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
a = (�h, 0, 0), b = (h, 0, 0)<latexit sha1_base64="+QjSsg+KUy+hArNokSuRb0knQiA=">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</latexit><latexit sha1_base64="+QjSsg+KUy+hArNokSuRb0knQiA=">AAACc3icbVHLSgMxFL0d3/VVddlNsAoKWmZE0I1Q7MalilWxUySTSW0wjzHJKGWYP3Hrj7jVH/BD3JtOC2r1wg2Hc87lJidRwpmxvv9R8iYmp6ZnZufK8wuLS8uVldVLo1JNaIsorvR1hA3lTNKWZZbT60RTLCJOr6L75kC/eqTaMCUvbD+hHYHvJOsygq2jVKUKGI5gC3ahBzvgF73tTgQv8ACpU2OHo8Lzw3Fbqfl1vyj0FwQjUINRnd6ulDbDWJFUUGkJx8a0g73EdjKsLSOc5uUwNTTB5B7f0baDEgtqOlnxwBxtOiZGXaVdS4sK9udEhoUxfRE5p8C2Z8a1Afmf1k5t97CTMZmklkoyXNRNObIKDdJCMdOUWN53ABPN3F0R6WGNiXWZlkNJn4gSAss4C8/zLBwsiKLsPM9/azff2s241vzWmnnugg3GY/wLLvfqgV8PzvZrjeNRxLNQhXX3SQEcQANO4BRaQOAZXuEN3kufXtVb9zaGVq80mlmDX+XtfgEXcrRu</latexit><latexit sha1_base64="+QjSsg+KUy+hArNokSuRb0knQiA=">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</latexit><latexit sha1_base64="+QjSsg+KUy+hArNokSuRb0knQiA=">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</latexit>
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
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