A linkage mechanismthat follows a discretesine-Gordon equation
SIDE13: Symmetries andIntegrability of Difference Equations
15 Nov. 2018
Shizuo KAJI
Joint withKenji Kajiwara and Hyeongki Park
(Kyushu Univ.)

Linkage mechanism
A (bar) linkage is a collection of rigid bars (links) connected by ball joints
so that it moves while keeping the pairwise distances of joints sharing a bar.
It is used to transfer/transform motion.

pantogprah(copier)

Linkage in R2Ex. Watt’s parallel motion transform
rotary motion to almost linear motion( actually, lemniscate )Linkage
Formally, a linkage L is a finite simple graph (V,E) whose edges are assigned lengths !
The moduli space Mm(L) is the space of isometric embeddings of L in Rm
modulo the global symmetry.

Each connected component corresponds to “motion” of the linkage
Bar linkage is well studied mathematically;e.g., by Thurston, Niemann, Kapovich-Millson, O’hara,…
Thm (Kempe's universality theorem)
Any bounded plane algebraic curve can be traced by some linkage

Moduli space example
Consider the linkage below with 3 joints in R3Let h>0 be a constant and the two ends

a = ( h, 0, 0),b = (h, 0, 0)x2<latexit sha1_base64="+QjSsg+KUy+hArNokSuRb0knQiA=">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</latexit>

are fixed to the wall. Then, the moduli space isx3x1abTopology changes according to the parameter !

2hKaleidocycle
Recreational maths
W. W. Rouse Ball (1939)“Mathematical recreations and essays”
D. Schattschneider and W. M. Walker (1985)“M. C. Escher Kaleidocycles”
Theory of Polytopes
Cauchy’s rigidity theorem
Bellows Theorem
Kinematics, Robotics
Bricard 6R linkage mechanism
violating Mobility formulaKaleidocycle as linkage

Kaleidocycle

2n joints
5n bars

Bricard linkage
n hingesmore convenient
n barsto work withHinged linkage
A hinged linkage is a finite simple graph (V,E) with edge labelling! ↦ # ! , % ! ∈ ℝ(Its moduli space is the set of configurations of the hinge axes in R3Sarrus
linkageHinged linkage ó Discrete framed curve
A linkage system is said to be closed when the underlying graph is a circle
in this casebars = common normals to adjacent hinges = polygon
hinges = binormals (up to sign)
Red edges are hinges

γ1γ2γ3

An n-Kaleidocycle is a closed hinged linkage with n hinges
in which the relative positions of adjacent hinges are all
equal; that is, it is made of congruent tetrahedra.

The moduli of KaleidocyclesBinormals determines a Kaleidocycle up to scaling:

Since the linkage is closed,

Since the angles between adjacent hinges are constant

The moduli M(n) of n-Kaleidocycles is defined to be the space ofthe real solutions to the above quadratic equations
modulo the global rotational symmetry

The moduli of Kaleidocycles
Consider the map M(n) à R assigning the constant twistand consider a connected component M(n;c) of its fibre overIt correspond to the space of motion of a particular Kaleidocycle.

example of a connected component of M(7,-0.29)

Dimension of generic fibres
For a generic c, let us count the dimension of M(n;c)
Fix two hinges to kill the global symmetry.
2We have degree two freedom for each b2 , b3 , . . . , bn 1 2 S
and there the angle constraints bi · bi+1 = c which contributes –(n–1)<latexit sha1_base64="MLROl0nxt4kHDMXznI3hG55Gngg=">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</latexit>

and the closing constraints

X<latexit sha1_base64="JGjii3mdRg4Oy/kHlttqCupN2Vs=">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</latexit><latexit sha1_base64="tXRDgFp6zsoyUo3JR7qieR5UoFw=">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</latexit>bi ⇥ bi+1 = 0

which contributes –3

In total, the dimension should be2(n–2) – (n–1) – 3 = n–6

n-Kaleidocycle has in general n-6
degrees of freedom

Dimension of singular fibres
Conjecture (K-Schoenke)
There is cn < 0 for each odd n such thatM(n;cn) is non-empty iff cn ≦ c ≦ 1Moreover,for each point, it is numerically checked thatthe ε-ball intersects exactly at two points

this S1 is the characteristic “everting” motion

We call an element of the singular slices the extreme n-Kaleidocycle
It is very rare for a linkage to have a degenerate but non-trivial configuration space

The “everting” motion of extreme Kaleidocycles

Deformation of discrete curves
Goal: model the everting motion of a Kaleidocycle
Inoguchi-Kajiwara-Matsuura-Ohta studied discrete deformation ofdiscrete curves with constant torsion which
preserves torsion and arc lengthis equidistant (every point travels the same distance at each step)and found a family of deformations governed by discrete mKdV
and discrete sine-Gordon equations

Framing of discrete curves
We start with $" ∈ & 'For each vertex i, we assign a framing

The signed curvature !" and the torsion #" is then defined byThis definition differs the standard Frenet-Serret framing by signTwist and writhe of a closed curve
There are two conformal invariants of a closed curve

For a Kaleidocycle in motion
Tw remains unchanged by definition
so Wr should remain unchanged as well

Calugareanu-White’s theorem#self-linking := Tw + Wr
is an integer and an isotopy invariant
This puts very strong topological constraint for a curve to be closed
Interestingly, there seems to be no Kaleidocycle with less than three half-twists.

Curve deformation and linkage motion
Consider deformation of a closed curve corresponding to a closed hinged linkage
Assume•arc length and torsion are preserved (ó bars are rigid)•velocity at each vertex lies in the osculating plane (è writhe is preserved)•equidistant (ó speed of motion is uniform at all vertices)

Theorem (KKP)
With the above assumptions, the deformation should satisfy
wheresatisfying

−1 # $#%#

Conservation laws
Corollary
If the deformation is governed by the sine-Gordon equation, we have

Extreme Kaleidocycles (n:odd and with the maximum torsion) have interesting properties
1-DOF: the linkage does not allowed to wobble, but it can just evert
no matter how many joints it has
The sum of signed curvature vanishes
We list some physically inspired quantities,
which we numerically verified to be constant under deformation

Bending energy
Imagine that all hinges are attached torsional springs.
The potential of the system isRemark: Safsten et al. 2016
analysed the energy andstable states for the classical K6

It is a discrete version of the bending energy

Conjecture
Ebend is constant under deformation

extreme Kaleidocycles require no force to evertCoulomb & dipole energies
Imagine the centres γ of hinges are electrically charged. The potential of the system is

Imagine the hinges are dipoles. The potential of the system is

Note that thesedepend on theglobal shape ofthe curve

Conjecture: Eclmb and Edipl are constant under deformation

again, extreme Kaleidocycles require no force to evertGalleryOpen question:
What is the limit of nà∞ ?

Summary
closed hinged linkage

discrete closed
framed curve
Framing differs from theFrenet-Serret by sign
Especially, curvature is signed

motion of linkage

length and torsion
preserving deformation
for closed curves, topology
poses strong constraints

everting motion ofKaleidocycle

flow on the moduli space defined by asemi-discrete sine-Gordon equation

codes for numerical simulation: https://github.com/shizuo-kaji/Kaleidocycle

Topology, Geometry, and Integrable systems
on one SIDE