Mobile Robot Localization (ch. 7)

• Mobile robot localization is the problem of determining the pose of a robot relative to a given map of the environment. Because,

• Unfortunately, the pose of a robot can not be sensed directly, at least for now. The pose has to be inferred from data.

• A single sensor measurement is enough?

• The importance of localization in robotics.

• Mobile robot localization can be seen as a problem of coordinate transformation. (One point of

view.)

Mobile Robot Localization• Localization techniques have been

developed for a broad set of map representations. – Feature based maps, location based maps,

occupancy grid maps, etc. (what exactly are they?) (See figure 7.2)

– (You can probably guess What is the mapping problem?)

• Remember, in localization problem, the map is given, known, available.

• Is it hard? Not really, because,

Mobile Robot Localization• Most localization algorithms are variants of

Bayes filter algorithm.

• Different representation of maps, sensor models, motion model, etc lead to different variant.

• Here is the agenda.

Mobile Robot Localization• We want to know different kinds of maps.

• We want to know different kinds of localization problems.

• We want to know how to solve localization problems, during which process, we also want to know how to get sensor model, motion model, etc.

Mobile Robot Localization(We want to know different kinds of maps. )

Different kinds of maps.

At a glance, ….

feature-based, location-based, metric, topological map, occupancy grid map, etc.

see figure 7.2– http://www.cs.cmu.edu/afs/cs/project/jair/pub/volume11/fox99a-html/node

23.html

Mobile Robot Localization – A Taxonomy

(We want to know different kinds of localization problems.)• Different kinds of Localization problems.

• A taxonomy in 4 dimensions– Local versus Global (initial knowledge)

– Static versus Dynamic (environment)

– Passive versus active (control of robots)

– Single robot or multi-robot

Mobile Robot Localization

• Solved already, the Bayes filter algorithm. How?

• The straightforward application of Bayes filters to the localization problem is called Markov localization.

• Here is the algorithm (abstract?)

Mobile Robot Localization

• Algorithm Bayes_filter ( )

• for all do

• • endfor• return

111^ )(),|()( tttttt dxxbelxuxpxbel

ttt zuxbel ,),( 1

)( txbel

)()|( )( ^tttt xbelxzpxbel

tx

Mobile Robot Localization

• Algorithm Markov Locatlization ( )

• for all do

• • endfor• return

The Markov Localization algorithm addresses the global localization problem, the position tracking problem, and the kidnapped robot problem in static environment.

111^ )(),,|()( tttttt dxxbelmxuxpxbel

mzuxbel ttt ,,),( 1

)( txbel

)(),|( )( ^tttt xbelmxzpxbel

tx

Mobile Robot Localization

• Revisit Figure 7.5 to see how Markov localization algorithm in working.

• The algorithm Markov Localization is still very abstract. To put it in work (eg. your project), we need a lot of more background knowledge to realize motion model, sensor model, etc….

• We start with Guassian Filter (also called Kalman filter)

Bayes Filter Implementations (1)

Kalman Filter(Gaussian filters)

(back to Ch.3)

Kalman Filter

• The discovery of KF --- one of the greatest discoveries in the history of statistical estimation theory and possibly the greatest in the 21th century.

• Has been used in many application areas ever since Richard Kalman discovered the idea in 1960.

• The KF has made it possible for humans to do things that would not have been possible without it.

• In modern technology, KFs have become as indispensable as silicon chips.

• Prediction

• Correction

Bayes Filter Reminder

111 )(),|()( tttttt dxxbelxuxpxbel

)()|()( tttt xbelxzpxbel

Gaussians

2

2)(

2

1

2

2

1)(

:),(~)(

x

exp

Nxp

-

Univariate

)()(2

1

2/12/

1

)2(

1)(

:)(~)(

μxΣμx

Σx

Σμx

t

ep

,Νp

d

Multivariate

),(~),(~ 22

2

abaNYbaXY

NX

Properties of Gaussians

22

21

222

21

21

122

21

22

212222

2111 1

,~)()(),(~

),(~

NXpXpNX

NX

• We stay in the “Gaussian world” as long as we start with Gaussians and perform only linear transformations.

• Review your probability textbook

),(~),(~ TAABANY

BAXY

NX

Multivariate Gaussians

12

11

221

11

21

221

222

111 1,~)()(

),(~

),(~

NXpXpNX

NX

http://en.wikipedia.org/wiki/Multivariate_normal_distribution

Kalman Filter

tttttt uBxAx 1

tttt xCz

Estimates the state x of a discrete-time controlled process that is governed by the linear stochastic difference equation

with a measurement

Components of a Kalman Filter

t

Matrix (nxn) that describes how the state evolves from t-1 to t without controls or noise.

tA

Matrix (nxl) that describes how the control ut changes the state from t-1 to t.tB

Matrix (kxn) that describes how to map the state xt to an observation zt.tC

t

Random variables representing the process and measurement noise that are assumed to be independent and normally distributed with covariance Rt and Qt respectively.

Kalman Filter Algorithm

1. Algorithm Kalman_filter( t-1, t-1, ut, zt):

2. Prediction:3. 4.

5. Correction:6. 7. 8.

9. Return t, t

ttttt uBA 1

tTtttt RAA 1

1)( tTttt

Tttt QCCCK

)( tttttt CzK

tttt CKI )(

Kalman Filter Updates in 1D

Kalman Filter Updates in 1D

1)(with )(

)()(

tTttt

Tttt

tttt

ttttttt QCCCK

CKI

CzKxbel

2,

2

2

22 with )1(

)()(

tobst

tt

ttt

tttttt K

K

zKxbel

Kalman Filter Updates in 1D

tTtttt

tttttt RAA

uBAxbel

1

1)(

2

,2221)(

tactttt

tttttt a

ubaxbel

Kalman Filter Updates

0000 ,;)( xNxbel

Linear Gaussian Systems: Initialization

• Initial belief is normally distributed:

• Dynamics are linear function of state and control plus additive noise:

tttttt uBxAx 1

Linear Gaussian Systems: Dynamics

ttttttttt RuBxAxNxuxp ,;),|( 11

1111

111

,;~,;~

)(),|()(

ttttttttt

tttttt

xNRuBxAxN

dxxbelxuxpxbel

Linear Gaussian Systems: Dynamics

tTtttt

tttttt

ttttT

tt

ttttttT

tttttt

ttttttttt

tttttt

RAA

uBAxbel

dxxx

uBxAxRuBxAxxbel

xNRuBxAxN

dxxbelxuxpxbel

1

1

1111111

11

1

1111

111

)(

)()(2

1exp

)()(2

1exp)(

,;~,;~

)(),|()(

• Observations are linear function of state plus additive noise:

tttt xCz

Linear Gaussian Systems: Observations

tttttt QxCzNxzp ,;)|(

ttttttt

tttt

xNQxCzN

xbelxzpxbel

,;~,;~

)()|()(

Linear Gaussian Systems: Observations

1

11

)(with )(

)()(

)()(2

1exp)()(

2

1exp)(

,;~,;~

)()|()(

tTttt

Tttt

tttt

ttttttt

tttT

ttttttT

tttt

ttttttt

tttt

QCCCKCKI

CzKxbel

xxxCzQxCzxbel

xNQxCzN

xbelxzpxbel

See page 45-54 for mathematical derivation.

The Prediction-Correction-Cycle

tTtttt

tttttt RAA

uBAxbel

1

1)(

2

,2221)(

tactttt

tttttt a

ubaxbel

Prediction

The Prediction-Correction-Cycle

1)(,)(

)()(

tTttt

Tttt

tttt

ttttttt QCCCK

CKI

CzKxbel

2,

2

2

22 ,)1(

)()(

tobst

tt

ttt

tttttt K

K

zKxbel

Correction

The Prediction-Correction-Cycle

1)(,)(

)()(

tTttt

Tttt

tttt

ttttttt QCCCK

CKI

CzKxbel

2,

2

2

22 ,)1(

)()(

tobst

tt

ttt

tttttt K

K

zKxbel

tTtttt

tttttt RAA

uBAxbel

1

1)(

2

,2221)(

tactttt

tttttt a

ubaxbel

Correction

Prediction

Kalman Filter Summary

• Highly efficient: Polynomial in measurement dimensionality k and state dimensionality n:

O(k2.376 + n2)

• Optimal for linear Gaussian systems!

• However, most robotics systems are nonlinear, unfortunately!