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Rigid Origami Simulation

Rigid Origami Simulation. Tomohiro Tachi The University of Tokyo http://www.tsg.ne.jp/TT/. About this presentation. For details, please refer to Tomohiro Tachi, "Simulation of Rigid Origami" in Origami^4 : proceedings of 4OSME (to appear). 1. Introduction. Rigid Origami?.

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Rigid Origami Simulation

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  1. Rigid Origami Simulation Tomohiro Tachi The University of Tokyo http://www.tsg.ne.jp/TT/

  2. About this presentation For details, please refer to • Tomohiro Tachi, "Simulation of Rigid Origami" in Origami^4 : proceedings of 4OSME (to appear)

  3. 1 Introduction

  4. Rigid Origami? • rigid panels + hinges • simulates 3 dimensional continuous transformation of origami • →engineering application: deployable structure, foldable structure

  5. Rigid Origami Simulator • Simulation system for origami from general crease pattern. • 3 dimensional and continuous transformation of origami • Designing origami structure from crease pattern.

  6. Software and galleries Software is available: http://www.tsg.ne.jp/TT/software/ flickr:tactom YouTube:tactom

  7. 2 Kinematics Single-vertex model Constraints Kinematics

  8. r 2 r 3 l 2 l 3 r 1 l 1 l 4 r 4 Model • Rigid origami model (rigid panel + hinge) • Origami configuration is represented by fold angles denoted as r between adjacent panels. • The configuration changes according to the mountain and valley assignment of fold lines. • The movement of panels are constrained around each vertex.

  9. r C ( ) 2 2 r C ( ) 3 3 B l 2 23 B 12 l 3 r C ( ) 1 1 l B 1 34 B 41 l 4 r C ( ) 4 4 Constraints of Single Vertex • single vertex rigid origami[Belcastro & Hull 2001] • equations represented by 3x3 rotating matrix

  10. Derivative of the equation 3x3=9 equations for each vertex F is orthogonal matrix: 3 of 9 equations are independent (6 is redundant)

  11. r 2 r 3 l 2 l 3 r 1 l 1 l 4 r 4 3 independent equations Derivative of orthogonal matrix F at F=I is skew-symmetric.Let denote direction cosine of li, then

  12. Constraints matrix constraints around vertex kis, For the entire model,

  13. Constraints of multi-vertex (general) origami single vertex: M vertex model: Iff N>rank(C), the model transforms, and the degree of freedom is N- rank(C) (If not singular, rank(C)=3M)

  14. Kinematics Constraints: When the model transforms, the equation has non-trivial solution. 　　represents the velocity of angle change when there are no constraints.

  15. numerical integration Euler integration >Accumulation of numeric error • Use residual of F corresponding to the global matrix elements.

  16. Euler Integration + Newton method Ideal trajectory Euler method + Newton method The solution is, Rawangle change Constrained angle change

  17. 3 System

  18. System • Input is 2D crease pattern in dxf or opx format • Real-time calculation of kinematics • Conjugate Gradient method • Runs interactively to • Local collision avoidance • penalty force avoids collision between adjacent facets • Implementation • C++, OpenGL, ATLAS • now availablehttp://www.tsg.ne.jp/TT/software/

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