7/31/2019 Qual Slides

1/12

Qualifying Oral Examination

Where am I?

7/31/2019 Qual Slides

2/12

Localization Mapping

Elfes & Moravec Occupancy GridsICRA$85, Computer %89

Kuipers & Byun ! Topological MapsRobotics & Autonomous Systems 1991

Map/Scan MatchingLu & Milios !!AR$97, Cox et al, IEEE Robotics & Automation 91

Geometric BeaconsLeonard & Durrant!

WhyteIEEE Robotics & Automation

%91

ParticleFilters

D. FoxPhDThesis

Bonn 1998

7/31/2019 Qual Slides

3/12

Simultaneous Localization And Mapping

p(st,|zt, u

t, n

t)

Robot Location

MapFeatures

Observations

Control Inputs

Data Associations

z1

z2

z3

zn

.

.

.

.

.

.

n1

n2

n3

nn

u1

u2

u3

un

.

.

.

123

n

.

.

.

[x,y,]

[p, q]

p(st,|zt, u

t, n

t)

Probability of robot being at position st

within environment represented as map

ut

zt

nt : f(zi) i

given knowledge of the observations

the control inputs 'commanded motion(

and Data Associations =

p(zt|st,, z

t1, ut, nt)p(st,|zt1, ut, nt)dst

= p(zt|st,, nt)

p(st|st1, u

t)p(st1,|zt1, ut1, nt1)dst1}

Measurement Model

}

Motion Model

Normalizing constantensures that the resulting posterior is a probability

p(st,|zt, u

t, n

t) Using Bayes Rule, Markov Assumption &Simplifying

7/31/2019 Qual Slides

4/12

t = [st, 1, 2, . . . , n]

t = E[tT

t ]

p(st,|zt, ut, nt) N(t,t)

N(,2) = 122

e(x)2

22

Kalman Filter approximates the posterior as a Gaussian

Mean 'state vector( contains robot location and mappoints

Covariance Matrix estimates uncertainties andrelationships between each element in state vector

O(n3)

O(nlog27) O(n2.807)Strictly speaking, matrix inversion is actually

p(xt) = {xi, wi}

7/31/2019 Qual Slides

5/12

p(st,|zt, u

t, n

t) p(st,|zt, ut, nt)

Estimates slightly di)erent posterior

Robot Trajectory or Paths instead of simply pose

This allows for a re!formulation as a Rao!Blackwellised ParticleFilter

p(st,|zt, ut, nt)= p(st|zt, ut, nt)N

n=1

p(n|st, z

t, u

t, n

t)}}

Particle FilterN Separate Kalman Filters

Map per Particle!

1

, T1

7/31/2019 Qual Slides

6/12

7/31/2019 Qual Slides

7/12

nt : f(zi) i

7/31/2019 Qual Slides

8/12

P(X Y) = P(X|Y)P(Y)

P(X Y) = P(Y|X)P(X)

P(X|Y) =P

(X Y

)P(Y)

P(Y) = 0

P(Y|X) =P X Y

P(X)P(X) = 0

P(X, Y) = P(X|Y)P(Y) = P(Y|X)P(X)

P(X|Y) =P(Y|X)P(X)

P(Y)

E[x

] =+

xp(x

)dx E

[[xE

[x

]]

r

] =+

(xE

[x

])

rp(x

)dx

E[c] = cE[E[x]] = E[x]E[x + y] = E[x] + E[y]

E[xy] = E[x]E[y]

Expected Value Rules

Generalized Central MomentMean

7/31/2019 Qual Slides

9/12

p(xt+1|x0, x1, . . . , xt) = p(xt+1|xt)

p(xt+1) =

p(xt+1|xt)p(xt)dxt

Probability of variable at time t+1 can be computed as

set of particles do notrelate to reality

informative particles nowlost due to resampling

not enough particles ortoo many copies of

same particleintroduced

7/31/2019 Qual Slides

10/12

R1 R2

L1 L2

R1 R2

L1 L2

Implicit Relationship

SEIF

Covariance Matrix Information Matrix

Eustice RSS05Map of Titanic

7/31/2019 Qual Slides

11/12

7/31/2019 Qual Slides

12/12

Courtesy of D. Fox

Courtesy of D. Hahnel