Known Poses Wolfram Burgard, Michael Ruhnke, Bastian Steder 1 Why - - PowerPoint PPT Presentation

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Known Poses Wolfram Burgard, Michael Ruhnke, Bastian Steder 1 Why - - PowerPoint PPT Presentation

Apr 14, 2023 202 likes •853 views

Introduction to Robot Mapping Mobile Robotics Grid Maps and Mapping With Known Poses Wolfram Burgard, Michael Ruhnke, Bastian Steder 1 Why Mapping? Learning maps is one of the fundamental problems in mobile robotics Maps allow

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Robot Mapping Grid Maps and Mapping With Known Poses

Wolfram Burgard, Michael Ruhnke, Bastian Steder

Introduction to Mobile Robotics

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Why Mapping?

  • Learning maps is one of the

fundamental problems in mobile robotics

  • Maps allow robots to efficiently carry
  • ut their tasks, allow localization …
  • Successful robot systems rely on maps

for localization, path planning, activity planning etc.

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The General Problem of Mapping

What does the environment look like?

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The General Problem of Mapping

  • Formally, mapping involves, given the

sensor data

  • to calculate the most likely map
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The General Problem of Mapping

  • Formally, mapping involves, given the

sensor data

  • to calculate the most likely map
  • Today we describe how to calculate

a map given the robot’s pose

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The General Problem of Mapping with Known Poses

  • Formally, mapping with known poses

involves, given the measurements and the poses

  • to calculate the most likely map
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Features vs. Volumetric Maps

Courtesy by E. Nebot

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Grid Maps

  • We discretize the world into cells
  • The grid structure is rigid
  • Each cell is assumed to be occupied or

free

  • It is a non-parametric model
  • It requires substantial memory

resources

  • It does not rely on a feature detector
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Example

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Assumption 1

  • The area that corresponds to a cell is

either completely free or occupied

free space

  • ccupied

space

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Representation

  • Each cell is a binary random

variable that models the occupancy

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Occupancy Probability

  • Each cell is a binary random

variable that models the occupancy

  • Cell is occupied
  • Cell is not occupied
  • No information
  • The environment is assumed to be

static

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Assumption 2

  • The cells (the random variables) are

independent of each other

no dependency between the cells

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Representation

  • The probability distribution of the map is

given by the product of the probability distributions of the individual cells

cell map

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Representation

  • The probability distribution of the map is

given by the product of the probability distributions of the individual cells

four-dimensional vector four independent cells

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Estimating a Map From Data

  • Given sensor data and the poses
  • f the sensor, estimate the map

binary random variable Binary Bayes filter (for a static state)

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Static State Binary Bayes Filter

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Static State Binary Bayes Filter

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Static State Binary Bayes Filter

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Static State Binary Bayes Filter

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Static State Binary Bayes Filter

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Static State Binary Bayes Filter

Do exactly the same for the opposite event:

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Static State Binary Bayes Filter

  • By computing the ratio of both

probabilities, we obtain:

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Static State Binary Bayes Filter

  • By computing the ratio of both

probabilities, we obtain:

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Static State Binary Bayes Filter

  • By computing the ratio of both

probabilities, we obtain:

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Occupancy Update Rule

  • Recursive rule
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Occupancy Update Rule

  • Recursive rule
  • Often written as
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Log Odds Notation

  • Log odds ratio is defined as
  • and with the ability to retrieve
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Occupancy Mapping in Log Odds Form

  • The product turns into a sum
  • or in short
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Occupancy Mapping Algorithm

highly efficient, only requires to compute sums

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Occupancy Grid Mapping

  • Developed in the mid 80’s by Moravec

and Elfes

  • Originally developed for noisy sonars
  • Also called “mapping with know poses”
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Inverse Sensor Model for Sonars Range Sensors

In the following, consider the cells along the optical axis (red line)

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Occupancy Value Depending on the Measured Distance

z+d1 z+d2 z+d3 z z-d1

measured dist.

prior distance between the cell and the sensor

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z+d1 z+d2 z+d3 z z-d1

Occupancy Value Depending on the Measured Distance

measured dist.

prior

“free”

distance between the cell and the sensor

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z+d1 z+d2 z+d3 z z-d1

Occupancy Value Depending on the Measured Distance

distance between the cell and the sensor

measured dist.

prior

“occ”

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Occupancy Value Depending on the Measured Distance

z+d1 z+d2 z+d3 z z-d1

measured dist.

prior

“no info”

distance between the cell and the sensor

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Update depends on the Measured Distance and Deviation from the Optical Axis

cell l

  • Linear in
  • Gaussian in
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Intensity of the Update

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Resulting Model

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Example: Incremental Updating

  • f Occupancy Grids
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Resulting Map Obtained with Ultrasound Sensors

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Resulting Occupancy and Maximum Likelihood Map

The maximum likelihood map is obtained by rounding the probability for each cell to 0 or 1.

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Inverse Sensor Model for Laser Range Finders

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Occupancy Grids From Laser Scans to Maps

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Example: MIT CSAIL 3rd Floor

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Uni Freiburg Building 106

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Alternative: Counting Model

  • For every cell count
  • hits(x,y): number of cases where a beam

ended at <x,y>

  • misses(x,y): number of cases where a

beam passed through <x,y>

  • Value of interest: P(reflects(x,y))
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The Measurement Model

  • Pose at time t:
  • Beam n of scan at time t:
  • Maximum range reading:
  • Beam reflected by an object:

0 1

measured

  • dist. in #cells
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The Measurement Model

  • Pose at time t:
  • Beam n of scan at time t:
  • Maximum range reading:
  • Beam reflected by an object:

0 1

max range: “first zt,n-1 cells covered by the beam must be free”

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The Measurement Model

  • Pose at time t:
  • Beam n of scan at time t:
  • Maximum range reading:
  • Beam reflected by an object:

0 1

  • therwise: “last cell reflected beam, all others free”

max range: “first zt,n-1 cells covered by the beam must be free”

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Computing the Most Likely Map

  • Compute values for m that maximize
  • Assuming a uniform prior probability for

P(m), this is equivalent to maximizing:

since independent and only depend on

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Computing the Most Likely Map

cells beams

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Computing the Most Likely Map

“beam n ends in cell j”

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Computing the Most Likely Map

“beam n ends in cell j” “beam n traversed cell j”

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Computing the Most Likely Map

Define

“beam n ends in cell j” “beam n traversed cell j”

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Meaning of aj and bj

Corresponds to the number of times a beam that is not a maximum range beam ended in cell j ( ) Corresponds to the number of times a beam traversed cell j without ending in it ( )

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Computing the Most Likely Map

Accordingly, we get If we set Computing the most likely map reduces to counting how often a cell has reflected a measurement and how often the cell was traversed by a beam. we obtain As the mj’s are independent we can maximize this sum by maximizing it for every j

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Difference between Occupancy Grid Maps and Counting

  • The counting model determines how
  • ften a cell reflects a beam.
  • The occupancy model represents

whether or not a cell is occupied by an

  • bject.
  • Although a cell might be occupied by

an object, the reflection probability of this object might be very small.

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Example Occupancy Map

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Example Reflection Map

glass panes

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Example

  • Out of n beams only 60% are reflected from a cell

and 40% intercept it without ending in it.

  • Accordingly, the reflection probability will be 0.6.
  • Suppose p(occ | z) = 0.55 when a beam ends in a

cell and p(occ | z) = 0.45 when a beam traverses a cell without ending in it.

  • Accordingly, after n measurements we will have
  • The reflection map yields a value of 0.6, while the
  • ccupancy grid value converges to 1 as n increases.

2 . * 4 . * 6 . * 4 . * 6 . *

9 11 9 11 * 9 11 55 . 45 . * 45 . 55 .

n n n n n

                               

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Summary (1)

  • Grid maps are a popular model for

representing the environment

  • Occupancy grid maps discretize the

space into independent cells

  • Each cell is a binary random variable

estimating if the cell is occupied

  • We estimate the state of every cell using

a binary Bayes filter

  • This leads to an efficient algorithm for

mapping with known poses

  • The log odds model is fast to compute
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Summary (2)

  • Reflection probability maps are an

alternative representation

  • The key idea of the sensor model is to

calculate for every cell the probability that it reflects a sensor beam

  • Given the this sensor model, counting

the number of times how often a measurement intercepts or ends in a cell yields the maximum likelihood model