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1 IGNSS 2013 Surfer s Paradise, Queensland, Australia Estimation of initial state and model parameters for autonomous GNSS orbit prediction Juha Ala-Luhtala, Mari Seppänen, Simo Ali-Löytty, Robert Piché & Henri Nurminen Tampere University of Technology Tampere Finland 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 a sat = a Earth + a Sun + a Moon + a SRP a Sun Earth gravity model: EGM08 Sun and moon considered as point masses a Earth a SRP a Moon 2-parameter SRP model SRP: Solar Radiation Pressure 3

4 Modeling uncertainty in the equation of motion a = a sat + R(r, v)w w N R(r, v) : Transformation from RTN to inertial coordinates w R w T w : Zero-mean Gaussian white noise, with diagonal covariance matrix 2 4 w R w T w N 3 5 {z } w Normal 2 4 q R q T q N 31 5A RTN: Radial, Tangential, Normal 4

5 State-space model d dt apple r v = apple v a(r, v, p) y k = T(p) + 0 apple r v apple 0 0 R(r, v) 0 + k w Model parameters: p = x p y p Nonlinear stochastic differential ECEF T(x p,y p ) Inertial equation T(x p,y p ) T State: position r and velocity v in inertial reference frame ECEF = Earth Centered, Earth Fixed Discrete measurements: ECEF broadcast positions e y ( 2 ) e s ( 1 ) 5

6 Computing measurements from broadcast ephemeris t oe = time of ephemeris y 1 y y K {z} t oe 1.5h t oe t=5min t oe +1.5h y k : ECEF positions computed using the 16 orbital elements: n, µ 0,e, p a, t oe, 0,i 0,!,, i, C rs,c rc,c is,c ic,c us,c uc Use positions from time interval: [t oe 1.5h, t oe +1.5h] 6

7 Estimation of the initial state and prediction Problem: Initial state: x(t K ) Predict 1) Estimate the initial state and model parameters 2) Predict the orbit x = 2 4 r v p 3 5 t K = t oe +1.5h x(t) t>t K Bayesian solution: 1) Filtering distribution: p(x(t K ) y 1:K ) 2) Predictive distribution: p(x(t) y 1:K ), t > t K 7

8 Approximative nonlinear filtering Exact computation of the posterior is intractable Estimate the posterior mean and covariance: Initial state E(x(tK ) y 1:K ) m(t K ) Var(x(t K ) y 1:K ) P(t K ) Prediction E(x(t) y1:k ) m(t) Var(x(t) y 1:K ) P(t),t>t K Three nonlinear filtering methods: Extended Kalman filter (EKF): linearization of nonlinear models Unscented Kalman filter (UKF) & Cubature Kalman filter (CKF): Deterministic sigma point approximations 8

9 3D orbit prediction errors 75 Quantiles 95% 3D error [m] 50 75% 25 EKF UKF CKF 50% length of prediction [days] 25% 5% 9

10 RTN orbit prediction errors for EKF 75 RTN: Radial, Tangential, Normal RTN error [m] R T N length of prediction [days] 10

11 Consistency of the predicted orbits Is the true position inside the predicted 95% probability ellipsoid? (r r PE ) T P 1 (r r PE ) apple F 1 2 (3) (0.95) True position from IGS precise ephemerides value of inverse chi-squared cdf with degrees of freedom 3 at p= % consistency Day 1 Day 2 Day 3 Day 4 Day 5 EKF UKF CKF Table 1: 95% consistencies of the predicted orbits 11

12 Conclusion EKF can be used to estimate initial state and model parameters for the orbit prediction algorithm Good performance for predictions of several days long Error is largest in the tangential direction Small errors in radial and normal directions Predicted covariance is consistent with actual prediction error Method can be extended for other GNSS systems with similar broadcast format: Galileo, BeiDou 12

Tampere University of Technology. Ala-Luhtala, Juha; Seppänen, Mari; Ali-Löytty, Simo; Piché, Robert; Nurminen, Henri

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