Transcript 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.

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?
• 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:tactom
YouTube:tactom
2
Kinematics
•Single-vertex model
•Constraints
•Kinematics
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.
r3
r2
l3
l2
l1
l4
r4
r1
Constraints of Single Vertex
1  C1B12
 2  C 2B 23
 3  C3B 34
 4  C 4B 41
• single vertex rigid origami
[Belcastro & Hull 2001]
1   n1 n  I
• equations represented by
3x3 rotating matrix
C2(r2)
C3(r3)
l3
B34
B23 l2
B12
B41
l4
C4(r4)
l1
C1(r1)
Derivative of the equation
Fr1 ,..., r n   1   n1 n   I

d
Fr1 ,..., r n 
dt
0 0 0 
F
F
F

r1   
r i   
r n  0 0 0
r1
ri
r n
0 0 0
F
 F

(
1
,
1
)

(
1
,
1
)
 r

r n
1

  r  0

F

F
1

(1,2) 
(1,2)     
 r1
     
r n
F is orthogonal matrix:

    


 r n  0 


3 of 9 equations are
 F (3,3)  F (3,3)
independent (6 is redundant)  r1

r n


3x3=9 equations for each
vertex
9n matrix
3 independent equations
Derivative of orthogonal matrix F at F=I is
skew-symmetric.
 li 
Let   i  denote direction cosine of li, then
 
 i 
 0
F 
  i
r i
 i
 i
0
li
i 
 li 
0 
r3
r2
l2
l3
l1
l4
r4
r1
Constraints matrix
constraints around vertex k is,
 lk1  lkn   r1  0

     0 


kn  
 k1
  
 k1   kn   r n  0
n is the number of fold lines from k 
For the entire model,
i

 lki
 
ki

  ki
 r1 
  

 
 0   r i 
 r1  0
 
 0      Ck     0
 r N  0
 0   r j 
 
  
 r 
 N
j
 fold line i is connected



 fold line j is not connected
 to vertex k



 N : number of fold lines 


Constraints of multi-vertex
(general) origami
single vertex:
M vertex model:
 r1  0





C k      0 
3 N  r
  N  0
C1    r1 

0 

     Cρ    


 
 
 C M    r N 
0


3 M  N matrix
Iff N>rank(C), the model transforms, and the
degree of freedom is N- rank(C)
(If not singular, rank(C)=3M)
Kinematics
Constraints:
Cρ  0
When the model transforms, the equation has nontrivial solution.


ρ  I N  CC ρ 0
 where C is the pseudo - inverse of C 





T
T 1
 if C is full - rank C  C CC



ρ 0 represents the velocity of angle change when there
are no constraints.
numerical integration
Euler integration
ρt  t   ρt   ρ t Δt

Δρ
>Accumulation of numeric error
• Use residual of F corresponding to the global matrix
elements.
 0
F 
  i
r i
 i
 i
0
li
 1
i 


 li  F   0  r
0 
0  r

0  r
1
0  rl
0  r 

0  rl 
1 
Euler method + Newton method
Euler
Integration
 CCρ0
 ra1 
r 
 b1 
Cρ  r where r   rc1 
 
  
rcM 
The solution is,





3M



 Cr
ρ0
ρ
+ Newton
method
r r
r0
Ideal trajectory

ρ  Cr  I N  CC ρ0
Constrained
angle change
Raw
angle change
3
System
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/