rigid origami simulation tomohiro tachi the university of tokyo
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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/
flickr:tactomYouTube:tactom
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/