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)

Introduction

1

Rigid Origami?• rigid panels + hinges• simulates 3 dimensional continuous

transformation of origami• →engineering application:

deployable structure, foldable structure

Rigid Origami Simulator

• Simulation system for origami from general crease pattern.

• 3 dimensional and continuous transformation of origami

• Designing origami structure from crease pattern.

Software and galleries

Software is available:http://www.tsg.ne.jp/TT/software/

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Kinematics

•Single-vertex model

•Constraints

•Kinematics

2

Model

• Rigid origami model (rigid panel + hinge)

• Origami configuration is represented by fold angles denoted as 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.

12

3

41

23

4

Constraints of Single Vertex

• single vertex rigid origami[Belcastro & Hull 2001]

• equations represented by 3x3 rotating matrix

1

2

3

4

B12B23

B34B41

C1(1)

C2(2)

C3(3)

C4(4)

I nn 114144

3433

2322

1211

BC

BC

BC

BC

Derivative of the equation

000

000

000

,...,

,...,

1

1

1

111

n

n

i

i

n

nnn

dt

d

FFFF

IF

0

0

)3,3()3,3(

)2,1()2,1(

)1,1()1,1(

1

matrix9

1

1

1

n

n

n

n

n

FF

FF

FF3x3=9 equations for each vertex

F is orthogonal matrix:3 of 9 equations are independent (6 is redundant)

3 independent equations

Derivative of orthogonal matrix F at F=I is skew-symmetric.

Let denote direction cosine of i, then

0

0

0

ii

ii

ii

i

F

i

i

i

1

23

4

1

23

4

Constraints matrix

from lines fold ofnumber theis

0

0

01

1

1

1

kn

nkn

kn

kn

k

k

k

constraints around vertex k is,

lines fold ofnumber :N

vertex to

connectednot is line fold

connected is line fold

0

0

0

0

0

0 1

1

k

j

i

ji

N

k

N

j

i

ki

ki

ki

C

For the entire model,

0

01

matrix3

1

ρC

C

C

N

NM

M

single vertex:

0

0

01

3N

N

k

C

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)

Kinematics

1TT0

rank -full is if

of inverse-pseudo theis where

CCCCC

CCρCCIρ N

Constraints:0ρC

When the model transforms, the equation has non-trivial solution.

0ρ represents the velocity of angle change when there are no constraints.

100

010

001

0

0

0

rr

rr

rr

ii

ii

ii

i

FF

numerical integration

Euler integration

Δρ

ρρρ Δttttt

Accumulation of numeric error

• Use residual of F corresponding to the global matrix elements.

Euler method + Newton method

0ρCCIrCρ N

M

cM

c

b

a

r

r

r

r

31

1

1

where

rrρC

The solution is,

0ρ

0ρCC

rC

ρ

0r

rr

Ideal trajectory

+ Newton method

Euler Integration

Constrained angle change

Rawangle change

System

3

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 available

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