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Probabilistic Robotics. Bayes Filter Implementations Particle filters. Sample-based Localization (sonar). Particle Filters. Represent belief by random samples Estimation of non-Gaussian, nonlinear processes

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probabilistic robotics

Probabilistic Robotics

Bayes Filter Implementations

Particle filters

particle filters
Particle Filters
  • Represent belief by random samples
  • Estimation of non-Gaussian, nonlinear processes
  • Monte Carlo filter, Survival of the fittest, Condensation, Bootstrap filter, Particle filter
  • Filtering: [Rubin, 88], [Gordon et al., 93], [Kitagawa 96]
  • Computer vision: [Isard and Blake 96, 98]
  • Dynamic Bayesian Networks: [Kanazawa et al., 95]d
importance sampling
Importance Sampling

Weight samples: w = f / g

distributions1
Distributions

Wanted: samples distributed according to p(x| z1, z2, z3)

this is easy
This is Easy!

We can draw samples from p(x|zl) by adding noise to the detection parameters.

importance sampling with resampling1
Importance Sampling with Resampling

Weighted samples

After resampling

particle filter algorithm
Particle Filter Algorithm
  • Algorithm particle_filter( St-1, ut-1 zt):
  • For Generate new samples
  • Sample index j(i) from the discrete distribution given by wt-1
  • Sample from using and
  • Compute importance weight
  • Update normalization factor
  • Insert
  • For
  • Normalize weights
particle filter algorithm1

draw xit-1from Bel(xt-1)

draw xitfrom p(xt | xit-1,ut-1)

Importance factor for xit:

Particle Filter Algorithm
resampling
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’.
resampling1

w1

wn

w2

Wn-1

w1

wn

w2

Wn-1

w3

w3

Resampling
  • Stochastic universal sampling
  • Systematic resampling
  • Linear time complexity
  • Easy to implement, low variance
  • Roulette wheel
  • Binary search, n log n
resampling algorithm
Resampling Algorithm
  • Algorithm systematic_resampling(S,n):
  • For Generate cdf
  • Initialize threshold
  • For Draw samples …
  • While ( ) Skip until next threshold reached
  • Insert
  • Increment threshold
  • ReturnS’

Also called stochastic universal sampling

proximity sensor model reminder
Proximity Sensor Model Reminder

Sonar sensor

Laser sensor

under a light
Under a Light

Measurement z:

P(z|x):

next to a light
Next to a Light

Measurement z:

P(z|x):

elsewhere
Elsewhere

Measurement z:

P(z|x):

application rhino and albert synchronized in munich and bonn
Application: Rhino and Albert Synchronized in Munich and Bonn

[Robotics And Automation Magazine, to appear]

limitations
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)?
approaches
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).
random samples vision based localization
Random SamplesVision-Based Localization

936 Images, 4MB, .6secs/image

Trajectory of the robot:

resulting trajectories
Resulting Trajectories

Position tracking:

resulting trajectories1
Resulting Trajectories

Global localization:

summary
Summary
  • Particle filters are an implementation of recursive Bayesian filtering
  • They represent the posterior by a set of weighted samples.
  • 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.
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