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Transcript of HCI / CprE / ComS 575: Computational Perception Instructor: Alexander Stoytchev...
![Page 1: HCI / CprE / ComS 575: Computational Perception Instructor: Alexander Stoytchev alexs/classes/2010_Spring_575](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649db05503460f94a9e7c4/html5/thumbnails/1.jpg)
HCI / CprE / ComS 575:
Computational Perception
Instructor: Alexander Stoytchevhttp://www.ece.iastate.edu/~alexs/classes/2010_Spring_575/
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Particle Filters
HCI/ComS 575X: Computational PerceptionIowa State UniversityCopyright © Alexander Stoytchev
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Sebastian Thrun, Wolfram Burgard and Dieter Fox (2005).
Probabilistic Robotics
MIT Press.
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F. Dellaert, D. Fox, W. Burgard, and S. Thrun (1999).
"Monte Carlo Localization for Mobile Robots", IEEE International Conference on Robotics
and Automation (ICRA99), May, 1999.
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Ioannis Rekleitis (2004).
A Particle Filter Tutorial for Mobile Robot Localization.
Technical Report TR-CIM-04-02, Centre for Intelligent Machines, McGill University,
Montreal, Quebec, Canada.
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Wednesday
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Next Week
• Preliminary Project Presentatons
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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
Particle Filters
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Example
![Page 10: HCI / CprE / ComS 575: Computational Perception Instructor: Alexander Stoytchev alexs/classes/2010_Spring_575](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649db05503460f94a9e7c4/html5/thumbnails/10.jpg)
Using Ceiling Maps for Localization
![Page 11: HCI / CprE / ComS 575: Computational Perception Instructor: Alexander Stoytchev alexs/classes/2010_Spring_575](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649db05503460f94a9e7c4/html5/thumbnails/11.jpg)
Vision-based Localization
P(z|x)
h(x)z
![Page 12: HCI / CprE / ComS 575: Computational Perception Instructor: Alexander Stoytchev alexs/classes/2010_Spring_575](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649db05503460f94a9e7c4/html5/thumbnails/12.jpg)
Under a LightMeasurement z: P(z|x):
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Next to a LightMeasurement z: P(z|x):
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ElsewhereMeasurement z: P(z|x):
![Page 15: HCI / CprE / ComS 575: Computational Perception Instructor: Alexander Stoytchev alexs/classes/2010_Spring_575](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649db05503460f94a9e7c4/html5/thumbnails/15.jpg)
Global Localization Using Vision
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Sample-based Localization (sonar)
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Example
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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 with Resampling
Weighted samples After resampling
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Quick review of Kalman Filters
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Conditional density of position based on measured value of z1
[Maybeck (1979)]
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Conditional density of position based on measured value of z1
[Maybeck (1979)]
position
measured position
uncertainty
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Conditional density of position based on measurement of z2 alone
[Maybeck (1979)]
![Page 28: HCI / CprE / ComS 575: Computational Perception Instructor: Alexander Stoytchev alexs/classes/2010_Spring_575](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649db05503460f94a9e7c4/html5/thumbnails/28.jpg)
Conditional density of position based on measurement of z2 alone
[Maybeck (1979)]measured position 2
uncertainty 2
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Conditional density of position based on data z1 and z2
[Maybeck (1979)]position estimate
uncertainty estimate
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Propagation of the conditional density
[Maybeck (1979)]
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Propagation of the conditional density
[Maybeck (1979)]
movement vector
expected position just prior to taking measurement 3
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Propagation of the conditional density
[Maybeck (1979)]
movement vector
expected position just prior to taking measurement 3
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Propagation of the conditional density
z3
σx(t3)
measured position 3
uncertainty 3
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Updating the conditional density after the third measurement
z3
σx(t3)
position uncertainty
position estimate
x(t3)
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Some Questions
• What if we don’t know the start position of the robot?
• What if somebody moves the robot without the robot’s knowledge?
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Robot Odometry Errors
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Raw range data, position indexed by odometry
[Thrun, Burgard & Fox (2005)]
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Resulting Occupancy Grid Map
[Thrun, Burgard & Fox (2005)]
![Page 40: HCI / CprE / ComS 575: Computational Perception Instructor: Alexander Stoytchev alexs/classes/2010_Spring_575](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649db05503460f94a9e7c4/html5/thumbnails/40.jpg)
Basic Idea Behind Particle Filters
x
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In 2D it looks like this
[http://www.ite.uni-karlsruhe.de/METZGER/DIPLOMARBEITEN/dipl2.html]
![Page 42: HCI / CprE / ComS 575: Computational Perception Instructor: Alexander Stoytchev alexs/classes/2010_Spring_575](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649db05503460f94a9e7c4/html5/thumbnails/42.jpg)
Robot Pose
![Page 43: HCI / CprE / ComS 575: Computational Perception Instructor: Alexander Stoytchev alexs/classes/2010_Spring_575](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649db05503460f94a9e7c4/html5/thumbnails/43.jpg)
Odometry Motion Model
![Page 44: HCI / CprE / ComS 575: Computational Perception Instructor: Alexander Stoytchev alexs/classes/2010_Spring_575](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649db05503460f94a9e7c4/html5/thumbnails/44.jpg)
Sampling From the Odometry Model
![Page 45: HCI / CprE / ComS 575: Computational Perception Instructor: Alexander Stoytchev alexs/classes/2010_Spring_575](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649db05503460f94a9e7c4/html5/thumbnails/45.jpg)
Motion Model
![Page 46: HCI / CprE / ComS 575: Computational Perception Instructor: Alexander Stoytchev alexs/classes/2010_Spring_575](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649db05503460f94a9e7c4/html5/thumbnails/46.jpg)
Motion Model
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Velocity model for different noise parameters
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Sampling from the velocity model
![Page 49: HCI / CprE / ComS 575: Computational Perception Instructor: Alexander Stoytchev alexs/classes/2010_Spring_575](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649db05503460f94a9e7c4/html5/thumbnails/49.jpg)
In Class Demoof Particle Filters
![Page 50: HCI / CprE / ComS 575: Computational Perception Instructor: Alexander Stoytchev alexs/classes/2010_Spring_575](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649db05503460f94a9e7c4/html5/thumbnails/50.jpg)
Example
[Thrun, Burgard & Fox (2005)]
![Page 51: HCI / CprE / ComS 575: Computational Perception Instructor: Alexander Stoytchev alexs/classes/2010_Spring_575](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649db05503460f94a9e7c4/html5/thumbnails/51.jpg)
Initially we don’t know the location of the robot so we have particles everywhere
![Page 52: HCI / CprE / ComS 575: Computational Perception Instructor: Alexander Stoytchev alexs/classes/2010_Spring_575](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649db05503460f94a9e7c4/html5/thumbnails/52.jpg)
Next, the robot senses that it is near a door
![Page 53: HCI / CprE / ComS 575: Computational Perception Instructor: Alexander Stoytchev alexs/classes/2010_Spring_575](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649db05503460f94a9e7c4/html5/thumbnails/53.jpg)
Since there are 3 identical doors the robot can be next any one of them
![Page 54: HCI / CprE / ComS 575: Computational Perception Instructor: Alexander Stoytchev alexs/classes/2010_Spring_575](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649db05503460f94a9e7c4/html5/thumbnails/54.jpg)
Therefore, we inflate balls (particles) that are next to doors and shrink all others
![Page 55: HCI / CprE / ComS 575: Computational Perception Instructor: Alexander Stoytchev alexs/classes/2010_Spring_575](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649db05503460f94a9e7c4/html5/thumbnails/55.jpg)
Therefore, we grow balls (particles) that are next to doors and shrink all others
![Page 56: HCI / CprE / ComS 575: Computational Perception Instructor: Alexander Stoytchev alexs/classes/2010_Spring_575](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649db05503460f94a9e7c4/html5/thumbnails/56.jpg)
Before we continue we have to make all ball to be of equal size. We need to resample.
![Page 57: HCI / CprE / ComS 575: Computational Perception Instructor: Alexander Stoytchev alexs/classes/2010_Spring_575](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649db05503460f94a9e7c4/html5/thumbnails/57.jpg)
Before we continue we have to make all ball to be of equal size. We need to resample.
![Page 58: HCI / CprE / ComS 575: Computational Perception Instructor: Alexander Stoytchev alexs/classes/2010_Spring_575](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649db05503460f94a9e7c4/html5/thumbnails/58.jpg)
Resampling Rules
=
=
=
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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’.
[From Thrun’s book “Probabilistik Robotics”]
![Page 60: HCI / CprE / ComS 575: Computational Perception Instructor: Alexander Stoytchev alexs/classes/2010_Spring_575](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649db05503460f94a9e7c4/html5/thumbnails/60.jpg)
w2
w3
w1wn
Wn-1
Roulette wheel Resampling
w2
w3
w1wn
Wn-1
• Roulette wheel
• Binary search, n log n
• Stochastic universal sampling
• Systematic resampling
• Linear time complexity
• Easy to implement, low variance
[From Thrun’s book “Probabilistik Robotics”]
![Page 61: HCI / CprE / ComS 575: Computational Perception Instructor: Alexander Stoytchev alexs/classes/2010_Spring_575](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649db05503460f94a9e7c4/html5/thumbnails/61.jpg)
1. Algorithm systematic_resampling(S,n):
2.
3. For Generate cdf4. 5. Initialize threshold
6. For Draw samples …7. While ( ) Skip until next threshold reached8. 9. Insert10. Increment threshold
11. Return S’
Resampling Algorithm
11,' wcS
ni 2i
ii wcc 1
1],,0]~ 11 inUu
nj 1
11
nuu jj
ij cu
1,'' nxSS i
1ii
Also called stochastic universal sampling
[From Thrun’s book “Probabilistik Robotics”]
![Page 62: HCI / CprE / ComS 575: Computational Perception Instructor: Alexander Stoytchev alexs/classes/2010_Spring_575](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649db05503460f94a9e7c4/html5/thumbnails/62.jpg)
Next, The robot moves to the right
![Page 63: HCI / CprE / ComS 575: Computational Perception Instructor: Alexander Stoytchev alexs/classes/2010_Spring_575](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649db05503460f94a9e7c4/html5/thumbnails/63.jpg)
… thus, we have to shift all balls (particles) to the right
![Page 64: HCI / CprE / ComS 575: Computational Perception Instructor: Alexander Stoytchev alexs/classes/2010_Spring_575](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649db05503460f94a9e7c4/html5/thumbnails/64.jpg)
… thus, we have to shift all balls (particles) to the right
![Page 65: HCI / CprE / ComS 575: Computational Perception Instructor: Alexander Stoytchev alexs/classes/2010_Spring_575](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649db05503460f94a9e7c4/html5/thumbnails/65.jpg)
… and add some position noise
![Page 66: HCI / CprE / ComS 575: Computational Perception Instructor: Alexander Stoytchev alexs/classes/2010_Spring_575](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649db05503460f94a9e7c4/html5/thumbnails/66.jpg)
… and add some position noise
![Page 67: HCI / CprE / ComS 575: Computational Perception Instructor: Alexander Stoytchev alexs/classes/2010_Spring_575](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649db05503460f94a9e7c4/html5/thumbnails/67.jpg)
Next, the robot senses that it is next to one of the three doors
![Page 68: HCI / CprE / ComS 575: Computational Perception Instructor: Alexander Stoytchev alexs/classes/2010_Spring_575](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649db05503460f94a9e7c4/html5/thumbnails/68.jpg)
Next, the robot senses that it is next to one of the three doors
![Page 69: HCI / CprE / ComS 575: Computational Perception Instructor: Alexander Stoytchev alexs/classes/2010_Spring_575](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649db05503460f94a9e7c4/html5/thumbnails/69.jpg)
Now we have to resample again
![Page 70: HCI / CprE / ComS 575: Computational Perception Instructor: Alexander Stoytchev alexs/classes/2010_Spring_575](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649db05503460f94a9e7c4/html5/thumbnails/70.jpg)
The robot moves again
![Page 71: HCI / CprE / ComS 575: Computational Perception Instructor: Alexander Stoytchev alexs/classes/2010_Spring_575](https://reader030.fdocuments.us/reader030/viewer/2022032600/56649db05503460f94a9e7c4/html5/thumbnails/71.jpg)
… so we must move all balls (particles) to the right again
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… and add some position noise
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And so on …
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Now Let’s Compare that With Some of the Other Methods
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Grid Localization
[Thrun, Burgard & Fox (2005)]
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Grid Localization
[Thrun, Burgard & Fox (2005)]
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Grid Localization
[Thrun, Burgard & Fox (2005)]
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Grid Localization
[Thrun, Burgard & Fox (2005)]
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Grid Localization
[Thrun, Burgard & Fox (2005)]
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Markov Localization
[Thrun, Burgard & Fox (2005)]
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Kalman Filter
[Thrun, Burgard & Fox (2005)]
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Particle Filter
[Thrun, Burgard & Fox (2005)]
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Importance Sampling
• Ideally, the particles would represent samples drawn from the distribution p(x|z).– In practice, we usually cannot get p(x|z) in
closed form; in any case, it would usually be difficult to draw samples from p(x|z).
• We use importance sampling:– Particles are drawn from an importance
distribution.– Particles are weighted by importance weights.
[ http://www.fulton.asu.edu/~morrell/581/ ]
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Monte Carlo Samples (Particles)
• The posterior distribution p(x|z) may be difficult or impossible to compute in closed form.
• An alternative is to represent p(x|z) using Monte Carlo samples (particles):– Each particle has a value and a weight
x
x
[ http://www.fulton.asu.edu/~morrell/581/ ]
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In 2D it looks like this
[http://www.ite.uni-karlsruhe.de/METZGER/DIPLOMARBEITEN/dipl2.html]
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Objective-Find p(xk|zk,…,z1)
• The objective of the particle filter is to compute the conditional distribution
p(xk|zk,…,z1)
• To do this analytically, we would use the Chapman-Kolmogorov equation and Bayes Theorem along with Markov model assumptions.
• The particle filter gives us an approximate computational technique.
[ http://www.fulton.asu.edu/~morrell/581/ ]
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Initial State Distribution
x0
x0
[ http://www.fulton.asu.edu/~morrell/581/ ]
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State Update
x0
x1 = f0 (x0, w0)
x1
[ http://www.fulton.asu.edu/~morrell/581/ ]
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Compute Weights
x1
x1
p(z1|x1)
x1
Before
After
[ http://www.fulton.asu.edu/~morrell/581/ ]
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Resample
x1
x1
[ http://www.fulton.asu.edu/~morrell/581/ ]
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THE END