Scientific Paper about mathematical Modeling of Rigid Origami.Calculability and representation of the systems.
Simulation of Rigid OrigamiTomohiroTachiThe University of
[email protected] paper presents a system
for computer based interactive simulation of
origami,basedonrigidorigami model.
Oursystemshowsthecontinuousprocessoffold-ing a piece of paper into
a folded shape by calculating the conguration from thecrease
pattern. The conguration of the model is represented by crease
angles,
andthetrajectoryiscalculatedbyprojectinganglesmotiontotheconstrainedspace.Additionally,
polygonsaretriangulatedandcreaselinesareadjustedinordertomake the
model more exible. Flexible motion and comprehensible
representationof origami help users to understand the structure of
the model and to fold the modelfrom a crease pattern.1
IntroductionSimulation of origami is very important for
representing or modeling origami in computer.It helps paper folders
to understand the structures of origami models, and can be used asa
tool for designers to draw diagrams. We propose a system for
simulating folding
processfromcreasepatterntofoldedbase,basedonrigidorigamisimulation(Figure1).Therehavebeenmanyapproachesfor
simulatingorigami. Onewayistosimulateorigami by a sequence of
simple folding steps as shown in [1] by Miyazaki et al. Since
thenal state of an origami model is represented by folding steps,
it is easy to reconstruct ananimationfromasheetofpapertothemodel.
However,theorigamimodelsthatcanberepresentedinthissystemarelimitedbythesimplicityoffoldingsteps,andnotsuitableformanycomplexorigami
modelswhosefoldingprocesscannotbedividedintosimplefoldingsteps.Figure1:
Screenshotsofthesimulationprogram. Left:
baseofKamehamehaWave(bytheauthor). Middle: pleatedhypar. Right:
baseofTachikoma(bytheauthor).Thus representing origami models by
crease pattern seems suitable. ORIPA is a
creasepatterneditorwithestimationoffoldedgure,developedbyMitani[2].
Finalstatewithstackingorderisestimated,
thoughsometimestheestimationfails. Thicknessof
paperisrepresentedforbetterunderstandingof thestructureof themodel,
buttherearenointermediatestatefromcreasepatterntothenalshape.It is
possible to simulate intermediate state of folding, by using rigid
origami model
(i.e.platesconnectedbyhingesconstrainedaroundvertices).
Balkcom[3]usesrigidorigamimodel fororigami simulation.
Hismethodisbasedonvirtual cuttingandcombinationof
forwardandinversekinematics. Thecalculationof thetrajectoryruns
fast inthismethod, butthemountain-valleyvaluesof
somecreasesarenottakenintoaccountforthefoldingmotion.Oursystemisbasedonrigidorigami
simulation, andusescreaseanglesof all foldlines as variables to
represent conguration of an origami model. The simulation
methodisbasedonprojectiontotheconstraintspace,
thuspossibletousefoldingmotionofallcreaselinesasdrivingforceof
folding. Thisresultsinarobustoverall motionwhichisindependent of
thepositions of thevirtual cutting. Additionally, our
systemcanaddexibility to origami models by adding and adjusting
crease lines. In this method,
foldingmotionofnonrigidlyfoldablemodelscanalsobesimulated.Thesystemprovidesasmoothandcomprehensibleinteractiveanimationof
foldingfromacreasepatterntothefoldedbase,
whichhelpsuserstounderstandthestructureoftheorigamimodelandthewaytofoldthemodelfromthecreasepattern.
Thishelpspaperfolderstofoldmodelsfromacreasepattern,andalsoencouragesdesignerstouseacreasepatternasamediumforpublishingorigamimodels.2
KinematicsofRigidOrigamiInourmodel,anorigamicongurationisrepresentedbyfoldinganglesofedges.
Foldingangleschangeaccordingtomountain-valleyvaluesofthefoldlines,
whenthepapergetfolded.
Thoseedgesareconnectedwithfacetsandformclosedloops,andforeachclosedloopwehaveaconstraintonanglemovement.
If thereisnoholeinsidetheloop,
thentheconstraintissatisedbyintersectionofconstraintsbysingleverticesinsidetheloop(Figure2).
Our model assumes that thepaper is without ahole.
Foldingmotionisnumericallycalculatedusinglinearapproximationofconstraintsbysinglevertices.Figure
2: Left: No hole is inside the loop and the constraint is satised
by the intersectionofsingle-vertexconstraintsinside. Right:
Aholeisinsidetheloopandwehavetothinkabouttheconstraintaroundthehole.Necessary
conditions for single-vertex rigid origami shown in [4] by
Belcastro and Hullareusedasconstraintsof multi-vertexrigidorigami
model. Supposetherearenedges
1, , n connected to the vertex. For a single-vertex origami, the
33 matrix constraintfunctionFoffoldangles1, , nisgivenbyF(1, , n) =
1 n1n= I (1)wherematrices1, , nrepresentrotationbyfoldlines1, ,
nrespectively(Figure3).
Derivativeoftheequationisgivenby1234B12B23B34B41C1(1)C2(2)C3(3)C4(4)X1=C1B12X2=C2B23X3=C3B34X4=C4B41Figure3:
Anexampleofrotationmatricesofasingle-vertexorigami.dFdt=F1 1 + +Fi
i + +Fn n=__0 0 00 0 00 0 0__(2)Weget9equations(i.e.
oneforeachelementof F)foranglesmovement1, , nrepresentedusing9
nmatrix.__F1 (1,1) Fn(1,1)F1 (1,2) Fn(1,2)......F1 (3,3) Fn(3,3)__.
.9nmatrix__ 1... n__ =__0...0__(3)However, since Fis
arotationmatrix(i.e. orthogonal matrix) the equations areredundant
and only 3 of 9 equations are independent. Partial derivative of an
orthogonalmatrixFatF =
Iisaskew-symmetricmatrixbecauseFi+_Fi_TF=I=FiFT+FFTiF=I=i_FFT_F=I=
0 (4)Partial
derivativeoftheconstraintfunctionisrepresentedinthefollowingformusing3independentvariablesa,
b, c.Fi=__0 a ca 0 bc b 0__(5)Wegeta3 nmatrixinsteadofa9
nmatrixshownin(3),__a1 anb1 bnc1 cn____ 1... n__ =__0...0__(6)Then
we rewrite the matrix using global edge number. Assume that edge i
is connectedandedgej isnotconnectedtovertexk.
ConstraintbyvertexkisrepresentedbythematrixCk, whose ithcolumnis
identical tothe columnof the previous
matrixthatcorrespondstotheedge,andjthcolumniszerovector.[Ck]__ 1...
N__ =ith jth__ aki 0 bki 0 cki 0 __. .3Nmatrix__ 1... i... j...
N__=__000__(7)whereNisthenumberofedges.
Globalconstraintsbymultipleofverticesaregivenbytheintersectionoftheconditions.C
=__[C1]...[CM]__. .3MNmatrix__ 1... N__
=__0...0__(8)whereMisthenumberofverticesinsidethepaper.IfandonlyiftherankofmatrixCislessthanN,thelinearequationhasnon-trivialsolutions.
Thesolutionofthislinearequationiscalculatedusingpseudo-inversematrixC+ofmatrixC.
=_IN C+C 0(9)where 0 is unconstrained value of the angles velocity
calculated by mountain-valley valueofthefoldedges.
Theideaistoprojectunconstrainedanglesmovementtoconstrainedspacebyusingorthogonal
projectionmatrix[IN C+C].
TrajectorycanbecalculatedbyEulerintegration. = (t)t =_IN
C+C0(10)However this Euler integration results in accumulation of
numerical error, so we modifyequation(8)usingtheresidualvectorr.C =
r (11)wherer =_r1ar1br1c rkarkbrkc rMarMbrMcT(12)where rka, rkb,
rkcare the residual of elements of F corresponding to ak, bk,
ckrespectively.F =__1 0 ra0 + rc0 + ra1 0 rb0 rc0 +
rb1__(13)Themodiedsolutionis = C+r +_IN
C+C0(14)Pseudo-inverseiscalculatedasC+=CT_CCT_1if Cisfull
rankand3M
