# Single degree of freedom everting ring linkages with nonorientable topology

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Edited by Howard A. Stone, Princeton University, Princeton, NJ, and approved November 15, 2018 (received for review June 8, 2018)

## Significance

Linkages are the basic functional elements of any machine. Known established linkages with a single degree of freedom, which facilitates control, have so far consisted of six or fewer links. We introduce “Möbius kaleidocycles,” a class of single-degree of freedom ring linkages containing nontrivial linkages having less mobility than expected. Möbius kaleidocycles consist of arbitrarily many (but at least seven) identical hinge-joined links and may serve as building blocks in deployable structures, robotics, or chemistry. These linkages are chiral and have a nonorientable topology equivalent to

## Abstract

Linkages are assemblies of rigid bodies connected through joints. They serve as the basis for force- and movement-managing devices ranging from ordinary pliers to high-precision robotic arms. Aside from planar mechanisms, like the well-known four-bar linkage, only a few linkages with a single internal degree of freedom—meaning that they can change shape in only one way and may thus be easily controlled—have been known to date. Here, we present “Möbius kaleidocycles,” a previously undiscovered class of single-internal degree of freedom ring linkages containing nontrivial examples of spatially underconstrained mechanisms. A Möbius kaleidocycle is made from seven or more identical links joined by revolute hinges. These links dictate a specific twist angle between neighboring hinges, and the hinge orientations induce a nonorientable topology equivalent to the topology of a

Linkages have been known since antiquity (1, 2). They can be found in nature, as in the powerful jaw mechanism of the parrotfish and the mammalian knee joint (3), in the vertebrate skull (4), in the raptorial appendages of the mantis shrimp (5), and in countless gadgets and machines (6). The latter range from simple manual tools (like bolt cutters) to deployable structures (like umbrellas, foldable camping gear, solar panels for spacecraft, and portable architecture) to intricate components of robots and prosthetic devices.

Designers of deployable structures have considerable interest in adopting notions derived from rigid origami as described, for example, by You (7) or Chen et al. (8). This design principle takes advantage of the folding and unfolding of structures made from flat rigid bodies connected by revolute hinges as exemplified by the famous folding of Miura (9). The resulting constructions belong to the general class of mechanisms made from rigid bodies connected by joints. You and Chen (10) note that all such mechanisms, which they call “motion structures,” combine a small set of fundamental building blocks: scissor-like elements, the Sarrus linkage, the Bennett linkage, and the Bricard linkage. Each of these linkages has one degree of freedom and except for the first, is overconstrained in the sense that it can move, although a simple mobility analysis dictates otherwise.

We present a class of ring linkages (also known as closed loop kinematic chains) that are fundamentally different from all previously known types. These linkages can have an unlimited number (greater than or equal to seven) of identical rigid bodies joined by hinges but still have only a single degree of freedom; an example is shown in Fig. 1. Except for the one with seven hinges, each of these objects is underconstrained, meaning that it has fewer degrees of freedom than a simple mobility analysis would suggest. Since they are rings and share the topology of a

## Classical and Möbius Kaleidocycles

### Classical Kaleidocycles.

A classical six-hinged kaleidocycle (K6) is a closed ring of six identical tetrahedra, the opposing edges of which serve as hinges. This object can be identified as the trihedral version of a general linkage invented by Bricard (11) in 1927, which is a closed loop kinematic chain consisting of six links connected by revolute hinges (and is known as a “6R Bricard linkage”). Fig. 2, *Upper* shows a paper model of a conventional K6 and a 3D printed realization of a 6R Bricard linkage that is kinematically equivalent to the paper model. A K6 possesses a single internal degree of freedom manifested by a cyclic everting motion, during which different tetrahedral faces are periodically exposed while a threefold rotational symmetry is preserved. In applications, the single degree of freedom affords controllability and is, therefore, a desirable property. Detailed kinematic analyses of a K6 are in, for example, Arponen et al. (12) and Fowler and Guest (13).

A classical eight-hinged kaleidocycle is made like six-hinged ones but with eight tetrahedra. This object is nevertheless markedly different from its six-hinged counterpart. It has two internal degrees of freedom; in any configuration, it can move in at least two independent directions as shown in Fig. 2, *Lower*. This raises a question: how many degrees of freedom does a ring of N tetrahedra linked by N revolute hinges generally have? If the

### Möbius Kaleidocycles.

The counting argument above shows that classical kaleidocycles made with more than seven tetrahedra usually have several degrees of freedom and thus, move in various ways with no prescribed regular internal motion. What all conventional kaleidocycles share is that the hinges of each tetrahedron are orthogonal. We use generalized tetrahedral shapes, leading to an inherently different class of kaleidocycles. These are disphenoids—or “twisted tetrahedra”—the four faces of which are congruent acute-angled triangles. With this generalization, the two hinges of each tetrahedral link are twisted by a necessarily acute “twist angle” α as illustrated in Fig. 3.

For a chain of identically twisted tetrahedra, there is a natural definition of orientation that allows the chain to be viewed as a twisted band with two “edges” and two “sides.” Each hinge is identified with a vector originating at one of its ends and pointing to the other end. The orientations of two consecutive hinge vectors shared by a tetrahedron are chosen so that the angle between them is the twist angle. The two edges of the twisted band are given by the line segments connecting either the origins (one edge) or the endpoints (the other edge) of the hinge vectors as illustrated in Fig. 4. With this notion of orientation, the closure of the chain induces a topology. The chain can be closed in two ways depending on how the terminal hinges are brought together: the corresponding vectors are either parallel and the closed band is orientable or antiparallel and the band is nonorientable. Hereafter, we consider the nonorientable case. Each such ring is topologically equivalent to a Möbius band.

There is an N-dependent critical twist angle

### Critical Twist Angle.

To show that there is a critical twist angle *SI Appendix*, section 1, we present the full mathematical description, discuss different solution methods, and detail how we determine

### Single Degree of Freedom.

One way to confirm that Möbius kaleidocycles have only a single degree of freedom is to show that the real solution set of the system is 1D, corresponding to a curve in the space of all hinge orientations. The algorithm for this is explained in detail in *SI Appendix*, section 1.

A different approach involves viewing kaleidocycles as assemblies of bars and pin joints and performing a kinematic analysis based on Calladine’s generalization of Maxwell’s rule for the stiffness of frames as presented by Pellegrino and Calladine (16). The idea is nicely illustrated with a simple example involving a chain of three bars, where the end joints are connected to the foundation as shown in Fig. 5. If the distance between the foundation joints is less than the sum of the bar lengths, we have a finite mechanism that can move (Fig. 5, *Left*). However, if the distance between the foundation joints equals the sum of the bar lengths, this finite mechanism disappears, and two infinitesimal mechanisms emerge that can be thought of as small displacements of the two internal joints in the direction orthogonal to the bars (Fig. 5, *Right*). Furthermore, a state of self-stress is possible, which here corresponds to a tension in the bars. This self-stress stiffens the two infinitesimal mechanisms. In *SI Appendix*, section 3 we describe the Maxwell–Calladine analysis for Möbius kaleidocycles in full detail. We find that an MKN has

### Topology of a 3 π -twist Möbius Band.

The topology of a Möbius kaleidocycle can be characterized by the linking number *SI Appendix*, section 1.

### Seven- and Nine-Hinged Möbius Kaleidocycles.

We discuss two representative Möbius kaleidocycle subtypes with very distinct features. A paper model of a seven-hinged Möbius kaleidocycle (MK7) is shown in Fig. 6. The motion of this MK7 is notably less regular than the motion of a K6. In any of its configurations, the tetrahedra appear to obstruct one another and thereby, prevent motion. This MK7 can nevertheless undergo a complete eversion (Movie S2). The false impression of obstruction arises, because the hinge lengths of the tetrahedra are maximized for the particular design shown. The hinge length is an independent geometric design parameter that specifies the size of the tetrahedra. The constraint forbidding collisions of the tetrahedra during the motion limits the hinge length as in the case of the K6. This leads to the existence of a maximum hinge length for every MKN. If the hinges for the MK7 paper design in Fig. 6 were any longer, then neighboring tetrahedra would collide in certain phases of the eversion.

An MK9 shares the threefold rotational symmetry of the K6, as shown in Fig. 6, and thus, it differs intrinsically from an MK7. If N is divisible by three, then an MKN also has that same symmetry. Due to its symmetry, the motion of an MK9 is much more regular than that of an MK7 as Movies S1, S4, and S5 show. The hinge length for the paper MK9 in Fig. 6 and Movie S4 is again maximal; were the hinges any longer, triples of tetrahedra would collide in the center during any attempted eversion.

### Deployable Structures from Möbius Kaleidocycles.

Möbius kaleidocycles can be connected to make deployable structures. To illustrate this, we combine six MK9 to form a hexagonal ring as shown in Fig. 7. Adjacent MK9 in this elementary unit cell have opposite chirality and are connected through a common hinge. The hexagonal elementary cell defined in this way can then be extended to form a 2D lattice. As all MK9 undergo an everting motion, one chiral group of MK9 moves in front of the other group, and the lattice contracts. The two chiral groups form individual triangular lattices that are always parallel, such as is evident from Fig. 7. The extended and contracted configurations of a prototypical hexagonal ring built from six 3D-printed MK9 are shown in Fig. 8. Another idea for a deployable structure is a tetrahedron formed by four MK9, where in each MK9, three symmetric hinges are extended to meet the corresponding hinges of the other MK9 in ball joints positioned at the corners of a tetrahedron.

### Energetics.

Möbius kaleidocycles subject to internal forces that depend on the current configuration of the cycle show interesting behavior. We attach a torsional spring of stiffness B at each hinge of an MKN,

The accurate calculation of *SI Appendix*, section 2. This parameterization allows us to express E as an explicit function of time and to obtain *SI Appendix*, section 4, we plot the energy evolution for an MK9 and in connection with this, the corresponding input-output relations for the kinematic variables (the joint angles

### Limit Surface and Limit Curve.

Since the critical twist angle

The smooth closed midline C, shown in Fig. 11, of S is a curve defined through the limit of the polygonal chain given by the tetrahedral midaxes defined in Fig. 3. Let

The edge *Bottom Right*. The linking number

## Discussion

We have introduced a class of ring linkages made from rigid bodies joined by revolute hinges. Each of these Möbius kaleidocycles is made from seven or more identical links but has only a single degree of freedom manifested by an everting motion more intricate, by far, than that of a classical six-hinged kaleidocycle (K6). This linkage class forms the basis for applications in many distinct fields. As individual objects, Möbius kaleidocycles may be used as robotic arms or as self-propelling rings that swim through liquids. Due to their highly irregular motions, the smaller linkages without threefold rotational symmetry (e.g., MK7, MK8, MK10, and MK11) seem destined for applications involving mixing and kneading or in light effect machines. We reiterate that there are no limits on the shapes of the links as long as collisions are avoided; moreover, a certain shape design may evoke collisions that intentionally result in only limited motions. The practically flat energy landscape of, for example, an MK7 augmented with torsional springs allows for the design of an openable ring that is equivalent to a straight elastic rod in its open form but in its closed form, undergoes a smooth everting motion without any perceivable counterforce. Aside from the two examples of deployable structures provided here, Möbius kaleidocycles can be connected rigidly via hinges, ball joints, or other linkages in limitless ways to create new mechanisms.

Synthetic chemistry is another field of potential applications for Möbius kaleidocycles. Building on the seminal contributions of Heilbronner (24), which among other things, showed that Möbius annulenes should exhibit novel electronic properties, Schaller et al. (25) recently synthesized an annulene with a topology of a

Finally, many fundamental questions in the fields of kinematics, geometry, and topology are prompted by Möbius kaleidocycles. Do other ring linkages with more than seven elements with only a single degree of freedom exist? Which ring geometries can undergo an everting motion and why? Which topologies are possible? The limit surface and its midline raise questions like the following. Which topologies can a closed, constantly twisted band have? What is the shape of a closed curve with constant nonzero torsion and minimal total curvature?

## Acknowledgments

We thank Michael Grunwald for producing image renderings and 3D-printed models, Gianni Furio Royer-Carfagni and Shizuo Kaji for illuminating discussions, John de Bryun and Gustavo Gioia for feedback on the first draft of this paper, and two anonymous referees for constructive criticism and suggestions. We also acknowledge support from the Okinawa Institute of Science and Technology Graduate University with subsidy funding from the Cabinet Office, Government of Japan.

## Footnotes

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^{1}To whom correspondence may be addressed. Email: johannes.schoenke{at}oist.jp or eliot.fried{at}oist.jp.

Author contributions: J.S. and E.F. designed research, performed research, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1809796115/-/DCSupplemental.

- Copyright © 2019 the Author(s). Published by PNAS.

This open access article is distributed under Creative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND).

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