Docsity
Docsity

Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity


Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan


Guidelines and tips
Guidelines and tips

Generalized Voronoi Graphs - Embedded Intelligent Robotics - Lecture Slides, Slides of Robotics

Course title is Embedded Intelligent Robotics. This course is for Electrical engineering students. Though good thing is everyone can learn about robotics in this course. This lecture includes: Generalized Voronoi Graphs, Sensor Based Exploration, Distance Function, Equidistant Face, Hierarchical Gvg, Gvg Tracing Function, Simulations, Hierarchical Voronoi Diagrams, Basic Motion Problem

Typology: Slides

2013/2014

Uploaded on 01/29/2014

surii
surii 🇮🇳

3.5

(13)

130 documents

1 / 28

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
Sensor Based Exploration:
Incremental Construction of the
Hierarchical Generalized
Voronoi Graph
docsity.com
pf3
pf4
pf5
pf8
pf9
pfa
pfd
pfe
pff
pf12
pf13
pf14
pf15
pf16
pf17
pf18
pf19
pf1a
pf1b
pf1c

Partial preview of the text

Download Generalized Voronoi Graphs - Embedded Intelligent Robotics - Lecture Slides and more Slides Robotics in PDF only on Docsity!

Sensor Based Exploration:

Incremental Construction of the

Hierarchical Generalized

Voronoi Graph

Why Sensor Based

  1. Classical work is based on the assumption that a robot has a

full knowledge of the world

  1. The problem: realistic deployment of robots into unknown

environments and into environments that are too difficult to

model

  1. Sensor based planning is important because:
    1. the robot often has no priori knowledge of the world or

may have only a coarse knowledge of the world

  1. the world model is bound to contain inaccuracies or

unexpected changes

Generalized

Voronoi Graphs

Generalized Voronoi Graph vs. Generalized

Voronoi Diagram

  • Generalized Voronoi Diagram

(GVD):for planar environment only.

  • Generalized Voronoi Graph

(GVG): a generalization of

the GVD into higher dimensions.

One dimensional.

A more concise representation

of the workspace or configuration space.

Equidistant Face

  • The building block of the GVG is two-equidistant face:
f { x R : 0 di ( x ) dj ( x ) dh ( x ) h i , j and di ( x ) dj ( x )}

m

ij         
  • Three-equidistant face:
  • By continuing intersection of the two-equidistant faces, a m-

equidistant face is formed, which is a one dimensional set of

points.

  • A m+1-equidistant face can be formed also, which is a

meet point.

  • The GVG is the collection of m-equidistant faces(edges) and

m+1-equidistant faces(meet point).

Equidistant Face

f ijk  fij  fik  fjk
  • Higher order GVG is defined to connect the GVG:

recursively defined on lower dimensional equidistant faces.

  • HGVG: the collection of all GVG and all higher order

GVG.

We will focus on R^3 only in the rest of this paper.

Hierarchical GVG

Second-Order GVG

  • Second order two-equidistant face:
f | { x fij : dh ( x ) dk ( x ) dl ( x ) di ( x ) dj ( x ) h i , j , k , l and dl ( x ) dk ( x )}

kl fij

GVG Tracing Function

  • Edge tracing:

trace the roots of the expression

as is varied.

x : point on the GVG.

z 1 : in the tangent direction of x.

At x , let the hyperplane

spanned by local coordinates

z 2 - z m be termed the normal

plane. The tracing function

This function assumes a zero

value only on the GVG.

G 1 ( y ,)  (^0) 

( )( , )

( )( , )

( )( , )

( , )

1

1 3

1 2

1

d d y

d d y

d d y

G y

m

GVG Edge Construction

  • edge construction:
  • predictor step: moves the robot for a small distance along the tangent

direction of the GVG

  • corrector step: find the intersection of the GVG and the correcting plane.

This is achieved through the Newton method:

It can be proved that the Jacobian matrix is always nonsingular.

( ) 1 ( , )

1 1

1 k k y

k k

y y G G y 

    

  • Meet point detection: by watching for an abrupt change in

the direction of the gradients to the m closest obstacles.

Some Details

Accessibility: using gradient ascent on the multi-object

distance function, moving in a direction to which the

sensor with the smallest value is facing.

Construction of the Second- Order GVG

  • This section applies the same tracing method for GVG to

trace the edges of second-order GVG.

  • The tracing function is:
  • Tangent direction: is the null space of the Jacobian of G 2 :

( )( , )

( )( , )

( )( , )

( , )

3

3 4

1 2

2

d d y

d d y

d d y

G y

m

19

  • Planar Simulations

Simulations