1

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.

2

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,

3

Mobile Robot Localization

•Most localization algorithms are variants of Bayes filter algorithm.

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

•Here is the agenda.

4

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.

5

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/node23.html

6

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

7

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?)

8

Mobile Robot Localization

• Algorithm Bayes_filter ( )

• for all do

•

• endfor

• return

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

ttt zuxbel ,),( 1

)( txbel

)()|( )( ^tttt xbelxzpxbel

tx

9

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

10

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)

11SA-1SA-1

Bayes Filter Implementations (1)

Kalman Filter(Gaussian filters)

(back to Ch.3)

12

•Prediction

•Correction

Bayes Filter Reminder

111 )(),|()( tttttt dxxbelxuxpxbel

)()|()( tttt xbelxzpxbel

13

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

14

),(~),(~ 22

2

abaNYbaXY

NX

Properties of Gaussians

22

21

222

21

21

122

21

22

212222

2111 1

,~)()(),(~

),(~

NXpXpNX

NX

15

• 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

16

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

17

Components of a Kalman Filter

t

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

tA

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

18

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 )(

19

Kalman Filter Updates in 1D

20

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

21

Kalman Filter Updates in 1D

tTtttt

tttttt RAA

uBAxbel

1

1)(

2

,2221)(

tactttt

tttttt a

ubaxbel

22

Kalman Filter Updates

23

0000 ,;)( xNxbel

Linear Gaussian Systems: Initialization

• Initial belief is normally distributed:

24

• 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

25

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)(

,;~,;~

)(),|()(

26

• Observations are linear function of state plus additive noise:

tttt xCz

Linear Gaussian Systems: Observations

tttttt QxCzNxzp ,;)|(

ttttttt

tttt

xNQxCzN

xbelxzpxbel

,;~,;~

)()|()(

27

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.

28

The Prediction-Correction-Cycle

tTtttt

tttttt RAA

uBAxbel

1

1)(

2

,2221)(

tactttt

tttttt a

ubaxbel

Prediction

29

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

30

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

31

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!

32

Nonlinear Dynamic Systems

•Most realistic robotic problems involve nonlinear functions

),( 1 ttt xugx

)( tt xhz

33

Linearity Assumption Revisited

34

Non-linear Function

35

EKF Linearization (1)

36

EKF Linearization (2)

37

EKF Linearization (3)

38

•Prediction:

•Correction:

EKF Linearization: First Order Taylor Series Expansion

)(),(),(

)(),(

),(),(

1111

111

111

ttttttt

ttt

tttttt

xGugxug

xx

ugugxug

)()()(

)()(

)()(

ttttt

ttt

ttt

xHhxh

xx

hhxh

39

EKF Algorithm

1. Extended_Kalman_filter( t-1, t-1, ut, zt):

2. Prediction:3. 4.

5. Correction:6. 7. 8.

9. Return t, t

),( 1 ttt ug

tTtttt RGG 1

1)( tTttt

Tttt QHHHK

))(( ttttt hzK

tttt HKI )(

1

1),(

t

ttt x

ugG

t

tt x

hH

)(

ttttt uBA 1

tTtttt RAA 1

1)( tTttt

Tttt QCCCK

)( tttttt CzK

tttt CKI )(

40SA-1SA-1

Bayes Filter Implementations (2)

Particle filters

41

Sample-based Localization (sonar)

42

Represent belief by random samples

Estimation of non-Gaussian, nonlinear processes

Monte Carlo filter, Survival of the fittest, Condensation, Bootstrap filter, Particle filter

Filtering: [Rubin, 88], [Gordon et al., 93], [Kitagawa 96]

Computer vision: [Isard and Blake 96, 98] Dynamic Bayesian Networks: [Kanazawa et al., 95]

Particle Filters

43

Particle Filters

44

)|()(

)()|()()|()(

xzpxBel

xBelxzpw

xBelxzpxBel

Sensor Information: Importance Sampling

45

'd)'()'|()( , xxBelxuxpxBel

Robot Motion

46

)|()(

)()|()()|()(

xzpxBel

xBelxzpw

xBelxzpxBel

Sensor Information: Importance Sampling

47

Robot Motion

'd)'()'|()( , xxBelxuxpxBel

48

1. Algorithm particle_filter( St-1, ut-1 zt):

2.

3. For Generate new samples

4. Sample index j(i) from the discrete distribution given by wt-

1

5. Sample from using and

6. Compute importance weight

7. Update normalization factor

8. Insert

9. For

10. Normalize weights

Particle Filter Algorithm

0, tS

ni ...1

},{ it

ittt wxSS

itw

itx ),|( 11 ttt uxxp )(

1ij

tx 1tu

)|( itt

it xzpw

ni ...1

/it

it ww

49

draw xit1 from Bel(xt1)

draw xit from p(xt | xi

t1,ut1)

Importance factor for xit:

)|()(),|(

)(),|()|(ondistributi proposal

ondistributitarget

111

111

tt

tttt

tttttt

it

xzpxBeluxxp

xBeluxxpxzp

w

1111 )(),|()|()( tttttttt dxxBeluxxpxzpxBel

Particle Filter Algorithm

50

Weight samples: w = f / g

Importance Sampling

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

51

Importance Sampling with Resampling

),...,,(

)()|(),...,,|( :fon distributiTarget

2121

n

kk

n zzzp

xpxzpzzzxp

)(

)()|()|( :gon distributi Sampling

l

ll zp

xpxzpzxp

),...,,(

)|()(

)|(

),...,,|( : w weightsImportance

21

21

n

lkkl

l

n

zzzp

xzpzp

zxp

zzzxp

g

f

52

Importance Sampling with Resampling

Weighted samples After resampling

53

Resampling

• Given: Set S of weighted samples.

• Wanted : Random sample, where the probability of drawing xi is given by wi.

• Typically done n times with replacement to generate new sample set S’.

54

w2

w3

w1wn

Wn-1

Resampling

w2

w3

w1wn

Wn-1

• Roulette wheel

• Binary search, n log n

• Stochastic universal sampling

• Systematic resampling

• Linear time complexity

• Easy to implement, low variance

55

1. Algorithm systematic_resampling(S,n):

2.

3. For Generate cdf4. 5. Initialize threshold

6. For Draw samples …7. While ( ) Skip until next threshold reached8. 9. Insert10. Increment threshold

11. Return S’

Resampling Algorithm

11,' wcS

ni ...2i

ii wcc 1

1],,0]~ 11 inUu

nj ...1

11

nuu jj

ij cu

1,'' nxSS i

1ii

Also called stochastic universal sampling

56

57

58

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

Sample-based Localization (sonar)

76

Initial Distribution

77

After Incorporating Ten Ultrasound Scans

78

After Incorporating 65 Ultrasound Scans

79

Estimated Path

80

Using Ceiling Maps for Localization

81

Vision-based Localization

P(z|x)

h(x)z

82

Under a Light

Measurement z: P(z|x):

83

Next to a Light

Measurement z: P(z|x):

84

Elsewhere

Measurement z: P(z|x):

85

Global Localization Using Vision

86

Robots in Action: Albert

87

Limitations

•The approach described so far is able to • track the pose of a mobile robot and to• globally localize the robot.

•How can we deal with localization errors (i.e., the kidnapped robot problem)?

88

Approaches

•Randomly insert samples (the robot can be teleported at any point in time).

• Insert random samples proportional to the average likelihood of the particles (the robot has been teleported with higher probability when the likelihood of its observations drops).