ABSTRACT
U u V v W w
p q r
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
=
+
+
+
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
Δ
Δ
Δ
, ⎥
⎥
⎥
⎥
=
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
=
+
+
+
⎡
⎣
Δ
Δ
Δ
Δ
Δ
Δ
p q r
, φ θ
ψ
φ φ θ θ
ψ ψ
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
(5.1)
where Ue, Ve and We represent the steady trimmed velocities of the aircraft and the steady-state body axis angular velocities are each assumed to be equal to 0. They are related to the trimmed angle of attack and the trimmed sideslip angle by the relations
W U V
V U= +
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
= ⎛
⎝
⎜
⎞
⎠
⎟
1and (5.2)
Hence,
V U W U V Ue e e e e e e e e e= ( ) = + ( ) = + ( )( ) ( )tan tan tan tan .β α β αand 2 2 21
(5.3)
Hence, the linearised equations of motion are
m u qW rV
m v rU pW
m w pV qU
Δ Δ Δ
Δ Δ Δ
Δ Δ Δ
+ −( ) + −( ) + −( )
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
= −( ) + ( ) −( ) + −( ) ⎡
⎣
⎢
⎢
⎢
F F T F F T TNB NBe BW e e AS ASe BI BIemgα β, 0 0 1⎢
⎤
⎦
⎥
⎥
⎥
⎥
(5.4)
and
I I I
I I
p q r
0 0 0
−
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
=
Δ
Δ
Δ
M B NBe BW e e AS ASe−( ) + ( ) −( )M T M Mα β, . (5.5)
Further, the gravitational force perturbation vector is
e e e eT T−( ) ⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
≈
−
−
0 0 1
0 0 0
cos cos sin
θ φ θ φ θcos sin
− −
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥sin cosφ θ φ θ
φ θ ψe e e ecos sin
. 0
Δ
Δ
Δ
(5.6)
In addition, we have the relations
Δ
Δ
Δ
φ θ ψ
φ θ φ θ φ φ
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
= −
1 0
sin tan cos tan sin
sin
p q rcos cos cos
. 0
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
Δ
Δ
Δ
(5.7)
Finally, the aforementioned equations must be complemented by perturbation equations for the vehicle position:
d dt
x y z
U u V v W w
Δ
Δ
Δ
Δ Δ Δ
Δ
Δ
Δ
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
= + + +( ) +
+
+
⎡
T ψ ψ θ θ φ φ, , ⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
− ( ) ⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
= ( )
T
T
U V W
u v
ψ θ φ
ψ θ φ
, ,
, , Δ
Δ
Δ
Δ
w
U V W
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
+ ( ) ⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
T ψ θ φ, , ,
(5.8)
where
TIB ψ θ φ ψ ψ ψ ψ
θ θ , ,
cos sin cos
cos
si ( ) =
− ⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
−
sin sin0 0
0 0 1
0 0 1 0 n cos
cos sin cos
, θ θ
φ φ φ φ0
1 0 0 0 0
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
sin
(5.9)
Δ Δ Δ ΔT T TIB e e e IB e e e IB e e eψ θ φ ψ ψ θ θ φ φ ψ θ φ, , , , , , ,( ) = + + +( ) − ( ) (5.10)
and Δ Δ Δu v w T ⎡
⎣
⎤
⎦
are the components of the aircraft perturbation velocity vector in the body axes.
