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```Probabilistic Robotics
Bayes Filter Implementations
Bayes Filter Reminder
•Prediction
bel ( xt )   p( xt | ut , xt 1 ) bel ( xt 1 ) dxt 1
•Correction
bel ( xt )   p( zt | xt ) bel ( xt )
Gaussians
p ( x) ~ N (  ,  2 ) :

p ( x) 

1
2 
2
e
1 ( x  )2
2 2
Univariate
-

p(x) ~ Ν (μ,Σ) :
p ( x) 
1
(2 )
d /2
Σ
1/ 2
e
Multivariate
1
 ( x μ ) t Σ 1 ( x μ )
2

Properties of Gaussians
X ~ N (  ,  2 )
2 2

Y
~
N
(
a


b
,
a
 )

Y  aX  b 
2
2
  22

X 1 ~ N ( 1 ,  1 ) 

1
1



p
(
X
)

p
(
X
)
~
N



,
1
2
1
2
2
2
2
2
2
2 
2 

1   2
1   2 
X 2 ~ N (  2 ,  2 )
 1   2
Multivariate Gaussians
X ~ N (  , ) 
T

Y
~
N
(
A


B
,
A

A
)

Y  AX  B 
X 1 ~ N ( 1 , 1 ) 
 2

1
1

1 
2 ,
  p( X 1 )  p( X 2 ) ~ N 
1
1 
X 2 ~ N (  2 ,  2 )
1   2
1   2 
 1   2
• We stay in the “Gaussian world” as long as we
transformations.
Discrete Kalman Filter
Estimates the state x of a discrete-time
controlled process that is governed by the
linear stochastic difference equation
xt  At xt 1  Bt ut   t
with a measurement
zt  Ct xt   t
6
Components of a Kalman Filter
At
Matrix (nxn) that describes how the state
evolves from t to t-1 without controls or
noise.
Bt
Matrix (nxm) that describes how the control
ut changes the state from t to t-1.
Ct
Matrix (kxn) that describes how to map the
state xt to an observation zt.
t
t
Random variables representing the process
and measurement noise that are assumed to
be independent and normally distributed
with covariance Rt and Qt respectively.
7
Bayes Filter Reminder
•Prediction
bel ( xt )   p( xt | ut , xt 1 ) bel ( xt 1 ) dxt 1
•Correction
bel ( xt )   p( zt | xt ) bel ( xt )
8
Kalman Filter Algorithm
1.
Algorithm Kalman_filter( t-1, t-1, ut, zt):
2.
3.
4.
Prediction:
 t  At t 1  Bt ut
5.
6.
7.
8.
Correction:
9.
Return t, t
t  At t 1 AtT  Rt
Kt  t CtT (Ct t CtT  Qt )1
t   t  Kt ( zt  Ct  t )
t  ( I  Kt Ct )t
9
Linear Gaussian Systems: Dynamics
• Dynamics are linear function of state and
xt  At xt 1  Bt ut   t
p( xt | ut , xt 1 )  N xt ; At xt 1  Bt ut , Rt 
bel ( xt )   p( xt | ut , xt 1 )

bel ( xt 1 ) dxt 1

~ N xt ; At xt 1  Bt ut , Rt  ~ N xt 1 ; t 1 ,  t 1 
10
Linear Gaussian Systems: Observations
• Observations are linear function of state
zt  Ct xt   t
p( zt | xt )  N zt ; Ct xt , Qt 
bel ( xt ) 

p( zt | xt )
bel ( xt )


~ N zt ; Ct xt , Qt 

~ N xt ;  t ,  t

11
Linear Gaussian Systems: Initialization
• Initial belief is normally distributed:
bel ( x0 )  N x0 ; 0 , 0 
12
13
14
 t  at t 1  bt ut
bel ( xt )   2
2 2
2


a



t t
act ,t
 t
 t  At t 1  Bt ut
bel ( xt )  
T


A

A
t t 1 t  Rt
 t
15
  t   t  K t ( zt   t )
bel ( xt )  
2
2


(
1

K
)

t
t
t

t  t  Kt ( zt  Ct t )
bel ( xt )  
 t  ( I  Kt Ct )t
with
with
 t2
Kt  2
2
 t   obs
,t
Kt  t CtT (Ct t CtT  Qt ) 1
16
Linear Gaussian Systems: Dynamics
bel ( xt )   p ( xt | ut , xt 1 )

bel ( xt 1 ) dxt 1

~ N  xt ; At xt 1  Bt ut , Rt  ~ N  xt 1 ; t 1 ,  t 1 

 1

bel ( xt )    exp  ( xt  At xt 1  Bt ut )T Rt1 ( xt  At xt 1  Bt ut )
 2

 1

T 1
exp  ( xt 1  t 1 )  t 1 ( xt 1  t 1 ) dxt 1
 2

 t  At t 1  Bt ut
bel ( xt )  
T


A

A
t
t t 1 t  Rt

17
Linear Gaussian Systems: Observations
bel ( xt ) 

p( zt | xt )
bel ( xt )


~ N zt ; Ct xt , Qt 

~ N xt ;  t ,  t


 1

 1

bel ( xt )   exp  ( zt  Ct xt )T Qt1 ( zt  Ct xt ) exp  ( xt  t )T t1 ( xt  t )
 2

 2

t  t  K t ( zt  Ct t )
bel ( xt )  
  t  ( I  K t Ct )  t
with
K t   t CtT (Ct  t CtT  Qt ) 1
18
The Prediction-Correction-Cycle
Prediction
 t  at t 1  bt ut
bel ( xt )   2
2
2
2
 t  at  t   act ,t
   At t 1  Bt ut
bel ( xt )   t
T
t  At  t 1 At  Rt
19
The Prediction-Correction-Cycle
     K t ( zt   t )
 t2
bel ( xt )   t 2 t
,
K

t
2
2
 t2   obs
  t  (1  K t ) t
,t
  t  K t ( zt  Ct t )
bel ( xt )   t
, K t  t CtT (Ct t CtT  Qt ) 1


(
I

K
C
)

t
t
t
t

Correction
20
The Prediction-Correction-Cycle
Prediction
     K t ( zt   t )
 t2
bel ( xt )   t 2 t
,
K

t
2
2
 t2   obs
  t  (1  K t ) t
,t
 t  at t 1  bt ut
bel ( xt )   2
2
2
2
 t  at  t   act ,t
  t  K t ( zt  Ct t )
bel ( xt )   t
, K t  t CtT (Ct t CtT  Qt ) 1


(
I

K
C
)

t
t
t
t

   At t 1  Bt ut
bel ( xt )   t
T
t  At  t 1 At  Rt
Correction
21
Kalman Filter Summary
• Highly efficient: Polynomial in
measurement dimensionality k and
state dimensionality n:
O(k2.376 + n2)
• Optimal for linear Gaussian systems!
• Most robotics systems are nonlinear!
22
Nonlinear Dynamic Systems
• Most realistic robotic problems involve
nonlinear functions
xt  g (ut , xt 1 )
zt  h( xt )
23
Linearity Assumption Revisited
24
Non-linear Function
25
EKF Linearization (1)
26
EKF Linearization (2)
27
EKF Linearization (3)
28
EKF Linearization: First Order
Taylor Series Expansion
• Prediction:
g (ut , t 1 )
g (ut , xt 1 )  g (ut , t 1 ) 
( xt 1  t 1 )
xt 1
g (ut , xt 1 )  g (ut , t 1 )  Gt ( xt 1  t 1 )
• Correction:
h( t )
h( xt )  h( t ) 
( xt  t )
xt
h( xt )  h( t )  H t ( xt  t )
29
EKF Algorithm
1. Extended_Kalman_filter( t-1, t-1, ut, zt):
2.
3.
4.
Prediction:
t  g (ut , t 1 )
 t  At t 1  Bt ut
t  Gt t 1GtT  Rt
t  At t 1 AtT  Rt
5.
6.
7.
8.
Correction:
Kt  t HtT ( Ht t HtT  Qt )1
t  t  Kt ( zt  h(t ))
9.
Return t, t
t  ( I  Kt H t )t
h( t )
Ht 
xt
Kt  t CtT (Ct t CtT  Qt )1
t   t  Kt ( zt  Ct  t )
t  ( I  Kt Ct )t
g (ut , t 1 )
Gt 
xt 1
30
EKF Summary
• Highly efficient: Polynomial in
measurement dimensionality k and
state dimensionality n:
O(k2.376 + n2)
• Not optimal!
• Can diverge if nonlinearities are large!
• Works surprisingly well even when all
assumptions are violated!
31
Unscented Transform
Sigma points
Weights
 
w 
0
i   
0
m

( n   )

i

n
wmi  wci 
w 
0
c
1
2(n   )

n
 (1   2   )
for i  1,...,2n
Pass sigma points through nonlinear function
 i  g ( i )
Recover mean and covariance
2n
 '   wmi  i
i 0
2n
'   wci ( i   )( i   )T
i 0
32
Linearization via Unscented
Transform
EKF
UKF
33
UKF Sigma-Point Estimate (2)
EKF
UKF
34
UKF Sigma-Point Estimate (3)
EKF
UKF
35
UKF Algorithm
36
UKF Summary
• Highly efficient: Same complexity as
EKF, with a constant factor slower in
typical practical applications
• Better linearization than EKF:
Accurate in first two terms of Taylor
expansion (EKF only first term)
• Derivative-free: No Jacobians needed
• Still not optimal!
37
Particle Filters


Represent belief by random samples

Monte Carlo filter, Survival of the fittest,
Condensation, Bootstrap filter, Particle filter



Filtering: [Rubin, 88], [Gordon et al., 93], [Kitagawa 96]
Estimation of non-Gaussian, nonlinear processes
Computer vision: [Isard and Blake 96, 98]
Dynamic Bayesian Networks: [Kanazawa et al., 95]d
38
Particle Filters
39
Particle Filter Algorithm
40
Resampling
• Given: Set S=(w1, w2, …, wM ) of weight
samples.
• Wanted : Random sample, where the
probability of drawing xi is given by wi.
• Typically done M times with replacement to
generate new sample set Xt.
41
Resampling Algorithm
42
Resampling
wn
Wn-1
wn
w1
w2
Wn-1
w3
• Roulette wheel
• Binary search, n log n
w1
w2
w3
• Stochastic universal sampling
• Systematic resampling
• Linear time complexity
• Easy to implement, low variance
43
Summary
• Particle filters are an implementation of
•
•
•
•
recursive Bayesian filtering
They represent the posterior by a set of
weighted samples.
In the context of localization, the particles
are propagated according to the motion
model.
They are then weighted according to the
likelihood of the observations.
In a re-sampling step, new particles are
drawn with a probability proportional to
the likelihood of the observation.
44
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