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

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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

2. The problem: realistic deployment of robots into unknown

environments and into environments that are too difficult to

model

3. 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

2. 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