lecture 8 map building 1

8/14/2019 Lecture 8 Map Building 1

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CpE 521

Part 2

Yan Meng

De artment of Electrical and Com uter En ineerin

Stevens Institute of Technology


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Toda s Content

Probabilistic map-based localization

Kalman filter localization

Other examples of localization system

Landmark-based localization

Position beacon systems Autonomous map building

Stochastic map technique


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General robot localization roblem and solution strate

Consider a mobile robot moving in a known environment.

As it starts to move, say from a precisely known location, it might

keep track of its location using odometry.

However, after a certain movement, the robot will get very uncertainabout its position.

update using an observation of its environment.

Observation leads to an estimate of the robots position which can be

fused with the odometric estimation to get the best possible update of

the robots actual position.


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Two-ste robot osition u date

Action update

with ot: encoder measurement, s

t-1: prior belief state

increases uncertainty

Perception update

perception model SEE

t

,t

decreases uncertainty


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Probabilistic Ma -Based Localization Problem Statement

Given

its covariance for time k,

the current control input

)( kkp)(ku

the current set of observations and

the map

)1( +kZ

)(kM

Compute the

its covariance )11( ++ kkp

Such a procedure usually involves five steps:


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The Five Ste s for Ma -Based LocalizationPrediction of

Measurement andEstimation

(fusion)positionestimate

Encoder

Ma

eature

ions matched predictions

and observationsdata base

redicted

observa

YES

Matching

1. Prediction based on previous estimate and odometry

-

Observationon-board sensorse

ption

raw sensor data orextracted features

.

3. Measurement prediction based on prediction and map

4. Matching of observation and map

-

Per

.


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Markov

Kalman Filter Localization

Markov localization Kalman filter localization

localization starting from anyunknown position

tracks the robot and is inherentlyvery precise and efficient.

situation.

However, to update the probability

,

robot becomes to large (e.g.

collision with an object) theKalman ilter will ail and the

state space at any time requires a

discrete representation of the

s ace rid . The re uired memor

position is definitively lost.

and calculation power can thus

become very important if a fine

grid is used.


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Markov Localization

Markov localization uses an explicit, discrete representation for the

.

This is usually done by representing the environment by a grid or a

topological graph with a finite number of possible states (positions).

During each update, the probability for each state (element) of the

entire space is updated.


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Probabilit theor

P(A): Probability that A is true.

. . t

We wish to compute the probability of each individual robot positiongiven actions and sensor measures.

P(A|B): Conditional probability of A given that we know B.

e.g. p(rt= l| it): probability that the robot is at position l given thet.

Product rule:

Bayes rule:


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Markov Localization

Bayes rule:

a rom a belie state and a sensor in ut to a re ined belie state SEE :

p(l): belief state before perceptual update process

o consult robots map, identify the probability of a certain sensor reading for each

possible position in the map

.


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Markov Localization

Bayes rule:

a rom a belie state and an action to new belie state ACT :

Summing over all possible ways in which the robot may have reached l.

Markov assumption: update only depends on previous state and its

most recent actions and perception.


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Markov Localization: Case Stud 1 - To olo ical Ma 1

1994 AAAI National Robot Contest: winner Dervish Robot

,

Topological Localization with Sonar


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Topological map of office-type environment


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Update of believe state for position n given the percept-pair i

p(ni): new likelihood for being in position n

p(n): current believe state

p(in): probability of seeing i in n (see table)

No action update !However, the robot is moving and therefore we can apply a combination

of action and perception update

t-i is used instead of t-1 because the topological distance between n and

n can very epen ng on e spec c opo og ca map


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The calculation is performed by multiplying the

probability of having failed to generate perceptual events at all nodesbetween n and n.


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Markov Localization: Case Stud 1 - To olo ical Ma 5 Example calculation Assume that the robot has two nonzero belie states

o p(1-2) = 1.0 ; p(2-3) = 0.2 *at that it is facing east with certainty

State 2-3 will progress potentially to 3 and3-4 to 4.

State 3 and3-4 can be eliminated because the likelihood of detecting an open dooris zero.

The likelihood of reaching state 4 is the product of the initial likelihood p(2-3)= 0.2,(a) the likelihood of not detecting anything at node 3; and (b) the likelihood ofdetecting a hallway on the left and a door on the right at node 4. (for simplicity we

assume t at t e e oo o etect ng not ng at no e - s . ) This leads to:

o 0.2 [0.60.4 + 0.4 0.05] 0.7[0.9 0.1] p(4) = 0.003.

o m ar ca cu at on or progress rom - p = . .

* Note that the probabilities do not sum up to one. For simplicity normalization was avoided in this example


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Markov Localization: Case Stud 2 Grid Ma 1

Finefixed decomposition grid (x,y, ), 15 cm x 15 cm x 1

Action update:

Sum over previous possible positions

and motion model

Discrete version of eq. 5.22

Perce tion u date:

Given perception i, what is the

probability to be at location l

W. Burgard


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The critical challenge is the calculation ofp(il)

p(il) is computed using a model of the robots sensor behavior, its position l, andthe local environment metric map around l.

ssump ons

o Measurement error can be described by a distribution with a mean

o Non-zero chance for any measurement

W. Burgard


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The 1D case

.

No knowledge at start, thus we havean uniform probability distribution.

.

Seeing only one pillar, the probability

being at pillar 1, 2 or 3 is equal.

3.Robot moves

Action model enables to estimate the

on the previous one and the motion.

4.Robot perceives second pillar

probability being at pillar 2 becomes

dominant


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Example 1: Office Building

Courtesy of

W. Burgard

Position 5

Position 3Position 4


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Example 2: MuseumCourtesy of

W. Burgard


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W. Burgard


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W. Burgard


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Markov Localization: Case Stud 2 Grid Ma 10 Finefixed decomposition grids result in a huge state space

Ver extensive rocessin ower needed

Large memory requirement

Reducing complexity

Various approac e ave een propose or re ucing comp exity

The main goal is to reduce the number of states that are updated in each

step Randomized Sampling / Particle Filter /Monte Carlo Algorithm

Approximated belief state by representing only a representative subseto all states (possible locations)

E.g update only 10% of all possible locations

The sampling process is typically weighted, e.g. put more samples

However, you have to ensure some less likely locations are still tracked,

otherwise the robot might get lost

lecture 8 map building 1

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