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Qualifying Oral Examination
Where am I?
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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
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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
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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}
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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
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nt : f(zi) i
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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
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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
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R1 R2
L1 L2
R1 R2
L1 L2
Implicit Relationship
SEIF
Covariance Matrix Information Matrix
Eustice RSS05Map of Titanic
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Courtesy of D. Fox
Courtesy of D. Hahnel
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