Loop Based Design and Classification of Planar Scissor Linkages

Scissor linkages have been used for several applications since ancient Greeks and Romans. In addition to simple scissor linkages with straight rods, linkages with angulated elements were introduced in the last decades. In the related literature, two methods seem to be used to design scissor linkages, one of which is based on scissor elements, and the other is based on assembling loops. This study presents a systematic classification of scissor linkages as assemblies of rhombus, kite, dart, parallelogram and anti-parallelogram loops using frieze patterns and long-short diagonal connections. After the loops are multiplied along a curve as a pattern, the linkages are obtained by selection of proper common link sections for adjacent loops. The resulting linkages are analyzed for their motion and they are classified as realizing scaling deployable, angular deployable or transformable motion. Some of the linkages obtained are novel. Totally 10 scalable deployable, 1 angular deployable and 8 transformable scissor linkages are listed. Designers in architecture and engineering can use this list of linkages as a library of scissor linkage topologies.


Introduction
Scissor linkages have been used for deployment for thousands of years. Folding tripods with a pair of scissor links on each side have been used by the Greeks and Romans starting from the times before Common Era (True and Hamma, 1994). Today, many deployable structures comprising scissor-like elements (SLEs) are used in wide range of applications such as household goods, lifts, architecture and outer-space structures. In the literature, the academic studies on use of scissor linkages as deployable structures are dated back to 1960s, where Piñero (1961) developed a movable theatre composed of rigid bars and cables. Using the principles of SLEs, Piñero (1962;1965) also proposed several structures for pavilions and retractable domes. Piñero's designs required to use some additional elements to lock the system and to provide the necessary stabilization after folding. The disadvantages that are inherent in his designs led other researchers to investigate scissor structures that are not require additional members for stabilization. Zeigler (1976) developed a self-supported dome structure and Clarke (1984) designed a deployable hemispherical dome composed of a novel spatial unit. Although specific configurations of his structure seemed to work fairly well, it allowed only limited geometric shapes and few applications.
The research on deployable structures was expanded by Escrig (1984;1985) who first presented the geometric conditions for deployability of scissor mechanisms composed of translational and polar SLEs. Escrig also developed new spherical grid structures and different types of deployable scissor structures including quadrilateral expandable umbrella, deployable polyhedral structure and compactly folded cylindrical, spherical and geodesic structures Valcárcel, 1987, 1993;Escrig and Sanchéz, 2006). The most notable application of the scissor structures developed by Escrig (1996) was a deployable roof structure for a swimming pool in San Pablo Sports Center in Seville, which consists of two identical rhomboid grid structures with spherical curvature.
Chuck Hoberman (1990) made a remarkable invention on scissor structures with the angulated scissor element (Fig. 1a). The discovery of this element extended the range of application of single degrees-of-freedom (DOF) scissor structures since it allows the structure to radially deploy from a center to the perimeter. Hoberman (1991) created impressive examples of scissor structures by using the angulated elements. Expanding Geodesic Dome, Hoberman Arch, Expanding Sphere, Expanding Icosahedron, Iris Dome and Expanding Helicoid are some of his interesting designs. You and Pellegrino (1997) investigated the conditions on the link lengths for which angulated elements subtend a constant angle and found two conditions leading to two types of generalized angulated elements (GAEs) (Fig. 1b,c). Hoberman's pioneering idea on the angulated element led Kassabian, You and Pellegrino (1999) make further progress on scissor structures and they discovered multi-angulated elements which comprise links with more than three joints. Based on the principles of multi-angulated elements, they developed a deployable structure mounted on pinned columns. Al Khayar and Lalvani (1998) studied the applications of angulated elements to polygonal hyperboloids and proposed many types of deployable hyperboloids by using the regular and semi-regular tessellation methods.  Gantes (1996) systematically investigated "snap-through" effect of the scissor structures that occurs at intermediate geometric configurations due to the geometric incompatibilities between the member lengths. He developed geometric design methodologies and determined deployability conditions for different types of scissor structures in order to achieve stable and stress-free states of such systems (Gantes et al., 1993;1994). Langbecker (1999) studied the foldability conditions of SLEs and presented a systematic method for kinematic analysis of the scissor structures. Using compatible translational SLEs, he also proposed foldable singly-curved barrel vaults and doublycurved synclastic structures (Langbecker, 2001). Kokawa (1997Kokawa ( , 2000 designed an expandable arch composed of scissor pairs and cables and also a retractable loop-dome consisting of 3D multi-angulated SLEs in lamella arrangement. Van Mele (2008) proposed a deployable roof structure in the shape of barrel vault by using scissor arches composed of angulated elements to cover a tennis arena. Rather than using a single arch that is pinned at one end, the scissor arches are cut in half and two halves are pinned to spectator area that are connected at a central hinge in the closed configuration. Rippmann (2008) developed a new scissor unit that has various intermediate hinge points.
By this means, he proposed a structure that can constitute different geometric shapes by switching the locations of the hinge points in his basic scissor unit. Although it seems that the structure provides the form flexibility, in fact, In his first patent about angulated elements, Hoberman (1990) described how identical angulated elements are paired to form angulated scissor-pairs. In the patent, the angle of an angulated element is called a "strut angle", the line connecting the left or right terminals of a pair of elements is called a "normal line" and the angle between two normal lines is called a "normal angle" (Fig. 1a). Although Hoberman (1990) describes an angulated scissor-pair as a module, he calls his mechanisms as "loop assemblies" implying that he constructs the mechanisms by assembling loops (in this case rhombuses). Also, the "normal line"s are normal to the curve to be approximated.
Later on in a lecture at MIT, Hoberman (2013) described his construction of expanding polygons as an assembly of "hinged rhombs" and calls the "normal lines" as "perpendicular bisectors" of polygonal sides. Although it is not mentioned in the patent, this latter study shows that Hoberman assembles rhombus loops to obtain his linkages.
Similar deployable structures are also issued by Liao and Li (2005) and Kiper and Söylemez (2010), but their procedure is not based on assembling loops. Scaling deployment contains linear deployment along a line and radial deployment (enlarging circular arc) as special cases. This classification for the transformation types is followed in this study as well.

Figure 2
Deployable v.s. transformable motion: a) scaling/dilation type deployable; b) angular deployable; c) transformable (Maden et al., 2019) This study classifies and systematically analyzes assembly of suitable loops for planar deployable structures comprising SLEs. According to the authors' best knowledge, this is the first full classification of scissor mechanisms considered as loop assemblies. As most of the assemblies are noted in the literature, there are several novel assemblies listed in this paper. In Section 2 loops are introduced. In Section 3, possible ways of assembling different loops are presented and the geometric transformation properties of the assemblies are examined. Section 4 concludes the paper.

Loops
Examining various scissor linkages in the literature, we see that the assemblies comprise either kite (deltoid) or parallelogram loops, or as a more special case rhombuses (Fig. 3). A kite is a quadrilateral with a pair of short adjacent equal sides and a pair of long adjacent equal sides. A parallelogram also comprises two pairs of equal sides, but equal sides are positioned opposite to each other. A rhombus is an equilateral quadrilateral, which can be considered as an equilateral kite or an equilateral parallelogram.  configuration where the two short links are inline (Fig. 5). For example, the kite loops in the assembly illustrated in   In a multi-loop linkage, the loops may constrain each other such that some or neither of the loops pass through singular configurations, and hence assembly mode change does not occur. Also assembly mode change may not be possible due to link collisions. A parallelogram loop is less likely to go through assembly mode change, because the mode change requires all links to be collinear and this results in link collisions unless there is a special constructional design. Therefore, the loop assemblies of rhombuses, kites, darts, parallelograms and antiparallelograms result in different mechanisms with different motion characteristics. Next section is devoted to systematically list possible loop assemblies composed of the five mentioned loop types.

Loop Assemblies
A deployable or transformable linkage can be obtained by assembling several loops at their vertices. When two loops are assembled at a common vertex, two pairs of adjacent sides are rigidly connected to each other to constitute a pair of links hinged at the common vertex. For instance, for the assembly of two rhombus loops illustrated in Fig. 7a, the adjacent sides of the loops can be connected to each other to obtain two possible types of Watt-type 6-link kinematic chains. In one of the chains, lower side of left loop is connected to the upper side of the right loop and vice versa (Fig. 7b). Due to the resulting shape this connection type shall be named as X-type connection. In the other alternative chain, upper and lower sides of left and right loops are connected to each other ( Fig. 7c). Due to the resulting shape this connection shall be named as V-type connection. Typically X-type connections are used in scissor linkages. loops. However the design approach in these latter two papers are unit-based design, not loop based design. In this paper we shall work on possible assemblies of identical type of loops, but not necessarily of the same size.
For assembling loops, we will consider a series of loops juxtaposed along a curve, which will be discretized into line segments. For a systematic classification of ways of assembling loops, we shall consider patterns along a line.
Afterwards, the line can be dissected into line segments representing a discretized version of a planar curve.
Patterns along a line are called frieze patterns and there are seven distinct such patterns (Conway et al. 2008). The first four frieze patterns for a general quadrilateral loop is depicted in Fig. 8b. In these patterns, two of the opposite corners of a loop are placed on the line. Unlike the other five patterns, TVR and THV patterns are obtained by repetition of a figure with four copies of the original shape, hence they will not be used in loop patterns. Also horizontal reflection operation results in overlapping loops, which is not desirable, hence TH and THV patterns will not be used in loop patterns. Therefore, only the first four frieze patterns will be used.
Besides the frieze patterns, an alternative way to obtain patterns of quadrilateral loops on a curve is by connecting long and short diagonals of the loop, which we shall name as long/short diagonal (LS) patterns. These type of connections can be seen in (Bai et al., 2014) for rhombus, kite, and parallelogram loops. In general four such possible patterns of quadrilateral loops can be obtained by rotating the loop clockwise (C) or counter-clockwise (CC) or combining the rotation with a horizontal reflection. Amount of rotation depends on the angle between the diagonals of the loops. Accompanied with a translation, the four possible patterns can be listed as TC, TCC, TCH and TCCH patterns as depicted in Fig. 8c. A rhombus loop has horizontal and vertical mirror symmetry, so all four Frieze patterns result in the same pattern.
Also all LS patterns result in the same pattern. Table 1 lists the possible two patterns on a line and examples of loop assemblies with X-type connections on a circular arc. As it is well known since Hoberman's (1990) patent, the rhombus loop assemblies with TI pattern can be used for scaling of any curve. In this pattern, all angulated elements are identical. The TC pattern comprises equilateral GAEs and also results in scaling deployment. These scaling assemblies are worked out by Bai et al. (2014). Kite loop assemblies can be investigated in two distinct subgroups: vertical and horizontal kite loop assemblies.
Since a vertical kite has vertical mirror symmetry, TI and TV patterns are identical. TG and TR patterns are also identical due to vertical mirror symmetry ( Table 2). The assemblies obtained from TI patterns with X-type connections go through a transformable motion. When the pattern is constructed on a straight line, the assembly can bend upwards and downwards such that the curve can switch from convex to concave from and vice versa.
Such transformable assemblies are issued by Yar et al. (2017). A special case is obtained when the pattern is constructed on a circular arc, in which case the linkage has angular deployable motion (Table 3).  A horizontal kite has horizontal mirror symmetry, so TI and TG patterns are identical. TV and TR patterns are also identical due to horizontal mirror symmetry (Table 4). Linkages obtained from TI pattern result in a transformable motion, whereas linkages obtained from TV pattern result in scaling deployment. These scaling deployable assemblies were also examined by Bai et al. (2014). LS patterns are already examined in Table 2. Vertical dart has vertical mirror symmetry, so TI and TV patterns are identical and also TG and TR patterns are identical (Table 5). Just like the vertical kite loop assemblies the assemblies obtained from TI patterns with X-type connections go through a transformable motion as noted by Yar et al. (2017) and the TG pattern of a vertical dart loop results in a linkage with scaling deployment. To the best knowledge of the authors, the scaling linkages connections of dart loops are kept out of scope in this study, because such assemblies result in too much link collisions.  A parallelogram does not possess neither vertical nor horizontal mirror symmetry, but it has a cyclic symmetry of order two, i.e. it has the same shape after half turns. Therefore, the following pairs of patterns are identical: TI and TR patterns; TG and TV patterns; TC and TCCH patterns; TCC and TCH patterns (Table 7). TI patterns with Xtype connections go through a scaling-deployable motion as also noted by Bai et al. (2014).   Gür et al. (2017). TG patterns with V-type connections might be considered as special interest and they result in a transformable motion. TC and TCC patterns with X-type connections also result in transformable motion. All possible assemblies with V-type connections are also investigated, but the resulting motions are generally not found to be of practical importance, as specified at the beginning of the section. Besides the TG pattern of the antiparallelograms depicted in Table 8, one of the rare cases of interest with V-type connections is presented in Figs. 9 and 10, where rhombus loops are assembled in a circle (Figs. 9b and 10b) with TI pattern in order to obtain hexagonal and orthogonal assemblies which are capable of scaling deployment. After the assembly, the outer links are straight rods (angulated elements with 180 kink angle) and the inner links are isosceles angulated elements. It can be seen from the motion that a pair of opposite joints in each rhombus loop move on fixed straight lines, which suggests that all links have the double-slider motion, that is the Cardan motion (see Kiper et al., 2008).

Conclusions
This study presents a systematic way to list possible scissor linkages obtained by assembling rhombus, kite, dart, parallelogram and anti-parallelogram loops using frieze patterns and long-short diagonal connections. For the dart and kite loops, assemblies of vertical and horizontal loops are evaluated separately. For each obtained linkage, the motion characteristics is specified as being scaling deployable or angular deployable or transformable. The linkages listed in this study may be used as a library of scissor linkage topologies. A summary of the results for Xtype connections may be seen in Table 9, where whether a scalable deployable motion or a transformable motion is obtained for the assemblies of a given loop type with a given pattern. Since angular deployment is only obtained for TI pattern of vertical kite loop assembly on a circular arc, it is not presented in Table 9. In Table 9, S stands for scalable deployable and T stands for transformable. Merged adjacent cells and also cells with same superscript (a, b, … h) correspond to the same assembly. Accordingly, 10 distinct scalable deployable and 8 distinct transformable assemblies are listed. Most of the obtained linkages already exist in the literature, but some novel linkages are also obtained. Since scaling linkages are of great importance, especially the vertical dart loop assembly obtained from TG patterns and parallelogram loop assemblies obtained from TG, TC and TCC patterns may be specified as important novel linkages presented in this study.
Dimensional synthesis of these scissor linkages as a general formulation or for specific tasks may be issued in future studies. The novel linkages have potential applications in kinetic architecture, outer-space applications, furniture design and machinery.

Declarations
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