Mobile Robotics II: Simultaneous localization and...

laboratory

Gerstner

Probability State estimation

Mobile Robotics II: Simultaneous localization andmapping

Introduction: probability theory, estimation

Miroslav Kulich

Intelligent and Mobile Robotics GroupGerstner Laboratory for Intelligent Decision Making and Control

Czech Technical University in Prague

Monday 23/07/2012


Course outline

• Introduction to probability and estimation

• Bayes filter and its implementations

• Bayes filter based localization

• Probabilistic approach to mapping

• Simultaneous localization and mapping

• EKF-SLAM

• Fast-SLAM

• Labs


Something to wake up...

Suppose a road between two pubs and a bus stop exactly at a halfway. A man goes every day at a random time to the bus stop,waits for the first bus and goes to the pub. The buses start atregular intervals (the same) from each pub. After one year, theman noticed that he drinks Pilsen twice frequently than Budweiser.How is it possible?

Budweiser bus stop Pilsen

house


Gentle introduction to probability theory

• Key idea: explicit representation of uncertainty using thecalculus of probability theory

• p(X=x) probability that the random variable X has the value x

• 0 ≤ p(x) ≤ 1

• p(true) = 1, p(false) = 0

• p(A ∨ B) = p(A) + p(B)− p(A ∧ B)

What is the probability of throwing an even number or a numberlower than five?


Discrete and continuous random variable

• Discrete: X is finite, i.e.X = x1, x2, . . . , xn

• p is called probability mass function

• Continuous: X takes on values in thecontinuum

• p is called probability density function

• Several distributions

• Mostly known: Normal distribution(Gaussian)

• p(x) = 1√2πσ

e−(x−µ)2

2σ2


Multivariete normal distribution

p(x) =1√

2π|Σ|e−

12

(x−µ)T Σ−1(x−µ)

• Eigenvectors and eigenvalues of covariance matrix determineelipses.


Joint and conditional probability

• p(X = x and Y = y) = p(x , y)

• If X and Y are independent then

p(x , y) = p(x)p(y)

• p(x |y) is the probability of x given y

p(x |y) = p(x , y)/p(y)

p(x , y) = p(x |y)/p(y)

• If X and Y are independent then

p(x |y) = p(x)


Law of Total probability, Marginals

Discrete case∑x

p(x) = 1

p(x) =∑y

p(x , y)

p(x) =∑y

p(x |y)p(y)

Continuous case∫xp(x)dx = 1

p(x) =

∫yp(x , y)dy

p(x) =

∫yp(x |y)p(y)dy


Exercise

A robot uses a range sensor that can measure ranges from 0m to3m. For simplicity, assume that actual ranges are distributeduniformly in this interval. Unfortunatelly, the sensor can be faulty.When the sensor is faulty, it constantly outputs a range bellow 1m,regardless of the actual range in the sensor’s measurements cone.We know that the prior probability for a sensor to be faulty isp = 0.01.Suppose the robot queried its sensor N times, and every single timthe measurement value is bellow 1m. What is the posteriorprobability of a sensor fault, for N = 1, 2, . . . . Formulate thecorresponding probabilistic model.


Simple example of state estimation

• Suppose a robot obtains measurement z

• What is p(open|z)?

• p(open|z) is diagnostic

• p(z |open) is causal

• Often causal knowledge is easier to obtain (countingfrequencies)

• Bayes rule allows us to use causal:

p(open|z) =p(z |open)p(open)

p(z)


Actions

• Often the world is dynamic since• actions carried out by the robot,• actions carried out by other agents,• or just the time passing by change the world (plants grow).

• Actions are never carried out with absolute certainty.

• In contrast to measurements, actions generally decrease theuncertainty.

• To incorporate the outcome of an action u into the current“belief”, we use the conditional pdf

p(x |u, x ′)

• This term specifies the pdf that executing u changes the statefrom x ′ to x .


Continuing the example - closing the door

p(x |u, x ′) for u = ”close door”

CLOSEDOPEN

0.9

0

10.1

p(x , u) =∑x ′

p(x |u, x ′)p(x ′)

If the door is open, the action ”close door” succeeds in 90% of allcases.



p(closed |u) =∑x ′

p(closed |u, x ′)p(x ′)

= p(closed |u, open)p(open)

+ p(closed |u, closed)p(closed)

= ??


ExerciseSuppose we live at a place where days are either sunny, cloudy, or rainy.The weather transition function is a Markov chain with the followingtransition table:

tomorrow will be . . .sunny cloudy rainy

today it’s . . .sunny 0.8 0.2 0cloudy 0.4 0.4 0.2rainy 0.2 0.6 0.2

• Draw state transition diagram.

• Suppose Day 1 is a sunny day. What is the probability if thefollowing sequence of days: Day 2=cloudy , Day 3=cloudy , Day4=rainy?

• What is the probability that a random day will be sunny, cloudy, orrainy?

• Compute the probability table of yesterday’s weather given today’sweather.


Exercise

Suppose that we cannot observe the weather directly, but insteadrely on sensor. The problem is that our sensor is noisy. Itsmeasurements are governed by the following measurement model:

our sensor tells us . . .sunny cloudy rainy

the actual weather is . . .sunny 0.6 0.4 0cloudy 0.3 0.7 0rainy 0 0 1

• Suppose Day 1 is sunny, and the subsequent four days oursensor observes cloudy , cloudy , rainy , sunny . What is theprobability that Day 5 is indeed sunny as predicted by oursensor?

Mobile Robotics II: Simultaneous localization and...

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