WiFi GraphSLAM in 1-D · WiFi Implementation Simulation Results Conclusions King Abdullah...

King Abdullah University of Science and Technology - Thuwal - KSA - 2011 – 1 / 28

WiFi GraphSLAM in 1-D

Jose Roberto Ayala Solares

August 2, 2011

Overview


Introduction

The GraphSLAM Algorithm

WiFi Implementation

Simulation Results

Conclusions

Introduction

Overview

Introduction

SLAM ProblemGraph-basedSLAM

WiFi GraphSLAM

The GraphSLAMAlgorithm

WiFiImplementation

Simulation Results

Conclusions


SLAM Problem

Overview

Introduction


WiFi GraphSLAM


WiFiImplementation

Simulation Results

Conclusions


Nowadays there is active research in the area of mobilerobotics that deals with the capacity of a robot to build amap of the environment and to simultaneoulsy localize itselfwithin this map in absence of external referencing systemssuch as GPS.

This scenario is the so-called simultaneous localization andmapping (SLAM) problem.

Solving the SLAM problem consists of estimating the robottrajectory and the map of the environment as the robotmoves in it.

One common approach to solving the SLAM problem is touse an Extended Kalman filter.

Graph-based SLAM

Overview

Introduction


WiFi GraphSLAM


WiFiImplementation

Simulation Results

Conclusions


A graph-based SLAM approach constructs a simplifiedestimation problem by abstracting the raw measuraments.

These raw measurements are replaced by the edges in agraphical model which can then be seen as “virtualmeasurements”.

Each edge between two nodes is labeled with a probabilitydistribution over the relative locations of the two poses,conditioned to their mutual measurements.

WiFi GraphSLAM

Overview

Introduction


WiFi GraphSLAM


WiFiImplementation

Simulation Results

Conclusions


The widespread deployment of wireless sensor networksprovide the opportunity for localization and mapping usingonly signal-strength measurements.

Figure 1: WiFi GraphSLAM scenario

The GraphSLAM Algorithm

Overview

Introduction


Description

The Algorithm:InitializationThe Algorithm:LinearizationThe Algorithm:ReductionThe Algorithm:Solution

WiFiImplementation

Simulation Results

Conclusions


Description

Overview

Introduction


Description


WiFiImplementation

Simulation Results

Conclusions


Figure 2: Illustration of the acquisition of the information matrix Ωin GraphSLAM [From Probabilistic Robotics, MIT Press]

Description

Overview

Introduction


Description


WiFiImplementation

Simulation Results

Conclusions



The Algorithm: Initialization

Overview

Introduction


Description


WiFiImplementation

Simulation Results

Conclusions


To build our initial information matrix Ω and informationvector ξ, we need an initial estimate µ0:t for all poses x0:t.

The Algorithm: Linearization

Overview

Introduction


Description


WiFiImplementation

Simulation Results

Conclusions


Prior

Ω← Ω0 (1)

ξ ← ~0 (2)

The Algorithm: Linearization

Overview

Introduction


Description


WiFiImplementation

Simulation Results

Conclusions


Controls

Ω← Ω+

(

−GTt

1

)

R−1

t

(

−Gt 1)

(3)

ξ ← ξ +

(

−GTt

1

)

R−1

t [g (ut, µt−1)−Gtµt−1] (4)

Measurements

Ω← Ω+H iTt Q−1

t H it (5)

ξ ← ξ +H iTt Q−1

t

[

zit − h(

µt, cit

)

+H itµt

]

(6)

The Algorithm: Reduction

Overview

Introduction


Description


WiFiImplementation

Simulation Results

Conclusions


Let us subdivide the matrix Ω and the vector ξ intosubmatrices:

Ω =

(

Ωx0:t, x0:tΩx0:t, m

Ωm, x0:tΩm,m

)

ξ =

(

ξx0:t

ξm

)

(7)

According to the marginalization lemma:

Ω = Ωx0:t, x0:t− Ωx0:t, mΩ−1

m, mΩm, x0:t(8)

ξ = ξx0:t− Ωx0:t, mΩ−1

m, mξm (9)

The Algorithm: Solution

Overview

Introduction


Description


WiFiImplementation

Simulation Results

Conclusions


The mean and covariance of the robot poses are given by

Σ = Ω−1 (10)

µ = Σξ (11)

The mean location estimate for all features in the map isgiven by

Σm = Ω−1

m,m (12)

µm = Σm

(

ξm +Ωm, x0:tξ)

(13)

WiFi Implementation

Overview

Introduction


WiFiImplementation

DataState-to-measurementmapping

GaussianInterpolation

Simulation Results

Conclusions


Data

Overview

Introduction


WiFiImplementation



Simulation Results

Conclusions


0 5 10 15 20 25 3030

35

40

45

50

55

60

Distance [blocks]

Sig

nal S

tren

gth

[dB

]

Data

Figure 5: WiFi signal-strength measurements

State-to-measurement mapping

Overview

Introduction


WiFiImplementation



Simulation Results

Conclusions


Gaussian interpolation weights have been used with successto interpolate WiFi signal strengths.

The sensor model is given by

h (µ) =1√2πσ

∑

i

si · exp[

−(|µj − µt| − xi)2

2σ2

]

(14)

(xi, si) is the data collected

(µj , µt) is the estimated position of the landmark and robot at t

σ is a length scale that determines the strenght of correlationbetween points

Gaussian Interpolation

Overview

Introduction


WiFiImplementation



Simulation Results

Conclusions


0 5 10 15 20 25 3030

35

40

45

50

55

60

Distance [blocks]

Sig

nal S

tren

gth

[dB

]

σ = 5

DataInterpolation

0 5 10 15 20 25 3030

35

40

45

50

55

60

Distance [blocks]

Sig

nal S

tren

gth

[dB

]

σ = 1

DataInterpolation

0 5 10 15 20 25 3030

35

40

45

50

55

60

Distance [blocks]

Sig

nal S

tren

gth

[dB

]

σ = 0.6

DataInterpolation

0 5 10 15 20 25 300

50

100

150

200

250

Distance [blocks]

Sig

nal S

tren

gth

[dB

]

σ = 0.1

DataInterpolation

Figure 6: Gaussian interpolation

Simulation Results

Overview

Introduction


WiFiImplementation

Simulation Results

Scenario

Results

Conclusions


Scenario

Overview

Introduction


WiFiImplementation

Simulation Results

Scenario

Results

Conclusions


System model: xt = xt−1 + 0.5

Process noise: R = 0.0625

Measurement noise: Q = 16

Number of routers: 2

Distance covered: 10 blocks

Results

Overview

Introduction


WiFiImplementation

Simulation Results

Scenario

Results

Conclusions


0 2 4 6 8 10 12 14 16 18 20−2

0

2

4

6

8

10

12

14

Iteration k

Pos

ition

of t

he r

obot

[blo

cks]

True valueEstimated valueConfidence interval

R1Location: 2.5584Variance: 0.7641


Figure 7: Simulation of the WiFi GraphSLAM algorithm

Results: Variation in the length scale σ

Overview

Introduction


WiFiImplementation

Simulation Results

Scenario

Results

Conclusions


0 2 4 6 8 10 12 14 16 18 20−2

0

2

4

6

8

10

12

Iteration k

Pos

ition

of t

he r

obot

[blo

cks]

σ = 0.1




0 2 4 6 8 10 12 14 16 18 20−2

0

2

4

6

8

10

12

14

Iteration k

Pos

ition

of t

he r

obot

[blo

cks]

σ = 5





Results: Variation in the process noise R

Overview

Introduction


WiFiImplementation

Simulation Results

Scenario

Results

Conclusions


0 2 4 6 8 10 12 14 16 18 20−2

0

2

4

6

8

10

12

Iteration k

Pos

ition

of t

he r

obot

[blo

cks]

R = 0.01




0 2 4 6 8 10 12 14 16 18 20−2

0

2

4

6

8

10

12

14

Iteration k

Pos

ition

of t

he r

obot

[blo

cks]

R = 1





Results: Variation in the measurement noise Q

Overview

Introduction


WiFiImplementation

Simulation Results

Scenario

Results

Conclusions


0 2 4 6 8 10 12 14 16 18 20−2

0

2

4

6

8

10

12

14

16

Iteration k

Pos

ition

of t

he r

obot

[blo

cks]

Q = 4




0 2 4 6 8 10 12 14 16 18 20−2

0

2

4

6

8

10

12

14

16

Iteration k

Pos

ition

of t

he r

obot

[blo

cks]

Q = 64





Conclusions

Overview

Introduction


WiFiImplementation

Simulation Results

Conclusions


Conclusions

Overview

Introduction


WiFiImplementation

Simulation Results

Conclusions


A WiFi GraphSLAM algorithm for localization and mappingin a one-dimensional environment was implemented.

The GraphSLAM algorithm is an intuitive and efficientalternative to solve the SLAM problem.

The obtention of real WiFi signal-strength measurementsprovided interesting and accurate results.

Variations on the parameters of the scenario providedinteresting results about their respective effect on the pathand the map estimates.

Further analysis is required to extend these results into amore realistic scenario (2D, fading, shadowing,...).

Questions?

Overview

Introduction


WiFiImplementation

Simulation Results

Conclusions


WiFi GraphSLAM in 1-D · WiFi Implementation Simulation Results Conclusions King Abdullah...

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