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. Introduction 1

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. Kinematics Single-vertex model Constraints Kinematics 2

8. 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. l 1 l 2 l 3 l 4 r 1 r 2 r 3 r 4

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

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

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

13. single vertex: M vertex model: Constraints of multi-vertex (general) origami 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 method + Newton method The solution is, Ideal trajectory + Newton method Euler Integration Constrained angle change Raw angle change

17. System 3

18. 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 available http://www.tsg.ne.jp/TT/software/