On Reconfiguring Radial Trees

Yoshiyuki Kusakari

Akita Prefectural University

JCDCG20022002.12.8(Sun.)

Linkages

bar

joint

A linkage is a collection of line segments possibly joined at their ends.

bar: movable segment

joint: movable point

A reconfiguration (a continuous motion of bars)

Wrong motions of a reconfiguration

Any bar can not be separated at the joint.

Any bar can become neither longer nor shorter.

Any bar can not move out of the plane.

Any two bars can not cross each other.

Wrong motions of a planar reconfiguration

A planar reconfiguration

the length of any bar is invariant,

all bars are in the plane, and

A planar reconfiguration is the motion from an initial configuration to the desired configurationsuch that, during the motion,

the topology of the linkage is invariant,

the configuration at any time is simple.

Applications

linkagerobot arm

motion planning reconfiguration

Designing a manipulator

A motion planning of robot arms

Straightenable manipulators are desired.

Fundamental questions１（ Polygonal chains）

Can any polygonal chains recongigure any other configuration in the plane?

the Carpenter's Rule Problem１

the Carpenter's Rule Problem 1'

Can any polygonal chains be "straighten" in the plane?

Any polygonal chains can be straighten.

Theorem １ [Connelly et al. ’00]

Known results 1

Can any tree linkages be "straighten" in the plane?

Problem 2

Fundamental questions2 （ Tree linkages）

There exists a tree linkage which can not be straighten.

Theorem 2 [Biedl et al. ’98]

Known results 2

Our problem

What kind of trees can be straighten?

Problem 3

Problem 2

Can any tree linkages be straighten?

Known results 3

rroot r

Any monotone trees can be straighten.

Theorem 3 [kusakari et al. ’02]

Monotone path and monotone tree

x-monotone path

x-monotone tree

rroot

Non-monotone path and non-monotone tree

non-monotone path (in x-direction)

rroot

non-monotone tree (in x-direction)

What kind of trees can be straighten?

Problem 3

In this talk:

We give a negative result.

There exists a radial tree which can not be straighten.

Theorem 4

A radial tree is a natural modification of a monotone tree.

Are there other classes of trees which can be straighten?

Radial path and radial tree

Radial path Radial tree

: root

Non-radial path and non-radial tree

Non-Radial path Non-Radial tree

: root

The previous example

This locked tree is not radial.

A locked radial tree

This radial tree can not be straighten.

C -componenti

This tree has six congruent C -components.i

=

6

Subcomponents

Ci

i

Li

Vi

-component

-component

-component

-component

Radial monotonicity

i-component

Lockableness 1

These bars can not swing out.

<

2

Vi

Vi+1

Lockableness 2

Vi

Vi+1

Ci-componentAny can not be squeezed.

Vi

Vi+1

Vi

Vi+1

Expanding the diagonal

Reducing the diagonal

Lockableness 3

Ci-componentAny can not be widened.

Conclusion

There exists a locked radial tree.

The classes of trees

class of treesStraightenablegeneralmonotoneradial

general

monotoneradial

counter example

Future works

Find a necessary and sufficient condition for straightenable tree in the plane.

Find a class of threes such that any trees in the class can be straighten in the 3D space.

A quadranglar linkage can be reconfigured to any quadranglar linkage.

Straightening the monotone tree 1

a pulling operation

r r

Any monotone tree can be straighen using only the pulling operations.

Order graph Ｇ　Tree Ｔ Ｔ

A vertex of the order graph G is a bar of tree T.

Edges of the order graph consists of two kind of edges:connecting edges and visible edges.

Order graph

Ｔ

Straightening the monotone tree 2

The order applying the pulling operations isa reverse topological order of the order graph.

Order graph Ｇ　Ｔ

1

3

2

45

6

78

910

11

Tree Ｔ

13

2

45

6

78

910

11

Straightening the monotone tree 3

Connecting edges

Tree Ｔ Connecting edges Ｅ　　con

Directed edges each of which consecutively appear on the path from the root to a leaf.

Visible edges Ｅ　　

Visible edges

Tree Ｔ vis

Directed edges from each bars to visible barsin x-direction.

Visible edges Ｅ　　visConnecting edges Ｅ　　con

Order Graph Ｇ　Ｔ

An property of the order graph

Order Graph Ｇ　Ｔ

The order graph of a monotone tree has no directed cycle.

The topological order can be found.