Download - 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
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Observationon-board sensorse
ption
raw sensor data orextracted features
.
3. Measurement prediction based on prediction and map
4. Matching of observation and map
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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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Markov Localization: Case Stud 1 - To olo ical Ma 2
Topological map of office-type environment
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Markov Localization: Case Stud 1 - To olo ical Ma 3
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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Markov Localization: Case Stud 1 - To olo ical Ma 4
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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Markov Localization: Case Stud 2 Grid Ma 2
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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Markov Localization: Case Stud 2 Grid Ma 3
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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Markov Localization: Case Stud 2 Grid Ma 4
Example 1: Office Building
Courtesy of
W. Burgard
Position 5
Position 3Position 4
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Markov Localization: Case Stud 2 Grid Ma 5
Example 2: MuseumCourtesy of
W. Burgard
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Markov Localization: Case Stud 2 Grid Ma 6
Example 2: MuseumCourtesy of
W. Burgard
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Markov Localization: Case Stud 2 Grid Ma 7
Example 2: MuseumCourtesy of
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