## Description

ECE 8540

Lab 6 – Particle Filter

1 Introduction

This lab report concerns the problem of applying a particle filter to a set of measurements.

The measurements are numerical data that has been recorded over time in order to provide

samples to filter. The measurements were recorded for a system that consists of a 1D

position, moving on a line. The motion pattern of the system is an oscillation over the line.

The sensor used in the system recoded a field strength that was the sum from two magnets

at fixed positions. The particle filter was used to actively filter the measurements read in

at each time step, allowing for more accurate state predictions. Unlike the previously used

Kalman filters, the particle filter lets us model systems where there is non-Gaussian noise.

The particle filter can filter data from a wider range of systems. It can handle systems where

the noise is believed to be Gaussian or non-Gaussian and it can also handle systems that

are linear or non-linear. Both the Kalman and extended Kalman filters are limited to the

range of problems they are able to filter. The particle filter improves upon these methods to

provide a solution that can be applied to a wider range of problems with proper knowledge

of the system. Since the particle filter has been introduced it has widely been considered the

best filtering method to accurately track system states.

2 Methods

The particle filter follows a cycle of prediction and update. In this cycle the next state is

calculated for each particle. Then the measurement from the current time step is obtained.

Using the new measurement the weight for each particle is updated. The weights are then

normalized so that they all sum to one. A desired output is then computed. The desired

output used in this lab was the expected value. Finally, there is a test to determine whether

or not resampling is required. Resampling is performed if necessary and the process is

repeated over the next time step.

When using the particle filter it is important to note that certain filter variables needed

to be initialized. For this specific implementation of the lab those variables included the

initial state and weight for each particle, the number of particles, the resampling threshold

and a variety of noise constraints. The initial state for each particle was set to be 0. The

initial weight for each particle was set to 1/M where M is the number of particles. The value

of M was experimented with but fell in the range from 100 to 1000. The steps mentioned in

the previous paragraph are further broken down below and the necessary equations are as

follows:

1

In this problem there was two state variables, position and velocity.

Xt =

xt

sx˙t

(1)

1. Calculate the next state for each particle m with the following state transition equation:

{x

(m)

t = f(x

(m)

t−1

, a

(m)

t

)}

M

m=1 (2)

Where f(x

(m)

t−1

, a

(m)

t

) is defined as the following:

f(xt

, at) =

xt+1 = xt + ˙xtT

x˙t+1 =

2 if xt < −20

x˙t + |at

| if −20 ≤ xt < 0

x˙t − |at

| if 0 ≤ xt ≤ 20

−2 if xt > 20

(3)

In equation 3 the input xt represents the state variable vector and at represents the dynamic noise. The dynamic noise, at was drawn from a zero-mean Gaussian distribution

N(0, σ2

a

). Where σa was defined to be 2−4

.

2. Obtain the measurement of magnetic strength from the sensor.

Yt =

yt

(4)

3. Calculate the ideal measurement using the observation equation for the model.

g(xt

, nt) = h

yt = √

1

2πσm

exp( −(xt−xm1)

2

2σ2m

) + √

1

2πσm

exp( −(xt−xm2)

2

2σ2m

) + nt

i

(5)

In equation 5, Xm1 = −10 and Xm2 = 10 representing the locations of the magnets

in the model. The value σm = 4. The input variable nt represents the measurement

noise and is pulled from the zero-mean Gaussian distribution N(0, σ2

n

). Where σn

was defined to be 2−8

. When solving for the ideal measurement of each particle it is

assumed that nt = 0 to represent no noise.

4. Use the ideal measurement calculated for each particle to compare against the actual

measurement from the model of the measurement noise. This comparison can be done

using the following equation:

p(yt

|x

(m)

t

) = 1

√

2πσn

exp(−(y

(m)

t − yt)

2

2σ

2

n

) (6)

5. Calculate the weight updates for each particle using the following equation:

w˜

(m)

t = w

(m)

t−1

· p(yt

|x

(m)

t

) (7)

2