introduction to mobile robotics bayes filter – particle filter and...
TRANSCRIPT
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Wolfram Burgard, Cyrill Stachniss,
Maren Bennewitz, Kai Arras
Bayes Filter – Particle Filter and Monte Carlo Localization
Introduction to Mobile Robotics
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§ Recall: Discrete filter § Discretize the continuous state space § High memory complexity § Fixed resolution (does not adapt to the belief)
§ Particle filters are a way to efficiently represent non-Gaussian distribution
§ Basic principle § Set of state hypotheses (“particles”) § Survival-of-the-fittest
Motivation
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Sample-based Localization (sonar)
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§ Set of weighted samples
Mathematical Description
§ The samples represent the posterior
State hypothesis Importance weight
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§ Particle sets can be used to approximate functions
Function Approximation
§ The more particles fall into an interval, the higher the probability of that interval
§ How to draw samples form a function/distribution?
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§ Let us assume that f(x)<1 for all x § Sample x from a uniform distribution § Sample c from [0,1] § if f(x) > c keep the sample
otherwise reject the sampe
Rejection Sampling
c
xf(x)
c’
x’
f(x’)
OK
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§ We can even use a different distribution g to generate samples from f
§ By introducing an importance weight w, we can account for the “differences between g and f ”
§ w = f / g § f is often called
target § g is often called
proposal § Pre-condition:
f(x)>0 à g(x)>0
Importance Sampling Principle
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Importance Sampling with Resampling: Landmark Detection Example
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Distributions
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Distributions
Wanted: samples distributed according to p(x| z1, z2, z3)
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This is Easy! We can draw samples from p(x|zl) by adding noise to the detection parameters.
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Importance Sampling
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Importance Sampling with Resampling
Weighted samples After resampling
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Particle Filters
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)|()(
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Sensor Information: Importance Sampling
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∫←− 'd)'()'|()( , xxBelxuxpxBel
Robot Motion
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)|()(
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Sensor Information: Importance Sampling
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Robot Motion
∫←− 'd)'()'|()( , xxBelxuxpxBel
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Particle Filter Algorithm
§ Sample the next generation for particles using the proposal distribution
§ Compute the importance weights : weight = target distribution / proposal distribution
§ Resampling: “Replace unlikely samples by more likely ones”
§ [Derivation of the MCL equations on the blackboard]
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1. Algorithm particle_filter( St-1, ut-1 zt): 2.
3. For Generate new samples 4. Sample index j(i) from the discrete distribution given by wt-1
5. Sample from using and
6. Compute importance weight 7. Update normalization factor
8. Insert 9. For
10. Normalize weights
Particle Filter Algorithm
0, =∅= ηtSni …1=
},{ ><∪= it
ittt wxSS
itw+=ηη
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)|( itt
it xzpw =
ni …1=
η/itit ww =
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draw xit-1 from Bel(xt-1)
draw xit from p(xt | xi
t-1,ut-1)
Importance factor for xit:
)|()(),|(
)(),|()|(ondistributi proposal
ondistributitarget
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Particle Filter Algorithm
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Resampling
§ Given: Set S of weighted samples.
§ Wanted : Random sample, where the probability of drawing xi is given by wi.
§ Typically done n times with replacement to generate new sample set S’.
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w2
w3
w1 wn
Wn-1
Resampling
w2
w3
w1 wn
Wn-1
§ Roulette wheel § Binary search, n log n
§ Stochastic universal sampling § Systematic resampling
§ Linear time complexity § Easy to implement, low variance
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1. Algorithm systematic_resampling(S,n):
2. 3. For Generate cdf 4. 5. Initialize threshold
6. For Draw samples … 7. While ( ) Skip until next threshold reached 8. 9. Insert 10. Increment threshold
11. Return S’
Resampling Algorithm
11,' wcS =∅=
ni …2=i
ii wcc += −1
1],,0]~ 11 =− inUu
nj …1=
11
−+ += nuu jj
ij cu >
{ }><∪= −1,'' nxSS i1+= ii
Also called stochastic universal sampling
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Mobile Robot Localization
§ Each particle is a potential pose of the robot
§ Proposal distribution is the motion model of the robot (prediction step)
§ The observation model is used to compute the importance weight (correction step)
[For details, see PDF file on the lecture web page]
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Start
Motion Model Reminder
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Proximity Sensor Model Reminder
Laser sensor Sonar sensor
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Sample-based Localization (sonar)
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Initial Distribution
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After Incorporating Ten Ultrasound Scans
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After Incorporating 65 Ultrasound Scans
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Estimated Path
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Localization for AIBO robots
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Using Ceiling Maps for Localization
[Dellaert et al. 99]
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Vision-based Localization
P(z|x)
h(x)
z
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Under a Light
Measurement z: P(z|x):
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Next to a Light
Measurement z: P(z|x):
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Elsewhere
Measurement z: P(z|x):
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Global Localization Using Vision
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Limitations
§ The approach described so far is able to § track the pose of a mobile robot and to § globally localize the robot.
§ How can we deal with localization errors (i.e., the kidnapped robot problem)?
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Approaches
§ Randomly insert samples (the robot can be teleported at any point in time).
§ Insert random samples proportional to the average likelihood of the particles (the robot has been teleported with higher probability when the likelihood of its observations drops).
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Summary – Particle Filters
§ Particle filters are an implementation of recursive Bayesian filtering
§ They represent the posterior by a set of weighted samples
§ They can model non-Gaussian distributions § Proposal to draw new samples § Weight to account for the differences
between the proposal and the target § Monte Carlo filter, Survival of the fittest,
Condensation, Bootstrap filter
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Summary – PF Localization
§ 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.