1. Mobile Robot controlled by Kalman Filter
Thanks for your attention!

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

3. 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

4. Example of estimation problem

5. 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!

6. 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?

7. Can we combine Measurements?
You make a measurement
Also, your friend makes a measurement
Question 1. Which one is the better?
Question 2. What’s the best way to combine these measurements

8. Example how to combine two measurements
Uncertainty is decreased by
combining the two pieces of
Information !!

9. 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,

10. Derivation of product of two PDFs from last slide
Given are two PDFs

11. What we have discussed
Two lectures ago we discussed product of probabilities on discrete examples
Last lecture we discussed product of PDFs - Gaussians

12. How to calculate the best estimate when you are moving?
Suppose you’re moving
u is a nominal velocity
w is a noisy term
The “noise” w will be modeled as a white Gaussian noise with a mean of zero and variance of
Best prediction of new position
Best estimate of new position takes into account new measurement
Nominal velocity
New variance

13. 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

14. What is a Filter by the way?

15. 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 didn’t miss the Moon!)

16. The “filtering Problem” in General (let’s 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
(not like river level example)
Measuring Devices
Estimator
Measurement
Error Sources
System state cannot be measured directly
Need to estimate “optimally” from measurements

17. FILTERS IN MOBILE ROBOTS

18. 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

19. Mobile Robot Localization uses landmarks.
REMINDER: Where am I?
Given a map, determine the robot’s location
Landmark locations are known,
but the robot’s 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?

20. 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?)

21. A Dynamic System x state p probability y observation h measurement
Most commonly - Available:
Initial State
Observations
System (motion) Model
Measurement (observation) Model

22. 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.

23. 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 -> inference task
Mapping -> learning task
Remember concepts of inference and learning
Inference can be Bayesian

24. Bayesian Estimation is based on Markov’s 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)

25. 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

26. Recursive Bayes Filter Implementations
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)

27. Example: State Representations for Robot Localization
These three are most often used, there are many others
Grid Based approaches (Markov localization)
Particle Filters (Monte Carlo localization)
Kalman Tracking

28. Example: Localization – Grid Based
Initialize Grid (Uniformly or according to prior knowledge)
At each time step:
For each grid cell
Use observation model to compute
Use motion model and probabilities to compute
Normalize
x state
P motion
y observation
H measurement

29. Bayesian Filter

30. 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 you’re working in mobile robotics, you’ll see it over and over!
Very important to understand and appreciate
Bayesian Filters are Efficient state estimators
Recursively compute the robot’s current state based on the previous state of the robot
What is the robot’s state?

31. 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:

32. What is a Posterior Distribution?

33. Derivation of the Bayesian Filter

34. 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 robot’s state given the data:
The robot’s data, Z, is expanded into two types: observations oi and actions ai
Invoking the Bayesian theorem

35. 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):

36. 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 robot’s data

37. 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

38. Localization Initial state detects nothing: Moves and
Reminder
Initial state
detects nothing:
Moves and
detects landmark:
Moves and
detects nothing:
Moves and
detects landmark:

39. Bayesian Filter : Requirements for Implementation
This applies to any Bayes filter
Representation for the belief function
Update equations
Motion model
Sensor model
Initial belief state
We have discussed all these components already

40. Representation of the Belief Function
Sample-based representations
e.g. Particle filters
Parametric representations
There can be many sample-based representations
There can be many parametric representations

41. References You can find useful materials about HMM from
CS570 AI Lecture Note(2003)
You can find useful materials about Kalman Filter from
Maybeck, 1979, “Stochastic models, estimation, and control”
Greg Welch, and Gray Bishop, 2001, “An introduction to the Kalman Filter”

42. Paul E. Rybski Haris Baltzakis
Sources