Mobile%20Robot%20controlled%20by%20Kalman%20Filter

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Mobile Robot controlled by Kalman Filter
Thanks for your attention!

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Overview

1. What could Kalman Filters be used for?
2. What is a Kalman Filter?
3. Conceptual Overview
4. The Theory of Kalman Filter (only the equations
   you need to use)
5. Simple Examples

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Most Generally WHAT IS Kalman Filter?

• What is the Kalman Filter?
• A technique that can be used to recursively
  estimate unobservable quantities called state
  variables, xt, from an observed time series
  yt.
• What is it used for?
• Tracking missiles
• Extracting lip motion from video
• Lots of computer vision applications
• Economics
• Navigation

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Example of estimation problem
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Example of Estimation Problem

• Estimating the location of a ship

• Suppose that you are lost at sea during the
  night and have no idea at all of your location.

Problem? Inherent measuring device
inaccuracies. Your measurement has some
uncertainty!
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How to model the Uncertainty of measurement?

• Let us write the conditional probability of the
  density of position based on measured value z1
• Assume Gaussian distribution

• z1 Measured position
• x Real position

Real position

• Q What can be a measure of uncertainty?

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Can we combine Measurements?

• You make a measurement
• Also, your friend makes a measurement

• Question 1. Which one is the better?

• Question 2. Whats the best way to combine these
  measurements

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Example how to combine two measurements

• Uncertainty is decreased by
• combining the two pieces of
• Information !!

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What does it mean for a robot?
We will use this in next slide

• Optimal estimate at t2, , is
  equal to the best prediction of its value before
  z2 is taken,
• Plus a correction term of an optimal weighting
  value
• times the difference between z2
• and the best prediction of its value
• before it is actually taken, .

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Derivation of product of two PDFs from last slide
Given are two PDFs
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What we have discussed

1. Two lectures ago we discussed product of
   probabilities on discrete examples
2. Last lecture we discussed product of PDFs -
   Gaussians

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How to calculate the best estimate when you are
moving?

• Suppose youre moving

• u is a nominal velocity
• w is a noisy term

• Best estimate of new position takes into account
  new measurement

• Best prediction of new position

Nominal velocity
New variance
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Summary on models, prediction and correction

• Process Model
• Describes how the state changes over time
• Measurement Model
• Where you are from what you see !!!
• Predictor-corrector
• Predicting the new state and its uncertainty
• Correcting with the new measurement

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What is a Filter by the way?
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What is a Filter by the way?

• Define mathematically what a filter is (make an
  analogy to a real filter)

• Other applications of Kalman Filtering (or
  Filtering in general)
• Your Car GPS (predict and update location)
• Surface to Air Missile (hitting the target)
• Ship or Rocket navigation (Appollo 11 used some
  sort of filtering to make sure it didnt miss the
  Moon!)

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The filtering Problem in General (lets get a
little more technical)

• Black Box

• Sometimes the system state and the measurement
  may be two different things

System Error Sources

• External Controls

System

• System State (desired but not known)

• Optimal Estimate of System State

• Observed Measurements

Measuring Devices

• Estimator

• Measurement
• Error Sources

• System state cannot be measured directly
• Need to estimate optimally from measurements

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• FILTERS IN MOBILE ROBOTS

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Problem Statement Mobile Robot Control

• Examples of systems in need for state
  prediction
• A mobile robot moving within its environment
• A vision based system tracking cars in a highway
• Common characteristics of these systems
• A state that changes dynamically
• State cannot be observed directly
• Uncertainty is due to noise in
• state
• way state changes
• observations

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Mobile Robot Localization uses landmarks.

• REMINDER Where am I?
• Given a map, determine the robots location
• Landmark locations are known,
• but the robots position is not known
• From sensor readings, the robot must be able to
  infer its most likely position on the field
• Example where are the AIBOs on the soccer
  field?

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Mobile Robot Mapping uses Landmarks

• What does the world look like?
• Robot is unaware of its environment
• The robot must explore the world and determine
  its structure
• Most often, this is combined with localization
• Robot must update its location with respect to
  the landmarks
• Known in the literature as Simultaneous
  Localization and Mapping, or Concurrent
  Localization and Mapping SLAM (CLM)
• Example AIBOs are placed in an unknown
  environment and must learn the locations of the
  landmarks
• (An interesting project idea?)

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A Dynamic System

• x state
• p probability
• y observation
• h measurement

• Most commonly - Available
• Initial State
• Observations
• System (motion) Model
• Measurement (observation) Model

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Filters must be optimal

• Filters compute the hidden state from
  observations
• Filters
• Terminology from signal processing
• Can be considered as a data processing algorithm.
  
• Filters are Computer Algorithms or hardware
  devices (FPGA)
• Filters do classification
• Classification Discrete time versus Continuous
  time
• Issues
• Sensor fusion
• Robustness to noise
• Wanted each filter to be optimal in some sense.

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Example Navigating Robot with odometry Input

• x state
• P probability
• y observation
• H measurement

• Motion model is done according to odometry.
• Observation model is done according to sensor
  measurements.

• Localization -gt inference task
• Mapping -gt learning task

• Remember concepts of inference and learning

• Inference can be Bayesian

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Bayesian Estimation is based on Markovs
assumption

• x state
• P probability
• y observation
• h measurement

Bayesian estimation Attempt to construct the
posterior distribution of the state given all
measurements.
Inference task (localization)Compute the
probability that the system is at state z at time
t given all observations up to time t Note
state only depends on previous state (first order
Markov assumption)
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Recursive Bayes Filter

• x state
• P probability
• y observation
• H measurement
• z data

• Bayes Filter
• Two steps Prediction Step - Update step
• Advantages over batch processing
• Online computation - Faster - Less memory -
  Easy adaptation
• Example of simple recursive Bayes Filter two
  states A,B

Possible states and other data

• It is like generalized flip-flop door open ,
  door closed from the past lecture

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Recursive Bayes FilterImplementations

• x state
• P probability
• y observation
• H measurement

Assuming Bayes Filter as here, from last slide
How is the prior distribution represented?
How is the posterior distribution calculated?

• Continuous representation
• Gaussian distributions Kalman filters (Kalman60)
• Discrete representation
• HMM Solve numerically
• Grid (Dynamic) Grid based approaches (e.g
  Markov localization - Burgard98)
• Samples Particle Filters (e.g. Monte Carlo
  localization - Fox99)

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Example State Representations for Robot
Localization

• These three are most often used, there are many
  others

Grid Based approaches (Markov localization)
Particle Filters (Monte Carlolocalization)
Kalman Tracking
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Example Localization Grid Based

1. Initialize Grid(Uniformly or according to prior
   knowledge)
2. At each time step
3. For each grid cell
4. Use observation model to compute
5. Use motion model and probabilities to compute
6. Normalize

• x state
• P motion
• y observation
• H measurement

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• Bayesian Filter

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Why Bayesian Filters are so important?

• Why should you care about Bayesian Filters?
• Robot and environmental state estimation is a
  fundamental problem in mobile robotics, and in
  our projects of GuideBot !
• Nearly all algorithms that exist for spatial
  reasoning make use of this approach
• If youre working in mobile robotics, youll see
  it over and over!
• Very important to understand and appreciate
• Bayesian Filters are Efficient state estimators
• Recursively compute the robots current state
  based on the previous state of the robot

• What is the robots state?

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Bayesian Filter link to known concepts

• x state
• P motion
• y observation
• H measurement
• Z data

• Estimate state x from data d
• What is the probability of the robot being at x?
• x could be robot location, map information,
  locations of targets, etc
• d could be sensor readings such as range,
  actions, odometry from encoders, etc)
• This is a general formalism that does not depend
  on the particular probability representation
• Bayes filter recursively computes the posterior
  distribution

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What is a Posterior Distribution?
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Derivation of the Bayesian Filter
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Derivation of the Bayesian Filter (slightly
different notation from before)

• x state
• P motion
• y observation
• H measurement
• data, Z
• observations oi
• actions ai

• Estimation of the robots state given the data

• The robots data, Z, is expanded into two types
  observations oi and actions ai

• Invoking the Bayesian theorem

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Derivation of the Bayesian Filter

• x state
• P motion
• y observation
• H measurement
• data, Z
• observations oi
• actions ai

review
Denominator is constant relative to xt
First-order Markov assumption shortens first term
Expanding the last term (theorem of total
probability)
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Derivation of the Bayesian Filter

• x state
• P probability
• y observation
• H measurement
• data, Z
• observations oi
• actions ai

review
First-order Markov assumption shortens middle
term
Finally, substituting the definition of Bel(xt-1)
The above is the probability distribution that
must be estimated from the robots data
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Iterating the Bayesian Filter
review

• Propagate the motion model
• Update the sensor model

• Compute the current state estimate before taking
  a sensor reading by integrating over all possible
  previous state estimates and applying the motion
  model

• Compute the current state estimate by taking a
  sensor reading and multiplying by the current
  estimate based on the most recent motion history

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Localization
Reminder
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Bayesian Filter Requirements for Implementation
This applies to any Bayes filter

1. Representation for the belief function
2. Update equations
3. Motion model
4. Sensor model
5. Initial belief state

We have discussed all these components already
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Representation of the Belief Function

• Parametric representations

Sample-based representations e.g. Particle filters

• There can be many sample-based representations

• There can be many parametric representations

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References

• You can find useful materials about HMM from
• CS570 AI Lecture Note(2003)
• http//www.idiap.ch/bengio/
• http//speech.chungbuk.ac.kr/owkwon/
• You can find useful materials about Kalman Filter
  from
• http//www.cs.unc.edu/welch/kalman
• Maybeck, 1979, Stochastic models, estimation,
  and control
• Greg Welch, and Gray Bishop, 2001, An
  introduction to the Kalman Filter

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Sources

• Paul E. Rybski
• Haris Baltzakis