT-2 Sin nT 1.58
n(Cos MT-l) -n(Sin nT) -Sin nT 2 Cos nT - 1
The development of the system by Tschauner and Hampel (Reference 1.4)
involves a substitution to simplify the matrix of coefficients. In addition,
the out-of-plane motion will be included and the set of six equations solved
with matrix notation for later reference. It is simplest to add the out-of-
plane coordinates to the set of four in-plane variables of Equation 1.48 as
18
1.55
1.56
1.57