On Reconfiguring Radial Trees Yoshiyuki Kusakari Akita Prefectural University JCDCG2002...
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Transcript of On Reconfiguring Radial Trees Yoshiyuki Kusakari Akita Prefectural University JCDCG2002...
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
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?
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?
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?
Lockableness 2
Vi
Vi+1
Ci-componentAny can not be squeezed.
Vi
Vi+1
Vi
Vi+1
Expanding the diagonal
Reducing the diagonal
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.
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.