An Empirical Solar Radiation Pressure Model for Autonomous GNSS Orbit Prediction

Transcription

1 Myrtle Beach, South Carolina An Empirical Solar Radiation Pressure Model for Autonomous GNSS Orbit Prediction Juha Ala-Luhtala, Mari Seppänen & Robert Piché Tampere University of Technology Tampere Finland e s e y with thanks for collaboration and support: 1

2 Autonomous GNSS orbit prediction In Assisted GPS (A-GPS), ephemeris information is downloaded over network to reduce time-to-first-fix (TTFF) For devices without network connection: Compute the satellite s orbit using initial conditions from broadcast ephemeris Use the resulting predicted ephemeris to reduce TTFF apple r0 v 0 apple rpred v pred 2

3 Satellite s equation of motion r sat = a Earth + a Sun + a Moon + a SRP Numerically integrate using 4-stage order 5 Runge-Kutta-Nyström method with step size 100 s a Sun a Moon Use initial position and velocity from broadcast ephemeris a Earth a SRP 3

4 Solar Radiation Pressure Model a SRP = (t) 1 K e s + 2 e y r 2 sun K constant common for all satellites r sun e y 1 satellite s scale parameter for direct solar radiation pressure e s 2 acceleration in the direction of the solar panel axis (y-bias acceleration) (t) accounts for the Earth s shadow 4

5 Ephemeris formats for GPS and GLONASS are different GPS Position and velocity calculated from 16 orbital elements Accurate for 4 hours error [m] 2 1 Median error in broadcast position GLONASS GPS GLONASS Time with respect to TOE [h] Position and velocity at one time instant Simple force model is used to calculate position and velocity at other times Accurate for 30 minutes 5

6 Position and velocity are transformed to inertial reference frame The equation of motion must be integrated in an inertial reference frame Transformation matrix R comprises Polar motion Earth rotation Nutation Precession ECEF R R T Inertial ECEF = Earth Centered, Earth Fixed Polar motion parameters are not included in the broadcast message 6

7 Initial state improvement Initial velocity from broadcast is not accurate enough for long term prediction Also, we need to solve for the polar motion parameters xp, yp f(x 0,t 2 ) f(x 0,t 1 ) Our approach is to fit the initial state by minimizing errors between model-predicted and broadcast states t 2 r BE (t 2 ) t 1 r BE (t 1 ) t 0 x 0 = time r BE (t 0 ) 2 4 v 0 x p y p 3 5 ˆx 0 = arg min x 0 kek 2 e = apple rbe (t 1 ) f(x 0,t 1 ) r BE (t 2 ) f(x 0,t 2 ) 7

8 Estimating solar radiation pressure parameters Scale and y-bias are estimated separately for each satellite using IGS precise ephemerides as reference Estimate parameters for several 7-day arcs using Extended Kalman filter, then take the median 1.5 Eclipse season 1 1 scale PRN=7 PRN=8 2 y bias (10 9 m/s 2 ) 0 PRN=8 PRN= GPS week GPS week 8

9 Estimated SRP model parameter values for GPS and GLONASS are different scale values y bias values scale 2 y bias (10 9 m/s 2 ) GPS GLONASS GPS GLONASS 9

10 Using y-bias significantly reduces GPS prediction errors 300 Error of predicted position for GPS constellation th percentile error [m] th percentile Scale and y-bias Scale only Length of prediction [days] median 25th percentile 5th percentile 10

11 Using y-bias reduces GLONASS prediction errors only slightly Error of predicted position for GLONASS constellation error [m] 50 Scale and y-bias 25 Scale only Length of prediction [days] 11

12 Prediction errors in RTN (Radial, Tangential, Normal) coordinate system Largest errors are in tangential (along-track) direction Errors are small in radial direction (which has the largest effect on the pseudorange error) 100 GPS prediction error components (95% quantile) 100 GLONASS prediction error components (95% quantile) error [m] T error [m] T 25 R N 25 R N Length of prediction [days] Length of prediction [days] 12

13 Conclusion Good accuracy in satellite orbit prediction can be achieved using equation of motion including gravitational forces of the Earth, the Sun and the Moon empirical solar radiation pressure model with 2 satellite specific parameters 95% of orbit errors in a 5-day long prediction are < 75 m for GPS and < 90 m for GLONASS Computational time for 4-day long prediction for 5 GPS satellites: 35 s (Nokia N900) Check out our continuously-updated 4-day orbit prediction at 13

14 Conclusion Good accuracy in satellite orbit prediction can be achieved using equation of motion including gravitational forces of the Earth, the Sun and the Moon empirical solar radiation pressure model with 2 satellite specific parameters 95% of orbit errors in a 5-day long prediction are < 75 m for GPS and < 90 m for GLONASS Computational time for 4-day long prediction for 5 GPS satellites: 35 s (Nokia N900) Check out our continuously-updated 4-day orbit prediction at 14