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GRAVIMETRY-AIDED INERTIAL NAVIGATION

GRAVIMETRY-AIDED INERTIAL NAVIGATION. Séminaire IHP 2 /3 décembre 2002. Christian MUSSO - ONERA/DTIM/IED christian.musso@onera.fr. From a project involving:. LRBA : Laboratoire de Recherches Balistiques et Aérodynamiques CRIL TECHNOLOGY

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GRAVIMETRY-AIDED INERTIAL NAVIGATION

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  1. GRAVIMETRY-AIDED INERTIAL NAVIGATION Séminaire IHP 2 /3 décembre 2002 Christian MUSSO - ONERA/DTIM/IED christian.musso@onera.fr From a project involving: • LRBA : Laboratoire de Recherches Balistiques et Aérodynamiques • CRIL TECHNOLOGY • - IRISA : Institut de Recherche en Informatique et Systèmes Aléatoires

  2. INERTIAL NAVIGATION SYSTEMPRINCIPLE Séminaire IHP 2 /3 décembre 2002

  3. INERTIAL NAVIGATION SYSTEMPRINCIPLE Séminaire IHP 2 /3 décembre 2002 Noise Absolute acceleration (what we want) Gravitational Drawback : position error grows with time External measurement system. Example : gravimetry

  4. GRAVITY ANOMALIES. GEOIDE If earth fluid : spheroïde Séminaire IHP 2 /3 décembre 2002

  5. GRAVITY ANOMALIES. GEOID Heterogeny distribution of the masses in the upper layers of the Earth Geoid Gravity corrected by the theorical one Séminaire IHP 2 /3 décembre 2002 Gravity anomaly - External measurement (mGal)

  6. INERTIAL ERROR MODELISATION Position and velocity inertial errors (d=6) Evolution of the error Séminaire IHP 2 /3 décembre 2002 Longitude/Latitude inertial error AIM : Estimate Estimate

  7. MEASUREMENT EQUATION Gravity anomaly measurement model Séminaire IHP 2 /3 décembre 2002 iid noise B(t) : unknown colored noise (Schuler oscillation for ex) iid noise

  8. DYNAMICAL EQUATION After discretisation Séminaire IHP 2 /3 décembre 2002 Models concatenation

  9. FILTER MODEL STATE Séminaire IHP 2 /3 décembre 2002 • Markov chain • Likelihood

  10. PARTICLE FILTER Compute the conditional law via samples (particles). But, - Unrealistic to estimate precisely (in particular in relative error) the density in any point in a high dimensional state Séminaire IHP 2 /3 décembre 2002 - Fortunately, generally the variables of interest are the first moments But this approach allows much flexibility

  11. REGULARIZED PARTICLE FILTER C. Musso, N.Oudjane, F. Le Gland (2001) Séminaire IHP 2 /3 décembre 2002 Useful when the dynamical noise is weak

  12. REGULARIZED PARTICLE FILTER : Algorithm Sample from (Multinomial) Séminaire IHP 2 /3 décembre 2002 (Epanechnikov) (Regularization)

  13. REGULARIZED PARTICLE FILTER : Algorithm Séminaire IHP 2 /3 décembre 2002 Resampling Entropy Filter parameters choice : relevant

  14. PARTICLE FILTER : Local error analysis Prior Posterior Likelihood Séminaire IHP 2 /3 décembre 2002 AIM : estimate ? Variance of SIR :

  15. Local error analysis : coherence prior/measurement Séminaire IHP 2 /3 décembre 2002 Give the quality of estimation

  16. Local error analysis : coherence prior/measurement (Liu, J.S and Chen (1998)) Séminaire IHP 2 /3 décembre 2002 - error with - h’ 0 : no information -density propagation (var of the prior) - flat terrain - peaked terrain - h’ : information (PCRB) - var (same results with rejection algorithms)

  17. Local error analysis : coherence prior/measurement • flat terrain : no observability • Dirac carpet terrain : bad estimation (only grid algorithms are possible) Séminaire IHP 2 /3 décembre 2002 TERRAIN SMOOTH BUT NOT TOO MUCH POSSIBILITY OF DIVERGENCE - MC FLUCTUATIONS

  18. POSTERIOR CRAMER-RAO BOUND : Useful tool for non-linear filtering P. Tichavsky, C. Muravchik and A. Nehorai (1998) : algorithms N. Bergman (2000) : application to altimetry navigation Information matrix (Fisher) Séminaire IHP 2 /3 décembre 2002 Cov matrix Estimator of

  19. POSTERIOR CRAMER-RAO BOUND : Useful tool for non-linear filtering • Evaluate the performance of the filter • Evaluate if the system/measurement is informative • Confidence ellipsoid (monomodal hyp) • Easy to implement • Trajectories optimisation Séminaire IHP 2 /3 décembre 2002

  20. POSTERIOR CRAMER-RAO BOUND : Application to nonlinear filtering Case Linear Dynamics Séminaire IHP 2 /3 décembre 2002 Loss of info due to the dynamics MC evaluation Gain of info due to the variation of h • Generalized Riccati equation • Informational Kalman formulation when h linear

  21. POSTERIOR CRAMER-RAO BOUND (PCRB) : Application to gravimetry = Séminaire IHP 2 /3 décembre 2002 Again particles !

  22. POSTERIOR CRAMER-RAO BOUND : Application to gravimetry Empirical ellipse PCRB ellipse Séminaire IHP 2 /3 décembre 2002 Estimated state True state : ellipse in the position plan

  23. SIMULATIONS : aircraft navigation Difficulties : • Multimodality • dim(state)/dim(meas) : high Séminaire IHP 2 /3 décembre 2002 Gravity anomalies simulated map

  24. SIMULATIONS : conditional density Séminaire IHP 2 /3 décembre 2002 Meas. Number 2 Meas. Number 12 Meas. Number 100

  25. SIMULATIONS : measurements Séminaire IHP 2 /3 décembre 2002

  26. SIMULATIONS : Results 10 MC Filter trials : 1 divergence Average of the state estimation error over 10 trials Séminaire IHP 2 /3 décembre 2002

  27. SIMULATIONS : Results Séminaire IHP 2 /3 décembre 2002

  28. CONCLUSIONS - PERSPECTIVES • Particle filter is adapted for the terrain aided navigation with some troubles • PCRB is a useful tool for nonlinear filtering • Improved Particle Filters (against MC fluctuations), work in progress with • Pham Dinh Tuan (CNRS/IMAG) • Karim Dahia, PhD student (ONERA) Séminaire IHP 2 /3 décembre 2002

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