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GRAVIMETRYAIDED INERTIAL NAVIGATION

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GEOID. Heterogeny distribution of the masses in the upper layers of the Earth. Geoid. Gravity corrected by the theorical one. Gravity anomaly - External measurement ... – PowerPoint PPT presentation

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Title: GRAVIMETRYAIDED INERTIAL NAVIGATION


1
GRAVIMETRY-AIDED INERTIAL NAVIGATION
  • Christian MUSSO - ONERA/DTIM/IED
  • christian.musso_at_onera.fr

Séminaire IHP 2 /3 décembre 2002
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 (d6)
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
Gain of info due to the variation of h
MC evaluation
  • 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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