Probabilistic Robotics

Bayes Filter Implementations

•Prediction

•Correction

Bayes Filter Reminder

111 )(),|()( tttttt dxxbelxuxpxbel

)()|()( tttt xbelxzpxbel

Gaussians

2

2)(

2

1

2

2

2

1)(

:),(~)(

x

exp

Nxp

-

Univariate

)()(2

1

2/12/

1

)2(

1)(

:)(~)(

μxΣμx

Σx

Σμx

t

ep

,Νp

d

Multivariate

),(~),(~ 22

2

abaNYbaXY

NX

Properties of Gaussians

22

21

222

21

21

122

21

22

212222

2111 1

,~)()(),(~

),(~

NXpXpNX

NX

• We stay in the “Gaussian world” as long as we start with Gaussians and perform only linear transformations.

),(~),(~ TAABANY

BAXY

NX

Multivariate Gaussians

12

11

221

11

21

221

222

111 1,~)()(

),(~

),(~

NXpXpNX

NX

6

Discrete 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

7

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 (nxm) 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.

8

•Prediction

•Correction

Bayes Filter Reminder

111 )(),|()( tttttt dxxbelxuxpxbel

)()|()( tttt xbelxzpxbel

9

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 )(

10

• 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

11

• Observations are linear function of state plus additive noise:

tttt xCz

Linear Gaussian Systems: Observations

tttttt QxCzNxzp ,;)|(

ttttttt

tttt

xNQxCzN

xbelxzpxbel

,;~,;~

)()|()(

12

0000 ,;)( xNxbel

Linear Gaussian Systems: Initialization

• Initial belief is normally distributed:

13

Kalman Filter Updates in 1D

14

Kalman Filter Updates

15

Kalman Filter Updates in 1D

tTtttt

tttttt RAA

uBAxbel

1

1)(

2

,2221)(

tactttt

tttttt a

ubaxbel

16

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

17

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)(

,;~,;~

)(),|()(

18

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

19

The Prediction-Correction-Cycle

tTtttt

tttttt RAA

uBAxbel

1

1)(

2

,2221)(

tactttt

tttttt a

ubaxbel

Prediction

20

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

21

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

22

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!

23

Nonlinear Dynamic Systems

•Most realistic robotic problems involve nonlinear functions

),( 1 ttt xugx

)( tt xhz

24

Linearity Assumption Revisited

25

Non-linear Function

26

EKF Linearization (1)

27

EKF Linearization (2)

28

EKF Linearization (3)

29

•Prediction:

•Correction:

EKF Linearization: First Order Taylor Series Expansion

)(),(),(

)(),(

),(),(

1111

111

111

ttttttt

ttt

tttttt

xGugxug

xx

ugugxug

)()()(

)()(

)()(

ttttt

ttt

ttt

xHhxh

xx

hhxh

30

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 )(

31

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!

32

Unscented Transform

nin

wwn

nw

nw

ic

imi

i

cm

2,...,1for )(2

1 )(

)1( 2000

Sigma points Weights

)( ii g

n

i

Tiiic

n

i

iim

w

w

2

0

2

0

))(('

'

Pass sigma points through nonlinear function

Recover mean and covariance

33

Linearization via Unscented Transform

EKF UKF

34

UKF Sigma-Point Estimate (2)

EKF UKF

35

UKF Sigma-Point Estimate (3)

EKF UKF

36

UKF Algorithm

37

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!

38

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]d

Particle Filters

39

Particle Filters

40

Particle Filter Algorithm

41

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.

42

Resampling Algorithm

43

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

44

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.