Introduction to Mobile Robotics Basics of LSQ Estimation, - - PowerPoint PPT Presentation

introduction to mobile robotics basics of lsq estimation
SMART_READER_LITE
LIVE PREVIEW

Introduction to Mobile Robotics Basics of LSQ Estimation, - - PowerPoint PPT Presentation

Introduction to Mobile Robotics Basics of LSQ Estimation, Geometric Feature Extraction Wolfram Burgard, Cyrill Stachniss, Maren Bennewitz, Giorgio Grisetti, Kai Arras Slides by Kai Arras Last update: June 2010 1 Feature Extraction:


slide-1
SLIDE 1

1

Wolfram Burgard, Cyrill Stachniss, Maren Bennewitz, Giorgio Grisetti, Kai Arras

Basics of LSQ Estimation, Geometric Feature Extraction Introduction to Mobile Robotics

Slides by Kai Arras Last update: June 2010

slide-2
SLIDE 2

2

Feature Extraction: Motivation

Landmarks for:

  • Localization
  • SLAM
  • Scene analysis

Examples:

  • Lines, corners, clusters: good for indoor
  • Circles, rocks, plants: good for outdoor
slide-3
SLIDE 3

Features: Properties

A feature/landmark is a physical object which is

  • static
  • perceptible
  • (at least locally) unique

Abstraction from the raw data...

  • type (range, image, vibration, etc.)
  • amount (sparse or dense)
  • origin (different sensors, map)

+ Compact, efficient, accurate, scales well, semantics − Not general

3

slide-4
SLIDE 4

4

Feature Extraction

Can be subdivided into two subproblems:

  • Segmentation: Which points contribute?
  • Fitting: How do the points contribute?

Segmentation Fitting

slide-5
SLIDE 5

5

Example: Local Map with Lines

Raw range data Line segments

slide-6
SLIDE 6

6

Example: Global Map with Lines

Expo.02 map

  • 315 m2
  • 44 Segments
  • 8 kbytes
  • 26 bytes / m2
  • Localization

accuracy ~1cm

slide-7
SLIDE 7

7

Example: Global Map w. Circles

Victoria Park, Sydney

  • Trees
slide-8
SLIDE 8

8

Split and Merge

Picture by J. Tardos

slide-9
SLIDE 9

9

Split and Merge

Algorithm

Split

  • Obtain the line passing by the two extreme points
  • Find the most distant point to the line
  • If distance > threshold, split and repeat with the left and

right point sets Merge

  • If two consecutive segments are close/collinear enough,
  • btain the common line and find the most distant point
  • If distance <= threshold, merge both segments
slide-10
SLIDE 10

10

Split and Merge: Improvements

  • Residual analysis before split

Split only if the break point provides a "better interpretation" in terms of the error sum

[Castellanos 1998] : start-, end-, break-point

slide-11
SLIDE 11

11

Split and Merge: Improvements

  • Merge non-consecutive segments

as a post-processing step

slide-12
SLIDE 12

12

Line Representation

Choice of the line representation matters!

Intercept-Slope Hessian model

Each model has advantages and drawbacks

slide-13
SLIDE 13

13

Fit Expressions

Given: A set of n points in polar coordinates Wanted: Line parameters ,

[Arras 1997]

slide-14
SLIDE 14

14

LSQ Estimation

Regression, Least Squares-Fitting Solve the non-linear equation system Solution (for points in Cartesian coordinates):

→ Solution on blackboard

slide-15
SLIDE 15

15

Can be formulated as a linear regression problem

Circle Extraction

Develop circle equation

slide-16
SLIDE 16

16

Circle Extraction

Solution via Pseudo-Inverse Leads to overdetermined equation system with vector of unknowns

(assuming that A has full rank)

slide-17
SLIDE 17

17

Fitting Curves to Points

Attention: Always know the errors that you minimize!

Algebraic versus geometric fit solutions

[Gander 1994]

slide-18
SLIDE 18

18

LSQ Estimation: Uncertainties?

How does the input uncertainty propagate

  • ver the fit expressions to the output?

X1, ..., Xn : Gaussian input random variables A, R : Gaussian output random variables

slide-19
SLIDE 19

19

Example: Line Extraction

Wanted: Parameter Covariance Matrix Simplified sensor model: all , independence Result: Gaussians in the parameter space

slide-20
SLIDE 20

20

Line Extraction in Real Time

  • Robot Pygmalion

EPFL, Lausanne

  • CPU: PowerPC

604e at 300 MHz Sensor: 2 SICK LMS

  • Line Extraction

Times: ~ 25 ms