1. Problem Statement
• Objective: Estimating user satellite’s drag coefficient in low orbit by its
trajectory estimated from GPS pseudoranges.
• Orbit determination problems.
• Question to ask: Comparing the measurement of 𝑐 𝐷 for different satellite initial conditions.
• Motivation: Drag is the largest non-gravitational force acting on satellite in
low orbit and affects the motion of satellite dramatically.
Satellite dynamics:
Including gravity and drag
(By using RK4 integrator )
Navigation model
(using GPS)
Orbit determination
𝑐 𝐷 𝑐 𝑐 Solving variation eqns to
get transition matrix
(By using RK4 integrator )
Initial Condition
By Teng-Hu Cheng

2. Dynamics
• Forces included: Gravity and Drag
• Integration approach: Runge-Kutta 4th-order method (RK4) for dynamics
• State vector (Geocentric coordinate)
𝑥 ≜ [ 𝑥 𝑧 𝑥 𝑧 𝑐 𝐷 𝑐 𝑐 ]
• Initial conditions:
Priori: 𝑥 = 7001 1 −1 0 0 8 2 0 0
True1: 𝑥 = [7002 0 0 0 0 8 1 0.029979 −2.99 × 10−6
]
Est.1: 𝑥 = [7002 0 0 0 0 8 1.02 0.03 −2.6 × 10−6
]
True2: 𝑥 = [7000 0 0 0 0 8 3 0.029979 −2.99 × 10−6
]
Est.2: 𝑥 = [7000 0 0 0 0 8 2.96 0.03 −2.8 × 10−6
]
Dev: 𝜎 0
< 0.5𝑚 𝜎 𝑣 0
< 0.001
𝑚
𝑠
𝜎𝑐 𝐷
=9× 10−3
𝜎𝑐 < 1.6 × 10−4
𝜎 𝑐 < 9 × 10−9
• EOM:
= −
𝐺𝑀⊕
3
−
1
2
𝑐 𝐷
𝐴
𝑚
𝜌 2
where = − 𝜔⊕ × .

3. Trajectory Plot
-1 -0.5 0 0.5 1
x 10
4
-8000
-6000
-4000
-2000
0
2000
4000
6000
8000
10000
x (km)
z(km)
 initial position
 earth
True trajectory
Traj. based on priori esti.
0 500 1000 1500 2000 2500 3000
-10
-8
-6
-4
-2
0
2
t - t0
(sec)
r
x
,r
y
,r
x
(km)
 rx
 ry
 rz
-8240 -8235 -8230 -8225 -8220 -8215 -8210 -8205 -8200
3355
3360
3365
3370
3375
3380
x (km)
z(km)
True trajectory
Traj. based on priori esti.

4. Navigation
• 24 GPS satellites are used for Navigation and estimation of drag coefficient.
• Pseudoranges between user satellite and GPS satellites in view are measured to
estimate the position of user satellite.
𝜌Φ 𝑖
= 𝑖 − + 𝑐 + 𝑁 0 0.0052
△ 𝜌Φ 𝑖
≈ 𝐻𝑖∆𝑥 + 𝜀 𝑊𝐿 : 𝑥 = = 𝑥 + ∆𝑥
0 500 1000 1500 2000 2500 3000
1
2
3
4
5
6
7
8
time (sec)
distance(m)
r
(m)
estimate error (m)

5. Orbit Determination
• State vector : 𝑥 ≜ 𝑐 𝐷 𝑐 𝑐
• Cost function: 𝐽 𝑥 = 𝑧 − 𝐻𝑥 𝑇
𝑊 𝑧 − 𝐻𝑥
𝑥 = 𝐹𝑥 𝑧 = 𝑔 𝑥 + 𝜀
△ 𝑧 = 𝑧 − 𝑔 𝑥 ≈
𝜕𝑔 𝑥
𝜕𝑥
𝜕𝑥
𝜕𝑥
△ 𝑥 + 𝜀
△ 𝑧 ≈ Γ △ 𝑥 + 𝜀 ⇒ 𝑙𝑒𝑎𝑠 − 𝑠𝑞𝑢𝑎 𝑒 𝑒𝑠 . 𝑜𝑓 △ 𝑥
• Solving variational eqns :
=
𝜕𝑥
𝜕𝑥
=
0
0 𝐼 0
0 0 𝐼
=
03×3 𝐼3×3
𝜕 𝑎 𝑥
𝜕
𝜕 𝑎 𝑥
𝜕
+
03×6 𝐼3×1
03×6
𝜕 𝑎 𝑥
𝜕 𝑝
𝑥 = 𝑥 +△ 𝑥
• Different Initial conditions are tested to examine the deviation of the parameters.
• Answers: The estimation doesn’t change dramatically with various initial conditions .
• What I have learned: Using measurement and dynamic model to conduct optimal estimation.