1

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

Paul Newman's mobile robot mapping MIT

What is Robotic Mapping?

• Sensors with different limitations

• Cameras, Sonar, Lasers, Radar,Compasses, GPS

2

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

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?

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

3

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 StochasticMapping

A Computationally Efficient Method for Large-ScaleConcurrent Mapping and Localization

John J. Leonard and Hands Jacob S. Feder, MIT, 2000

Introduction

• DSM: Feature based approach to CML

• Previous solutions are O(n2), where n is thenumber of features– Results from the number of correlations

between the vehicles and features

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 tutorialpresented at ACM SIGGRAPH 2001 by GregWelch and Gary Bishop (UNC).

– The slides of the tutorial are available athttp://www.cs.unc.edu/~tracker/ref/s2001/kalman/index.html

– More information (papers, software, links , etc) isavailable 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

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Kalman Filter Example2D Position-Only (e.g., 2D Tablet)

Process Model:

Measurement Model:†

xk

yk

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˚ ˙ =

1 00 1

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xk-1

yk-1

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Î Í

˘

˚ ˙ +

~ xk-1

~ yk-1

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Î Í

˘

˚ ˙

†

uk

vk

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˚ ˙ =

Hx 00 Hy

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xk

yk

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˚ ˙ +

~ uk

~ vk

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state statetransition state noise

measurement measurementmatrix state 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 00 1

È

Î Í

˘

˚ ˙

†

Q = E w * w T{ } =Qxx 00 Qyy

È

Î Í

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R = E v *v T{ } =Rxx 00 Ryy

È

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x 0 = H-1z 0

P0 =e 00 e

È

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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

Kalman Filter Example

Predict Correct

†

x k- = Ax k-1

Pk- = APk-1A

T + Q

†

K = Pk-HT HPk

-HT + R( )-1

x k = x k- + K z k - Hx k

-( )Pk = I - KH( )P-

Kalman Filter ExampleExtend example to 2D Position-Velocity

Process model:

Measurement model:

†

state transition state

1 0 dt 00 1 0 dt0 0 1 00 0 0 1

È

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xy

dxdt

dydt

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˙ ˙ ˙ ˙ ˙

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measurement matrix state

Hx 0 0 00 Hy 0 0

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xy

dxdt

dydt

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Í Í Í Í Í

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Kalman Filter

• But, Kalman filter is not enough !!!

– Only matrix operations allowed (only works forlinear 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.)

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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

Stochastic Mapping

• Size-varying Kalman filter

• Add and Update of representation

• Build a map through spatial relationship

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 f v[ ]T

and x f k[ ]T= x 1 k[ ]T ...x N k[ ]T[ ], such that xi = xi yi[ ]T

†

P k[ ] =Prr k[ ] Prf k[ ]Pfr k[ ] Pff k[ ]

È

Î Í

˘

˚ ˙

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[ ] = df du[ ]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

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How do we move from map tomap?

Cross-map relocationA B

Cross-map updatingA 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[ ]

È

Î Í

˘

˚ ˙ ,PB k[ ] ¨

PrrA k[ ] + Prr

B j[ ] PrfB j[ ]

PfrB j[ ] Pff

B j[ ]

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Î Í

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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: Usestate as measurement and covariance in A asprediction for state in B.

†

x B k-[ ] ¨f B

x fB k[ ]

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Î Í

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˚ ˙ ,PB k-[ ] ¨

PrrB j[ ] + FB Prf

B j[ ]Pfr

B j[ ] 2PffB j[ ]

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†

x rA k[ ]

†

PrrA k[ ]

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K = PB k-[ ]HT HPB k-[ ]HT + PrrA k[ ]( )

-1

x B k +[ ] ¨ x B k-[ ] + K z - Hx B k-[ ]( )PB k +[ ] ¨ I - KH( )PB k-[ ] I - KH( )T

+ KPrrA k[ ]KT

†

z

Methods Comparison

Full covariance ASM

Single-pass DSM

Multi-pass DSM

Testing

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Limitations

• Sensor noise modeled by gaussian process

• Limited map dimensionality

Hybrid ApproachesA 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 using2D laser range finders

• Mapping: Fast scan-matching• Localization: Sample-based probabilities• Motivation: 3D-Maps and large cyclic

environments

Benefits

• Computation is all real-time• Builds 3D maps• Handles cycles• Accurate map generation in the absence of

odometric data

Background

• Incremental Localization

• Expectation Maximization

Incremental Localization

• Iterate localization for each new sensor scan• Can be done in real-time• Fail on cyclic environments as error grows

unbounded

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Expectation Maximization (EM)

• Search most-likely map while consideringall past scans

• Probabilistically, iterate and refine the map• Can handle cyclic environments• Batch algorithms - not real-time

Goal

• Combine IL and EM in a real-timealgorithm that can handle cycles

• Use posterior estimation like in EM• Incremental map construction with

maximum likelihood estimators as in IL

Mapping

• A map is a collection of scans and poses

mt = { ot ,st }t =0,1,...,t

Map likelihood

P(m | dt ) = hP(m) L P(ott =0

t

’ÚÚ | m,st )

⋅ P(st +1 | at ,stt = 0

t -1

’ )ds1...dst

arg maxm

P(m | dt )dt ={so,ao ,s1,a1, ...,st}

•The most likely map:where:

Mapping

P(s | a, ¢ s )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 futurescans will show obstacles in that space

• Darker regions indicate lower probability of anobstacle

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Maximizing Map Likelihood

• Infeasible to maximize while robot ismoving in real-time

• In the past, the robot had to stop (EM) orrisk unbounded error (IL)

Conventional Incremental Map

• Given a scan and odometry reading,determine the most likely pose.

• Use that pose to increment the map. Nevergo back to change it.

ˆ s t = argmax P(st | ot , at -1, ˆ s t -1)

mt +1 = mt { ot , ˆ s tU }

Conventional Incremental Map

• This approach works in non-cyclicenvironments

• 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 fullposterior over robot poses

• Probability distribution over poses based onsensor 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 ) = hP(ot | st ,mt -1)

P(st | at -1, st -1)Bel(st -1 )dst-1Ú

s t = argmaxst

Bel(st ) mt +1 = mt { ot ,s tU }

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Posterior Incremental Mapping

• Use the posterior belief to determine themost 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 cango back and correct our pose estimates

• Distribute the error ∆st among all poses inthe loop

• Use gradient descent for all poses in theloop to maximize likelihood

D st= s t - ˆ s t

Handling a Cycle

Multi-Robot Extensions

• Using posterior estimation extends naturallyto environments with multiple robots

• Each robot need not know any other robot’sinitial pose

• BUT every robot localize itself within themap 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

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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

Results - Mapping w/out 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

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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 meaningfulmaps 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