L ECTURE 25: B AYESIAN F ILTERS M ONTE C ARLO L OCALIZATION (PF) I - - PowerPoint PPT Presentation

16-311-Q INTRODUCTION TO ROBOTICS FALL’17

LECTURE 25:

BAYESIAN FILTERS MONTE CARLO LOCALIZATION (PF)

INSTRUCTOR: GIANNI A. DI CARO

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P R O B A B I L I S T I C I N F E R E N C E

Measurement noise Process noise

Localization is an instance of the more general problem of state estimation in a noisy (feedback- based) controlled system

• Probabilistic inference is the problem of estimating the hidden variables (states or

parameters) of a system in an optimal and consistent fashion (using probability theory), given noisy or incomplete observations.

Given z, what can we infer about x?

For a robot typically the system evolves over time → Sequential probabilistic inference: Estimate xk given z1:k and information about system’s dynamics and about how

• bservations are obtained

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P R O B A B I L I S T I C I N F E R E N C E

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M A R K O V A S S U M P T I O N

State is a sufficient statistics Static world Independent noise

)

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B AY E S I A N F I LT E R

Use Bayes rule for performing Prediction and Correction updates Posterior probability is called the belief

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N O N - PA R A M E T R I C V S . G A U S S I A N F I LT E R S

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B AY E S F O R M U L A

evidence prior likelihood ) ( ) ( ) | ( ) ( ) ( ) | ( ) ( ) | ( ) , ( ⋅ = = ⇒ = = y P x P x y P y x P x P x y P y P y x P y x P

) ( ) | ( 1 ) ( ) ( ) | ( ) ( ) ( ) | ( ) (

1

x P x y P y P x P x y P y P x P x y P y x P

x

∑

= = = =

−

η η Normalization

y x x y x y x

y x P x x P x y P x

| | |

aux ) | ( : aux 1 ) ( ) | ( aux : η η = ∀ = = ∀

∑

Algorithm

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T O TA L P R O B A B I L I T Y A N D C O N D I T I O N I N G

• Total probability:

∫ ∫ ∫

= = = dz z P z y x P y x P dz z P z x P x P dz z x P x P ) ( ) , | ( ) ( ) ( ) | ( ) ( ) , ( ) (

) | ( ) | ( ) , ( z y P z x P z y x P = ) , | ( ) ( y z x P z x P = ) , | ( ) ( x z y P z y P =

equivalent to and

• Conditional independence

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B AY E S F I LT E R S

• Given:
• Stream of observations z and action data u:
• Sensor model P(z|x).
• Action model P(x|u,x’).
• Prior probability of the system state P(x).
• Wanted:
• Estimate of the state X of a dynamical system.
• The posterior of the state is also called Belief:

) , , , | ( ) (

1 1 t t t t

z u z u x P x Bel ! = } , , , {

1 1 t t t

z u z u d ! =

) , | ( ) , , | (

1 : 1 : 1 1 : 1 t t t t t t t

u x x p u z x x p

− −

=

) | ( ) , , | (

: 1 : 1 : t t t t t t

x z p u z x z p =

Markov assumption:

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N E X T …

Follow up on the slides from Cyrill Stachniss (check course website):

• Bayes filters
• Particle filter