A closed linkage mechanism having the shape of a discrete Möbius strip
TTranslated from the Symposium Proceedings of the 2018 Spring meeting of the Japan Society for Precision Engineering, pp. 62–65, 1 Mar. 2018.
A closed linkage mechanism having the shape of a discrete M ¨obius strip
Shizuo KAJI ∗ A closed linkage mechanism in three-dimensional space is an object comprising rigid bodies connected with hinges in acircular form like a rosary. Such linkages include Bricard6R and Bennett4R. To design such a closed linkage, it is necessaryto solve a high-degree algebraic equation, which is generally difficult. In this lecture, the author proposes a new family ofclosed linkage mechanisms with an arbitrary number of hinges as an extension of a certain Bricard6R. They have singularproperties, such as one-dimensional degree of freedom (1-DOF), and certain energies taking a constant value regardlessof the state. These linkage mechanisms can be regarded as discrete M¨obius strips and may be of interest in the context ofpure mathematics as well. However, many of the properties described here have been confirmed only numerically, with norigorous mathematical proof, and should be interpreted with caution.
Key words: (5 - 10 words) linkage mechanism, kinematic chain, deployable structure, Kaleidocycle, rotating ring oftetrahedra, Kirchhoff elastic rod, curve and ribbon theory
1. Introduction
A joint is called a hinge (or revolute joint) if it has a rotation axisand can freely change its direction around this axis. Let us call aset of rigid bodies connected by hinges a linkage mechanism . Oneexample is the Bricard6R linkage (Fig. , left), a linkage consist-ing of six hinges. One of the most important subjects of the studyof linkage mechanisms is to determine all possible states, i.e., theconfiguration space of the linkage mechanism. Mathematically, be-cause the state of a hinge can be identified with the unit circle S := [0 , π ) using the rotation angle, the configuration space of asystem consisting of n hinges will be a subspace of the torus ( S ) n . Fig. 1 : Bricard6R and -KaleidocycleLinkage mechanisms have various applications, such as in robotarms, folding and deployable mechanisms, and power transmis-sion. Details of the classification and analysis of linkage mecha-nisms can be found in [10]. In the context of mathematics, study-ing the configuration space of linkage mechanisms is a topic in thearea of topology [4, 5], which is the specialty of the author.The most important property of a configuration space, whichshould be considered first, is its dimension. This is equal to thedegree of freedom ( DOF ) of the motion of the mechanism . The mobility formula (also called the Chebychev-Gr¨ubler-Kutzbach cri-terion) M = 6( N − − n ) + n (cid:88) i =1 f i (1) ∗ Yamaguchi University / JST, PRESTO More precisely, S is obtained by identifying the two ends { , π } ofthe closed space [0 , π ] . A configuration space can have a singularity. When we discuss thedimension of a configuration space, however, we will neglect them. Wewill consider the degrees of freedom assuming the mechanism is in genericstates that cover almost all possible states. gives M , an estimation of the DOF of the system, by simply count-ing the number of variables and constraints involved. Here, N isthe number of rigid bodies involved, n is the number of joints,and f i is the DOF of each joint . If some of the constraints areredundant and not independent, the system possesses degrees offreedom larger than this estimation; such a system is called over-constrained . Some famous mechanisms, including the Bricard6Rand Bennett4R linkages, are over-constrained. In contrast, othersystems have degrees of freedom smaller than this simple estima-tion. This occurs, as discussed later, as a result of the system beinga real algebraic variety. In this lecture, we will consider closedlinkage mechanisms, each of which has n hinges for an arbitrarynatural number n ≥ that are connected in a cyclic manner, andhas just one DOF. In particular, the n = 6 case corresponds to avariation of the Bricard6R linkage, and hence, the n ≥ case canbe regarded as its generalisation.
2. Kaleidocycle
The cyclic, closed linkage mechanism described here is based on atoy called a Kaleidocycle [8](
Fig.1 , right). A Kaleidocycle , whichcan be made by folding a sheet of paper, is a cyclic object con-sisting of n congruent tetrahedrons, where two opposite sides ofeach tetrahedron serve as hinges to connect the adjacent tetrahe-drons, and you can continue rotating the object . Let us call thislinkage with n hinges for n ≥ the n -Kaleidocycle. The mainsubject of this lecture is to show that the DOF of a Kaleidocyclecan be made one-dimensional (rotating motion) by selecting a par-ticular shape for the tetrahedron, although a Kaleidocycle generallyhas a larger DOF, such as bending motion in addition to rotating,as n becomes larger. We first describe a mathematical frameworkfor Kaleidocycles using the Denavit-Hartenberg parameters, whichare the standard parameters for the analysis of linkage mechanisms(e.g., [9, Ex 5.2, Ex 8.13]).We first consider a serial linkage mechanism obtained whenopening a closed linkage by cutting one of its hinges, and we will For a hinge, f i = 1 A Kaleidocycle is also referred to as a rotating ring of tetrahedra insome articles; be careful when performing a literature search. Hereinafter, this motion will be referred to as rotating motion; it mightalso be called everting motion. a r X i v : . [ m a t h . HO ] S e p reat conditions for the serial linkage to be closed as an inversekinematics problem. Let the direction of the i -th (0 ≤ i ≤ n ) hinge be b i ∈ S := { ( x, y, z ) ∈ R | x + y + z = 1 } and theposition of its centre be γ i ∈ R . Here, the n -th hinge correspondsto the end that emerges as a result of opening up a ring. The seriallinkage considered will close if the opposite end overlaps with this,and there are two kinds of those closed cases. Let us say that sucha closed linkage is • Oriented if b n = b , γ n = γ • Non-oriented if b n = − b , γ n = γ .Because, for the linkage mechanism we are now considering ( n -Kaleidocycle), two adjacent hinges are fixed with a rigid body, andthe twist angle for two adjacent hinges is constant across all sets oftwo adjacent hinges (as all the tetrahedrons are congruent), we canwrite b i − · b i = c , using a parameter c ∈ [ − , that stands forthe cosine of the twist angle, and the inner product of b i − and b i .If b i − and b i are not parallel, γ i can inductively be determined as γ i − + b i − × b i using the cross product . If all hinges are parallel(i.e., c = ± ), the corresponding linkage practically represents aplanar one, which may not be of interest; such a case will be calleda trivial case and neglected hereafter.Summarising the above, the configuration space of an n -Kaleidocycle can be identified with the solution space (set of realsolutions) of the simultaneous quadratic equations n (cid:88) i =1 b i − × b i = (0 , , , b i − · b i = c, b i · b i = 1 (1 ≤ i ≤ n ) . (2)Contrary to their appearance, studying this solution space is noteasy (cf. [6]).It is worth noting that the parameter c cannot be freely chosen.For some tetrahedrons, it is impossible to connect copies of themin a way such that both ends meet to make a cyclic object. Forexample, when n = 6 , the above simultaneous equations have so-lutions only when c = ± for oriented cases, or c = 0 for non-oriented cases. The non-oriented case with c = 0 corresponds tothe Bricard6R linkage and has a configuration space homeomor-phic to S , i.e., this linkage has one degree of freedom. For each n and each of the oriented and non-oriented cases, the range of c within which solutions exist can be written as follows, using acertain constant c n : • Oriented cases with even n : [ − , • Oriented cases with odd n : [ − c n , • Non-oriented cases with even n : [ − c n , c n ] • Non-oriented cases with odd n : [ − , c n ] n -Kaleidocycles with a non-trivial boundary value c = ± c n inthe above range are called extreme n -Kaleidocycles, which are themain topic of this lecture. The oriented and non-oriented extremecases with an odd value of n are transformed to each other if every For the purpose of discussing the DOF of a system alone, the center ofeach hinge does not have to be physically positioned at γ i , and the hinge isallowed to be at any position in the line that passes through γ i and possessesorientation vector b i (see Fig. , right). This is an important remark whenconsidering applications to folding structures for which each hinge can slidealong its axis. other hinge in one linkage is oriented toward the opposite direction;the c = ± c n cases with an even value of n are also transformedto each other if every other hinge in one linkage is oriented towardthe opposite direction. Therefore, there is essentially only one extreme n -Kaleidocycle for each n . Fig. 2 : Oriented (left) and non-oriented (right) extreme 7-KaleidocyclesThe value of c n , which is a solution of a high-degree polynomialequation, cannot be obtained algebraically , but it can be easily ob-tained numerically as a solution of a constrained optimisation prob-lem. Values of c n obtained by numerical computation are shown in Table1 .
3. extreme Kaleidocycle
In the previous section, we saw that the set of all possible statesof n -Kaleidocycles can be identified with the solution space of thesimultaneous quadratic equations (2). Considering the rotationaland reflectional symmetries of the entire system, we can choose b = (0 , , and b = (0 , √ − c , c ) . With this, let us countthe DOF. Because the number of variables (except c ) is n − ,and the number of equations is n −
1) + ( n − , the DOFof the system is expected to be n − , which is consistent with (1).Although the expected DOF for a -Kaleidocycle with c = 0 is , the actual DOF is , and this system is an over-constrained one(to account for this, [1] explores improvements for Eq. (1)). Thephenomenon where the expected DOF differs from the actual DOFoccurs as a singular property for the extreme n -Kaleidocycle, andthe DOF is in fact , regardless of n . n -Kaleidocycles other thanthe extreme ones can undergo bending motions in addition to therotating motion, and generically have ( n − -DOF, according to(1). Although the author is not very knowledgeable about linkagemechanisms, he presumes that the extreme n -Kaleidocycles mightrepresent a very rare family of linkage mechanisms, as he could notfind another example of a mechanism for which the DOF is always no matter how many hinges the linkage has. Remark:
The set of all possible Kaleidocycles can be identifiedwith the space of all real solutions for the simultaneous equations(2) with the parameter c being a variable. For each fixed value of c , a slice is determined in this solution space, and the connectedcomponent of this slice provides the configuration space of the cor-responding Kaleidocycle. The configuration space of the extreme n -Kaleidocycle corresponds to the slice where c has the boundaryvalue, and the slice determined by c reduces to a one-dimensionalspace only there . Because this is an important point, let us illus-trate the situation with a very simple example by devoting some There also exists the mirror for a c . When n = 7 , c n can be determined algebraically, as it is a solution ofa cubic equation. Thinking of the real algebraic variety determined by (2) as the moduli ages. For simplicity, in this example only, we consider joints, nothinges, that can orient in any direction (i.e., have the same DOF as S ). Fig. 3 : A simple example in which the DOF change according toparameter values.As shown in Fig. , we consider a system consisting of threejoints and two rods, in which each of the two rods is connected toa wall with a joint and the opposite ends of the two rods are con-nected to each other with another joint p in the three-dimensionalspace . The configuration space of this system can be identifiedwith the space of all possible locations of the junction p , which isequal to the space of solutions of the simultaneous quadratic equa-tions { p ∈ R | | p − a | = l , | p − b | = l } . Regarding thisspace, there are three possible cases, according to the length of therod l : • When l > h , the configuration space is a circle (in the planenormal to the page). • When l = h , the configuration space is a point. • When l < h , the configuration space is an empty set.Note that the dimension of the configuration space reduces whenthe parameter l has the boundary value h/ . Real solutions exhibitmore complicated behaviours compared with complex solutions ofa system of algebraic equations.Let us describe some of the intriguing properties of an extreme n -Kaleidocycle, other than the DOF being one. In [7], a systemwas considered in which a torsional spring that has an energy po-tential proportional to the square of its rotation angle is attached toeach hinge of a -Kaleidocycle. One subject studied in the refer-ence was to determine equilibrium points for this -Kaleidocyclebecause the energy of this linkage changes according to its stateduring its rotating motion. Meanwhile, an extreme n -Kaleidocyclewith seven or more hinges, in contrast, has a peculiar property: theenergy is constant, i.e., when n ≥ , E bend := n (cid:88) i =1 arccos (cid:18) ( b i − × b i ) · ( b i × b i +1 ) | b i − × b i || b i × b i +1 | (cid:19) is almost constant in the configuration space of an extreme n -Kaleidocycle. This is interesting with respect to both theory andapplication, as it means that although the angle of each hinge willhave various values as the Kaleidocycle rotates, the total energypotential remains constant, and no force is required to rotate thelinkage.An extreme n-Kaleidocycle ( n ≥ ) has a constant value alsofor the following two energies in its configuration space. One is space for a Kaleidocycle, you are seeing a singular fibre for the projectionmap onto the c axis. Equation (1) cannot be applied to this system without modificationsbecause the walls are fixed. the Coulomb potential, when the centre of each hinge is electricallycharged: E clmb := (cid:88) i If considering all Kaleidocycles, including non-extreme cases, E bend appears to have a minimum value when the positions of γ i are as planar as possible and have a higher symmetry , and anextreme n -Kaleidocycle (except when n = 7 ) does not provideeven a local minimum. 4. Discrete Strips As inferred in the title of this article, extreme n -Kaleidocycles havesome geometrical aspects, which are the topic of this last section.The author apologises for the description being a little bit rough,partly because of the limited number of pages. Consider a closedcurve γ : S → R , | ˙ γ ( s ) | = 1 with a length of π in the three-dimensional space. When a map b : [0 , π ] → S such that ∀ s, ˙ γ ( s ) · b ( s ) = 0 is given, we say that the set of γ and b is a strip with γ being its centre line . Specifically, a strip is a curvefor which a direction normal to the tangent has been determined ateach point on the curve. Imagine a M¨obius strip-like object thatis made by gluing both ends of a long and narrow sheet of papertogether after twisting it several times.For a given strip, we consider the number of half-twists for thestrip; although its meaning may be intuitively clear, it is preciselydetermined as T w + W r ) using the Cˇalugˇareanu-White theorem. For example, when n is even, E bend has a minimum value for theKaleidocycle with c = 0 in which every other hinge is parallel to the z -axis and they form a regular polygon when viewed from the direction of the z -axis. ere, T w := 12 π (cid:90) π ˙ b ( s ) · ( ˙ γ ( s ) × b ( s )) dsW r := 14 π (cid:90) π (cid:90) π ( ˙ γ ( s ) × ˙ γ ( s )) · ( ˙ γ ( s ) − ˙ γ ( s )) | ˙ γ ( s ) − ˙ γ ( s ) | ds ds and these quantities are called the twist and writhe, respectively.Each of these quantities has various geometrical interpretations.For example, T w measures how much the vector obtained byparallel-transporting b (0) along the centre line one round differsfrom b (2 π ) , while W r represents the average number of self-intersections in the image of the centre line projected in every di-rection. In particular, the latter is a quantity that does not dependon b , only on the centre line. Intuitively, T w and W r represents theextent of the strip winding around the centre line and the extent ofthe centre line running irregularly, respectively, and the sum makesthe twisting number for the whole system.A Kaleidocycle can be regarded as a strip such that its centreline is the polygonal line connecting γ i and there are associatedvectors interpolating b i and b i +1 with a constant angular velocityon the line segment γ i γ i +1 (Fig. , left). For a continuous curve,the direction in which the tangent changes is called the normal,and the cross product of the tangent and normal is called the bi-normal ; as b i is normal to both of the two neighbouring “tangents” γ i − γ i , γ i γ i +1 , it can be thought of as the discrete binormal upto sign. Furthermore, the rate of change of a binormal is called the torsion ; ± c can be thought of as its discrete version up to a constantfactor.In this setting, calculation shows T w = π (cid:80) ni =1 arccos( b i − · b i ) = n arccos( c )2 π , and W r for the central polygonal line is given,e.g., in [2]. For an extreme n -Kaleidocycle, the number of half-twists is for c = c n or n − for c = − c n . The c = c n casegives, by definition, the least twisted strip among those with an oddnumber of half-twists, and it seems mysterious that the number ofhalf-twists for this case is not , but . As seen from Table , T w for the c = c n case appears to monotonically converge to a con-stant as n increases; if this is the case, T w + W r ) = 3 requiresthat W r also monotonically converges to a constant. It is unknownwhat the strip obtained in the limit and its centre line are. With itsrotating motion, its centre line also moves, but because T w is con-stant, W r is also constant. It is interesting to see this from the pointof view of the curve on the sphere drawn by the tangent moving onthe centre line, i.e., the Gauss map s (cid:55)→ ˙ γ ( s ) . Here, the discreteGauss map is supposed to be yielded by the curve connecting thevectors on the sphere with arcs of great circles. From the Gauss–Bonnet theorem, the area cut off by this curve from the sphere onthe left hand is equal to πW r modulo π . E bend is equal to thesquared sum of the length of each arc. According to the rotatingmotion, the image of the Gauss map moves in such a way that boththe area cut off and the squared sum of the length of each arc areconstant. Although the author hasn’t been able to write down thetime evolution of this yet , results of numerical experiments indi- The rotating motion takes the form of so-called falling cat motion, themotion resulting in a different direction as a whole after rotating once whilepreserving the angular momentum. Therefore, applications to, e.g., anten-nas that can change their direction under the condition of zero gravity mightbe possible. cate soliton-like structures, which are expected to provide a key tounderstanding the constancy of the various energies.The motion of the centre line can be interpreted also in the con-text of an elastic rod problem. If we consider the tangent and b andtheir cross product at each point on the centre line of a strip, wehave a family of orthonormal systems ( adapted frame ) along thecentre line. A curve with this adapted frame is often studied as amodel of an elastic rod (cf. [3]). For an extreme n -Kaleidocyclewith n ≥ , as both b i − · b i and E bend are constant, we can see thatthe isotropic Kirchhoff elastic energy also remains constant duringthe rotating motion. Implications for the theory in [3] may also beof much interest. Fig. 4 : Strip representations for the extreme -Kaleidocycles cor-responding to the c = c n and c = − c n cases (left and centre, re-spectively). The -Kaleidocycle (right) made by sliding the centreof a hinge of the extreme -Kaleidocycle along the hinge with acertain offset and extending it one more round can be regarded as aM¨obius strip cut along its centre line, which gives a rotating trifo-liate knot (cf. [11]).Finally, because the number of half-twists is always an integer,two solutions of (2) with different numbers of half-twists belong todifferent connected components. The properties of those connectedcomponents may be of mathematical interest; the author numeri-cally searched for n -Kaleidocycles with c having a boundary valueon the components other than those with the number of half-twistsbeing or n − and found that at least they seem to have no sin-gular properties, similar to those for the extreme cases describedabove. Acknowledgements This lecture is partially based on a collaborative research with E.Fried, M. Grunwald, and J. Sch¨onke (OIST). The author is sup-ported by JST PRESTO (Grant Number: JPMJPR16E3). References [1] P. W. Fowler and S. D. Guest, A symmetry analysis of mechanisms inrotating rings of tetrahedra , Proc. R. Soc. A 461, 1829–1846 (2005).[2] K. Klenin and J. Langowski, Computation of writhe in modeling ofsupercoiled DNA , Biopolymers 54, 307–317 (2000).[3] J. Langer and D. Singer, Lagrangian Aspects of the Kirchhoff ElasticRod , SIAM Reviews 38, 605–618 (1996).[4] R. J. Milgram and J. C. Trinkle, The geometry of configuration spacesfor closed chains in two and three dimensions , Homology, Homotopyand Applications 6.1, 237–267 (2004).[5] P. Paveˇsi´c, A Topologist’s View of Kinematic Maps and ManipulationComplexity , arXiv:1707.03899.[6] M. Raghavan and B. Roth, Inverse Kinematics of the General 6R Ma-nipulator and Related Linkages , J. Mech. Des 115(3), 502–508 (1993).[7] C. 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