Abstract
A linkage mechanism consists of rigid bodies assembled by joints which can be used to translate and transfer motion from one form in one place to another. In this paper, we are particularly interested in a family of spacial linkage mechanisms which consist of copies of a rigid body joined together by hinges to form a ring. Each hinge joint has its own axis of revolution and rigid bodies joined to it can be freely rotated around the axis. The family includes the famous threefold symmetric Bricard6R linkage also known as the Kaleidocycle, which exhibits a characteristic “turning over” motion. We can model such a linkage as a discrete closed curve in with a constant torsion up to sign. Then, its motion is described as the deformation of the curve preserving torsion and arc length. We describe certain motions of this object that are governed by the semidiscrete mKdV equations, where infinitesimally the motion of each vertex is confined in the osculating plane.
Linkage Mechanisms Governed by Integrable Deformations of Discrete Space Curves
[5mm] Shizuo Kaji, Kenji Kajiwara
[1mm] Institute of Mathematics for Industry, Kyushu University
744 Motooka, Fukuoka 8190395, Japan
email: skaji@imi.kyushuu.ac.jp, kaji@imi.kyushuu.ac.jp
[3mm] Hyeongki Park
[1mm] Graduate School of Mathematics, Kyushu University
744 Motooka, Fukuoka 8190395, Japan
email: hpark@math.kyushuu.ac.jp
[2mm]
1 Introduction
A linkage is a mechanical system consisting of rigid bodies (called links) joined together by joints. They are used to transform one motion to another as in the famous Watt parallel motion and a lot of examples are found in engineering as well as in natural creatures (see, for example, [7]).
Mathematical study of linkage dates back to Euler, Chebyshev, Sylvester, Kempe, and Cayley and since then the topology and the geometry of the configuration space have attracted many researchers (see [12, 24, 31] for a survey). Most of the research focuses on pin joint linkages, which consist of only one type of joint called pin joints. A pin joint constrains the positions of ends of adjacent links to stay together. To a pin joint linkage we can associate a graph whose vertices are joints and edges are links, where edges are assigned its length. The state of a pin joint linkage is effectively specified by the coordinates of the joint positions, where the distance of two joints connected by a link is constrained to its length. Thus, its configuration space can be modelled by the space of isometric imbeddings of the corresponding graph to some Euclidean space. Note that in practice joints and links have sizes and they collide to have limited mobility, but here we consider ideal linkages with which joints and links can pass through each other.
While the configuration spaces of (especially planar) pin joint linkages are well studied, there are other types of linkages which are not so popular. In this paper, we are mainly interested in linkages consisting of hinges (revolute joints). To set up a framework to study linkages with various types of joints, we first introduce a mathematical model of general linkages as graphs decorated with groups (§2.1), extending previous approaches (see [34] and references therein). This formulation can be viewed as a special type of constraint network (e.g., [14]). Then in §2.2, we focus on linkages consisting of hinges. Unlike a pin joint which constrains only the relative positions of connected links, a hinge has an axis so that it also constrains the relative orientation of connected links.
We are particularly interested in a simple case when links in are joined by hinges to form a circle (§3). Such a linkage can be roughly thought of as a discrete closed space curve, where hinge axes are identified with the lines spanned by the binormal vectors. Properties of such linkages can thus be translated and stated in the language of discrete curve. An example of such linkage is the threefold symmetric Bricard 6R linkage consisting of six hinges (Fig.1), which exhibits a turning over motion and has the configuration space homeomorphic to a circle. As a generalisation to the threefold symmetric Bricard 6R linkage, we consider a family of linkages consisting of copies of an identical links connected by hinges, which we call Kaleidocycles, and they are characterised as discrete curves with constant speed and constant torsion.
The theory of discrete space curves has been studied by many authors. The simplest way to discretise a space curve is by a polygon, that is, an ordered sequence of points . Deformation of a curve is a timeparametrised sequence of curves , where runs through (an interval of) the real numbers. Deformations of a given smooth/discrete space curve can be described by introducing an appropriate frame such as the Frenet frame, which satisfies the system of linear partial differential/differentialdifference equations. The compatibility condition gives rise to nonlinear partial differential/differentialdifference equation(s), which are often integrable. It is sometimes possible to construct deformations for the space curves using integrable systems which preserve some geometric properties of the space curve such as length, curvature, and torsion. For example, a deformation is said to be isoperimetric if the deformation preserves the arc length. In this case, the modified Kortewegde Vries (mKdV) or the nonlinear Schrödinger equation and their hierarchies naturally arise as the compatibility condition [6, 10, 15, 18, 27, 28, 37, 41]. Various continuous deformations for the discrete space curves have been studied in [11, 16, 18, 36, 38], where the deformations are described by the differentialdifference analogue of the mKdV and the NLS equations.
The motion of Kaleidocycles corresponds to isoperimetric and torsionpreserving deformation of discrete closed space curves with constant torsion. In §4, we define a flow on the configuration space of a Kaleidocycle by the differentialdifference analogue of the mKdV equation (semidiscrete mKdV equation). This flow generates the characteristic turning over motion of the Kaleidocycle.
Kaleidocycles exhibit interesting properties and pose some topological and geometrical questions. In §5 we indicate some directions of further study to close this exposition.
We list some more preceding work in different fields which are relevant to our topic in some ways.
Mobility analysis of a linkage mechanism studies how many degreeoffreedoms a particular state of the linkage has, which corresponds to determination of the local dimension at a point in the configuration space (see, for example, [35]). On the other hand, rigidity of linkages consisting of hinges are studied in the context of the bodyhinge framework (see, for example, [20, 25]). The main focus of the study is to give a characterisation for a generic linkage to have no mobility. That is, people ask when the configuration space is homeomorphic to a point or isolated points.
Sato and Tanaka [42] study the motion of a certain linkage mechanism with constrained degreeoffreedom and observed soliton solutions appear.
2 A mathematical model of linkage
The purpose of this section is to set up a general mathematical model of linkages. This section is almost independent of later sections, and can be skipped if the reader is concerned only with our main results on the motion of Kaleidocycles.
2.1 A group theoretic model of linkage
We define an abstract linkage as a decorated graph, and its realisation as a certain imbedding of the graph in a Euclidean space. Our definition generalises the usual graphical model of a pin joint linkage to allow different types of joint.
Denote by the group of orientation preserving linear isometries of the dimensional Euclidean space . An elements of is identified with a sequence of dimensional column vectors which are mutually orthogonal and have unit length with respect to the standard inner product of . Denote by the group of dimensional orientation preserving Euclidean transformations. That is, it consists of the affine transformations which preserves the standard metric. We represent the elements of by homogeneous matrices acting on
by multiplication from the left. For example, an element of is represented by a matrix
The vector is called the translation part. The upperleft block of is called the linear part and denoted by . So the action on is also written by .
Definition 1.
An dimensional abstract linkage consists of the following data:

a connected oriented finite graph

a subgroup assigned to each , which defines the joint symmetry

an element assigned to each , which defines the link constraint.
In practical applications, we are interested in the case when or . When linkages are said to be planar, and when linkages are said to be spacial.
We say a linkage is homogeneous if for any pair , the following conditions are satisfied:

there exists a graph automorphism which maps to (i.e., acts transitively on ),

,

and for any .
A state or realisation of an abstract linkage is an assignment of a coset to each vertex
such that for each edge , the following condition is satisfied:
(2.1) 
where cosets are identified with subsets of .
Let us give an intuitive description of (2.1). Imagine a reference joint sitting at the origin in a reference orientation. The subset consists of all the rigid transformations which maps the reference joint to the joint at with a specified position and an orientation up to the joint symmetry . The two subsets and intersects if and only if the joint at can be aligned to that at by the transformation .
Example 1.
The usual pin joints connected by a barshaped link of length are represented by and being any translation by . Note that . It is easy to see that (2.1) amounts to saying the difference in the translation part of and should have the norm equal to .
Two revolute joints (hinges) in connected by a link of length making an angle are represented by being the group generated by rotations around the axis and the rotation around the axis, and being the rotation by around axis followed by the translation along axis by ; that is
Note that is the space of based lines (i.e., lines with specified origins) in , and the line is identified with the axis of the hinge.
The space of all realisations of a given linkage admits an action of defined by for . The quotient of by is denoted by and called the configuration space of . Each connected component of corresponds to the mobility of the linkage in a certain state. When a connected component is a manifold, its dimension is what mechanists call the (internal) degreesoffreedom (DOF, for short). Given a pair of points on , the problem of finding an explicit path connecting the points is called motion planning and has been one of the main topics in mechanics [30]. In a similar manner, many questions about a linkage can be phrased in terms of the topology and the geometry of its configuration space.
Example 2.
Consider the following spacial linkages consisting of pin joints depicted in Figure 2. In the latter, we assume the two joints and are fixed to the wall. Up to the action of the global rigid transformation , these two linkages are equivalent and share the same configuration space ; in the left linkage, the global action is killed by fixing the positions of three joints except for .
The topology of changes with respect to the parameter which is the length of the bars. Namely, we have
This seemingly trivial example is indeed related to a deeper and subtle question on the topology of the configuration space; the space is identified with the real solutions to a system of algebraic equations.
2.2 Hinged linkage in three space
Now, we focus on a class of spacial linkages consisting of hinges, known also as three dimensional bodyhinge frameworks [20]. In this case, the definition in the previous section can be reduced to a simpler form.
Notice that in a pair of hinges connected by a link can be modelled by a tetrahedron. A hinge is a based line, which is a line with a specified base point on it. Given a pair of hinges, unitlength segments on the hinges containing the base points in the center span a tetrahedron (or a quadrilateral when the two hinges are parallel). We call the line segment connecting two base points the midaxis (see Fig. 3 Left). It is sometimes convenient to decompose the link constraint into three parts; a translation along the hinge direction at , a screw motion along an axis perpendicular to the both hinges, and a translation along the hinge direction at . This corresponds to a common presentation among mechanists called the Denavit–Hartenberg parameters [9]. We can find the decomposition geometrically as follows: Find a line segment which is perpendicular to the both hinges connected by the link , which we call the core segment. It is unique unless the hinges are parallel. The intersection points of the core segment and the hinges are called the marked points. Form a tetrahedron from the line segments on hinges containing the marked points in the center. By construction, this tetrahedron has a special shape that the line connecting the center of two hinge edges (the core segment) is perpendicular to the hinge edges. Such a tetrahedron is called a disphenoid. The shape of the disphenoid defines a screw motion along the core segment up to a rotation. The translations along the hinge directions are to match the marked points to the base points (see Fig. 3). To sum up, a spacial hinged linkage can be considered as a collection of lines connected by disphenoids at marked points.
Thus, we arrive in the following definition.
Definition 2.
A hinged network consists of

a connected oriented finite graph ,

two edge labels called the torsion angle and called the segment length,

and a vertex label called the marking, where is the set of edges adjacent to .
A state of a hinged network is an assignment to each vertex of an isometric embedding such that for any

, where

and

, where the angle is measured in the righthand screw manner with respect to .
Intuitively, is the line spanned by the hinges, and the first two conditions demands that the marked points are connected by the core segments , whereas the last condition dictates the torsion angle of adjacent hinges and .
A hinged network is said to be serial when the graph is a line graph; i.e., a connected graph of the shape . It is said to be closed when the graph is a circle graph; i.e., a connected finite graph with every vertex having outgoing degree one and incoming degree one. A hinged network is homogeneous if

acts on transitively,

, and do not depend on and . That is, it is made of congruent tetrahedral links.
Example 3.
A planar pin joint linkage is a special type of hinged network with for all and for all . That is, all hinges are parallel and marked points are all at the origin. On the other hand, any hinged network can be thought of as a spacial pin joint linkage by replacing every tetrahedral link with four bar links connected by four pin joints forming the tetrahedron. Therefore, hinged networks form an intermediate class of linkages which sits between planar pin joint linkages and spacial pin joint linkages.
Example 4.
Example 5.
Closed hinged networks with for all provide a linkage model for discrete developable strips studied recently by K. Naokawa and C. Müller (see Fig. 6). They are made of (planar) quadrilaterals joined together by the pair of nonadjacent edges as hinges.
3 Hinged network and discrete space curve
In this section, we describe a connection between spacial closed hinged networks and discrete closed space curves. This connection is the key idea which provides a way to study certain linkages using tools in discrete differential geometry.
First, we briefly review the basic formulation of discrete space curves (see, for example, [18]). A discrete space curve is a map
For simplicity, in this paper we always assume that for any and that three points , and are not colinear. The tangent vector is defined by
(3.1) 
We say has a constant speed of if for all . A discrete space curve with a constant speed is sometimes referred to as an arc length parametrised curve [17]. The normal vector and the binormal vector are defined by
(3.2)  
(3.3) 
respectively. Then, is called the Frenet frame of . For our purpose, it is more convenient to use a modified version of the ordinary Frenet frame, which we define as follows. Set and define recursively so that and . Then, , where (see Fig. 7).
For , are defined by
(3.4) 
There exist and
(3.5) 
which is referred to as the (discrete) FrenetSerret formula. We call the signed curvature angle and the torsion angle. Fig. 8 illustrates how to obtain from by (3.5). Note that we have
(3.6) 
The reason why we introduce the modified frame is that the ordinary Frenet frame behaves discontinuously under deformation when the ordinary curvature angle vanishes at a point. During the turning over motion of a Kaleidocycle, it goes through such a state at some points, and the above modified frame behaves consistently even under the situation.
Fix a natural number . A discrete space curve is said to be closed of length if for any . Unlike the ordinary Frenet frame, closedness does not imply but it can be rotated by around . We say is oriented (resp. antioriented) if (resp. ) for all .
We can consider a discrete version of the Darboux form [8, 46], which gives a correspondence between spherical curves and space curves. Given with for all and , we can associate a discrete space curve satisfying
(3.7) 
which we denote by . The curve is closed of length if
(3.8) 
for all .
Notice that a serial (resp. closed) hinged network with for all (see Def. 2) can be modelled by an open (resp. a closed) discrete space curve; its base points form the curve and hinge directions are identified with (see Fig. 9). This is the crucial observation of this paper.
Now we introduce our main object Kaleidocycles, which are homogeneous closed hinged network. We model them as constant speed discrete space curves with constant torsion angle . They are a generalisation to a popular paper toy called the Kaleidocycles (see, e.g., [4, 43]). A serial hinged network similar to our Kaleidocycle is proposed in [33].
Definition 3.
Fix and . An Kaleidocycle with speed and torsion angle is a closed discrete space curve of length which has constant speed and constant torsion angle . It is said to be oriented (resp. antioriented) when associated is oriented (resp. antioriented).
When is either or , the corresponding Kaleidocyles are planar, and we call them degenerate.
For fixed and , an oriented (resp. antioriented) nondegenerate Kaleidocycle with torsion angle is determined by the Darboux form by a map satisfying p

(resp. ),

,

.
We use and interchangeably to represent a Kaleidocycle.
Consider the real algebraic variety defined by the following system of quadratic equations ([44, Ex. 5.2, 8.13]):
(3.9) 
where is considered as an indeterminate. The orthogonal group acts on ’s in the standard way, and hence, on . Denote by the quotient of by the action of . The variety serves as the configuration space of all Kaleidocycles with varying . It decomposes into two disjoint subspaces consisting of all oriented Kaleidocycles () and consisting of antioriented ones ().
As (resp. ) is a closed variety, its image under the projection onto the axis is a union of closed intervals. Notice that the image does not coincide with the whole interval ; means are all equal so we cannot have . The fibre consists of Kaleidocycles with a fixed . With a generic value of , a simple dimension counting in (3.9) shows that . Hence, the degree of freedom (DOF) of the Kaleidocycle with torsion angle is generally . For , a generic Kaleidocycle is reconfigurable meaning that it can continuously change its shape. We will investigate a particular series of reconfiguration in the next section.
4 Deformation of discrete curves
4.1 Continuous isoperimetric deformations on discrete curves
Kaleidocycles exhibits a characteristic turning over motion (see Fig. 11 and see [22] for some animations). In general, an Kaleidocycle has degreesoffreedom so that it wobbles in addition to turning over. With a special values of torsion angle, however, the DOF of Kaleidocycle seems to degenerate to exactly one, leaving only the turning over motion as we will discuss in §5. In this case, the motion of the core segment looks to be orthogonal to the hinge directions. In the following, we would like to model the motion explicitly. It turns out that we can construct a motion of any Kaleidocycle using a semidiscrete mKdV equation with which it turns over.
In this section, we consider certain continuous deformations of discrete space curves which corresponds to motion of homogeneous serial and closed hinged networks. Our approach is to construct a flow on the configuration space by differentialdifference equations. We use the same notations as in Section 3. Observe that a hinged network moves in such a way that its tetrahedral links are not distorted. In the language of discrete space curves, the motion corresponds to a deformation which preserves the speed and the torsion angle for all , and the deformation at each point of the discrete curve is confined to the osculating plane.
Let be an (open) discrete space curve which has a constant speed and a constant torsion angle . Given a family of functions with the deformation parameter and a constant , we consider a family of discrete space curves defined by
(4.1) 
That is, the motion of each point is confined in the osculating plane and its speed depends only on the length of the segment . We say a deformation is isoperimetric if the segment length does not depend on for all , We would like to find conditions on under which the above deformation is isoperimetric. From (3.1), (3.5) and (4.1), we have
Therefore, for each , if and only if
(4.2) 
which yields
(4.3) 
of
(4.4) 
We consider a deformation when (4.3) (resp. (4.4)) simultaneously holds for all . Note that in this case for all is determined once is given.
Those deformations are characterized by the following propositions:
Proposition 1.
Let be a discrete space curve with a constant speed and a constant torsion angle . Let be its deformation according to (4.1) with satisfying the condition (4.3). Then we have:

The speed and the torsion angle do not depend on nor . That is, and for all and .

The signed curvature angle and satisfy
(4.5) where .

The deformation of the frame is given by
(4.6)
Proposition 2.
Let be a discrete space curve with a constant speed and a constant torsion angle . Let be its deformation according to (4.1) with satisfying the condition (4.4). Then we have:

The speed and the torsion angle do not depend on nor . That is, and for all and .

The signed curvature angle and satisfy
(4.7) where .

The deformation of the frame is given by
(4.8)
Proof.
We give the proof of Proposition 1 only, since Proposition 2 can be proved in the same manner. We first show the second and the third statements. We denote , and for simplicity. Since is a constant by the preceding argument, the deformation of can be computed from (4.1) and (4.4) as
(4.9) 
Differentiating with respect to , we have
(4.10) 
Noting
(4.11) 
and
(4.12) 
(4.13) 
which is equivalent to (4.7). This proves the second statement. Next we derive . We have from the definition of
(4.14) 
Noting
(4.15) 
(4.16) 
and
(4.17) 
(4.18) 
We immediately obtain from (4.9) and (4.14) as
(4.19) 
Then we have (4.8) from (4.9), (4.14) and (4.19), which proves the third statement. Finally, differentiating with respect to , it follows from (4.18) and (4.2) that
which implies . This completes the proof of the first statement. ∎
Remark 2.
The condition (4.3) suggests the potential function in Proposition 1 as
(4.20) 
Then (4.5) is rewritten as
(4.21) 
To the best of the authors’ knowledge, this is a novel form of the semidiscrete potential mKdV equation. In fact, the continuum limit , , , yields the potential mKdV equation
(4.22) 
Similarly, introducing the potential function in Proposition 2 such that
(4.23) 
suggested by (4.4), (4.7) is rewritten as
(4.24) 
which is nothing but the semidiscrete sineGordon equation [3, 39, 40].
Remark 3.
In the above argument, we assume that the speed of the deformation in (4.1) is a constant and does not depend on . Then, by demanding the deformation to preserve arc length ((4.3) or (4.4)), it followed that the torsion angle is also preserved. Conversely, it seems to be the case that for the deformation to preserve both of arc length and torsion angle, the speed is required not to depend on .
Remark 4 (Continuum limit).
The isoperimetric torsionpreserving discrete deformations for the discrete space curves with constant torsion have been considered in [18], where the deformations are governed by the discrete sineGordon and the discerte mKdV equations. It is possible to obtain the continuous deformations discussed in this Chapter by suitable continuum limits from those discrete deformations. More precisely, let () be a family of discrete curves obtained by applying the discrete deformations times to , where is the discrete curve with a constant speed and a constant torsion angle . Then the above discrete deformation is given by
(4.25) 
Then if is constant for all , the isoperimetric condition and the compatibility condition of the Frenet frame yields the discrete mKdV equation
(4.26) 
when , and the discrete sineGordon equation
(4.27) 
when with
(4.28) 
For the discrete mKdV equation (4.26), taking the limit
(4.29) 
(4.26) is reduced to the semidiscrete mKdV equation (4.5). Similarly, the discrete sineGordon equation (4.27) is reduced to the semidiscrete sineGordon equation (4.7) by the limit
(4.30) 
Obviously, the discrete deformation equation of the discrete curve (4.25) is reduced to the continuous deformation equation (4.1). Moreover, it is easily verified that the discrete deformation equations of the Frenet frame in [18] is reduced to (4.6) and (4.8).
4.2 Turning over motion of Kaleidocycles
Now, let us consider an Kaleidocycle with an odd . If it is antioriented , we can define an oriented Kaleidocycle by taking its “mirrored image” which conforms the definition 3.
The Kaleidocycle corresponds to a closed discrete curve of length having a constant speed and a constant torsion angle whose is oriented. Since is closed, for (4.1) to define a deformation of , should be periodic for any . The equation (4.3) together with form a linear system for which is regular since is odd. Therefore, we can find uniquely as the solution to the system. Then, the equation (4.1) generates a deformation of which preserves the segment length, the torsion angle, while remains closed. That is, the turning over motion of the Kaleidocycle is governed by the semidiscrete mKdV equation (see Fig. 12). Note that by (4.5), the total curvature angle is also preserved.
Currently, we do not know how to generate a motion when is even.
5 Extreme Kaleidocycles
We defined Kaleidocycles in Def. 3 and saw the torsion angle cannot be chosen arbitrarily. A natural question is for what torsion angle there exists an Kaleidocycle for each . It seems there are no Kaleidocycles with . For , we conducted numerical experiments with [22] and found that there exists which satisfy the following. Recall that is the projection of the configuration space onto the axis, where .

When is odd, and .

When is even, and .
Moreover, converges monotonously to a constant, where takes the principal value in . Interestingly, at the boundary values the fibre of seem to be exactly onedimensional for any . This means, they are exactly the onedimensional orbits defined in §4.2 when is odd.
We summarise our numerical findings.
Conjecture 1.
Let . We have the following:

The space is a circle. Moreover, the involution defined by induces isomorphisms when is odd and when is even.