maneuvering Target Tracking using cost reference Particle Filtering [Bugallo et al 2004]

Monica F. Bugallo, Shanshan Xu, Joaquin Miguez, Peter M. djuric.

This paper discuss a new method CRPF (Cost Reference Particle Filtering) compared them to SRPF (Statistical Reference Particle Filtering). Eliminate all probabilistic assumption.

Work compared to SMC method

System Modelling with MATLAB , Basic [NIse, Norman S. 2000]

SYSTEM MODELING

• Use for analysis and design of feedback control system
• Consist of two approaches:

1. Classical or frequency-domain method which converting a differential equation system to transfer function using Laplace transform or z transform
o Simplify the representation of individual subsystem and modeling interconnection subsystems
o Disadvantage: limited applicability only to linear, Time-invariant system
o Advantage: rapidly provide stability and transient response information therefore can immediately see the effect of varying system parameters until an acceptable design is met
o Few quick calculation or graphic representation of data rapidly yields the physical interpretation

2. State-space method or modern or time-domain approach
o Increase in space exploration and involves modeling using linear, time-invariant differential equation.
o MIMO (multiple output multiple output) system is compactly represented similar to single input-output system
o Can handle system with non-zero initial condition
o This time-domain approaches can be used to represent system via digital computer for digital simulation where system response can be obtained for changes in system parameters.
o Numerous state-space software packages offered for PC
o Disadvantage: Designer engaged in several calculations before the physical representation of the model is apparent.

• LINEAR
• TIME VARIANT
  

• LAPLACE TRANSFORM
• Use table and theorem for transformation from time domain to frequency domain
• Partial fraction expansion can be applied to simplify a complicated function by braking it down to a sum of simpler term
• F(S)=N(S)/D(S)
• The order of N(S) must be less than D(S) if not N(S) must be divided by D(S)
• Case 1: Roots of the Denominator of F(S) Are Real and Distinct
• Case 2: Roots of the Denominator of F(S) Are Real and Repeat
• Case 3: Roots of the Denominator of F(S) Are Complex or Imaginary

• TRANSFER FUNCTION
• H(S)=C(S)/Y(S) r = reference input, c = control output
• Differential equation (time domain) transform using Laplace (frequency domain) therefore allow separation of input, system, and output.
• Algebraically relates a system’s output to its input.
• Allow separation of the input, system, and output into three separate and distinct parts.
• Algebraically combine mathematical representations of subsystems to yield a total system representation.
  

• LINERIZATION
• Obtained linear approximation for nonlinear system in order to obtained transfer function

• STATE SPACE

MATLAB CODE TRANSFER FUNCTION STATE SPACE

An Introduction to Kalman Filter [Greg Welch and Gary Bishop, 2006]

Kalman Filter: an efficient recursive filters that estimates the state of a process by minimizing the mean squared error from a series if incomplete and noisy measurements.

1. DISCRETE KALMAN FILTER

• state estimation governed by the linear stochastic difference equation
• state eqn
  
• xk = Axk-1 + Buk-1 + wk-1
• Measurement eqn
• zk = Hxk-1 + vk-1
• Process and measurement noise with probability distribution
  
• p(w) ~ N(0,Q) , p(v) ~ N(0,R)
• priori and posteriori estimate error
  
• 
  
• priori and posteriori estimate covariance error

1. PREDICT
• 
  
2. CORRECT
   

2. EXTENDED KALMAN FILTER

• state estimation governed by the non-linear stochastic difference equation

Sequential Monte Cralo Particle Filtering

Arnaud Doucet, Nando de Freitas, and Neil Gordon

Little background on SMC.

Monte Carlo Method originally known as ‘method of statistical sampling’

ESTIMATION GENERAL CONCEPT

• Estimating unknown quantities from given observations. I.e.: Prior knowledge available.
• Able to formulate Bayesian Model which is the prior distribution for the unknown quantities and the likelihood function relating these quantities to the observations.
• Inferences on the unknown quantities are made from the posterior distribution obtained from Bayes’ Theorem.
• Often observations arrive sequentially in time and able to perform inference on-line. Therefore necessary to update the posterior distribution as data become available.
• Goal: Computational simplicity allows not having to store all data.
• Example
• Data model using linear Gaussian state space: derive the posterior distribution using Kalman Filter.
• Data model using partially observed state space Markov Chain: obtained analytical solution using Hidden Markov Model HMM Filter
• Estimation method such as Kalman Filter and Gaussian sum approximation is based on normal distribution fail to cover the non-Gaussianity and nonlinearity. While Grid-based filter based on deterministic numerical integration method is too computationally expensive to be used in high dimension.

SMC SOLUTION

• Able to handle very complex data, typically involving non-Gaussianity, nonlinearity which condition usually preclude analytic solution.
• Flexible, easy to implement, parallelizable and applicable in very general setting.
• Closely related algorithm: bootstrap filters, condensation, particle filters, Monte Carlo filters, interacting particle approximations, and survival of the fittest.

An Overview of Recent Developments in Target Tracking, in the Active Airbone Sonar Networks Domain

Francis Martineri and Sebastian Brisson, 1993

ABSTRACT

1. Address Target Tracking problem from the measurements made on a passive sonars activated by an active sonars (multistatic networks)
2. recalls on principles of classic tracking approaches in the specific context of distributed sensors measuring range, doppler, and azimuth
3. Include improvements on simultaneous tracking of a target and calibration of the sensor field
4. Introduce improvement on 'recently introduced approach in target tracking' and target motion analysis which is based on a non-deterministic modelisation of the target evolution.
   
5. Method based on Hidden Markov Modelling leads to multiple and maneuvering target tracking with few Hypotheses

CASE INTRODUCTION

• The increase in level of difficulties in detection and localisation from basic sensor due to low level of submarine radiated noise, develop interest in distributed sonar networks.
• Network consist of active sonar acting as a transmitter/receiver, and N passive sensors acting as receiver
• Sensors and targets assume to be at the same plane
• Targets are characterised by position and speed in Cartesian coordinates
• X(t) = (x(t),y(t),vx(t),vy(t))
• Sensor positions are known Xs = ( (xs1,ys1), ... , (xsN,ysN)
• False alarm assumed uniform
• Measurements carried in discrete time: a signal emitted by the active sensor will give rise to detection on the passive sensor after it has impinged the targets.
• The range, doppler, or azimuth measurements corresponding to a detection on sensor are extracted. (equations given)
  

CLASSIC APPROACHES IN TRACKING

• Come from a classic data association and tracking method and based on 2 steps

1. Extracting and Data Association
• Extracting the measurement corresponding to each single target received from the sensors. Usually Extraction perform by track formation at the sensor level. (Based on the block diagram of track extraction, I presume they use a centralized fusion techniques)
  
• Track to track association where tracks from different sensors corresponding to the same targets have to be associated.
  
• often perform manually by an operator. However at this point automatic algorithms based on Generalized Likelihood Ratio Test have been developed and validated
2. Target Parameter Estimate

• M = M ( X(t), Xs ) + B where
  
• M is a global vector of measure corresponding to the same target
• M ( X(t), Xs ) composed of measurements at the various time instant computed according the range, doppler, or azimuth equations.
• B is a zero mean Gaussian noise vector of known covariance ∑B.
• Assuming target motion model in known (in most cases rectilinear unaccelerated), the target characteristic are next derived from the measurement vector M with the help of maximum likelihood estimators (MLE)
• MLE are based on the identification of the target characteristic which minimize a least square criterion on the measurements. In this case is as follow

• J = [M-M(X(t),Xs),Xs]^T∙∑B∙[M-M(X(t),Xs)] + [Xs-Xap]^T∙∑ap∙[Xs-Xap]
• Xap represents supposed sensor position
• ∑ap represents the covariance of this uncertainty

• J take into account the imperfect knowledge of the sonar position, therefore calibration of the network can be perform simultaneously to the target characteristic estimation by considering Xs as an unknown and Xap as a parameter.
• Classic MLE - Gaussian Newton (GN) algorithm and EKF are extended to this case with few modification.
• Difficulties occur for EKF when the initial parameter is to be set.
• Results of measuring range and azimuth for both methods are provided.
• EKF able to track maneuvering target provided the target noise model covariance is well chosen
• GN algorithm is significantly degraded due to the fact that it is implicitly non adaptive
• Accurate calibration is performed jointly with an unbiased target motion estimation provided at least one sensor position is exactly known and azimuth measurements are made by at least one sensor.
• Classic approach has limitation on maneuvering targets
  

THE DIAMANT ALGORITHM

• DIAMANT ( DIscrete Association, Motion ANalysis and Tracking) differs compare to classic sonar tracking method.
• Use Hidden Markov Chain method

Note: No further reading has been made. (seem irrelevant at a.t.m)

Covariance Matrix

Reference for Covariance and Covariance Matrices from wiki.

Variance : measure how much a single random variable varies
Covariance: measure how much two random variable varies together
Covariance Matrix: covariance in the form of matrix for higher dimension of scalar-valued random variable

Refernce for matrix from wiki

Overview of Problems and Techniques in Target Tracking

Wolfgang Koch

SENSOR SYSTEM

• Sensing hardware of sensor or sensor networks collecting information on time varying quantities (waveform). This information is a series of sensor data set (also called scans or data frames) that received at discrete instants of time.
  
• Via a detector device, the data go through a data rate reduction process and forwarded for signal processing.
  
• This data set is used to estimate the state of a stochastically driven dynamical system. Therefore the signal processing results in estimates of the waveform parameters and produced the final sensor reports (measured quantities) and become the input of tracking system

TRACKING SYSTEM

• Tracking results from the data association and estimation algorithm techniques (sensor data processing) which used to exploit efficiently the data from sensor resources and also to obtain information that not directly produce by the sensor reports.
• i.e: tracks mean estimate of state trajectories which statistically represent the quantities or object considered along with their temporal history.
• Sensor reports that can be associated to existing tracks are used for track maintenance
• While remaining reports that non-associated to any existing track are used to initiate new track or multiple frame track extraction.(track initiation)
  
• Both track initiation and track maintenance required a prior knowledge of the sensor performance, object characteristic and object environment. This prior knowledge available in the form of statistical modeling assumptions.
• Plot-to-track unit is important when dealing with multiple target tracking system
  
• Stored data (in Track File Storage) is extracted during track processing and used for track termination /conformation, object classification/identification, and track to track fusion (fusion of track representing identical information).
• Results are displayed through man-machined interface. Other purpose is for interaction function where available information on sensor, object of interest, and the environment can be specified, updated, or corrected by direct Human interaction or the track processor itself.

CHALLENGE

• High false return background. Sensor can't compress data
• Ambiguous correlation between new and existing track become a problem for closely-spaced moving object. Plus false return or unwanted object, identity of the individual object tracks might get lost
• Limited sensor resolution capability make the data association harder bcoz the closely-spaced object may continuously change from being resolved to unresolved and back again. Besides, sensor returns having poor quality, low SNR, or fading phenomena. Scan rate maybe low in certain application ex: long-range air surveillance.
• The underlying dynamics models are restricted to one particular sample out of several sets of alternatives. Sudden switches between the underlying dynamics models do occur and tracks can get lost in such critical situation.

BAYESIAN APPROACH

• Most mathematical techniques of tracking system essentially make use of Bayes' Rule
• Tracking algorithm is an iterative updating scheme for conditional probability densities that describe the object states given both the accumulated sensor data and all available prior information
• Optimal state estimators may be derived related to various risk function provided that filtering, density iteration, has been done correctly.
• Generalization of standard smoothing algorithms, retrodiction, provides a backwards iteration scheme for calculating the probability densities of the past objects states given all information accumulated up to the current scan.
• Track maintenance and data acquisition are closely related
• exist feedback of tracking information to the sensor system
• Tracking algorithm must be initiated by appropriately chosen prior densities.
  

SENSOR FUSION ASPECTS

• Network of homogeneous or heterogeneous sensors are preferred compare to single sensors
• Networks' advantages:
  
• Total coverage of suitably distributed sensors are much larger
• Low-cost sensor network
• Redundancy of overlapping fields of view increased data rates and observation under several aspect
• Multiple-sited networks provide that is on principle not available by corresponding single-sited sensor
• More robust against failure or destructive of individual components
• Centralised data fusion: sensor reports are transmitted to a processing centre without significant delay.
  
• Issue of single-sited sensor or sensor networks is irrelevant.
• Practical realization is difficult bcoz of limited data links between the sensors and the fusion centre, synchronisation problems, or misalignment errors.
• Decentralised fusion architecture or hybrid solution is proposed.
• Data of the sensor system are pre-processed at their individual sites
• Fusion centre receives higher-level information
• i.e: sensor-individual track which are to be fused with other tracks resulting in a central track.

EXPERIMENTAL EXAMPLE

• Real radar data from single-sited radar.
• Data compared using IMM-MHT Retroiction and MMSE-MHT Retrodiction
• IMM-MHT perform better especially in a verysmooth condition (accurate speed and heading information)
• Both filtering and retrodiction produce better results when using model histories more than length n = 1 (delay the frame for MMES-MHT)