A linkage mechanism that follows a discrete sine-Gordon equation

Shizuo KAJI

SIDE13: Symmetries and Integrability of Difference Equations

15 Nov. 2018

Joint withKenji Kajiwara and Hyeongki Park

(Kyushu Univ.)

Linkage mechanismA (bar) linkage is a collection of rigid bars (links) connected by ball jointsso that it moves while keeping the pairwise distances of joints sharing a bar.

It is used to transfer/transform motion.

Ex. Watt’s parallel motion transform rotary motion to almost linear motion

( actually, lemniscate )

pantogprah(copier) Linkage in R2

Linkage

The moduli space Mm(L) is the space of isometric embeddings of L in Rm

modulo the global symmetry.

Formally, a linkage L is a finite simple graph (V,E) whose edges are assigned lengths !

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

Each connected component corresponds to “motion” of the linkage

Moduli space example

2h

x2

a b

x1 x3

Consider the linkage below with 3 joints in R3

Let h>0 be a constant and the two ends

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are fixed to the wall. Then, the moduli space is

Topology changes according to the parameter !

KaleidocycleRecreational mathsW. W. Rouse Ball (1939) “Mathematical recreations and essays”D. Schattschneider and W. M. Walker (1985)“M. C. Escher Kaleidocycles”

Theory of PolytopesCauchy’s rigidity theoremBellows Theorem

Kinematics, RoboticsBricard 6R linkage mechanismviolating Mobility formula

Kaleidocycle as linkage

Kaleidocycle

2n joints5n bars

Bricard linkagen hingesn barsmore convenient

to 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 R3

Sarruslinkage

Hinged linkage ó Discrete framed curve

γ1

γ2 γ3

Red edges are hinges

bars = common normals to adjacent hinges = polygonhinges = binormals (up to sign)

A linkage system is said to be closed when the underlying graph is a circlein this case

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

Binormals determines a Kaleidocycle up to scaling:

Since the linkage is closed,

Since the angles between adjacent hinges are constant

The moduli of Kaleidocycles

The moduli M(n) of n-Kaleidocycles is defined to be the space of the 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 twist

and consider a connected component M(n;c) of its fibre over

It correspond to the space of motion of a particular Kaleidocycle.

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

Dimension of generic fibresFor a generic c, let us count the dimension of M(n;c)

Fix two hinges to kill the global symmetry.

We have degree two freedom for eachand there the angle constraints which contributes –(n–1)

and the closing constraints which contributes –3

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In total, the dimension should be

2(n–2) – (n–1) – 3 = n–6 n-Kaleidocycle has in general n-6 degrees of freedom

Conjecture (K-Schoenke)

There is cn < 0 for each odd n such thatM(n;cn) is non-empty iff cn ≦ c ≦ 1

Moreover,

Dimension of singular fibres

We call an element of the singular slices the extreme n-Kaleidocycle

for each point, it is numerically checked that the ε-ball intersects exactly at two points

It is very rare for a linkage to have a degenerate but non-trivial configuration space

this S1 is the characteristic “everting” motion

The “everting” motion of extreme Kaleidocycles

Deformation of discrete curves

Inoguchi-Kajiwara-Matsuura-Ohta studied discrete deformation of discrete curves with constant torsion which

preserves torsion and arc length

is equidistant (every point travels the same distance at each step)

and found a family of deformations governed by discrete mKdVand discrete sine-Gordon equations

Goal: model the everting motion of a Kaleidocycle

Framing of discrete curves

For each vertex i, we assign a framing

This definition differs the standard Frenet-Serret framing by sign

The signed curvature !" and the torsion #" is then defined by

We start with $" ∈ &'

Twist and writhe of a closed curve

Interestingly, there seems to be no Kaleidocycle with less than three half-twists.

There are two conformal invariants of a closed curve

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

For a Kaleidocycle in motionTw remains unchanged by definition

so Wr should remain unchanged as well

Curve deformation and linkage motionConsider 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 satisfywhere

satisfying

−1 #$# %#

Conservation laws

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 evertno matter how many joints it has

The sum of signed curvature vanishes

CorollaryIf the deformation is governed by the sine-Gordon equation, we have

We list some physically inspired quantities, which we numerically verified to be constant under deformation

Bending energyImagine that all hinges are attached torsional springs.The potential of the system is

It is a discrete version of the bending energy

extreme Kaleidocycles require no force to evert

Remark: Safsten et al. 2016analysed the energy and stable states for the classical K6

ConjectureEbend is constant under deformation

Coulomb & dipole energies

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

Imagine the centres γ of hinges are electrically charged. The potential of the system is

again, extreme Kaleidocycles require no force to evert

Conjecture: Eclmb and Edipl are constant under deformation

Note that these depend on the global shape of

the curve

GalleryOpen question:What is the limit of nà∞ ?

Summary

closed hinged linkagediscrete closed framed curve

Framing differs from the Frenet-Serret by sign

Especially, curvature is signed

motion of linkagelength and torsion

preserving deformation

everting motion of Kaleidocycle

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

for closed curves, topology poses strong constraints

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

Topology, Geometry, and Integrable systems on one SIDE