SA-1 Probabilistic Robotics Probabilistic Sensor Models Beam-based Scan-based Landmarks.
Probabilistic Robotics
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Probabilistic Robotics
SLAM
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Given:
• The robot’s controls (U1:t)
• Observations of nearby features (Z1:t)
Estimate:
• Map of features (m)
• Pose / Path of the robot (xt)
The SLAM Problem
A robot is exploring an unknown, static environment.
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Why is SLAM a hard problem?
• In the real world, the mapping between observations and landmarks is unknown
• Picking wrong data associations can have catastrophic consequences
Robot poseuncertainty
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Nature of the SLAM Problem
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Continuous Location of objects in component the map
Robot’s own pose
Discrete Correspondence component
Object is the same or not
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SLAM: Simultaneous Localization and Mapping
•Full SLAM:
Estimates Entire pose (x1:t) and map (m)
Given Previous knowledge (Z1:t-1, U1:t-1) Current measurement (Zt, Ut)
),|,( t:1t:1t:1 uzmxp
Estimates entire path and map!
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Graphical Model of Full SLAM:
),|,( :1:1:1 ttt uzmxp
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SLAM: Simultaneous Localization and Mapping
•Online SLAM:
Estimates Most recent pose (xt) and map (m)
Given Previous knowledge (Z1:t-1, U1:t-1) Current measurement (Zt, Ut)
),|,( t:1t:1t uzmxp
Estimates most recent pose and map!
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Graphical Model of Online SLAM:
121:1:1:1:1:1 ...),|,(),|,( ttttttt dxdxdxuzmxpuzmxp Integrations typically done one at a time
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SLAM with Extended Kalman Filter
•Pre-requisites
• Maps are feature-based (landmarks)
small number (< 1000)
• Assumption - Gaussian Noise
• Process only positive sightings
No landmark = negative
Landmark = positive
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EKF-SLAM with known correspondences
Correspondence Data association problem
Landmarks can’t be uniquely identified
Correspondence variable (Cit)
between feature (fit) and real landmark
Titit
it
it Srf
True identity of observed feature
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EKF-SLAM with known correspondences
Signature Numerical value (average color)
Characterize type of landmark (integer)
Multidimensional vector
(height and color)
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EKF-SLAM with known correspondences
Similar development to EKF localization
Diff robot pose + coordinates of all landmarks
Combined state vector
TNyNxNyxt
t SmmSmmyxm
xy ,,1,1,1 ...
(3N + 3)
),|( :1:1 ttt uzyp Online posterior
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Motion update
Mean
Covariance
Iteration through measurements
Test for new landmarks
Initialization of elements
Expected measurement
Filter is updated
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EKF-SLAM with known correspondences
Observing a landmark improves robot pose estimate
eliminates some uncertainty of other landmarks
Improves position estimates of the landmark + other landmarks
We don’t need to model past poses explicitly
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EKF-SLAM with known correspondences
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Example:
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EKF-SLAM with known correspondences
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Example:
• Uncertainty of landmarks are mainly due to robot’s pose uncertainty (persist over time)
Estimated location of landmarks are correlated
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EKF-SLAM with unknown correspondences
• No correspondences for landmarks
• Uses an incremental maximum likelihood (ML) estimator
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Determines most likely value of the correspondence variable
Takes this value for granted later on
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EKF-SLAM with unknown correspondences
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Motion update
Hypotheses of new landmark
Mean
Covariance
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General Problem
Gaussian noise assumption Unrealistic
Spurious measurements Fake landmarks
Outliers
Affect robot’s localization
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Solutions to General Problem
Provisional landmark list
New landmarks do not augment the map
Not considered to adjust robot’s pose
Consistent observation regular map
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Solutions to General Problem
Landmark Existence Probability
Landmark is observed
Observable variable (o) increased by fixed value
Landmark is NOT observed when it should
Observable variable decreased
Removed from map when (o) drops below threshold
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Problem with Maximum Likelihood (ML)
Once ML estimator determines likelihood of correspondence, it takes value for granted
always correct
Makes EKF susceptible to landmark confusion
Wrong results
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Solutions to ML Problem
Spatial arrangement
Greater distance between landmarks
Less likely confusion will exist
Trade off:
few landmarks harder to localize
Little is known about optimal density of landmarks
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Signatures
Give landmarks a very perceptual distinctiveness
(e,g, color, shape, …)
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EKF-SLAM Limitations
• Selection of appropriate landmarks
• Reduces sensor reading utilization to presence or absence of those landmarks
Lots of sensor data is discarded
• Quadratic update time
Limits algorithm to scarce maps (< 1000 features)
• Low dimensionality of maps harder data association problem
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EKF-SLAM Limitations
• Fundamental Dilemma of EKF-SLAM
It might work well with dense maps (millions of features)
It is brittle with scarce maps
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BUT
It needs scarce maps because of complexity of the algorithm (update process)
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SLAM Techniques
•EKF SLAM (chapter 10)
•Graph-SLAM (chapter 11)
•SEIF (chapter 12)
•Fast-SLAM (chapter 13)
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Graph-SLAM
• Solves full SLAM problem
• Represents info as a graph of soft constraints
• Accumulates information into its graph without resolving it (lazy SLAM)
• Computationally cheap
• At the other end of EKF-SLAM
Process information right away (proactive
SLAM)
Computationally expensive
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Graph-SLAM
• Calculates posteriors over robot path (not incremental)
• Has access to the full data
• Uses inference to create map using stored data
Offline algorithm
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Sparse Extended Information Filter (SEIF)
• Implements a solution to online SLAM problem
• Calculates current pose and map (as EKF)
• Stores information representation of all knowledge (as Graph-SLAM)
Runs Online and is computationally efficient
• Applicable to multi-robot SLAM problem
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FastSLAM Algorithm• Particle filter approach to the SLAM problem
• Maintain a set of particles
• Particles contain a sampled robot path and a map
• The features of the map are represented by own local Gaussian
• Map is created as a set of separate Gaussians
Map features are conditionally independent given the path
Factoring out the path (1 per particle)
Map feature become independent
Eliminates the need to maintain correlation among them
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FastSLAM Algorithm
• Updating in FastSLAM
Sample new pose update the observed features
• Update can be performed online
• Solves both online and offline SLAM problem
• Instances Feature-based maps
Grid-based algorithm
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• Local submaps [Leonard et al.99, Bosse et al. 02, Newman et al. 03]
• Sparse links (correlations) [Lu & Milios 97, Guivant & Nebot 01]
• Sparse extended information filters [Frese et al. 01, Thrun et al. 02]
• Thin junction tree filters [Paskin 03]
• Rao-Blackwellisation (FastSLAM) [Murphy 99, Montemerlo et al. 02, Eliazar et al. 03, Haehnel et al. 03]
Approximations for SLAM Problem
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Thanks !!