probabilistic robotics
DESCRIPTION
Probabilistic Robotics. Bayes Filter Implementations Gaussian filters. Bayes Filter Reminder. Prediction Correction. m. Univariate. - s. s. m. Multivariate. Gaussians. Properties of Gaussians. Multivariate Gaussians. - PowerPoint PPT PresentationTRANSCRIPT
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Probabilistic Robotics
Bayes Filter Implementations
Gaussian filters
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•Prediction
•Correction
Bayes Filter Reminder
111 )(),|()( tttttt dxxbelxuxpxbel
)()|()( tttt xbelxzpxbel
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Gaussians
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Multivariate
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Properties of Gaussians
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• We stay in the “Gaussian world” as long as we start with Gaussians and perform only linear transformations.
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Multivariate Gaussians
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6
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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7
Components of a Kalman Filter
t
Matrix (nxn) that describes how the state evolves from t to t-1 without controls or noise.
tA
Matrix (nxl) that describes how the control ut changes the state from t to t-1.tB
Matrix (kxn) that describes how to map the state xt to an observation zt.tC
t
Random variables representing the process and measurement noise that are assumed to be independent and normally distributed with covariance Rt and Qt respectively.
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Kalman Filter Updates in 1D
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9
Kalman Filter Updates in 1D
1)(with )(
)()(
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K
zKxbel
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10
Kalman Filter Updates in 1D
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Kalman Filter Updates
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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
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ttttttttt
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Linear Gaussian Systems: Dynamics
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,;~,;~
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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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,;~,;~
)()|()(
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16
Linear Gaussian Systems: Observations
1
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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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The Prediction-Correction-Cycle
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Prediction
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The Prediction-Correction-Cycle
1)(,)(
)()(
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zKxbel
Correction
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20
The Prediction-Correction-Cycle
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Correction
Prediction
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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
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1)( tTttt
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)( 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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1. EKF_localization ( t-1, t-1, ut, zt, m):
Prediction:
2.
3.
4.
5.
6.
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Motion noise
Jacobian of g w.r.t location
Predicted mean
Predicted covariance
Jacobian of g w.r.t control
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1. EKF_localization ( t-1, t-1, ut, zt, m):
Correction:
2.
3.
4.
5.
6.
7.
8.
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Predicted measurement mean
Pred. measurement covariance
Kalman gain
Updated mean
Updated covariance
Jacobian of h w.r.t location
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EKF Prediction Step
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EKF Observation Prediction Step
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EKF Correction Step
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Estimation Sequence (1)
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Estimation Sequence (2)
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Comparison to GroundTruth
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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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Linearization via Unscented Transform
EKF UKF
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UKF Sigma-Point Estimate (2)
EKF UKF
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UKF Sigma-Point Estimate (3)
EKF UKF
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Unscented Transform
nin
wwn
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cm
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1 )(
)1( 2000
Sigma points Weights
)( ii g
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Pass sigma points through nonlinear function
Recover mean and covariance
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UKF_localization ( t-1, t-1, ut, zt, m):
Prediction:
2
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xti
imt w
2
0,
Motion noise
Measurement noise
Augmented state mean
Augmented covariance
Sigma points
Prediction of sigma points
Predicted mean
Predicted covariance
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UKF_localization ( t-1, t-1, ut, zt, m):
Correction:
zt
xtt h
L
iti
imt wz
2
0,ˆ
Measurement sigma points
Predicted measurement mean
Pred. measurement covariance
Cross-covariance
Kalman gain
Updated mean
Updated covariance
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Tttttt KSK
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1. EKF_localization ( t-1, t-1, ut, zt, m):
Correction:
2.
3.
4.
5.
6.
7.
8.
)ˆ( ttttt zzK
tttt HKI
,
,
,
,
,
,),(
t
t
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yt
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yt
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xt
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mhH
,,,
2,
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,2atanˆ
txtxyty
ytyxtxt
mm
mmz
tTtttt QHHS
1 tTttt SHK
2
2
0
0
r
rtQ
Predicted measurement mean
Pred. measurement covariance
Kalman gain
Updated mean
Updated covariance
Jacobian of h w.r.t location
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UKF Prediction Step
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UKF Observation Prediction Step
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UKF Correction Step
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EKF Correction Step
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Estimation Sequence
EKF PF UKF
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Estimation Sequence
EKF UKF
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Prediction Quality
EKF UKF
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UKF Summary
•Highly efficient: Same complexity as EKF, with a constant factor slower in typical practical applications
•Better linearization than EKF: Accurate in first two terms of Taylor expansion (EKF only first term)
•Derivative-free: No Jacobians needed
•Still not optimal!
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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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• Hypotheses are extracted from LRF scans
• Each hypothesis has probability of being the correct one:
• Hypothesis probability is computed using Bayes’ rule
• Hypotheses with low probability are deleted.
• New candidates are extracted from LRF scans.
MHT: Implemented System (1)
)}(,,ˆ{ iiii HPxH
},{ jjj RzC
)(
)()|()|(
sP
HPHsPsHP ii
i
[Jensfelt et al. ’00]
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MHT: Implemented System (2)
Courtesy of P. Jensfelt and S. Kristensen
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MHT: Implemented System (3)Example run
Map and trajectory
# hypotheses
#hypotheses vs. time
P(Hbest)
Courtesy of P. Jensfelt and S. Kristensen