Post on 11-Jan-2016
transcript
Robotic Mapping
6.834 Student Lecture
Itamar Kahn, Thomas Lin, Yuval Mazor
Outline
Introduction (Tom)
Kalman Filtering (Itamar)J.J. Leonard and H.J.S. Feder. A computationally efficient method for large-scale concurrent mapping andlocalization. In J. Hollerbach and D. Koditschek, editors, Proceedings of the Ninth International Symposium onRobotics Research, Salt Lake City, Utah, 1999
Hybrid Mapping Approaches (Yuval)S. Thrun, W. Burgard, and D. Fox. A real-time algorithm for mobile robot mapping with applications to multi-robot and 3D mapping. In Proceedings of the IEEE Internatinoal Conference on Robotics and Automation(ICRA), San Francisco, CA, 2000. IEEE
Conclusion (Tom)
Vision / Steps
Truly autonomous mobile robots
Sense the environment Acquiring models of the environment Reason Act on environment
State of the Art
20 years of research
Do well on static, structured, limited size
Difficulty with dynamic, unstructured,
large scale
Simulated versus Real-life
What is Robotic Mapping?
Acquiring spatial models of physicalenvironments with robots
Qui ckTi me™ and a Graphi cs decompressor are needed to see thi s pi cture.
P a ul N e w m a n' s mobil e robo t m a pping M I T
What is Robotic Mapping?
Sensors with different limitations
Cameras, Sonar, Lasers, Radar,Compasses, GPS
Main Challenges
Noise
High Dimensionality
Correspondence Problem
Changing Environments
Robotic Exploration Planning
Challenges - Noise
Measurement errors accumulate over time
Odometry error will accumulate and throw off an entire map [Thrun 2002]
Challenges - High Dimensionality
3-D visual maps can take millions ofnumbers
Challenges - Correspondence Problem
Do these sensor readings from differenttimes correspond to the same object?
Is the blue object the same one it sensed earlier, or it a different object that seemslike it's in the same location because of accumulated sensor noise?
[Thrun 2002]
Challenges - Changing Environments
Moving furniture, moving doors
Even faster: Moving cars, moving people
Hard to distinguish sensor noise and
moving items
Challenges - Robotic Exploration Planning
How robots should explore usingincomplete maps
Today's Methods
All Probabilistic
Better models uncertainty, sensor noise
Kalman Filtering (Itamar will present), Hybrid
Methods (Yuval will present)
EM, Occupancy Grids, Multi-Planar Maps
(not presenting)
Decoupled Stochastic Mapping
A Computationally Efficient Method for Large-Scale Concurrent Mapping and Localization
John J. Leonard and Hands Jacob S. Feder, MIT, 2000
Robotic Mapping Problem
• Identify features in the environment– E.g., landmarks, distinctive objects or shapes in the
environment, etc.
• Estimate the robot location in reference to the features
• Correct for noise (error in estimation) contributed by the sensors and controls
Acquire a spatial model of a robot’s environmentAcquire a spatial model of a robot’s environment
What is DSM?
• SM: Use Extended Kalman filtering (EKF) to build a map through spatial relationship of features
– PROBLEM: EKF based solutions are O(n2), where n is the number of features
• Results from the number of correlations between the vehicles and features
– SOLUTION: Break into submaps and apply SM only on submaps
Feature based approach to Concurrent Mapping and Localization (CML)
Feature based approach to Concurrent Mapping and Localization (CML)
What is a Feature?
• Determine the relevant visual features– These maybe specific to the to be mapped environment (e.g.,
walls in a room, obstacles in an under water envirnoment, etc.)
A map is obtained by defining visual features dynamics and observation function
A map is obtained by defining visual features dynamics and observation function
QuickTime™ and a JPEG 2000 decompressor are needed to see this picture. QuickTime™ and a JPEG 2000 decompressor are needed to see this picture.
F[4]: the whole wall is a single feature in the mapF[4]: the whole wall is a single feature in the map
Overview
• Kalman and Extended Kalman Filters
• Conventional Stochastic Mapping
• Decoupled Stochastic Mapping
• Algorithm Testing
Kalman Filter Mini Tutorial
• The mini tutorial is an adaptation of a tutorial presented at ACM SIGGRAPH 2001 by Greg Welch and Gary Bishop (UNC).
– The slides of the tutorial are available at http://www.cs.unc.edu/~tracker/ref/s2001/kalman/index.html
– More information (papers, software, links , etc) is available athttp://www.cs.unc.edu/~welch/kalman/index.html
Kalman Filter• KF operates by
– Predicting the new state and its uncertainty– Correcting with the new measurement
• IN: Noisy data --> OUT:less noisy
Kalman Filter Example2D Position-Only (e.g., 2D Tablet)
Process Model:
Measurement Model:
xk
yk
1 0
0 1
xk 1
yk 1
~ xk 1
~ yk 1
uk
vk
Hx 0
0 Hy
xk
yk
~ uk
~ vk
statestate
transition state noise
measurementmeasurement
matrixstate noise
x k Ax k 1 w k 1
z k Hx k v k
Kalman Filter ExamplePreparation and Initialization
State transition:
Process Noise Covariance:
Measurement Noise Covariance:
Initialization:
A 1 0
0 1
QE w * w T Qxx 0
0 Qyy
R E v *v T Rxx 0
0 Ryy
x 0 H 1z 0
P0 0
0
state at t0
error covariance estimate at t0
Kalman Filter ExamplePredict
Correct
x k Ax k 1
Pk APk 1A
T Q
x k x k K z k Hx k
Pk I KH P
K Pk HT HPk
HT R 1
predict next state
predicted error covariance
correct for the discrepancy between predicted and actual measurement
minimize the a posteriori error covariance (Kalman gain)
correct state and error covariance
Kalman Filter
Predict Correct
(1) Project the state ahead
x k Ax k 1
(2) Project the error covariance ahead
Pk APk 1A
T Q
(1) Computer the Kalman gain
K Pk HT HPk
HT R 1
(2) Update the estimate with measurement z k
x k x k K z k Hx k
(3) Updtae the error covariance
Pk I KH P
Kalman Filter ExampleExtend example to 2D Position-Velocity
Process model:
Measurement model:
state transition state
1 0 dt 0
0 1 0 dt
0 0 1 0
0 0 0 1
x
ydx
dtdy
dt
measurement matrix state
Hx 0 0 0
0 Hy 0 0
x
ydx
dtdy
dt
Kalman Filter
• But, Kalman filter is not enough !!!
– Only matrix operations allowed (only works for linear systems)
– Measurement is a linear function of state– Next state is linear function of previous state– Can’t estimate non-linear variables (e.g., gain,
rotation, projection, etc.)
Extended Kalman Filter
• Nonlinear Process (Model)– Process dynamics: A becomes a(x)
– Measurement: H becomes h(x)
• Filter Reformulation– Use functions instead of matrices
– Use Jacobians to project forward, and to relate measurement to state (first order Taylor expansion)
(1) Project the state ahead
x k f x k 1,uk,0
(2) Project the error covariance ahead
Pk APk 1A
T WkQk 1WkT
Extended Kalman Filter
Predict Correct
(1) Computer the Kalman gain
K Pk Hk
T HkPk Hk
T VkRkVkT 1
(2) Update the estimate with measurement z k
x k x k K z k h x k
,0 (3) Updtae the error covariance
Pk I KkHk Pk
• A is the Jacobian matrix of partial derivatives of f with respect to x• W is the Jacobian matrix of partial derivatives of f with respect to w• H is the Jacobian matrix of partial derivatives of h with respect to x• H is the Jacobian matrix of partial derivatives of h with respect to v
• A is the Jacobian matrix of partial derivatives of f with respect to x• W is the Jacobian matrix of partial derivatives of f with respect to w• H is the Jacobian matrix of partial derivatives of h with respect to x• H is the Jacobian matrix of partial derivatives of h with respect to v
Stochastic Mapping
• Size-varying Kalman filter
• Add and Update of representation
• Build a map through spatial relationship
Use SM to generate maps (solve CML)Use SM to generate maps (solve CML)
Stochastic Mapping• Estimated locations of the robot and the features in
the map
• Estimated error covariance
x k x r k T x f k T T where x r xr yr v T
and x f k T x 1 k T ...x N k T , such that x i x i y i T
P k Prr k Prf k Pfr k Pff k
Stochastic Mapping• The dynamic model of the robot is given by
• The observation model for the system is given by
x k +1 = f x k , u k + dx u k where u k T
z k = h x k + dz
Augmented Stochastic Mapping
• Given these assumptions, an extended Kalman filter (EKF) is employed to estimate the state and covariance .
x
P
Decoupled Stochastic Mapping
• Stochastic Mapping: complexity O(n2)
• Solution: DSM– Divide the environment into multiple submaps– Each submap has a vehicle position estimate
and a set of features estimates
Decoupled Stochastic Mapping
dependencies are local
Map of landmarks Inverse covariance matrix
the map is divided in 4 sub maps
DSM: Divide the map into smaller submapsDSM: Divide the map into smaller submaps
How do we move from map to map?
Cross-map relocation
A B
Cross-map updating
A B
Single-pass vs. Multi-pass DSM
Decoupled Stochastic Mapping
• Vehicle travels to a previously visited area:Cross-map relocation
x B k x r
A k x r
B j
,P B k
PrrA k Prr
B j PrfB j
PfrB j Pff
B j
Decoupled Stochastic Mapping
• Facilitate spatial convergence by bringing more accurate vehicle estimates from lower to higher maps:Cross-map updating
Using EKF, estimate vehicle location in submap B: Use state as measurement and covariance in A, as predictions for state in B.
x B k B
x fB k
,PB k Prr
B j B PrfB j
PfrB j 2Pff
B j
x rA k
PrrA k
z
Methods Comparison
Full covariance ASM
Single-pass DSM
Multi-pass DSM
Testing
QuickTime™ and a YUV420 codec decompressor are needed to see this picture.
Limitations
• Sensor noise modeled by Gaussian process
• Limited map dimensionality
Hybrid Approaches
A Real-Time Algorithm for Mobile Mapping with Applications to Multi-Robot and 3D Mapping
Sebastian Thrun, Carnegie Mellon UniversityWolfram Burgard, University of FreibergDieter Fox, Carnegie Mellon University
Overview
• Concurrent mapping and localization using 2D laser range finders
• Mapping: Fast scan-matching
• Localization: Sample-basedprobabilities
• Motivation: 3D-Maps and large cyclic environments
Benefits
• Computation is all real-time
• Builds 3D maps
• Handles cycles in a map
• Accurate map generation in the absence of odometric data
Mapping Basics
• A map is a collection of sensor scans, o, and robot positions (poses), s
• For every time, t, a new data scan and pose is added to the map:
mt { o ,s } 0,1,...,t
Map likelihood
P(m | dt ) P(m) P(o 0
t
| m,s )
P(s1 | a ,s 0
t 1
)ds1...dst
• The most likely map:
where: dt {so,ao ,s1,a1, ...,st}
arg maxm
P(m | dt )
Mapping
Posterior pose, s, after moving distance a from s’:
• The PDF has an elliptical/banana shape
PDF Intuition
• If a scan shows free space it is unlikely that future scans will show obstacles in that space
• Darker regions indicate lower probability of an obstacle
Maximizing Map Likelihood
• Goal: Find the most-likely map given all the data the robot has seen
• Infeasible to maximize in real-time• Two possibilities:
– Have the robot stop and calculate after every scan (not real-time)
– Assume map is correct, add new data (large error growth)
Background Methods
• Incremental Localization
• Expectation Maximization
Incremental Localization (IL)
• Assume previous map and localizations are accurate
• Append new sensor scans to the old map• Localize based on updated map• Can be done in real-time• Fail on cyclic environments as error grows
unbounded
Incremental Localization
• IL never corrects old errors based on new information
• Errors can grow unbounded
• While traversing a cycle in a map, error growth leads the robot to “get lost” and the map breaks down
Expectation Maximization (EM)
• Store scans and pose data probabilistically
• Search through all possible previous maps (from times 0-t) and find the most likely maps
• After each scan, or set number of scans, recalculate
Expectation Maximization
• Can handle cyclic environments
• Batch algorithms - not real-time
Goal
• Combine IL and EM in a real-time algorithm that can handle maps with cycles
• Use posterior estimation like in EM
• Incremental map construction with maximum likelihood estimators as in IL
Conventional Incremental Map
• Given a scan and odometry reading, determine the most likely pose.
• Use that pose to increment the map. Never go back to change it.
ˆ s t argmax P(st | ot , at 1, ˆ s t 1)
mt1 mt { ot , ˆ s t }
Conventional Incremental Map
• This approach works in non-cyclic environments
• Pose errors necessarily grow
• Past poses cannot be revised
• Search algorithms cannot find solutions to close loops
Incremental Map Problem
Posterior Incremental Mapping
• Basic premise:Use Markov localization to compute the full posterior over robot poses
• Probability distribution over poses based on sensor data:
Bel(st )P(st | dt , mt 1)
Posterior Incremental Mapping
• Posterior is where the robot believes it is.
• Can be incrementally updated over time
• Updated pose and maps:
Bel(st )P(ot | st ,mt 1)
P(st | at 1, st 1)Bel(st 1 )dst 1
s t argmaxst
Bel(st ) mt1 mt { ot ,s t }
Posterior Incremental Mapping
• Use the posterior belief to determine the most likely pose
• Uncertainty grows during a loop
• The robot has a larger window to search to close the loop
Implementation Details
• Take samples of posterior beliefs
• Save computation and easier to generalize
• Use gradient descent on each sample to find globally maximum likelihood function.
Backwards Correction
• When a loops closes successfully, we can go back and correct our pose estimates
• Distribute the error ∆st among all poses in
the loop
• Use gradient descent for all poses in the loop to maximize likelihood
sts t ˆ s t
Handling a Cycle
Multi-Robot Extensions
• Using posterior estimation extends naturally to environments with multiple robots
• Each robot need not know any other robot’s initial pose
• BUT every robot localize itself within the map of an initial Team Leader robot
Multi-Robot Extensions
• Use Monte Carlo Localization
• Initially any location is likely
• Posterior estimation localizes the robot in the Team Leader’s map
Results - Cycle Mapping
• Groundrules:– Every scan used for localization– Scans appended to map every two meters
• Random odometric errors (30˚ or 1 meter)• Error generates large error during the cycle
but within acceptable range of “true” pose• Posterior estimation finds the true pose and
corrects prior beliefs
Mapping without Odometry
• Same as before but with no odometric data
• Traversing the cycles leads to very large error growth
• Once again, on cycle completion the errors are found and fixed
• Final map is virtually identical to map generated with odometric data
Limitations
• Non-optimal
• Nested cycles
• Dynamic environments
• Changing the map backwards in time can be dangerous
• Pseudo-Real Time
Brief Comparison
Kalman Filtering Hybrid MethodsRepresentation landmark locations point obstaclesSensor Noise Gaussian any
Map Dimensionality limited unlimitedDynamic Env's limited no
Scenario 1 - Infinite Corridor at Night
Which algorithm is better for a robotmapping the infinite corridor late at night,when one janitor is walking around? Vote Kalman Filtering Hybrid Approaches Don't Know
Scenario 1 - Infinite Corridor at Night
Changing environment problem Kalman - good! (Itamar will explain)
Infinite corridor has few features Can handle janitor (limited dynamics) Hybrid - bad! (Yuval will explain)
Can't handle dynamic environments
Scenario 2 - Airport Parking Lot
Which algorithm is better for a robotmapping an airport parking lot withhundreds of cars but no people? Vote Kalman Filtering Hybrid Approaches Don't Know
Scenario 2 - Airport Parking Lot
High dimensionality problem Kalman - bad! (Itamar will explain)
Only handles limited map dimensionality Hybrid - good! (Yuval will explain)
Nothing moving Handles unlimited map dimensionality
Scenario 3 - Amusement Park
Which algorithm is better for a robotmapping a busy amusement park duringChristmas? Vote Kalman Filtering Hybrid Approaches Don't Know
Scenario 3 - Amusement Park
Both fail Kalman - bad! (Itamar will explain)
Only does limited dynamics Hybrid - bad! (Yuval will explain)
Can't handle such a dynamic environment Almost no algorithms learn meaningful
maps in such a dynamic environment
Recap
The Mapping Problem
Main Challenges
Kalman Filtering
Hybrid Methods
Comparison
Contributions
Provided overview of robotic mapping
Presented Kalman Filtering in depth
Presented Hybrid Methods in depth