CHAPTER 15 SECTION 1 – 2 Markov Models. Outline Probabilistic Inference Bayes Rule Markov Chains.
Markov Localization & Bayes Filtering
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Transcript of Markov Localization & Bayes Filtering
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Markov Localization & Bayes Filtering
1
with
Kalman Filters
Discrete Filters
Particle Filters
Slides adapted from Thrun et al.,
Probabilistic Robotics
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Faculties of Engineering & Computer Science 2
Autonomous RoboticsCSCI 6905 / Mech 6905 – Section 3
Dalhousie Fall 2011 / 2012 Academic Term
• Introduction • Motion• Perception• Control• Concluding Remarks• LEGO Mindstorms
Control Scheme for AutonomousMobile Robot
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Faculties of Engineering & Computer Science 3
Autonomous RoboticsCSCI 6905 / Mech 6905 – Section 3
Dalhousie Fall 2011 / 2012 Academic Term
• Introduction • Motion• Perception• Control• Concluding Remarks• LEGO Mindstorms
Control Scheme for AutonomousMobile Robot – the plan
– Thomas will cover generalized Bayesian filters for localization next week
– Mae sets up the background for him, today, by discussing motion and sensor models as well as robot control
– Mae then follows on Bayesian filters to do a specific example, underwater SLAM
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Markov Localization
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The robot doesn’t know where it is. Thus, a reasonable initial believe of it’s position is a uniform distribution.
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Markov Localization
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A sensor reading is made (USE SENSOR MODEL) indicating a door at certain locations (USE MAP). This sensor reading should be integrated with prior believe to update our believe (USE BAYES).
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Markov Localization
6The robot is moving (USE MOTION MODEL) which adds noise.
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Markov Localization
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A new sensor reading (USE SENSOR MODEL) indicates a door at certain locations (USE MAP). This sensor reading should be integrated with prior believe to update our believe (USE BAYES).
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Markov Localization
8The robot is moving (USE MOTION MODEL) which adds noise. …
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Bayes Formula
evidence
prior likelihood
)(
)()|()(
)()|()()|(),(
yP
xPxyPyxP
xPxyPyPyxPyxP
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Bayes Rule with Background Knowledge
)|(
)|(),|(),|(
zyP
zxPzxyPzyxP
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Normalization
)()|(
1)(
)()|()(
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1
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x
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|
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Algorithm:
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Recursive Bayesian Updating
),,|(
),,|(),,,|(),,|(
11
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nnnn
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Markov assumption: zn is independent of z1,...,zn-1 if we know x.
)()|(
),,|()|(
),,|(
),,|()|(),,|(
...1...1
11
11
111
xPxzP
zzxPxzP
zzzP
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ni
in
nn
nn
nnn
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Putting oberservations and actions together: Bayes Filters• Given:
• Stream of observations z and action data u:
• Sensor model P(z|x).• Action model P(x|u,x’).• Prior probability of the system state P(x).
• Wanted: • Estimate of the state X of a dynamical system.• The posterior of the state is also called Belief:
),,,|()( 11 tttt zuzuxPxBel
},,,{ 11 ttt zuzud
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Graphical Representation and Markov Assumption
Underlying Assumptions• Static world• Independent noise• Perfect model, no approximation errors
),|(),,|( 1:1:11:1 ttttttt uxxpuzxxp )|(),,|( :1:1:0 tttttt xzpuzxzp
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15111 )(),|()|( ttttttt dxxBelxuxPxzP
Bayes Filters
),,,|(),,,,|( 1111 ttttt uzuxPuzuxzP Bayes
z = observationu = actionx = state
),,,|()( 11 tttt zuzuxPxBel
Markov ),,,|()|( 11 tttt uzuxPxzP
Markov11111 ),,,|(),|()|( tttttttt dxuzuxPxuxPxzP
1111
111
),,,|(
),,,,|()|(
ttt
ttttt
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xuzuxPxzP
Total prob.
Markov111111 ),,,|(),|()|( tttttttt dxzzuxPxuxPxzP
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•Prediction
•Correction
111 )(),|()( tttttt dxxbelxuxpxbel
)()|()( tttt xbelxzpxbel
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Bayes Filter Algorithm
1. Algorithm Bayes_filter( Bel(x),d ):2. 0
3. If d is a perceptual data item z then4. For all x do5. 6. 7. For all x do8.
9. Else if d is an action data item u then10. For all x do11.
12. Return Bel’(x)
)()|()(' xBelxzPxBel )(' xBel
)(')(' 1 xBelxBel
')'()',|()(' dxxBelxuxPxBel
111 )(),|()|()( tttttttt dxxBelxuxPxzPxBel
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Bayes Filters are Familiar!
• Kalman filters
• Particle filters
• Hidden Markov models
• Dynamic Bayesian networks
• Partially Observable Markov Decision Processes (POMDPs)
111 )(),|()|()( tttttttt dxxBelxuxPxzPxBel
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SA-1
Probabilistic Robotics
Bayes Filter Implementations
Gaussian filters
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),(~),(~ 22
2
abaNYbaXY
NX
Linear transform of Gaussians
2
2)(
2
1
2
2
1)(
:),(~)(
x
exp
Nxp
-
Univariate
Gaussians
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• We stay in the “Gaussian world” as long as we start with Gaussians and perform only linear transformations.
),(~),(~ TAABANY
BAXY
NX
Multivariate Gaussians
12
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),(~
),(~
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NX
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Discrete Kalman Filter
tttttt uBxAx 1
tttt xCz
Estimates the state x of a discrete-time controlled process that is governed by the
linear stochastic difference equation
with a measurement
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0000 ,;)( xNxbel
Linear Gaussian Systems: Initialization
• Initial belief is normally distributed:
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• Dynamics are linear function of state and control plus additive noise:
tttttt uBxAx 1
Linear Gaussian Systems: Dynamics
ttttttttt RuBxAxNxuxp ,;),|( 11
1111
111
,;~,;~
)(),|()(
ttttttttt
tttttt
xNRuBxAxN
dxxbelxuxpxbel
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• Observations are linear function of state plus additive noise:
tttt xCz
Linear Gaussian Systems: Observations
tttttt QxCzNxzp ,;)|(
ttttttt
tttt
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xbelxzpxbel
,;~,;~
)()|()(
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Kalman Filter Algorithm
1. Algorithm Kalman_filter( t-1, t-1, ut, zt):
2. Prediction:3. 4.
5. Correction:6. 7. 8.
9. Return t, t
ttttt uBA 1
tTtttt RAA 1
1)( tTttt
Tttt QCCCK
)( tttttt CzK
tttt CKI )(
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Kalman Filter Summary
•Highly efficient: Polynomial in measurement dimensionality k and state dimensionality n: O(k2.376 + n2)
•Optimal for linear Gaussian systems!
•Most robotics systems are nonlinear!
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Nonlinear Dynamic Systems
•Most realistic robotic problems involve nonlinear functions
),( 1 ttt xugx
)( tt xhz
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Linearity Assumption Revisited
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Non-linear Function
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EKF Linearization (1)
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EKF Linearization (2)
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EKF Linearization (3)
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•Prediction:
•Correction:
EKF Linearization: First Order Taylor Series Expansion
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EKF Algorithm
1. Extended_Kalman_filter( t-1, t-1, ut, zt):
2. Prediction:3. 4.
5. Correction:6. 7. 8.
9. Return t, t
),( 1 ttt ug
tTtttt RGG 1
1)( tTttt
Tttt QHHHK
))(( ttttt hzK
tttt HKI )(
1
1),(
t
ttt x
ugG
t
tt x
hH
)(
ttttt uBA 1
tTtttt RAA 1
1)( tTttt
Tttt QCCCK
)( tttttt CzK
tttt CKI )(
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Localization
• Given • Map of the environment.• Sequence of sensor measurements.
• Wanted• Estimate of the robot’s position.
• Problem classes• Position tracking• Global localization• Kidnapped robot problem (recovery)
“Using sensory information to locate the robot in its environment is the most fundamental problem to providing a mobile robot with
autonomous capabilities.” [Cox ’91]
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Landmark-based Localization
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EKF Summary
•Highly efficient: Polynomial in measurement dimensionality k and state dimensionality n: O(k2.376 + n2)
•Not optimal!•Can diverge if nonlinearities are large!•Works surprisingly well even when all
assumptions are violated!
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• [Arras et al. 98]:
• Laser range-finder and vision
• High precision (<1cm accuracy)
Kalman Filter-based System
[Courtesy of Kai Arras]
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Multi-hypothesisTracking
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• Belief is represented by multiple hypotheses
• Each hypothesis is tracked by a Kalman filter
• Additional problems:
• Data association: Which observation
corresponds to which hypothesis?
• Hypothesis management: When to add / delete
hypotheses?
• Huge body of literature on target tracking, motion
correspondence etc.
Localization With MHT
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MHT: Implemented System (2)
Courtesy of P. Jensfelt and S. Kristensen
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SA-1
Probabilistic Robotics
Bayes Filter Implementations
Discrete filters
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Piecewise Constant
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Discrete Bayes Filter Algorithm
1. Algorithm Discrete_Bayes_filter( Bel(x),d ):2. 0
3. If d is a perceptual data item z then4. For all x do5. 6. 7. For all x do8.
9. Else if d is an action data item u then10. For all x do11.
12. Return Bel’(x)
)()|()(' xBelxzPxBel )(' xBel
)(')(' 1 xBelxBel
'
)'()',|()('x
xBelxuxPxBel
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Grid-based Localization
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Sonars and Occupancy Grid Map
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SA-1
Probabilistic Robotics
Bayes Filter Implementations
Particle filters
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Sample-based Localization (sonar)
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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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Weight samples: w = f / g
Importance Sampling
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Importance Sampling with Resampling:Landmark Detection Example
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Particle Filters
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)|()(
)()|()()|()(
xzpxBel
xBelxzpw
xBelxzpxBel
Sensor Information: Importance Sampling
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'd)'()'|()( , xxBelxuxpxBel
Robot Motion
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)|()(
)()|()()|()(
xzpxBel
xBelxzpw
xBelxzpxBel
Sensor Information: Importance Sampling
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Robot Motion
'd)'()'|()( , xxBelxuxpxBel
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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, tS
ni 1
},{ it
ittt wxSS
itw
itx ),|( 11 ttt uxxp )(
1ij
tx 1tu
)|( itt
it xzpw
ni 1
/it
it ww
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draw xit1 from Bel(xt1)
draw xit from p(xt | xi
t1,ut1)
Importance factor for xit:
)|()(),|(
)(),|()|(ondistributi proposal
ondistributitarget
111
111
tt
tttt
tttttt
it
xzpxBeluxxp
xBeluxxpxzp
w
1111 )(),|()|()( tttttttt dxxBeluxxpxzpxBel
Particle Filter Algorithm
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Start
Motion Model Reminder
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Proximity Sensor Model Reminder
Laser sensor Sonar sensor
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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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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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Global Localization
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Kidnapping the Robot
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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.