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Rinki Imada
Department of General Systems Studies,
Graduate School of Arts and Sciences,
The University of Tokyo,
Meguro-ku, Tokyo 153-8902, Japan
e-mail: r-imada@g.ecc.u-tokyo.ac.jp
Tomohiro Tachi
Department of General Systems Studies,
Graduate School of Arts and Sciences,
The University of Tokyo,
Meguro-ku, Tokyo 153-8902, Japan
e-mail: tachi@idea.c.u-tokyo.ac.jp
Geometry and Kinematics
of Cylindrical Waterbomb
Tessellation
Folded surfaces of origami tessellations have attracted much attention because they often
exhibit nontrivial behaviors. It is known that cylindrical folded surfaces of waterbomb tes-
sellation called waterbomb tube can transform into peculiar wave-like surfaces, but the the-
oretical reason why wave-like surfaces arise has been unclear. In this paper, we provide a
kinematic model of waterbomb tube by parameterizing the geometry of a module of water-
bomb tessellation and derive a recurrence relation between the modules. Through the visu-
alization of the configurations of waterbomb tubes under the proposed kinematic model, we
classify solutions into three classes: cylinder solution, wave-like solution, and finite solu-
tion. Through the stability and bifurcation analyses of the dynamical system, we investigate
how the behavior of waterbomb tube changes when the crease pattern is changed. Further-
more, we prove the existence of a wave-like solution around one of the cylinder solutions.
[DOI: 10.1115/1.4054478]
Keywords: dynamics, folding and origami, mechanism design
1 Introduction
Origami tessellations are origami obtained by tiling translational
copies of a modular crease pattern. Origami tessellations can be
folded from flat sheets of paper and transform into various
shapes. Even though origami tessellations are corrugated polyhedral
surfaces, they sometimes approximate smooth surfaces of various
curvatures, including synclastic and anticlastic surfaces [1]. Such
macroscopic surfaces exhibit nontrivial behaviors we cannot
expect from the periodicity of its crease pattern and recently
attracted much attention of scientists and engineers. For example,
Schenk and Guest [2] showed that Miura-ori forms anticlastic sur-
faces while egg-box pattern forms synclastic surfaces through
numerical modeling using bar-and-hinge model. Nasser et al.
[3,4] analyzed the approximated surface of Miura-ori and egg-box
pattern based on the knowledge of differential geometry.
In this paper, we focus on folded surfaces of waterbomb tessella-
tion, specifically, cylindrical folded surfaces called waterbomb
tube. Waterbomb tube is known as origami work “Namako”[5]
or more commonly, the name “Magic Ball.”Based on the property
of waterbomb tube, various engineering applications are attempted,
such as origami-stent graft [6], earthworm-like robot [7], soft
gripper [8], and robot wheel [9] using the property that the radius
of the tube can vary. The dynamics of the stent graft and the
wheel inspired by waterbomb tube has been analyzed [10,11].
Also, the kinematics and structural deformation of waterbomb
tube has been studied [12–14]. One of the most interesting phenom-
ena reported on waterbomb tube is that it can form wave-like sur-
faces [15,16] like in Fig. 1. This is the unique phenomenon not
yet observed in the folded surfaces of other tessellations such as
Miura-ori and egg-box pattern; however, the theoretical reason
why wave-like surfaces arise has been unclear.
Here, our objective is to know “why”waterbomb tube produces
wave-like surfaces, i.e., to clarify the mathematics behind the beha-
vior. In this paper, we model waterbomb tube as the sequence of
modules and focus on its recurrence behavior instead of simulta-
neously solving the network of 6R spherical linkages as in
[12,13,15,17]. We first provide the kinematic model of each
module of waterbomb tube and the relation with adjacent
modules to obtain the recurrence relation dominating the folded
states of modules based on the rigid origami model with symmetry
assumption (Sec. 2). Then, through computation and visualization
of the folded states of waterbomb tube using the recurrence relation,
we observe that the solutions fall into three types: cylinder solution,
wave-like solution, and finite solution (Sec. 3). Furthermore, by
applying the stability and bifurcation analyses of the dynamical
system of waterbomb, we illustrate how the behavior of the
system, i.e., whether waterbomb tube can be wave-like surface or
not, changes when the crease-pattern parameters are changed
(Sec. 4). Finally, we prove the existence of wave-like solution by
applying theorems of discrete dynamical systems (Sec. 5).
2 Model
2.1 Definition. First, we introduce parameters of the crease
pattern of waterbomb tessellation. We can obtain the crease
pattern of waterbomb tessellation by tiling translational copies of
a unit module. The entire pattern is controlled by four parameters
(Fig. 2): a unit module shown in Fig. 2, left, is controlled by two
parameters α,β(α∈(0, 90°), β∈(0, 180° −α)), and the repeating
number of modules in column and row directions is given by two
integers Nand M(N∈Z>0,M∈Z>0).
We consider a waterbomb tube as the rigid folded state of the
crease pattern of waterbomb tessellation, such that the left and
right sides in Fig. 2are connected to form a cylindrical form. In
our model, we assume N-fold symmetry about an axis and mirror-
symmetry about Nplanes passing through the axis as in the existing
research [13] (see Fig. 3). However, unlike the previous research
[13], we do not assume mirror symmetry with respect to a plane per-
pendicular to the axis. Here, the behavior of waterbomb tube is gov-
erned by three parameters (α,β,N), because Mspecifies the finite
subset interval of infinitely continuing waterbomb tube.
Fig. 1 Wave-like surface of waterbomb tessellation
Contributed by the Mechanisms and Robotics Committee of ASME for publication
in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received October 20, 2021;
final manuscript received April 13, 2022; published online May 26, 2022. Assoc.
Editor: Chin-Hsing Kuo.
Journal of Mechanisms and Robotics AUGUST 2022, Vol. 14 / 041009-1
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2.2 Kinematics of Module. First, we consider the degrees of
freedom (DOF) of each module. Because the kinematic DOF of
n-valent vertex in general is n−3[18,19], the DOF of waterbomb
module (without symmetry) is 3. With the assumed mirror symme-
try, the DOF drops by one more degree. Therefore, the module has
6−3−1=2 DOF.
Here, we focus on the module with correct mountain and valley
(MV) assignment and pop-up pop-down assignment, where we
assume that the center vertex of the module is popped-down, i.e.,
the sum of fold angles (positive for valley and negative for moun-
tain) are positive, and other vertices are popped-down. The defini-
tion of pop-up and pop-down follows that of [20], and it is
known that rigid origami cannot continuously transform from
popped-up and popped-down state without being completely flat.
To represent the folded states of modules, we consider the isos-
celes trapezoid formed by two pairs of mirror reflected vertices of
the module (Figure 4, left). The parallel bottom and top edges are
aligned in the hoop direction, while the hypotenuses are along the
longitudinal direction. The length of hypotenuses of the trapezoid
is fixed at 2cosα, while the half lengths of bottom and top edges,
denoted by xand y, change in (0, sinα) by folding the module.
For each given pair of xand yin (0, sinα), there exists eight different
states of a module (see Fig. 5). However, only one of the folded
state has the correct MV and pop-up/pop-down assignment for
the following reason, thus we may safely use xand yas the param-
eters to uniquely represent the folded state of the module.
To show the uniqueness, first notice that the top and bottom
modules illustrated in the same column in Fig. 5are the mirror
reflection of each other through the plane containing the isosceles
trapezoid, so the top states are popped-down and the bottom
states are popped-up. Therefore, the folded state needs to be one
of the four top states. Within the top row we compare the first
and second left configurations. Vertex P′is the reflection of
vertex Pthrough the plane defined by vertices O,A, and B,so
crease OP is mountain and OP′is valley, thus we choose vertex
Fig. 2 Crease pattern of waterbomb tessellation and enlarged view of its module
((α,β,N,M)=(45°, 36°, 5, 5)). We fix the scale of the module by setting the length
of the diagonal lines to 1.
Fig. 3 Because of the N-fold symmetry and mirror symmetry of
waterbomb tube, each module is mirror symmetric (left figure),
and modules belonging to the same hoop (right figure) has the
same configuration. Note that because of the N-fold symmetry,
we can assume N≥3 reasonably.
Fig. 4 Folded states of the module (left), and the entire shape
(right) is parameterized by the pair of (x,y), and the sequence
(xm,ym)m=0,...,M−1, respectively. xand yare the half-length of the
parallel bottom and top edges of the isosceles trapezoid in the
left figure.
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P(counterclockwise side of cycle OAB) is selected as the correct
side. In a similar manner, vertex Qis chosen over Q′. Thus, the
top-left configuration with Pand Qis chosen. Finally, we can
show that the top-left configuration has indeed correct MV-
assignment also for other creases; this can be verified by checking
that P,D, and Care on the counterclockwise side of OAB and
that Pis on the symmetry plane between Cand D. Therefore,
only the top-left configuration has the correct MV-assignment.
2.3 Kinematics of Entire Tube. Next, we consider represent-
ing the folded states of an entire waterbomb tube. First, we take out
the mth hoop of the waterbomb tube. Because of rotational symme-
try, the configuration of the mth hoop can be represented by using
parameter x
m
,y
m
for each congruent module and the common dihe-
dral angles between isosceles trapezoids 2ρ
m
(Fig. 4, right). In order
that the mth hoop to be a closed cylinder, we have the following
constraint (see Appendix Afor derivation).
cos ρm=
−(xm−ym)2sec2α+4 sin2π
N
4−(xm−ym)2sec2α
(1)
Because 2ρ
m
can be derived as the function of x
m
and y
m
, the con-
figuration of each hoop can be uniquely represented by parameters
x
m
,y
m
. However, note that not all values of (x
m
,y
m
) represent valid
states as cosρ
m
may become nonreal number for certain given pair
of values.
Therefore, we can represent folded states of waterbomb tube by
the following sequence (Figure 4, right):
(xm)m=0,1,...,M−1,(ym)m=0,1,...,M−1(2)
In order that two consecutive hoops are compatible, there are con-
straints between x
m
,y
m
and x
m+1
,y
m+1
that need to be satisfied. The
constraints result in the significant property of waterbomb tube that
the state of mth hoop uniquely determines that of m+1th hoop. This
is confirmed by the fact that any vertex of m+1th modules can be
solved by sequentially applying three-sphere-intersections three
times (see Fig. 6). In each step of three-sphere-intersection,
position-unknown vertex Xincident to three position-known verti-
ces V
1
,V
2
,V
3
are fixed by constructing three spheres S
1
,S
2
,S
3
of
radii XV
1
,XV
2
,XV
3
centering at V
1
,V
2
,V
3
, respectively, and
taking their intersection. In each step, there are at most two candi-
dates for X, denoted by Xand X′. They are the mirror reflection
of each other through the plane defined by V
1
,V
2
, and V
3
. Thus,
MV-assignments of creases XV
3
and X′V
3
are the inversions of
each other, and we choose the one consistent with MV-assignments
shown in Fig. 2(for more details, see Appendix B). Therefore,
the m+1th module with consistent MV-assignment is uniquely
determined from the mth module. Note that the existence of a solu-
tion is not guaranteed because intersections of three spheres may be
empty.
When the solution exists, the relationship between mth hoop and
m+1th hoop can be formulated by following recurrence relation:
xm+1=f(xm,ym;α,β,N)
ym+1=g(xm,ym;α,β,N)
(3)
where function fand gare complex nonlinear function which
parameters are α,β, and N, but can be analytically derived (see
Appendix B). Consequently, waterbomb tube can be interpreted
as a 2-DOF mechanism for an arbitrary M, whose entire shape is
determined if initial values x
0
,y
0
(0 < x
0
,y
0
< sinα) are given.
Fig. 5 The figure shows the example of the eight different states
of a module corresponding to the same isosceles trapezoid
ABCD. The accompanying figure of the developed module
shows the actual MV-assignment of each configuration. The
upper left one is the only one with the correct pop-up/down
and MV-assignment.
Fig. 6 The figure shows the procedure to identify the state of m+1th hoop modules under the
fixed mth hoop modules. The top and bottom figures show the two solutions given by
three-sphere-intersection; top solution has consistent MV-assignment.
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2.4 Kinematics Interpreted As Dynamical Systems. Hereaf-
ter, we clarify the mathematics behind wave-like surfaces by
interpreting the recurrence relation (3) as a discrete dynamical
system. Recurrence relation (3) is categorized as a nonlinear dis-
crete dynamical system in two dimension, as functions f,gare non-
linear functions of two variables x
m
,y
m
. Also, the system (3) is
autonomous system because the function f,gis independent of
indices m.
An important property of the system is reversibility. Because the
module of waterbomb tessellation is point symmetric about the
central vertex (see Fig. 2, left), the following equation holds
(m=1,...,M−1):
xm−1=g(ym,xm)
ym−1=f(ym,xm)
(4)
From the above formula, if we define the map F:(x,y)7!
(f(x,y),g(x,y)) and linear involution G:(x,y)7! (y,x), the follow-
ing one holds:
G=F◦G◦F(5)
Like Eq. (5), the system where exist some involution reversing time
direction called reversible dynamical system [21], and involution G
is the symmetry of the system.
3 Visualization of Configuration
To understand the 2-DOF kinematics of waterbomb tube, we
computed the sequences (2) under different pairs of initial values
(x
0
,y
0
) and visualized their configurations as the sequence of
points in x,y-plane, i.e., the phase space. The sequence {(xm,ym)=
Fm(x0,y0)|m∈Z} is called orbit of (x
0
,y
0
) and the plot of them is
called phase diagram. Here, we fix the crease-pattern parameters (α,
β,N,M)=(45°, 34.6154°, 24, 100) and show the phase diagram
under this parameter in Fig. 7. Under this parameter, we can
observe all possible types of solutions we explain below.
3.1 Three Types of Solutions. It can be observed that the
characteristics of the solution change depending on the given
initial values: the solutions can be classified into three types; cylin-
der solution where it forms a constant radius cylinder, wave-like
solution where the corresponding waterbomb tube form wave-like
surface, and finite solution where the system fails to obtain a solu-
tion after some iterations. Note that the behavior of the system (3)
can change under different crease-pattern parameters as we
discuss in Sec. 4(i.e., we cannot observe some types of solutions
in different parameters). However, we can still classify solutions
under different parameters into these three types.
3.1 Cylinder Solution. The top and bottom of waterbomb tube
in Fig. 7, left, forms a constant-radius cylinder that corresponds to
fixed two points shown in black on the phase space. In this type of
Fig. 7 Under crease-pattern parameters (α,β,N,M)=(45°, 34.6154°, 24, 100), plots of sequences (xm,ym)m=0,1,...,M−1(0 < x
m
,y
m
<
sin α) under different initial values are shown above. Each sequence is computed numerically and plotted in different colors
while the parts of corresponding waterbomb tubes are placed on the left side of the plot. Also, the first three points (x
0
,y
0
),
(x
1
,y
1
) and (x
2
,y
2
) of each sequence are highlighted in triangle, square, and pentagon markers, respectively. The disk
region indicates the set of initial values that yield solutions for any m.
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solution folded states of modules composing waterbomb tube are
fixed (i.e. (x0,y0)=(x1,y1)=...=(xM−1,yM−1)), and correspond-
ing isosceles trapezoids become rectangles. Hence, if (w,w)(0 < w
< sin α) represents this uniform state, this type of solution can be
represented or defined by following equation:
f(w,w)=g(w,w)=w(6)
Hereafter, we call (w,w) satisfying Eq. (6) cylinder solution of
crease pattern under parameters (α,β,N,M). Remarkably, cylinder
solutions defined by Eq. (6) are called symmetric fixed points of the
system (3) which is invariant under the map Fand G.
Cylinder solutions shown in black in Fig. 7are numerically cal-
culated by solving Eq. (6) under crease-pattern parameter (α,β,N,
M)=(45°, 34.6154°, 24, 100), which two solutions exist and each
have different radius.
3.1 Wave-Like Solution. The second case corresponds to third
to fifth from the top of waterbomb tube shown in Fig. 7, left. The
wave-like folded state corresponds to the sequence of points on
the phase space moving in a clockwise direction. We call such solu-
tion a wave-like solution. In a wave-like solution, the point contin-
ues rotating around the same closed curve without divergence or
convergence. Note that the points along the closed curve does not
coincide, thus the folded state is not periodic. However, the
overall folded shape seems to approximate a smooth periodic wave-
like surface. These sequences and corresponding cycles are nested
around one of two cylinder solutions (smaller one).
In addition, these plots of nested closed curves seemed to be sym-
metric about the graph of y
m
=x
m
. The symmetry of the orbits is the
result of the reversibility of the system. Generally, the orbit of
reversible systems initiated from x
0
=(x
0
,y
0
), that is defined as
the set {x=Fm(x0)|m∈Z}, is invariant under the symmetry G
when the orbit has a point which is invariant under G[21].
For this reason, the orbits shown in Fig. 7, which initial terms x
0
are invariant under G, are actually symmetric about the graph of
y
m
=x
m
.
3.1 Finite Solution. The second waterbomb tube from the top
in Fig. 7, left, corresponds to the third type, where its points are
plotted just a little outside of the above-mentioned concentric
plots. The reason plots stop at the middle is that, at some index
m,(x
m
,y
m
) deviate from the region (0, sin α) × (0, sinα), that is,
there is no state of modules corresponding to parameter (x
m
,y
m
).
In other words, finite solution appears in the case that the intersec-
tion of three spheres become empty at some step, when computing
vertices of modules as shown in Fig. 6. Specifically, in the sequence
corresponding to second waterbomb tube in Fig. 7, the numerical
value of ninth term (x
8
,y
8
) is (0.689573, 0.724059), which y
8
is
greater than sin α=sin 45° ≈0.707107. So, these solution termi-
nates at some point, and only a finite portion of the paper can be
folded along this solution.
3.2 Kinematics. From this visualization, we can observe a
single disk region in the phase space as the set of initial values yield-
ing solutions for any m(the gray region in Fig. 7). The region is the
union of all wave-like solutions and the smaller cylinder solution.
As the region is the configuration space of the mechanism with
m→∞, there exists a 2-DOF rigid folding motion. The motion
can be represented by the “amplitude”and “phase”of wave-like
surfaces. As the initial configuration gets closer to the cylinder solu-
tion, the amplitude of wave shapes gets smaller. We can change the
phase by rotating the initial configuration along the closed curve
(Supporting Movie).
1
In the example state shown in Fig. 7, each
initial value is taken along x
0
=y
0
, so the left-most module forms
the “valley”of the wave.
3.3 Outline of Subsequent Analysis. From Fig. 7, we found
that the system behaves differently around the different cylinder
solutions. In Sec. 4, we perform a stability analysis to classify the
symmetric fixed points, i.e., cylinder solutions, by the behavior of
the system around them. We also investigate how the cylinder solu-
tions emerge and disappear and how the stability changes when the
crease-pattern parameters are changed.
We conjecture that quasi-periodic solutions exist around a neu-
trally stable cylinder solution, which explains the nested closed
solutions representing wave-like solutions. In Sec. 5, we give a
proof for the conjecture for a particular crease pattern.
4 Stability and Bifurcation Analysis
In this section, we investigate how the behavior of the system
around cylinder solutions changes when the crease-pattern parame-
ters are changed. For this, we numerically analyze the stability of
the cylinder solutions and visualize the changes of the stability
using the bifurcation diagram. Figure 8shows the concept of the
analysis in this section. From the bifurcation diagram, we found
that there are some critical values of crease-pattern parameters
where the stability or the behavior of the system changes. Note
that the result of this section is obtained numerically (not symboli-
cally) using MATHEMATICA based on the explicit form of system (3).
4.1 Stability Analysis. First, we introduce the concept of sta-
bility and classify cylinder solutions. In order to know the stability
of the cylinder solutions, we perform the linear stability analysis.
The linear stability analysis is the stability analysis on the following
linearized system at the cylinder solutions w=(w,w):
xm+1−w≈DF(w)(xm−w)(7)
where DF(w) is the Jacobian matrix of the system at the cylinder
solutions defined by
DF(x,y)=
∂xf(x,y)∂yf(x,y)
∂xg(x,y)∂yg(x,y)
(8)
The linearized behavior of the nonlinear system around the fixed
point can be characterized by using the eigenvalues of the Jacobian
matrix at that point. In a typical case of the system (3), we found that
the eigenvalues of the Jacobian matrix at the symmetric fixed point
(w,w) can be classified into two types. In the first type, the eigen-
values are complex conjugate whose magnitude equals to 1 at w.
Such a point is called elliptic fixed point, where the linear system
(7) has concentric closed elliptical orbits around the origin. The
linear stability around elliptic fixed point is neutrally stable,
meaning that the orbit around the fixed point keeps some distance
from that point called center. In the second type, the eigenvalues
are real, and one absolute value is less than 1 and the other is
greater than 1. Such a point is called a hyperbolic fixed point. The
linear stability at a hyperbolic fixed point is unstable, and the
fixed point is called a saddle, i.e., meaning that it has both stable
and unstable directions.
2
In Fig. 8, the top-left figure shows the two different cylinder con-
figurations with larger and smaller radius under (α,β,N)=(45°,
40°, 8) represented by the numerically calculated cylinder solutions
(w
1
,w
1
) and (w
2
,w
2
), respectively. The linear stability of (w
1
,w
1
)
and (w
2
,w
2
) are unstable saddle and neutrally stable center,
respectively.
Afixed point classified as an unstable saddle point in the linear-
ized system (7) is also an unstable saddle point in the original non-
linear system (see Sec. 5). However, a fixed point classified as a
neutrally stable fixed point in the linearized system is not necessar-
ily neutrally stable in the original system. Therefore, the linear
1
https://www.youtube.com/watch?v=H328vi3gk8A
2
Here, stable or unstable refers to the stability of the dynamical systems as m
increases. It does not refer to mechanical stability.
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stability analysis alone does not guarantee the existence of nested
closed solutions. Nevertheless, from the phase diagram in Fig. 8,
we found that there are nested closed plots around (w
2
,w
2
). This
leads us to conjecture that if the linearized system (7) has a neutrally
stable symmetric fixed point, it is also a neutrally stable symmetric
fixed point in the original system, around which there are quasi-
periodic solutions. We show the conjecture is correct for a particular
crease pattern in Sec. 5.
4.2 Bifurcation Analysis. Next, we analyze how the linear
stability of cylinder solutions changes when βchanges under
fixed αand N. For the visualization of the result, we use a
bifurcation diagram. The bottom of Fig. 8shows the bifurcation
diagram for α=45°, N=8.
To create bifurcation diagram for given αand N, we move the
remaining parameter βin the range (0, 180° −α), and compute
the cylinder solutions (w,w) of the crease pattern by solving
Eq. (6) at each value. Here, we transform parameter wto
Θ≡arcsin(w/( sin α)) ∈(0,90◦), the half of the dihedral angle
formed by the two isosceles triangles (Fig. 8), so that the plot
range is independent of α. Then, we perform the linear stability
analysis and plot the cylinder solutions in β–Θplanes by solid
and dotted line if it is neutrally stable and unstable, respectively.
In Fig. 8, the vertical line β=40° intersects with the bifurcation
diagram at two different points, (40°, Θ
1
) on the dotted line and
Fig. 8 The top-left, top-right, and bottom figure shows the two cylinder configurations, the phase diagram under (α,β,N)=
(45°, 40°, 8), and the bifurcation diagram (α=45°, N=8), respectively
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(40°, Θ
2
) on the solid line, shown in red and blue, respectively. The
red and the blue points correspond to the unstable cylinder with a
larger radius and the neutrally stable cylinder with a smaller
radius, respectively.
4.3 Result. Now, we describe how the linear stability of the
cylinder solutions depends on the parameters using the bifurcation
diagram. First, we observe the bifurcation diagram for fixed N=8.
By varying α, we found the critical value α*≈60.4° at which the
appearance of the bifurcation diagram changes. So, we use the rep-
resentative cases α=45° < α* and α=65° > α* for further observ-
ing the bifurcation diagram for these cases. We found some
critical values of βwhere the number of cylinder solutions or
their linear stability change, i.e., whether waterbomb tube can be
wave-like surface changes. We also explain the differences
between the two cases. Finally, we show that we can apply the
observations of the case of N=8 for the other N.
Here, we fixN=8. Under fixed N, we can create the 3D bifurca-
tion diagram by varying the value αin range (0, 90°) and arranging
the 2D bifurcation diagrams for each α(right in Fig. 9). From the 3D
diagram, we found the critical value α*≈60.4° where the behavior
of the bifurcation diagram changes. The behaviors of the bifurcation
diagram for α<α* are represented by (α,N)=(45°, N=8) (blue
bifurcation diagram on the top-left in Fig. 9), while the bifurcation
diagram for α>α* is represented by (α,N)=(65°, N=8) (red bifur-
cation diagram on the bottom-left in Fig. 9).
First, we describe the behavior of the representative case of
(α,N)=(45°, 8) described in the blue bifurcation diagram on the
top-left in Fig. 9. As we change the remaining variable β, there
are some critical value of βdenoted by β*
1
,β*
2
, where the linear
stability of a cylinder solution changes. As we increase βfrom 0,
at β=β*
1
≈35.8°, the graph of y=f(w,w) and y=wtouches and
the fixed point that is saddle and center generated (saddle-node
bifurcation). The branch of the fixed point that is saddle extends
from the bottom of the graph at β=45°. Then, at β=β*
2
≈46.1°,
saddle-node bifurcation occurs again where the center and the
saddle coincide and disappear. Thus, the system (3) has no cylinder
solutions under β∈(0, β*
1
), two cylinder solutions (center and
saddle) under β∈(β*
1
, 45°), three cylinder solutions (center and
two saddle) under β∈(45°, β*
2
), and only one cylinder solution
(saddle) under β∈(β*
2
, 135°). Here, Fig. 10 shows the part of the
bifurcation diagram and the phase diagrams for β=35°, 40°,
45.5°, and 50° belonging to (0, β*
1
), (β*
1
, 45°), (45°, β*
2
), and
(β*
2
, 135°), respectively. Under β=40° ∈(β*
1
, 45°) and β=
45.5° ∈(45°, β
2
°), there are nested cyclic plots around the center,
which suggests the existence of a wave-like configurations. On
the other hand, there are no cyclic plots in the other two diagrams,
i.e., all solutions are finite solutions; therefore, waterbomb tube
under α=45◦,N=8,β∈(0,β∗
1)(β∗
2,135◦) cannot be wave-like
surfaces. Thus, whether waterbomb tube can be wave-like surface
or not changes at the bifurcation values.
Next, for the case of (α,N)=(65°, 8), the red diagram in Fig. 9
shows that there are three critical values for β:β*
1
,β*
2
, and β*. As
we increase βfrom 0, saddle-node bifurcation occurs at β=β*
1
≈
27.8° in the same way as the α=45°; however, as βis further
increased, the linear stability changes at β=β*≈52.3°, which is
not observed in the case of α<α*. Furthermore, the branch of the
center fixed point extends from the bottom of the graph at β=
65°. Finally, at β=β*
2
≈65.5°, saddle-node bifurcation occurs
again where the saddle and the center coincide and disappear.
From the above, in the case of α=65°, the system (3) has zero,
two (center and saddle), two (two saddles), three (two saddles and
center), and one (saddle) cylinder solutions when β∈(0, β*
1
), (β*
1
,
β*), (β*, 65°), (65°, β*
2
), and (β*
2
, 115°), respectively. Figure 11
shows the part of the bifurcation diagram and the phase diagrams
for β=25°, 40°, 49°, 57°, 65.1°, and 70° belonging to (0, β*
1
),
(β*
1
,β*), (β*
1
,β*), (β*, 65°), (65°, β*
2
), and (β*
2
, 115°), respec-
tively. In the top-right, middle-left, and bottom-left phase diagrams,
there are the plots of wave-like solutions around the center; there-
fore, waterbomb tube has wave-like configurations. However, the
wave-like configurations placed in each diagram self-intersect at
some parts in the same way as the configuration in the bottom on
the left side of Fig. 11, or this is caused by too small radius as for
Fig. 9 The top-left, bottom-left, and the right figure shows the 2D bifurcation diagram under α=45°, N=8, the one under α=
65°, N=8, and the 3D one under N=8, respectively. The intersection of 3D diagram and the plane α=45° and α=65°, i.e., the
top-left and bottom-left 2D diagrams are highlighted in the right figure.
Journal of Mechanisms and Robotics AUGUST 2022, Vol. 14 / 041009-7
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β=65.1°, which means these wave-like configurations are not real-
izable. As for the middle-left diagram, we found the unstable non-
symmetric fixed points (x,y)≈(0.698488, 0.375731), (x,y)≈
(0.375731, 0.698488) satisfying f(x,y)=g(x,y)=(x,y), which is
not observed in the case of α=45°, but the configurations of the
module and the tube corresponding to the nonsymmetric fixed
points are self-intersecting complicatedly as shown in Fig. 11.
The other three diagrams, top-left, middle-right, and bottom-right
on the right side of Fig. 11, having no wave-like solutions, which
suggests that waterbomb tube cannot be wave-like surfaces if
β∈(0,β∗
1)(β∗,65◦)(β∗
2,115◦). Hence, bifurcation values
are closely related to the possible forms of waterbomb tubes as in
the case of α=45°.
For cases other than N=8, the top, middle, and bottom figure
in Fig. 12 shows the 3D bifurcation diagrams for N=6, N=30,
and N=100, respectively. Comparing 3D diagrams under the dif-
ferent values of Nin Figs. 9and 12, the shape of the diagrams
and the critical values α* depends on the values of N. However,
the pattern formed by the dotted and solid lines is the same
for the four different values of N; therefore, we can apply the fea-
tures described for N=8, i.e, the existence of critical values such
as α*, β*
1
,β
2
*, β* and their relation with the behavior of water-
bomb tube for N=6, 30, and 100.
5 Proof of Wave-Like Solution
In this section, we claim that there are nested wave-like solution
around the neutrally stable cylinder solution and they are quasi-
periodic orbits of the system; and we give the proof of existence
for fixed crease-pattern parameters (α,β,N)=(45°, 45°, 6). We
also show the system behaves differently around the saddle cylinder
solution. For the proof of quasi-periodic orbits, we use the
Kolmogorov–Arnold–Moser (KAM) theorem that guarantees the
equivalence of the behaviors in the nonlinear and linearized
systems under some conditions as described later.
Figure 13 visualizes the solutions under (α,β,N)=(45°, 45°, 6).
The function funder this crease pattern is given as follows:
f(x,y)=1
4(y2−1)((x−y)2−2)
×(x(3
√(
(2y2−1)(x2+y2−1)
+1)
+2
√y
(2y2−1)(2(x−y)2−1)
)
−y(2
√y(
(2y2−1)(2(x−y)2−1)
−2
(2(x−y)2−1)(x2+y2−1)
)
+3
√(
(2y2−1)(x2+y2−1)
−3) +33
√y2)
+2
√(
(2y2−1)(2(x−y)2−1)
−
(2(x−y)2−1)(x2+y2−1)
)+(−3
√)x2y)(9)
We derived the symbolic forms of cylinder solutions, Jacobian
matrix, and its eigenvalues for linear stability analysis by using
Mathematica. We obtain two cylinder solutions analytically as the
following by solving Eq. (6) for w(0 < w< sin 45°):
w=1
4
10 −3
3
ñ
−1+43
√
(10)
Hereafter, let w
1
be the solution with a larger radius, and w
2
be
another one with a smaller radius.
Fig. 10 Assuming α=45° and N=8, the figure on the left side shows the part of the top-left figure in Fig. 9. On the right side,
the top-left, top-right, bottom-left, and bottom-right figure shows the phase diagram under β=35°, 40°, 45.5°, and 50°, respec-
tively. We plotted the orbits which initial value (x
0
,y
0
) lies on the line x
m
=y
m
and highlight unstable, neutrally stable cylinder
solutions in red and blue point, respectively. In the top-right and bottom-left phase diagrams, one of the wave-like solutions is
highlighted and its corresponding wave-like configuration is placed.
041009-8 / Vol. 14, AUGUST 2022 Transactions of the ASME
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Next, we derive the Jacobian matrix DF(x,y) at cylinder solutions
(x,y)=(w
i
,w
i
)(i=1, 2). Note that ∂
x
g(w
i
,w
i
) and ∂
y
g(w
i
,w
i
) can
be computed from symbolic forms of ∂
x
f(w
i
,w
i
), ∂
y
f(w
i
,w
i
) using
chain rule:
∂g
∂x(wi,wi)=−∂f
∂y(wi,wi)(11)
∂g
∂y(wi,wi)=
1−(∂f
∂y(wi,wi))2
∂f
∂x(wi,wi)
(12)
The elements of the Jacobian matrices DF(w
i
,w
i
)(i=1, 2) are com-
puted as
(DF(w1,w1))1,1=26
24 3
√+
384 3
√+609
+47
(DF(w1,w1))1,2=1
8−15 3
√−
5112 3
√−8845
+28
(DF(w1,w1))2,1=1
815 3
√+
5112 3
√−8845
−28
(DF(w1,w1))2,2=1
104 575 3
√
+
40388432 3
√−69922045
−688(13)
Fig. 11 Assuming α=65° and N=8, the top figure on the left side shows the part of the bottom-left figure in Fig. 9. On the right
side, the top-left, top-right, middle-left, middle-right, bottom-left and bottom-right figure shows the phase diagram under β=
25°, 40°, 49°, 57°, 65.1°, and 70°, respectively under the same setting as Fig. 10. In the middle-left figure, the two unstable non-
symmetric fixed points are highlighted in the red triangle. The bottom figure on the left side shows the self-intersecting module
and the tube corresponding to these nonsymmetric fixed points.
Journal of Mechanisms and Robotics AUGUST 2022, Vol. 14 / 041009-9
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and
(DF(w2,w2))1,1=1
8−33
√+
96 3
√−165
+8
(DF(w2,w2))1,2=1
8−15 3
√+
5112 3
√−8845
+28
(DF(w2,w2))2,1=1
815
3
√−
5112 3
√−8845
−28
(DF(w2,w2))2,2=1
104 575 3
√
−
40388432
3
√−69922045
−688(14)
Finally, we consider eigenvalues of the derived Jacobian matrices
(13),(14) and the linear stability of corresponding cylinder solu-
tions. Table 1shows the numerical values of the eigenvalues and
the linear stability of the cylinder solutions. The eigenvalues for
both cylinder solutions are given as the roots of the following poly-
nomial:
13w8+292w7+161868w6−1083460w5+1704206w4
−1083460w3+161868w2+292w+13
(15)
The eigenvalues for larger (x,y)=(w
1
,w
1
) are real, the absolute
values of which are greater than 1 and smaller than 1, respectively;
therefore, the type of cylinder solution (w
1
,w
1
) as the fixed point
of the system is saddle. Hence, both linear and nonlinear stabilities
of (w
1
,w
1
) are unstable, so there are no cyclic solutions around
(w
1
,w
1
).
On the other hand, at the smaller cylinder solution (x,y)=(w
2
,
w
2
) with a smaller radius, DF(w
2
,w
2
) has complex conjugate eigen-
values whose magnitudes are exactly 1. Hence, the type of the cyl-
inder solution (w
2
,w
2
) is center and linear stability of (w
2
,w
2
)is
neutrally stable. Nevertheless, because (w
2
,w
2
) is an elliptic fixed
point, it is not yet guaranteed that the original nonlinear system
(3) has nested closed orbits around (w
2
,w
2
) as we mentioned in
Sec. 4.1.
Now, to prove the existence of the nested closed symmetric
quasi-periodic orbits around (w
2
,w
2
) in the original system, we
use the one derived form of KAM theorem. The KAM theorem
guarantees the existence of such orbits called KAM curve around
afixed point of the reversible dynamical system if the fixed point
is elliptic symmetric fixed point and nonresonant, that is, the eigen-
values are not being roots of unity [21,22]. We already know that
the system is reversible and (w
2
,w
2
) is an elliptic symmetric
Fig. 12 The top, middle, and bottom figure shows the 3D bifur-
cation diagrams under N=6, N=30, and N=100, respectively.
Fig. 13 Visualization of solutions under the crease-pattern
parameter (α,β,N,M)=(45°, 45°, 6, 10000) under the same
setting as Fig. 7. The gray region contains all wave-like solutions.
041009-10 / Vol. 14, AUGUST 2022 Transactions of the ASME
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fixed point. Also, (w
2
,w
2
) is nonresonant because of the following
reason. The eigenvalues of DF(w
2
,w
2
) are the roots of the irreduc-
ible polynomial (15). Then, some coefficients of the minimal poly-
nomial, obtained as the monic form of (15), are not integers (e.g.,
coefficients for seventh-order term is 292/13), so the eigenvalues
of (w
2
,w
2
) are not algebraic integer, which implies that the eigen-
values of (w
2
,w
2
) are not the roots of unity. Thus, we can apply
KAM theorem to the system at (w
2
,w
2
), and the existence of wave-
like solution around the point is proved.
6 Conclusion
In this paper, we revealed a part of the mathematical structure
behind wave-like surfaces of waterbomb tube. First, we have
described the kinematic model representing the configuration of
each module of waterbomb tube and its entire shape to derive recur-
rence relation of two variables dominating its kinematics. Then,
based on the observation of the plots on the phase space correspond-
ing to the folded states, we classified solutions into three types to
identify the wave-like solutions surrounding one of two cylinder
solutions. We applied the knowledge of discrete dynamical
systems to the recurrence relation, and investigated how the stability
of the symmetric fixed point of the system, i.e., the behavior around
cylinder solutions, changes when we change the crease-pattern
parameters. Then, we observed that around the neutrally stable cyl-
inder solution, there exists nested wave-like solutions, and found
critical values where the behavior of the system changes. Finally,
we gave the proof of the existence of quasi-periodic solutions,
i.e., wave-like solutions, around the neutrally stable cylinder
solution.
Note that we gave proof of the existence of quasi-periodic solu-
tions only for fixed crease-pattern parameters; characterizing the
existence of such solutions is a future work of this paper. Also,
the physical interpretation, i.e., the relation between the mathemat-
ical properties of solutions and the mechanical properties are still
left to be investigated.
More generally, our proposed approach based on the geometry of
a module and applying the knowledge of dynamical system to the
recurrence relation can be useful to explain behaviors of various
origami tessellations. The analysis of tessellations consisting of
modules with higher or lower DOF is of our interest.
Acknowledgment
This work is supported by JST PRESTO (Grant No.
JPMJPR1927).
Appendix A: Derivation of Equation Imposing Cylinder
Constraint
Here, we derive Eq. (6) imposing cylinder constraint. Assuming
that values of crease-pattern parameters are given, we represent
some distance of vertices or angle using (α,β,N,M) based on
the preservation of edge length.
We derive some quantities for considering the cylinder con-
straint. First, edge length l
1
,l
2
in Fig. 14 can be written in the
following form:
l1=sin α
sin (α+β),l2=cos α−sin α
tan (α+β)
Next, we consider the dihedral angle ρ
m
. The notation of vertices is
defined as Fig. 15. Because of the N-fold symmetry of waterbomb
tube, vertices A1
m,0,A1
m,1,...,A1
m,N−1form regular N-sided polygon.
Based on this, we derive Eq. (6). Let θ
m,n
,M
m,n
,P
m,n
,Q
m,n
be the
base angle ∠A1
m,nB1
m,nB2
m,n=∠A2
m,nB1
m,nB1
m,n, a midpoint of
A1
m,nB1
m,n, a foot of perpendicular from point M
m,n
to edge
A1
m,nA2
m,n, and a foot of perpendicular from point M
m,n
to edge
B1
m,nB2
m,n, respectively. Here, following equations hold:
cos θm,n=xm−ym
2 cos α
|Mm,nPm,n|=|Mm,nQm,n|=xmsin θm,n
Because ∠Mm,n+1A1
m,nMm,n=π(N−2)/N, by using cosine theorem
in △Mm,n+1A1
m,nMm,n,
|Mm,nMm,n+1|=
2x2
m−2x2
mcos π(N−2)
N
Thus, by using cosine theorem in △Mm,n+1Pm,nMm,n,
cos 2ρm=cos ∠Mm,n+1Pm,nMm,n
=2|Mm,nPm,n|2−|Mm,nMm,n+1|2
2|Mm,nPm,n|2
By using half angle formula, we can get the following equation
equivalent to Eq. (6):
cos ρm=
−(xm−ym)2sec2α+4 sin2π
N
4−(xm−ym)2sec2α
Table 1 Eigenvalues and stability at two cylinder solutions
CYLINDER SOLUTION EIGENVALUE TYPE STABILITY
w1=1
4
10 −33
√+
−1+43
√
λ1
||
=4.69 ...
||
>1
λ2
||=0.212 ...||<1
λ1,λ2∈R()
Saddle Unstable
w2=1
4
10 −33
√−
−1+43
√
λ
||=0.855 ...±i0.517 ...
||
=1
λ∈C
()
Center Neutrally
Stable
Fig. 14 A
i
,B
i
,C
i
(i=1, 2), and Odenote vertices of a module of
waterbomb tessellation
Journal of Mechanisms and Robotics AUGUST 2022, Vol. 14 / 041009-11
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Appendix B: Derivation of Function f
Here, we derive the function fand gassuming that values of
crease-pattern parameters and the state of modules belonging to
mth hoop, i.e., x
m
and y
m
, are given. First, we represent some quan-
tities such as distance between vertices or angles by crease-pattern
parameters and x
m
and y
m
. Next, we derive the function fand gby
introducing the coordinate system and representing x
m+1
and y
m+1
by the coordinates of some vertices of waterbomb tube.
Some Necessary Quantities. First, we represent some quantities
using (α,β,N,M) and (x
m
,y
m
) based on the preservation of edge
length. The notation of vertices is defined in Fig. 16, and for simpli-
city, we omit subscripts of x
m
and y
m
.
Let θ
1
,θ
2
be the base angle ∠B1A1A2=∠B2B1A1,
∠A2B2B1=∠A1A2B2, respectively (0 < θ
i
<π). Their cosine and
sine can be represented as follows:
cos θ1=x−y
|A1A2|=x−y
2 cos α,sin θ1=
1−cos2θ1
(A1)
cos θ2=y−x
|A1A2|=y−x
2 cos α,sin θ2=
1−cos2θ2
(A2)
Next, we calculate h:=|OO′|, where O′corresponds to the circum-
center of isosceles trapezoid A
1
B
1
B
2
A
2
because point Ois equidis-
tant from A
i
,B
i
(i=1, 2); therefore, if we represent the length of
diagonals of A
1
B
1
B
2
A
2
, radius of circumcircle of A
1
B
1
B
2
A
2
as d,
r, respectively, d,r,hcan be expressed as follows:
d=|A2B1|
=
|B1A1|2+|A1A2|2−2|B1A1A1A2|cos θ1
=2
xy +cos2α
r=|A1O′|=|A2B1|
2 sin ∠B1A1A2=d
2 sin θ1
h=|OO′|=
|A1O′|2−|A1O′|2
=
1−r2
√
Fig. 15 Three of isosceles trapezoids formed by modules belonging to raw m, and two of
ones belonging to raw m+1 are highlighted in the figure. 2ρ
m
is the dihedral angle between
isosceles trapezoids A1
m,nB1
m,nB2
m,nA2
m,nand A1
m,n+1B1
m,n+1B2
m,n+1A2
m,n+1(m=0,1,...,M−1,
n=0,1,...,N−1(mod ∼N))
Fig. 16 In this figure, M
i
,R
i
,T
i
denote a midpoint of edge A
i
B
i
,a
foot of perpendicular from vertex C
i
to edge M
1
M2, a foot of per-
pendicular from vertex C
i
to edge A
1
A2, respectively (i=1, 2).
Additionally, O′denotes a foot of perpendicular from vertex O
to edge M
1
M
2
.
041009-12 / Vol. 14, AUGUST 2022 Transactions of the ASME
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On the other hand, if ψi=∠OMiCi,ζi=∠OMiO′,ξi=ψi−
ζi(|ξi|=∠RiMiCi)(i=1, 2), the following equations hold:
cos ψi=|MiCi|2+|OMi|2−|CiO|2
2|MiCiOMi|
=
(l2
1−x2)+(1−x2)−l2
2
2
l2
1−x2
√
1−x2
√(i=1)
(l2
1−y2)+(1−y2)−l2
2
2
l2
1−y2
√
1−y2
√(i=2)
⎧
⎪
⎨
⎪
⎩
sin ψi=
1−cos2ψi
sin ζi=|OO′|
|OMi|=
h
1−x2
√(i=1)
h
1−y2
√(i=2)
⎧
⎨
⎩
cos ζi=
1−cos2ζi
cos ξi=cos ψicos ζi+sin ψisin ζi
sin ξi=sin ψicos ζi−cos ψisin ζi(A3)
Derivation of x
m+1
and y
m+1
.Next, we derive x
m+1
and y
m+1
,
i.e., the functions fand g. Here, we take the coordinate system
shown in Fig. 17 and represent the coordinates of each vertex as
the functions of x
m
and y
m
. Then, we obtain x
m+1
and y
m+1
, i.e.,
the functions fand g, by using the coordinates of vertices. In
Figs. 17 and 18, the coordinate system is taken so that the rotation
axis of symmetry of the waterbomb tube is the X-axis and one of the
modules belonging to the mth hoop which vertices are denoted as
□
m,0
is mirror-symmetric for the XZ-plane. In the following, the
coordinate components of vertex Vare denoted as V
x
,V
y
, and V
z
,
respectively, and R
y
(η) and tdenote the rotation matrix of angle
ηwith Y-axis as the rotation axis and translation vector defined as
t≡[0, 0, x
m
/tan (π/N)]
T
. Here, we can represent the coordinates
of each vertex as follows using Eqs. (A1),(A3):
[A1
m,0x,A1
m,0y,A1
m,0z]T=Ry(η)[0,−xm,0]T+t
[B1
m,0x,B1
m,0y,B1
m,0z]T=Ry(η)[0,xm,0]T+t
[A2
m,0x,A2
m,0y,A2
m,0z]T=Ry(η)[2 cos αsin θ1
m,0,−ym,0]T+t
[B2
m,0x,B2
m,0y,B2
m,0z]T=Ry(η)[2 cos αsin θ1
m,0,ym,0]T+t
[Om,0x,Om,0y,Om,0z]T=Ry(η)[
r2
m,0−x2
m
,0,−hm,0]T+t
[C1
m,0x,C1
m,0y,C1
m,0z]T=Ry(η)[
l2
1−x2
m
cos ξ1
m,0,0,
l2
1−x2
m
sin ξ1
m,0]T+t
[C2
m,0x,C2
m,0y,C2
m,0z]T=Ry(η)[
l2
1−y2
m
cos ξ2
m,0,0,
l2
1−y2
m
sin ξ2
m,0]T+t
Note that for the elements of the matrix R
y
(η) we have
sin η=−A2
m,0z−A1
m,0z
2 cos αsin θ1
m,0
,cos η=
1−sin2η
Using the coordinates of the vertices, we can obtain the coordinate
of the vertices of the other modules belonging to the mth hoop by
rotating the corresponding vertex by 2nπ/Naround the X-axis. For
example,
[C2
m,1x,C2
m,1y,C2
m,1z]T=Rx(2π/N)[C2
m,0x,C2
m,0y,C2
m,0z]T
Here, xm+1=[C2
m,0x,C2
m,0y,C2
m,0z]T·[0,cos π
N,sin π
N]T, which gives
the function f(Fig. 6).
Next, we derive the function gby finding y
m+1
(Fig. 6). To do so,
we first find the coordinates of vertex O
m+1,1
, and then the coordi-
nates of vertex B2
m+1,1.FromFig.6, vertex O
m+1,1
is located at a dis-
tance 1, 1, l
2
from the vertices C2
m,0,C2
m,1,C2
m+1,1, respectively.
Vertices C2
m,0and C2
m,1are mirror-symmetric across plane Pobtained
by rotating the XZ-plane by π/Naround X-axis. Hence, vertex O
m+1,1
is contained in the intersection of the two circles on Pcentered at P
and C1
m+1,1which radius are |PO1
m+1,1|=
1−|PCm,0|2
=
1−x2
m+1
and l
2
, respectively. Using x
1
:=[P
x
,P
y
,P
z
]
T
,x2:=
[C1
m+1,1xC1
m+1,1y,C1
m+1,1z]T,R1:=
1−x2
m+1
,R2:=l2,andn:=
R
x
(π/N)[0, 1, 0]
T
which is the normal vector of P, we can represent
Fig. 18 P,Qis the midpoint of vertices C2
m,0,C2
m,1and O
m+1,0
,
O
m+1,1
, respectively
Fig. 17 In this figure, Ai
m,0,Bi
m,0,Ci
m,0(i=1,2) are the vertices of
the module, and Mi
m,0(i=1,2) is the midpoint of Ai
m,0and Bi
m,0
Journal of Mechanisms and Robotics AUGUST 2022, Vol. 14 / 041009-13
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the intersection of the two circles as follows:
x1+Rn,±arccos R2
1+x1−x22−R2
2
2R1x1−x2
R1(x2−x1)
x2−x1(A4)
Here, R(v,θ)(v∈R3,θ∈R) is the rotation matrix in which rotation
axis and rotation angle are vand θ, respectively. The signs +,−of the
angle of rotation results crease C1
m+1,1Om+1,1to mountain-folded and
valley-folded, respectively; therefore, in this case, +sign is consistent
with the prescribed MV-assignment.
Next, we derive the coordinates of vertex B2
m+1,1. Vertex B2
m+1,1is
located at a distance 1, 1, 2 cos αfrom vertices
Om+1,0,Om+1,1,C2
m,0, respectively. Here, since the vertices O
m+1,0
,
O
m+1,1
are the mirror-reflection with respect to XZ-plane,
the vertex B2
m+1,1is contained in the intersection of a circle of
radius
1−(Om+1,1y)2
centered at the vertex Qon the XZ-
plane and a circle of radius 2cosαcentered at the vertex C2
m,0.
We can obtain the intersection of the circles by Eq. (A4) where
x1=[C2
m,0x,C2
m,0y,C2
m,0z]T,R1=2 cos α,x2=[Qx,Qy,Qz]T,R2=
1−(Om+1,1y)2
,n=[0,1,0]T. In this case, the positive sign is
consistent with the prescribed MV-assignment of C2
m,0B2
m+1,1.
Therefore, we obtain the coordinates of vertex B2
m+1. Since
ym+1=[B2
m+1x,B2
m+1y,B2
m+1z]T·[0,cos π
N,sin π
N]T, the function g
is obtained.
Conflict of Interest
There are no conflicts of interest.
Data Availability Statement
The datasets generated and supporting the findings of this article
are obtainable from the corresponding author upon reasonable
request.
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041009-14 / Vol. 14, AUGUST 2022 Transactions of the ASME
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