sensors kalman filters

8/14/2019 Sensors Kalman Filters

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KALMANFILTERINGOFSENSORDATA

- Nilofer Mehta


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INTRODUCTION

Why do we need sensors in UbiComp?

Because we need to understand context Humans interacting with humans have a complex

contextual understanding

On the other hand when computers interact with

humans, the complex context needs to be understood

by the computer for taking accurate actions

Sensors are used for this purpose


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INTRODUCTION

Measured sensor data usually consists of noise

which reduces accuracy In order to get accurate estimates of the true

value, the sensor data needs to be filtered


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FILTERING

A good filtering algorithm can eliminate noise

from the data and retain useful informationVarious types of filters are used in UbiComp

Example:

Kalman Filter

Hidden Markov Model


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KALMANFILTER

It is an optimal recursive data processing

algorithm Optimaldepends on the criteria chosen to

evaluate the performance

Recursivesince it does not require storage of all

previous data. It also does not require the

previous data to be reprocessed


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KALMANFILTERBASICS

The underlying concept of Kalman filter is a

discrete time linear dynamic system This system depends on the following two

equations:


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Xkis the state of the system at time k. It is based on the

state of the system at time k-1.

Xkis simply defined by its position, velocity and

acceleration.

A is in the form of a matrix. It is an operation used to

calculate the current state of the system from the

previous state with assumption of constant acceleration.


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Zkis the measured value of the system and it relates to

the calculated value Xk.

In a perfect world, Zk= Xkand B is an identity matrix.

But in real life applications there is always a noise

factor called the Gaussian noise (vkand wk).


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Kalman filter has two phases:

PREDICTPredicts the state estimate of the current time using

the state estimate of the previous time. The current

measured values are not considered.

UPDATE

Update phase combines the current time state

estimate with the current measured values and

updates the current state estimate.


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SIMPLEEXAMPLE

Suppose we want to measure the temperature in

this room. We think it is about 22 degrees 2. We also have a thermometer that gives a result

within the range of 5 degrees of the true

temperature. The thermometer reading shows

that it is 25 degrees. What is the best estimate of

the true temperature?

Kalman filters use a weighted average to pick apoint between our guess and the thermometers

reading i.e. between 22 and 25 degrees.


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CALCULATION

GIVEN:

Temperature variance tempvar = 22= 4 Thermometer variance thermvar = 52 = 25

Guess G = 22

Measured M = 25

FIND: Optimal Estimate E?


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CALCULATION

To calculate the optimal estimate we first

calculate the weight w:w = tempvar/ (tempvar + thermvar)

= 4/ (4+25)

= 0.14

This means we trust our guess more than wetrust the thermometer reading.

w 1 means thermometer measurement is moretrustworthy.

w 0 means our guess is more trustworthy.


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CALCULATION

Now calculate the estimate:

E = G + W(MG)= 22 + 0.14(25-22)

= 22.42

The estimate is pretty close to our guess because

we put more trust on our guess as opposed to the

thermometer reading.

How confident are we about our estimate? For

this we need to calculate the estimate variance

evar?


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CALCULATION

evar = (tempvar x thermvar)/(tempvar + thermvar)

= (4 x 25) / (4+25)= 3.44

The deviation in this case is 1.86

Hence, our current guess is 22.42 1.86.

i.e. G = 22.42, tempvar = 1.862

In a Kalman filter, we predict and update and repeatthe process as required. In this case we may take anew measurement M = 21 degrees and update ourestimate.


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ANSWERS

w = 3.44/ (3.44 + 25) = 0.12

E = 22.42 + 0.12(2122.42) = 22.25 evar = (3.44 x 25)/(3.44 + 25) = 3.02

Hence, after the second measurement we update

the estimate to be 22.25 1.74 degrees.

Predict Update


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EXAMPLE- FLYPER

FLYPERFlying Performing Robot

Research conducted by Dr Benjamin N. Passowat De Montfort University.

It uses an Inertial Measurement Unit to

maneuver the helicopter

IMU consists of gyroscopes and accelerometers to

report on the helicopters orientation, velocity

and gravity.

The data from the IMU is filtered and fused

using Kalman filter


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EXAMPLE- FLYPER


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CONCLUSION

Kalman Filter combines measured data as well

as previous knowledge about the system andmeasuring devices to produce an optimal

estimate of the desired variables.

Compared to other filters Kalman filter

minimizes the error significantly


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QUESTIONS?


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QUIZ


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QUIZRESULTS

1. Context

2. Filtered3. Eliminate, Retain

4. It requires no prior data storage except previous

state estimate and current observed

measurement

5. Predict

6. Update


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Annotated Bibliography

Antoniou, Constantinos, Moshe Ben-Akiva and Haris N. Koutsopoulos. "Kalman Filter Applications for

Traffic Management." n.d. July 2012. .

This is a very interesting research paper. It deals with real time estimation and prediction of

traffic conditions. The real time traffic management model is non-linear. Since the basic

assumptions of Kalman filter are derived for a linear model, this paper shows how the Kalman

filter can be modified for a non-linear system. The paper discusses Extended Kalman Filter (EKF),

Limiting EKF and Unscented Kalman filter all of which have various advantages for non-linear

systems.

Bletsas, Aggelos. "Evaluation of Kalman Filtering for Network Time Keeping." IEEE Transactions on

Ultrasonics, Ferroelectrics, and Frequency Control52.9 (2005): 1452-1460.

This research paper proposes a novel Kalman Filtering algorithm for time tracking between a

client computer and a server computer by sending messages as packets over the network. It

evaluates this algorithm by comparing it with two other algorithms namely Linear Programming

and Averaged Time Differences. It concludes that Linear Programming is better than both other

filtering techniques. Kalman filter only works well when the number of packets is increased. This

paper helped me understand the inapplicability of Kalman filter for certain situations.

Funk, Nathan.A Study of the Kalman Filter applied to Visual Tracking. Edmonton, 7 December 2003.

This project report analyzes Kalman Filter as a probabilistic prediction method for visual

tracking. The author conducts a detailed literature survey to identify problems associated with

Kalman Filter. The relationship between Hidden Markov Model and Kalman Filter is also

described. This helps in establishing Kalman Filter as a probabilistic model. Overall, this was a

well written project report.

Martnez, Brbara Valenciano. Speech Enhancement using Kalman Filtering. November 2008.

This project report evaluates Kalman filtering technique applied to enhance speech on a

telephonic conversation. The experiments are first performed using white noise as well as

colored noise. The report describes the basics of Kalman filtering and how it applies to speechfiltering for both kinds of noise. It also discusses the advantages and disadvantages of the

Kalman filter. This was useful for my understanding. In particular, the distinguished advantage of

Kalman filter is that it optimally predicts the state of a dynamic system. One of the

disadvantages of Kalman filter is that it is necessary to have previous knowledge about the

system and measuring devices. The report finally concludes that using Kalman filter helps in

minimizing noise overall.


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Maybeck, Peter S. "Stochastic Models, Estimation, and Control, Volume 1." New York: Academic Press,

1979. 1-23.

The first chapter in this book helped in understanding the basic principles of Kalman filtering. It

was essential to understand how Kalman filtering works before researching its applications. This

book explains Kalman filtering using an example about estimating ones location using areference point.

Passow, Benjamin N. Flyper - An Autonomous Helicopter for Intelligent Acoustic Sensing Research. n.d.

July 2012. .

This research describes the development of an autonomous flying robot called as FLYPER. The

research shows the improvement in stability of the robot after application of the Kalman filter.

The research includes a video that demonstrates the difference between the outputs from an

unfiltered reading versus Kalman filtered reading for an Inertial Measurement Unit. This was

useful for my seminar demonstration.

Plessis, Roger Du. Poor Man's Explanation of Kalman Filtering: or How I Stopped Worrying & Learned to

Love Matrix Inversion. California: North American Rockwell Electronics Group, 1967.

This paper has been written with the intent of providing the most simple and intuitive

explanation of the Kalman Filter. In the writers opinion, all attempts at explaining Kalman filter

have gotten complicated to a certain extent. This paper helped clarify many uncertainties. The

paper takes an example of estimating the resistance of a resistor. The example used in my

seminar is derived from the example given in this paper with some modifications. This paper

also gives a second, more complicated example later in the text.

Schumitsch, Brad, et al. The Identity Management Kalman Filter. Stanford, n.d. July 2012.

This paper presents a new Kalman Filter, called the Identity Management Kalman Filter (IMKF),

for the purpose of tracking of multiple objects. It compares the IMKF to the multi-hypothesis

Kalman filter that has been used for tracking multiple objects. Apart from the standard statistics

used in the basic Kalman Filter, the IMKF maintains only an additional statistic. The paper

concludes that this is much more efficient than the multi-hypothesis Kalman Filter. This was a

noteworthy paper and the findings were valuable for understanding the vast scope of

application of the Kalman filter.

sensors kalman filters

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