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Novel approach to nonlinearnonGaussian Bayesian state estimation

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Title: Novel approach to nonlinearnonGaussian Bayesian state estimation

1
Novel approach to nonlinear/non-Gaussian Bayesian
state estimation

Presenter Tri Tran 7-2005
2
Outline

3
Motivations

In many applications of positioning, navigation
and tracking, we want to estimate the position of
a moving object at discrete times.
Positioning position of an object is to be
estimated when an navigation system is used to
provide measurements of movement.
Navigation besides the position also velocity,
attitude and heading, accelerating and angular
rates are included in the problem.
Target tracking another objects position is to
be estimated based on measurement of relative
range and angles to ones own position.

4
Motivations (cont.)

The problems can be described by state space
models where the state vector contains the
position and derivatives of the position.
Bayesian methods is to construct a probability
density function (PDF) of the state based on all
the available information.
For linear/Gaussian models, the required PDF
remains Gaussian at every iteration of the
filter, Kalman filter can be used to propagate
and update the mean and covariance of the
distribution.
For nonlinear/non-Gaussian models, there is no
general analytic expression for the required PDF
? a new way of representing and recursively
generating an approximation to the state PDF.
Central idea represent the required PDF as a set
of random samples, rather than as a function over
state space.

5
Particle Filters

Recursive methods to estimate the dynamic state
(a set of random samples) of a moving object in a
discrete time nonlinear model based on posterior
distributions.

6
Recursive Bayesian estimation

System model
xk1 fk(xk, wk)
where fk RnxRm?Rn the system transition function
wk is white noise sequence of known PDF,
independent of past and current states.
Measurement model (observation equation)
yk hk(xk, vk)
where hk RnxRr?Rp the system transition function
vk is another white noise sequence of known PDF,
independent of past and current states.
Assume that the initial PDF p(x1D0) of the state
vector, the functional forms fi and hi for i1,,
k are known.
At time step k, available information is the set
of measurements Dk yki1,,k
Need to construct the PDF of the current state
xk, given all the available information p(xkDk)

7
Bayesian bootstrap filter

A set of random samples xk-1(i) i1,,N from
the PDF p(xk-1Dk-1).
Propagate and update the set of random samples to
obtain xk(i) i1,,N which approximates as
p(xkDk)
Prediction each sample is passed through the
system model to obtain samples from the prior at
time step k
xk(i) fk-1(xk-1(i), wk-1(i))
Update with the measurement yk, evaluate a
normalized weight for each sample
Thus define a discrete distribution over xk(i)
i1,,N with probability qi associated with
element i
Resample N times from the discrete distribution
to generate samples xk(i) i1,,N, so that for
any j, Prxk(j)xk(i) qi

8
Bootstrap filter

Advantages no restrictions on the functions fk
and hk or on the distributions of the system or
measurement noise.
Requirements
(a) p(x1) is available for sampling
(b) the likelihood p(ykxk) is a known functional
form
(c) p(wk) is available for sampling
The number of random samples N depends on
(a) the dimension of the state space
(b) the typical overlap between the prior and
the likelihood
(c) the required number of time steps

9
Evaluations of the bootstrap filter algorithm

One-dimensional nonlinear example
Consider the following nonlinear model
Xk 0.5xk-125xk-1/(1x2k-1)8cos(1.2(k-1))wk
Ykx2k/20vk
Wk and vk are zero-mean Gaussian white noise with
variances 10.0 and 1.0, respectively.
K50, x0 0.1, N500

10
Evaluations of the bootstrap filter algorithm

Bearings-only tracking example
Target moves within the x-y plan
A fixed observer at the origin of the plane takes
noisy measurements zk of the target bearing
k24, N4000, initial state
x1(-0.05, 0.001, 0.7, -0.055)

11
Evaluations of the bootstrap filter algorithm

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Summary

A bootstrap filter for implementing recursive
Bayesian filters
The method is applied for nonlinear models and
approximates the posterior distribution as a set
of random samples.

13
References

N.J Gordon, D.J. Salmond and A.F.M. Smith, Novel
approach to nonlinear/non-Gaussian Bayesian state
estimation, in IEE Processinds-F, Vol. 140,
1993.
Fredrik Gustafsson, Fredrik Gunnarsson, Niclas
Bergman, Urban Forssell, Jonas Jansson, Rickard
karlsson, Per-Johan Nordlund, Prticle Filters
for Positioning, Navigation and Tracking, IEEE
Transactions on Signal Processing, vol. 50, pp.
425- 435, Feb. 2002