An Introduction to Sensor Fusion

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An Introduction to Sensor Fusion

CSE398/498 11 Feb 05
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Seminar Announcement Today at 410PM,
PL466

• Dr. Metin Sitti, "Robotics at the Micro and Nano
  Scales"

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Administration

• Lab sessions are ongoing
• Still have a few bugs to iron out with Linux
• Change to course format
• Team Challenges are added
• Depending on progress these may migrate to full
  scrimmages
• Course grade distribution adjusted to reflect
  this

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Clarifications to Team Challenge 1

• Obstacle heights will be at a height gt the
  Aibos nose
• The maze will be sparse no walls, just
  obstacles
• If you need more time because of hardware issues,
  let me know ASAP
• Any other questions?

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Coordinate Transformations Conclusion
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Homogeneous Transformation Matrices in Three
Dimensions

• The representation readily extends to three
  dimensions

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Coordinate Transformations Example
The distance from the camera in the dogs nose
to the origin of the head frame is 8 cm, as is
the body frame from the head frame. If only the
neck joint is tilted by an angle f, write the
homogenous transform relating the nose position
in the head frame with the body frame.
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Composing Homogenous Transformations

• Perhaps the strongest point for the homogeneous
  transformation representation is the ability to
  compose multiple transformations across multiple
  frames
• Suppose we would like to estimate the position of
  a point that has seen coordinate transformations
  across 2 frames

3p
1A2
3y
3x
2A3
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Composing Homogenous Transformations
2p
3p
1A2
3y
1p
3x
2A3

• This generalizes for n frames
• NOTE The transformations are done LOCAL to the
  current frame

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Homework Exercise Due 18 Feb 05 at the
BEGINNING of Class

• The Aibo sees the soccer ball in the center of
  the camera image
• The ball is estimated to be 1 meter away from the
  camera frame
• Its head is tilted up 20 degrees
• Its head is panned to the left 45 degrees
• Its neck is tilted down 30 degrees
• Q1 What are the necessary homogeneous
  transformations to calculate the balls position
  in the body frame?
• Q2 What is the balls position in the body
  frame?
• Make certain that you show all your work!

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SummaryCoordinate Transformations

• Points are defined with respect to a specific
  coordinate frame
• Often, it is convenient to measure a point with
  respect to one frame (e.g. an objects position
  in the sensor frame), but it must be transformed
  to another frame for other reasons (e.g.
  navigational convenience)
• Coordinate transformations provide this mechanism
• The transformation necessary to align coordinate
  frame F1 with frame F2 is also the same
  transformation necessary to convert points from
  frame F2 to frame F1
• Homogeneous coordinates provide a convenient
  means for representing and composing rigid
  transformations

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An Introduction to Sensor Fusion
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Motivation

• Lets say our Aibo detects an opponent at a
  distance of 100 cm with its nose IR, but the same
  opponent reads at a distance of 80 cm from its
  chest IR.
• What is our estimate of the distance?

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Why arent the Sensor Measurements Correct?

• Anyone test the accuracy of the IRs on the Aibo?
• How was the ambient light?
• What color was the target?
• Are the errors the same for all ranges?
• What about the orientation of the target surface?
• Is it sensor noise?
• What about bias?
• There are many reasons
• These are often lumped (incorrectly) into the
  term sensor noise

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How should we Merge the Measurements?

• Cant we just average them?

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Merging Data from Heterogeneous Sensors

• Lets say that instead you have 2 different (not
  very accurate) sensor measuring the distance to a
  target over time
• How could you combine the data to get the best
  estimate possible for the true range to target?

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Merging Data from Heterogeneous Sensors (contd)

• Why not just average them?
• That works OK, but not really well
• Can we do more?

mean err 1.71
mean err 1.75
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Fusing Sensor Data

• Our ability to do something intelligent with
  the sensor data is a function of what we know
  about the sensors themselves
• The more accurate sensor model we have, the
  better our estimation performance will be
• These models are typically generated in a time
  consuming, empirical fashion
• Sensor models can vary dramatically as a function
  of the operational environment (recall the Aibo
  example)
• Often sensor models are represented compactly in
  the terms of probability density
  functions/distributions

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A Brief Probability Review
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Basic Concepts

• p(x) denotes the probability that outcome x will
  occur (or that proposition x is true)
• Random Variables
• A random variable x can take on any values
  associated with its sample space X
• Boolean true, false, heads, tails
• Multi-valued 1,2,3,4,5,6 the sides of a die
• Continuous a x b

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A Quick Primer/Refresher

• The expected value for a random variable X is
  (i.e. the mean) defined as
• The variance of X about the mean is defined as

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Probability Density Functions (PDFs)

• A probability density function pX(x) is a
  mapping of probability values to all elements x ?
  X
• A pdf can be discrete or continuous

multi-modal
unimodal
uniform
sample continuous pdfs
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Properties of PDFs

• p(x) 0
• The sum of probabilities over the sample space
  equals to 1, i.e.
• For continuous distributions, p(x)0 for x ? X
• Rather than look at the probability of a point
  for a continuous distribution, we look at the
  probability over intervals, i.e.

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Some Important PDFs

• The uniform distribution is defined as
• Our friend, the Gaussian distribution
• Particle distributions

1/(b-a)
a
b
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Probability Distribution Function

• The probability distribution is defined as
• Some properties of the probability distribution

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The Gaussian Distribution

• A 1-D Gaussian distribution is defined as
• In 2-D (assuming uncorrelated variables) this
  becomes
• In n dimensions, it generalizes to

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The Gaussian Distribution (contd)

• Gaussian distributions are popular for modeling
  sensor noise
• This is often justified by the central limit
  theorem which states that given a set of n
  independent variates with arbitrary PDFs of
  finite variance, then the average of these tends
  to be normal
• The ulterior motive is that Gaussian functions
  have many useful mathematical properties which
  allow them to be used to obtain elegant
  theoretical results
• The convolution of 2 Gaussian functions is a
  Gaussian
• The Fourier transform of a Gaussian function is a
  Gaussian
• Separability

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A Simple Sensor Fusion Examplewith a Gaussian
Noise Model

• Consider a ship sailing east with a perfect
  compass trying to estimate its position.
• You estimate the position x from the stars as
  z1100 with a precision of sx4 miles

100
x
Borrowed from Maybeck, 1979
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A Simple Example (contd)

• Along comes a more experienced navigator, and she
  takes her own sighting z2
• She estimates the position x z2 125 with a
  precision of sx3 miles
• How do you merge her estimate with your own?

x
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A Simple Example (contd)
x
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A Simple Example (contd)

• With the distributions being Gaussian, the best
  estimate for the state is the mean of the
  distribution, so
• or alternately

Correction Term
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Lets Apply this to Our Example

• Lets assume that we knew that the standard
  deviation of our sensors was 2 meters (blue) and
  4 meters (red)

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What if we had Merely Averaged the Data Over
Time Instead

• The time average is in effect assuming that the
  variance of the two sensors are the same
• If we know better, we can use this to obtain a
  more accurate estimate

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Summary

• Sensor fusion refers to the techniques applied in
  (intelligently) merging multiple sensor
  measurements across different sensors and/or time
  in order to obtain a more accurate estimate of
  some parameter of interest (e.g. the position of
  a target)
• The quality of your fused estimate will be a
  direct function of the accuracy of your sensor
  model
• In practice, the best models are often obtained
  empirically
• These empirical data are often fit to a Gaussian
  model when possible for mathematical convenience
• The primary motivation for this is the Kalman
  Filter and Extended Kalman Filter (EKF)
  algorithms that rely upon the Gaussian
  assumption, and work extremely well in theory and
  practice