probabilistic robotics robot localization. 2 localization given map of the environment. sequence of...
TRANSCRIPT
2
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]
12
•Prediction:
•Correction:
EKF Linearization: First Order Taylor Series Expansion
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13
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
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1. EKF_localization ( t-1, t-1, ut, zt, m):
Prediction:
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Motion noise
Jacobian of g w.r.t location
Predicted mean
Predicted covariance
Jacobian of g w.r.t control
15
1. EKF_localization ( t-1, t-1, ut, zt, m):
Correction:
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Predicted measurement mean
Pred. measurement covariance
Kalman gain
Updated mean
Updated covariance
Jacobian of h w.r.t location
26
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!
30
Unscented Transform
média a relação em espalhados estão points sigma os distantes
quão determinam que parâmetros a iguais sendok e com ,)(
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Sigma points Weights
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Pass sigma points through nonlinear function
Recover mean and covariance
31
UKF_localization ( t-1, t-1, ut, zt, m):
Prediction:
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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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40
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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Monte Carlo (Particle Filter) Localization
1. Algorithm sample_normal_distribution(b):
2. return
1. Algorithm sample_triangular_distribution(b):
2. return
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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