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

– p. 1/12

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:

– p. 2/12

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

– p. 3/12

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)

– p. 4/12

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!

– p. 5/12

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.

– p. 6/12

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

– p. 7/12

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)

– p. 8/12

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

– p. 9/12

Tests

1 2 3

20

40

60

80

100

Length of prediction [days]

[m]

Satellite position error (95% quantile)

LSQ-fitted BE

Precise initialconditions

– p. 10/12

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

– p. 11/12

HTML

– p. 12/12