mari seppänen, tommi perälä and robert pichémath.tut.fi/posgroup/itm2011slides_final.pdf ·...
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TA M P E R E U N I V E R S I T Y O F T E C H N O L O G Y
M a t h e m a t i c s
AutonomousSatellite Orbit Prediction
Mari Seppänen, Tommi Perälä and Robert Pichéhttp://math.tut.fi/posgroup
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Autonomous satellite orbit prediction
What? A method to predict satellite orbits in a positioning deviceworking without any network connection
Why? To provide fast Time To First Fix (TTFF) when AssistedGPS is not available
GPS
Load broadcast
message from
visible GPS
satellites.
Compute position
using predicted
satellite positions.
+
broadcast
TTFF ~ 30 s
TTFF ~ 5s
GPS
GPSGPS
Today:
Later:
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Force Model
FSUN
FMOON
FSRP
FEarth
SRP = Solar Radiation Pressure
ΣF = FEarth + FMoon + FSun + FSRP
d2r
dt2= a =
ΣF
m
r0
v0
=satellite’s
initial state
r(t) = r0 +
∫ t
t0
(
v0 +
∫ t
t0
adt)
dt
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Initial state from broadcast message
Position can be computed using the 16 ephemeris parametersi.e. ∆n, µ0, e,
√a, toe, Ω0, i0, ω, Ω, i, Crs, Crc, Cis, Cic, Cus, Cuc
Velocity can be evaluated by differentiating these parameterswith respect to time
−4 −2 0 2 40
5
10
Time with respect to toe
[h]
[m]
Broadcast position error(50% quantile)
−4 −2 0 2 40
1
2
3 x 10−3
Time with respect to toe
[h]
[m/s]
Broadcast velocity error(50% quantile)
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Reference frames
Satellite’s initial state has to be transformed to the inertialreference frame used by the equation of motion
rTRS = W ·G ·N ·P · rCRS
W = Ry(−xp)Rx(−yp)
−10 −5 0 5
5
10
15
x [m]
y [m]
1.1.2005
1.1.2006
1.1.2007
1.1.2008
1.1.2009
Polar motion (W): Earth’saxis of rotation moves withrespect to the Earth’s crust
There is no model for longterm prediction of polar motion parameters!
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Summary of the prediction steps:
Compute r0 and v0 from broadcast
Transform to an inertial reference frame
Integrate the orbit r(t) = r0 +∫ t
t0
(
v0 +∫ t
t0adt
)
dt
Transform back to ECEF
For evaluation purposes we compute the prediction error usingNGA’s precise ephemeris (PE) as a reference.
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Inaccurate BE and unknown polar motion parameters
After the first simulations we encountered two problems:
Initial conditions from PE ⇒ Results goodInitial conditions from BE ⇒ Results much worse
Polar motion parameters xp and yp were loaded from theinternet ⇒ Not possible in the device
1 2 3
100
300
500
Length of prediction [days]
[m]
Satellite position error(95% quantile)
Broadcast initial conditions
Precise initial conditions
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Least squares fitting - the basic idea
t2t1 toe
r (v(t ))
time
rBE(t1) rBE(t2)
rFM(t1) = rBE(t2) +∫ t1
t2
(
v2 +∫ t1
t2adt
)
dt
Fitting the velocity:
v2 = argminv2
‖p(v2)‖2
where the residual function is
p(v2) = rBE(t1)− rFM(t1)
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Least squares fitting - whole algorithm
t2t1 toe
f(x,rBE(t2))
rBE(t2)
vBE(t2)
rBE(t1)
vBE(t1)y=
time
x =
xp
yp
vBE(t2)
Fitting vBE(t2) i.e. the velocities of several satellites at t2 and thepolar motion parameters xp and yp:
x = argminx
∑
i
p2i (x) = argminx
‖p(x)‖2
where the residual function p is
p(x) =
fr(x, rBE(t2))− yr
(fv(x, rBE(t2))− yv) · 1000
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Tests
1 2 3
20
40
60
80
100
Length of prediction [days]
[m]
Satellite position error (95% quantile)
LSQ-fitted BE
Precise initialconditions
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Conclusion
Received broadcast ephemeris data together with satellite’sequation of motion can be used to solve the polar motionparameters while improving the satellite initial statesimultaneously
Algorithm can be used to improve TTFF when Assisted GPSis not available
Satellite orbit predictions (continuously updated): see
http://math.tut.fi/posgroup
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HTML
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