Probabilistic Mapping

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Efficient Approaches to Mapping with
Rao-Blackwellized Particle Filters
Wolfram Burgard
Department of Computer Science University of
Freiburg, Germany
Special thanks to Austin Eliazar, Giorgio
Grisetti, Dirk Hähnel, Mike Montemerlo, Ronald
Parr, Cyrill Stachniss,
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Dimensions of Robot Mapping
SLAM
localization
mapping
integrated approaches
active localization
exploration
motion control
Makarenko et al., 02
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Types of SLAM-Problems

• Grid maps or scans
• Lu Milios, 97 Gutmann, 98 Thrun 98
  Burgard, 99 Konolige Gutmann, 00 Thrun, 00
  Arras, 99 Haehnel, 01
• Landmark-based

Leonard et al., 98 Castelanos et al., 99
Dissanayake et al., 2001 Montemerlo et al.,
2002
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Why is SLAM Hard Ambiguity
End
Start
Courtesy of Eliazar Parr
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Properties of Standard EKF-SLAM

• Requires pre-defined landmarks/features
• Complexity O(n2) where n is the number of
  landmarks
• Data association problem
• How can we solve the SLAM problem for not
  feature-based representations?

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Occupancy Grid Maps

• Introduced by Moravec and Elfes in 1985
• Represent environment by a grid.
• Estimate the probability that a location is
  occupied by an obstacle.
• Key assumptions
• Occupancy of individual cells (mxy) is
  independent
• Robot positions are known!

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Updating Occupancy Grid Maps

• Update the map cells using the inverse sensor
  model
• Or use the log-odds representation

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Typical Sensor Model for Occupancy Grid Maps

• Combination of a linear function and a Gaussian

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Key Parameters of the Model
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Occupancy Value Depending on the Measured Distance
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Deviation from the Prior Belief(the sphere of
influence of the sensors)
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Calculating the Occupancy Probability Based on
Single Observations
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Incremental Updating of Occupancy Grids
(Example)
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Resulting Map Obtained with Ultrasound Sensors
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Mapping with Raw Odometry
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Techniques for Generating Consistent Maps

• Scan matching (online)
• Probabilistic mapping with a single map and a
  posterior about poses Mapping Localization
  (online)
• EKF SLAM (online, mostly landmarks or features
  only)
• EM techniques (offline)
• Lu and Milios (offline)
• Rao-Blackwellized particle filters (landmarks and
  grids)

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Scan Matching

• Maximize the likelihood of the i-th pose and map
  relative to the (i-1)-th pose and map.

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Scan Matching Example
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Key Problems

• How to maintain multiple map and pose hypotheses
  during mapping?
• How to control the robot?

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Rao-Blackwellized Mapping

• Observation
• Given the true trajectory of the robot, all
  measurements are independent.
• Idea
• Use a particle filter to represent potential
  trajectories of the robot (multiple hypotheses).
• For each particle we can analytically compute the
  map of the environment (mapping with known
  poses).
• Each particle survives with a probability that is
  proportional to the likelihood of the observation
  given that particle and its map.

Murphy et al., 99
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Rao-Blackwellized Mapping (2)
Compute a posterior over the map and possible
trajectories of the robot
map and trajectory
measurements
robot motion
map
trajectory
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A Graphical Model of Rao-Blackwellized Mapping
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FastSLAM
Robot Pose
2 x 2 Kalman Filters

Particle M
Begin courtesy of Mike Montemerlo
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FastSLAM
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FastSLAM Simulation

• Up to 100,000 landmarks
• 100 particles
• 103 times fewer parameters than EKF SLAM

Blue line true robot path Red line estimated
robot path Black dashed line odometry
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Victoria Park Results

• 4 km traverse
• 100 particles
• Uses negative evidence to remove spurious
  landmarks

Blue path odometry Red path estimated path
End courtesy of Mike Montemerlo
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Key Questions

• Can we apply Rao-Blackwellized particle filters
  to mapping with large grid-maps?
• How can we compactly represent the individual
  maps carried by the particles?
• How can we reduce the number of particles needed?

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Tasks to be Solved

• Mapping (occupancy grids)
• Each particle carries its own map m.
• The history of each particle represents a
  potential trajectory of the robot.
• Localization
• Propagate the particles according to the motion
  model (draw from p(xu,x)).
• Compute importance weight according to the
  likelihood of the observation z given the pose x
  and the map m of the particle.

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Computing the Likelihood of a Measurement Ray
Casting

1. Determine the distance to the closest obstacle in
   the direction of the measurement (ray-casting).
2. Approximate the likelihood p(z m, x) by the
   likelihood p(z d) of z given the expected
   measurement d for x.

Fox et al., 98
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Mixture Approximation of p(z d)
Choset et al., to appear, Thrun et al., to
appear
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Computing the Likelihood of a Measurement
Correlation Models

• Determine the cell xy a beam ends in.
• Approximate the likelihood p(z m, x) by the
  occupancy probability Bel(m xy) contained in
  mxy (correlation model).Konolige, 99
• Smoothing of Bel(mxy) yields a better gradient
  and improves the robustness.Thrun, 01
  (likelihood fields).

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RPBF with Grid Maps
3 particles
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Map Maintenance Challenges

• High resolution maps are big
• Typically 100s or 1000s of particles are needed
• One full map per particle requires
• O(mn) work (re-sampling)
• Gigabytes of memory movement
• Anecdotal reports Tried, but impractical(see
  later)

Begin courtesy of Eliazar Parr
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DP-SLAM Distributed Particle Mapping

• Exploit sampling/re-sampling steps of PF
• Common ancestry Redundant map sections
• History representation Ancestry Tree
• Leaves correspond to current particles
• New map Representation
• Store multiple maps in a single grid

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Ancestry Trees
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Ancestry Trees
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Ancestry Trees
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Ancestry Trees
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Ancestry Trees
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Ancestry Trees
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Ancestry Trees
Ancestors with no children can be removed
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Ancestry Trees
Ancestors with only one child can be merged
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Ancestry Trees
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Ancestry Trees

• Maintain a minimal tree (improves complexity)
• Exactly n leaves
• Branching factor at least 2
• Depth no more than n
• Explicitly store the ancestry info
• Node Ancestor particle with unique ID
• Stores parent link, map updates

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Map Representation

• Map is an occupancy grid
• Avoid one map per particle

Naïve Map Representation
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DP-Mapping

• Distribute particles over a single map
• Each grid square stores
• ID of each ancestry node that has seen this
  square
• Associated observations
• No redundant data
• No unnecessary data

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Localization

• For each laser cast of the current particle
• Trace laser cast through grid
• For each grid square return map occupancy
• Store observations as balanced trees (keyed on
  IDs)
• Linear storage ancestry
• Logarithmic access/updates

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Complexity

• Localization O(An2)
• n particles check A grid squares
• Worst case cost n to check occupancy(harder
  than it sounds)
• Map Maintenance O(Anlogn)
• Additions, Deletions O(Anlogn)
• Ancestry Tree Maintenance O(Anlogn)
• Amortized analysis (see papers by EliazarParr)

A Area observed n Number of
particles m Map size
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Complexity Summary

• Total Time O(An2)
• Compare to O(mn)
• m gtgt An
• Linear in observation size
• Independent of map size

A Area observed n Number of
particles m Map size
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DP-SLAM Results
scale 3cm
Run at real-time speed on 2.4GHz Pentium 4 at
10cm/s
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Consistency
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Results obtained with DP-SLAM 2.0 (offline)
Eliazar Parr, 04
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Close up
End courtesy of Eliazar Parr
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Observations

• DP-SLAM is an efficient and elegant way to store
  the individual maps assigned to the particles.
• Complexity O(An2) where n is the number of
  particles
• How can we reduce the number of particles?

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Techniques to Reduce the Number of Particles
Needed

• Better proposals (put the particles in the right
  place in the prediction step).
• Avoid particle depletion (re-sample only when
  needed).

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Generating better Proposals

• Use scan-matching to compute highly accurate
  odometry measurements from consecutive range
  scans.
• Use the improved odometry in the prediction step
  to get highly accurate proposal distributions.

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Motion Model for Scan Matching
Raw Odometry
Scan Matching
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Graphical Model for Mapping with Improved Odometry
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Rao-Blackwellized Mapping with Scan-Matching
Map Intel Research Lab Seattle
Loop Closure
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RBPF Mapping with Scan-Matching
Map Intel Research Lab Seattle
Loop Closure
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Rao-Blackwellized Mapping with Scan-Matching
Map Intel Research Lab Seattle
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Comparison to Previous Techniques

• Standard Rao-Blackwellized mapping with grid maps
  (Intel Research Lab data set)
• Wean Hall (32m x 10m), noise added to odometry
  (simulation)
• Scan Matching
• Single map plus posterior about poses

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Comparison to the Original Approach

• Same model for observations
• Odometry instead of scan matching results
• Number of particles varying from 500 to 2.000
• Typical result

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Dynamically Adapting the Motion Model

• The previous approach used a constant motion
  model p(xu, x).
• It needs to be more peaked than the model for raw
  odometry.
• Accordingly, it will fail in situations in which
  scan matching yields bad results (e.g., in wide
  open spaces)
• Goal better proposal distribution

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The Optimal Proposal Distribution
Arulampalam et al., 01
For lasers is extremely
peaked and dominates the product.
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Resulting Proposal Distribution
Gaussian approximation
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Estimating the Parameters of the Gaussian for
each Particle

• xj are a set of sample points around the point x
  the scan matching has converged to.
• ? is a normalizing constant

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Computing the Importance Weight
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Selective Re-sampling

• Re-sampling is dangerous, since important samples
  might get lost(particle depletion problem)
• In case of suboptimal proposal distributions
  re-sampling is necessary to achieve convergence.
• Key question When should we re-sample?

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Number of Effective Particles

• Empirical measure of how well the goal
  distribution is approximated by samples drawn
  from the proposal
• We only re-sample when neff drops below a given
  threshold (n/2)
• See Doucet, 98 Arulampalam, 01

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Typical Evolution of neff
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Example (Intel Lab)

• 15 particles
• four times faster than real-timeP4, 2.8GHz
• 5cm resolution during scan matching
• 1cm resolution in final map

Courtesy by Giorgio Grisetti Cyrill Stachniss
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Outdoor Campus Map

• 30 particles
• 250x250m2
• 1.75 km (odometry)
• 20cm resolution during scan matching
• 30cm resolution in final map

• 30 particles
• 250x250m2
• 1.088 miles (odometry)
• 20cm resolution during scan matching
• 30cm resolution in final map

Courtesy by Giorgio Grisetti Cyrill Stachniss
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Exploration

• The approaches seen so far are purely passive.
• By reasoning about control, the mapping process
  can be made much more effective.

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Where to Move Next?
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Combining Rao-Blackwellized Mapping with
Exploration
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Exploration

• Given
• Unknown environment.
• Question
• How to control the robot so that it efficiently
  learns a map.

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Decision-Theoretic Formulation of Exploration
cost (path length)
reward (expected information gain)
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Naïve Approach to Combine Exploration and Mapping

• Learn the map using a Rao-Blackwellized particle
  filter.
• Apply an exploration approach that minimizes the
  map uncertainty.

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Disadvantage of the Naïve Approach

• Exploration techniques only consider the map
  uncertainty for generating controls.
• They avoid re-visiting known areas.
• Data association becomes harder.
• More particles are needed to learn a correct map.

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Application Example
Path estimated by the particle filter
True map and trajectory
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Map and Pose Uncertainty
pose uncertainty
map uncertainty
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Goal

• Integrated approach that considers
• exploratory actions,
• place revisiting actions, and
• loop closing actions
• to control the robot.

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Dual Representation for Loop Detection

• Trajectory graph stores the path traversed by the
  robot.
• Grid map represents the space covered by the
  sensors.
• Loops correspond to long paths in the trajectory
  graph and short paths in the geometric map.

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Dual Representation for Loop Detection
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Application Example
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Real Exploration Example
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Corridor Exploration
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Comparison
Map uncertainty only
Map and pose uncertainty
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Example Entropy Evolution
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Summary

• Rao-Blackwellization is well-suited for
  maintaining multiple hypotheses during occupancy
  grid mapping.
• Grid-based approaches can be scaled to larger
  environments by
• using appropriate data structures for the maps
  carried by the individual particles (DPSLAM), by
• using improved motion models (better proposals),
  by
• using adaptive re-sampling schemes, and by
• actively controlling the actions of the robot.

92
Potential Projects

• Dynamic environments
• Detection of errors
• Recovery from errors
• Three-dimensional maps
• Objects in maps
• Adaptive models (motion, sensor, )