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On Reconfiguring Radial Trees

On Reconfiguring Radial Trees. JCDCG2002 2002.12.8(Sun.). Akita Prefectural University. Yoshiyuki Kusakari. Linkages. joint. bar. A linkage is a collection of line segments possibly joined at their ends. bar: movable segment. joint: movable point.

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On Reconfiguring Radial Trees

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  1. On Reconfiguring Radial Trees JCDCG2002 2002.12.8(Sun.) Akita Prefectural University Yoshiyuki Kusakari

  2. Linkages joint bar A linkage is a collection of line segments possibly joined at their ends. bar: movable segment joint: movable point

  3. A reconfiguration (a continuous motion of bars)

  4. Wrong motions of a reconfiguration Any bar cannot be separated at the joint. Any bar can become neither longer nor shorter.

  5. Wrong motions of a planar reconfiguration Any bar can not move out of the plane. Any two bars can not cross each other.

  6. A planar reconfiguration A planar reconfiguration is the motion from an initial configuration to the desired configuration such that, during the motion, the topology of the linkage is invariant, the length of any bar is invariant, all bars are in the plane, and the configuration at any time is simple.

  7. Applications linkage robot arm motion planning reconfiguration A motion planning of robot arms Designing a manipulator Straightenable manipulators are desired.

  8. Fundamental questions１（Polygonal chains） the Carpenter's Rule Problem１ Can any polygonal chains recongigure any other configuration in the plane? ？ the Carpenter's Rule Problem 1' Can any polygonal chains be "straighten" in the plane? ？

  9. Known results 1 Theorem １[Connelly et al. ’00] Any polygonal chains can be straighten. ＯＫ

  10. Fundamental questions2 （Tree linkages） Problem 2 Can any tree linkages be "straighten" in the plane? ？

  11. Known results 2 Theorem 2 [Biedl et al. ’98] There exists a tree linkage which can not be straighten.

  12. Our problem Problem 2 Can any tree linkages be straighten? ＮＯ Problem 3 What kind of trees can be straighten?

  13. Known results 3 Theorem 3 [kusakari et al. ’02] Any monotone trees can be straighten. ＯＫ r root r

  14. Monotone path and monotone tree root r x-monotone path x-monotone tree

  15. Non-monotone path and non-monotone tree non-monotone path (in x-direction) non-monotone tree (in x-direction) root r

  16. In this talk: Problem 3 What kind of trees can be straighten? Are there other classes of trees which can be straighten? We give a negative result. Theorem 4 There exists a radial tree which can not be straighten. A radial tree is a natural modification of a monotone tree.

  17. Radial path and radial tree Radial path Radial tree : root

  18. Non-radial path and non-radial tree Non-Radial path Non-Radial tree : root

  19. The previous example This locked tree is not radial.

  20. A locked radial tree

  21. This radial tree can not be straighten.

  22. C -component 2p = 6 i This tree has six congruent C -components. i

  23. Subcomponents V i C L i i G i -component -component -component -component

  24. Radial monotonicity G i -component

  25. Lockableness 1 V i p V < i+1 2 These bars can not swing out.

  26. Lockableness 2 V V V i i i V V V i+1 i+1 i+1 C i Any -component can not be squeezed. Expanding the diagonal Reducing the diagonal

  27. Lockableness 3 C i Any -component can not be widened.

  28. Conclusion There exists a locked radial tree.

  29. The classes of trees general radial monotone counter example

  30. 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.

  31. END

  32. A quadranglar linkage can be reconfigured to any quadranglar linkage.

  33. Straightening the monotone tree 1 a pulling operation r r Any monotone tree can be straighen using only the pulling operations.

  34. Straightening the monotone tree 2 Order graph Tree Ｔ Order graph Ｇ　 Ｔ A vertex of the order graph G is a bar of tree T. Ｔ Edges of the order graph consists of two kind of edges: connecting edges and visible edges.

  35. The order applying the pulling operations isa reverse topological order of the order graph. Straightening the monotone tree 3 5 4 5 11 11 4 3 7 7 8 8 10 6 3 1 6 1 9 2 10 9 2 Tree Ｔ Order graph Ｇ　 Ｔ

  36. Connecting edges Tree Ｔ Connecting edges Ｅ con Directed edges each of which consecutively appear on the path from the root to a leaf.

  37. Visible edges Tree Ｔ Visible edges Ｅ　　 vis Directed edges from each bars to visible bars in x-direction.

  38. Connecting edges Ｅ Visible edges Ｅ　　 con vis Order Graph Ｇ　 Ｔ

  39. An property of the order graph The order graph of a monotone tree has no directed cycle. Order Graph Ｇ　 Ｔ The topological order can be found.

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