SPORE Mission Design - Georgia Tech SSDL - Georgia Institute of ...

⃗⃗⃗ [ ̇ ̇ ̇ ] (29)
⃗ ⃗ (30)
Inertial Fixed
As seen in Equation 31 and 32, rotate the inertial coordinate system by angle α to transform velocity and
position into a fixed coordinate system. The following method was used for PESST to find the
atmospheric relative position and velocity.
⃗⃗⃗⃗ [ ] ( ⃗⃗⃗ ̂ ⃗⃗⃗⃗ ) (31)
⃗⃗⃗⃗ [ ] ⃗⃗⃗⃗ (32)
The angle α is given by the following equation.
Here α 0 is the initial angle between the inertial and fixed x-axes (at time t 0 ), Ω E is the rotation of the
earth in degrees per second (~0.0042), and t is the amount of time elapsed in seconds since t 0 . For a
start time of 9 Feb 2011 17:00:00.000 UTCG as specified in the STK model, α 0 is equal to 34.294°. The
above transformation does not take into account the effects of precession and nutation. Therefore it is
not equivalent to the STK transformation.
Fixed Inertial
Use the transpose of the rotation matrix given in Equation 32 to transfer between fixed and inertial
axes. These relations are given in Equation 33 and 34.
(16)
⃗⃗⃗⃗ [ ] ⃗⃗⃗⃗ (33)
⃗⃗⃗ [ ] ( ⃗⃗⃗⃗ ̂ ⃗⃗⃗⃗ ) (34)
The transformation in Equation 33 and 34 does not take into account the effects of precession and
nutation. Therefore it is not equivalent to the STK transformation.
4.2 Baseline Orbital Trajectories
The LEO and GTO trajectories can be found in Table 6, Table 7, and