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Appendix D / Point Mass Dynamics
_ _
17.2 Transformation Time Derivatives
Consider the transform from frame (I) to frame (R) at time (t), and again
(∆t) seconds later,
T
R
I
R
R
( t + ∆t
) : = T ( t , t + ∆t
) ⋅ T ( t )
If the rotation of frame (R) during the time (∆t) is small,
T
R
I
17-4
I
R(
t + ∆t
)
R
( t + ∆t
) : = ( I − [ ∆E
× ] ) ⋅ T ( t )
3
R(
t )
Replacing the Euler angle triplet with average body rate over ∆t,
R
( t + ∆t
) − TI
( t ) R
: = − [ ω × ] T ( t )
R
TI R
I,
R ⋅ I
∆t
As ∆t tends to zero, the transformation matrix derivative at time (t),
R
I
R
R
[ ω × ] T
T& : = − ⋅
Consider a 2 nd rotating frame (C), and using the same process,
T &
I
C
: =
T
I
C
⋅
I,
R
C [ ω × ]
I,
C
I
I
I
Equation 17.2-1
Equation 17.2-2
Equation 17.2-3
Equation 17.2-4
Equation 17.2-5
The time rate of change of the transform between the two rotating frames,
T &
R
C
: =
−
R
C
R
C
R
I
I
C
T : = T ⋅ T
T&
: = T&
⋅ T + T ⋅ T&
R
I
I
C
R
I
R R I R I C
[ ω × ] ⋅ T ⋅ T + T ⋅ T ⋅ [ ω × ]
I,
R
I
C
I
I
C
C
I,
C
Equation 17.2-6
Equation 17.2-7
Equation 17.2-8