Journ ee AS 67 M ethodes Particulaires Applications du Filtrage Particulaire ENST 3 December 2003 brPage 2br Summary 1 What is the BearingsOnly Tracking problem 2 Solving using particle 64257ltering algorithm 3 From cartesian to modi64257ed polar c ID: 22491 Download Pdf

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Journ ee AS 67 M ethodes Particulaires Applications du Filtrage Particulaire ENST 3 December 2003 brPage 2br Summary 1 What is the BearingsOnly Tracking problem 2 Solving using particle 64257ltering algorithm 3 From cartesian to modi64257ed polar c

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Bearings-Only Tracking in Modiﬁed Polar Coordinate System : Initialization of the Particle Filter and Posterior Cram er-Rao Bound Thomas Brehard (IRISA/CNRS), Jean-Pierre Le Cadre (IRISA/CNRS). Journ ee AS 67 M ethodes Particulaires Applications du Filtrage Particulaire ENST 3 December 2003

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Summary 1. What is the Bearings-Only Tracking problem ? 2. Solving using particle ﬁltering algorithm... 3. From cartesian to modiﬁed polar coordinate system. 4. Initialization of the particle ﬁltering algorithm. 5. Posterior Cram er-Rao bound. 6.

Conclusion.

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1. What is the Bearings-Only Tracking problem ? (1/2) IG . 1 Trajectories of the observer (pink) and the target (blue) and simulated bearing measurements. )= tan is the target state at time composed of relative velocity and position of the target in the plane is the bearing measurement received at time Problem : Estimate using , .. . ,Z

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1. What is the bearings-only tracking problem ?(2/2) The stochastic system : +1 AX HU ) + where : is the target state at time composed of relative velocity and position of the target in the plane, the bearing

measurement received at time is the known difference between observer velocity at time + 1 and has a center normal distribution with variance known. has a center normal distribution with covariance matrix known. This is a non-linear ﬁltering problem which can be solved using particle ﬁltering algorithm !

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2. Solving using particle ﬁltering algorithm... At each step of time : 1. Propagating the set of particles using the state equation. 2. Weighting each of the particles using the measurement equation. 3. Resampling step. Reference : Doucet et al. (2001)

Problem Particles must be properly initialized !

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3. From cartesian to modiﬁed polar coordinate system (1/3) There is an unobservability problem hidden in the cartesian formulation of the system : +1 AX HU ) + Problem The range is unobservable until the observer has maneuvered. Solution A coordinate system more suited to the problem : the modiﬁed polar coordinate system Reference : Aidala and Hammel (1983)

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3. From cartesian to modiﬁed polar coordinate system (2/3) The modiﬁed polar coordinates ) = We can show that : is unobservable

until the observer has not maneuvered. ) = is always observable.

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3. From cartesian to modiﬁed polar coordinate system (3/3) Before the observer maneuvers, the stochastic system in modiﬁed polar coordinate system is +1 + 1) = ) + where and has a center normal distribution with covariance matrix known. An interesting model : This is a non linear ﬁltering problem with unknown covariance state is unknown !).

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4. Initialization of the particle ﬁltering algorithm (1/7) 2 key ideas We can proove that if the target has a deterministic

trajectory then for all ) = ) + tan k 1 + k It is just an optimization problem. The observable components can be estimated using the set of measurements , .. . ,Z We only assume a prior information on the unobservable component max min

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4. Initialization of the particle ﬁltering algorithm (2/7) Initialization of the particle ﬁltering algorithm Wait until time , the particule is initialized by 1. CA where is computed using a Gauss-Newton optimization algorithm (initialized by linear regression). CA is the conﬁdence area of (approximated by an

hyperellipsoid). 2. min , R max

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4. Initialization of the particle ﬁltering algorithm (3/7) Two important points Before the observer maneuvers, the observable component of the particles and the unobservable component of the particles must be resampled independently ! How the initialization time K can be ﬁxed ? ”The particle ﬁltering algorithm is initialized as soon as the volume of the conﬁdence area for is sufﬁciently small to be ﬁlled by N particles”.

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4. Initialization of the particle ﬁltering algorithm (4/7)

Simulation scenario target 10 ms ms 10000 , X obs ms ms 10000 The observer follows a leg-by-leg trajectory. His velocity vector is constant on each leg and modiﬁed at the two following instants : obs (2100) obs (2100) 10 ms ms obs (3900) obs (3900) 10 ms ms =1800 s t =3600 s t =5400 Trajectories of the observer (pink) and the target (blue).

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4. Initialization of the particle ﬁltering algorithm (5/7) Simulation scenario Simulated bearing measurements. The measurement standard deviation is 0.05 rad (about 3 deg). Parameters for the particle ﬁltering

algorithm Number of particles : 10000, Sampling threshold : 0.5, The single assumption : min =1000 m and max =40000 m.

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4. Initialization of the particle ﬁltering algorithm (6/7) Simulation results in modiﬁed polar coordinates True value (blue), Estimates (red), Conﬁdence (green)

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4. Initialization of the particle ﬁltering algorithm (7/7) Simulation results in cartesian system =1800 s t =3600 s t =5400 Trajectory of the observer (pink), trajectory of the true target (blue), particles (red), conﬁdence for the estimate (green).

Conclusion We have initialized the particle ﬁltering algorithm using a weak prior on range. Performance analysis in polar modiﬁed coordinate system.

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5. Posterior Cram er-Rao Bound (1/9) Next step The performance analysis in modiﬁed polar coordinate system. Tool The Posterior Cram er-Rao Bound (PCRB). ”The PCRB gives a lower bound for the Mean Square Error

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5. Posterior Cram er-Rao Bound (2/9) Notations : 0: , .. . ,Y and 0: , .. . ,Z The bias 0: ) = 0: 0: )) 0: where 0: is the estimator of 0: The asymptotic bias assumption lim →Y 0:

0: ) = lim →Y 0: 0: ∈{ , .. . ,n and ∈{ , .. . ,t where is the state space of for all in , .. . ,t and are the endpoints of

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5. Posterior Cram er-Rao Bound (3/9) Proposition 1 Under the asymptotic bias assumption, ECM 0: 0: where ECM 0: 0: 0: ))( 0: 0: )) 0: { 0: 0: ln 0: , Y 0: and 0: is the estimator of 0: In the ﬁltering context, we only want to approximate the right lower block of 0: noted

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5. Posterior Cram er-Rao Bound (4/9) Using Tichavsky et al. results, we have +1 22 21 11 12 where 11 { ln +1 ln +1 }} 21 { +1 ln +1 ln +1 12 {

+1 ln +1 +1 ln +1 22 { +1 ln +1 +1 ln +1 { +1 ln +1 +1 +1 ln +1 +1 Conclusion can be computed recursively . Rq : 11 12 22 21 are approximated using Monte Carlo method and is obtained by a Cram er-Rao Bound.

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5. Posterior Cram er-Rao Bound (5/9) Simulation results (Mean square error in blue, PCRB in red) Problem The PCRB seems over optimistic... Hypothesis Is the asymptotic bias assumption true ? Is 0: ill-conditionned due to range unobservability ?

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5. Posterior Cram er-Rao Bound (6/9) Is the asymptotic bias assumption true ? A more general bound : ECM 0: 0:

0: 0: where 0: 0: 0: 0: ln 0: , Y 0: Remarks : 0: can be approximated using Monte Carlo approximation. The recursive formulation of the PCRB is no longer valid.

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5. Posterior Cram er-Rao Bound (7/9) Is the asymptotic bias assumption true ? Simulation results (Mean square error in blue, classical PCRB in red, new PCRB in green) Solution The asymptotic bias assumption is not true in the bearings-only tracking problem.

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5. Posterior Cram er-Rao Bound (8/9) Problem The posterior Cram er-Rao bound seems over optimistic... Hypothesis Is the asymptotic bias

assumption true ? No Is 0: ill-conditionned due to range unobservability ?

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5. Posterior Cram er-Rao Bound (9/9) Is 0: ill-conditionned due to range unobservability ? Idea : We only construct a bound for the observable components of the state We can proove that : ECM 0: 0: )) 0: )) 0: )) where 0: )) = { 0: 0: ln 0: , Y 0: 0: )) = 0: 0: 0: ln 0: , Y 0: It is possible to approximate 0: )) and 0: ))

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6. Conclusion We have proposed : An initialization method for particle ﬁltering algorithm using modifed polar coordinates using a weak prior on range. A new

interpretation of the Bearings-Only Tracking problem. A realistic posterior Cram er-Rao bound in modiﬁed polar coordinate system. Perspectives 3D target tracking. Maneuvering target.

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