Title: Probabilistic Robotics
1Probabilistic Robotics
Bayes Filter Implementations Particle filters
2Sample-based Localization (sonar)
3Particle 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., 95d
4Importance Sampling
Weight samples w f / g
5Importance Sampling with ResamplingLandmark
Detection Example
6Distributions
7Distributions
Wanted samples distributed according to p(x z1,
z2, z3)
8This is Easy!
We can draw samples from p(xzl) by adding noise
to the detection parameters.
9Importance Sampling with Resampling
10Importance Sampling with Resampling
Weighted samples
After resampling
11Particle Filters
12Sensor Information Importance Sampling
13Robot Motion
14Sensor Information Importance Sampling
15Robot Motion
16Particle 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
17Particle Filter Algorithm
18Resampling
- 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.
19Resampling
- Stochastic universal sampling
- Systematic resampling
- Linear time complexity
- Easy to implement, low variance
- Roulette wheel
- Binary search, n log n
20Resampling Algorithm
- Algorithm systematic_resampling(S,n)
-
- For Generate cdf
-
- Initialize threshold
- For Draw samples
- While ( ) Skip until next threshold
reached -
- Insert
-
Increment threshold - Return S
Also called stochastic universal sampling
21Motion Model Reminder
Start
22Proximity Sensor Model Reminder
Sonar sensor
Laser sensor
23(No Transcript)
24(No Transcript)
25(No Transcript)
26(No Transcript)
27(No Transcript)
28(No Transcript)
29(No Transcript)
30(No Transcript)
31(No Transcript)
32(No Transcript)
33(No Transcript)
34(No Transcript)
35(No Transcript)
36(No Transcript)
37(No Transcript)
38(No Transcript)
39(No Transcript)
40(No Transcript)
41(No Transcript)
42Sample-based Localization (sonar)
43Initial Distribution
44After Incorporating Ten Ultrasound Scans
45After Incorporating 65 Ultrasound Scans
46Estimated Path
47Using Ceiling Maps for Localization
48Vision-based Localization
49Under a Light
Measurement z
P(zx)
50Next to a Light
Measurement z
P(zx)
51Elsewhere
Measurement z
P(zx)
52Global Localization Using Vision
53Robots in Action Albert
54Application Rhino and Albert Synchronized in
Munich and Bonn
Robotics And Automation Magazine, to appear
55Localization for AIBO robots
56Limitations
- 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)?
57Approaches
- 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).
58Random SamplesVision-Based Localization
- 936 Images, 4MB, .6secs/image
- Trajectory of the robot
59Odometry Information
60Image Sequence
61Resulting Trajectories
Position tracking
62Resulting Trajectories
Global localization
63Global Localization
64Kidnapping the Robot
65Recovery from Failure
66Summary
- 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.