Deployable scissor grids consisting of translational units

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Deployable scissor grids consisting of translational units Kelvin Roovers∗, Niels De Temmerman Department of Architectural Engineering, Vrije Universiteit Brussel (VUB), Pleinlaan 2, 1050 Brussels, Belgium

a r t i c l e

i n f o

Article history: Received 20 October 2016 Revised 28 March 2017 Available online xxx Keywords: Deployable structure Double-layer grid Mechanism Translational unit Joint Geometry Kinematics Design

a b s t r a c t Deployable scissor grids can quickly transform between different configurations, making them particularly fit for mobile and temporary applications. Their ability to deploy typically comes along with a high design complexity and a limited freedom of shape. However, we’ve found that by using so-called translational scissor units it is possible to generate a myriad of curved spatial grids through a design process that can be simplified into a set of two-dimensional problems. The resulting scissor grids are mechanisms with a smooth and stress-free deployment behaviour. Due to their qualities they have formed the topic of previous research, but nevertheless we’ve noticed that a large part of their design potential has remained unexplored. By for the first time unravelling the general principles that govern the motion and shape of this scissor grid type, we’ve managed to reveal various new and interesting design possibilities. This paper presents these new proposals together with the existing ones in order to form a comprehensive overview of the geometric potential and kinematic behaviour of deployable scissor grids consisting of translational scissor units. It covers the mathematical concepts needed to analyse and generate this scissor grid type, ranging from a single scissor unit to large assemblies. In addition, the paper introduces multiple methods to include joints in the line models without modifying their deployment behaviour. This work therefore broadens the design space and compiles the main characteristics of this scissor grid type in order to improve their accessibility and applicability in design. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction Deployable structures can rapidly transform in shape and volume in order to answer to changing needs. Scissor grids are a type of deployable structure consisting of articulated bars (Fig. 1). They have enjoyed much attention in past research for their ability to achieve large volume expansions through an easy to control deployment process. This feature makes them broadly applicable in architecture, engineering and design. Applications include mobile and temporary structures for recreational purposes or disaster relief, portable furniture and panels, deployable solar arrays or antennae for outer space, and adaptable roof and shading systems in static constructions (Alegria Mira et al., 2014; Gantes, 2001; Bernhardt et al., 2008; Buhl et al., 2004). A deployable scissor grid can be considered as a kinematic linkage of scissor units, also known as pantographs or scissor-like elements (SLEs). A scissor unit consists of a pair of bars crosswise interconnected by a revolute joint, allowing a relative rotation about an axis normal to the unit plane (i.e. the plane containing the scissor bars). The imaginary lines running through the upper and ∗

Corresponding author. E-mail addresses: [email protected] (K. Roovers), niels.de.temmerman@ vub.be (N. De Temmerman).

lower end point at both sides of the unit are called the unit lines. Depending on how these lines vary during deployment, different types of scissor units can be distinguished (Fig. 2), each offering a unique range of geometric possibilities and kinematic behaviour. The scissor units considered in this work consist of straight bars and have unit lines that are parallel throughout deployment. They are commonly referred to as translational scissor units. Other scissor grid concepts make use of scissor units with intersecting unit lines (e.g. the polar unit of Fig. 2b) and can even consist of kinked bars (e.g. the angulated unit of Fig. 2c). Scissor grids are created by interconnecting multiple scissor units at their end points. First, scissor units are combined to form single closed loops, called scissor modules (Fig. 3). Afterwards, multiple scissor modules are combined to form scissor grids. Stacking modules of an equal amount of units gives rise to single-layer grids. Well-known examples include the Iris Dome by Hoberman (1991) (Fig. 4a) and the multi-angulated grids by You and Pellegrino (1997b). Double-layer grids are formed by tessellating modules along a surface. They exist in a broader variety of shapes and are interesting for their ability to deploy towards compact bundles of bars (Fig. 4b) (Hanaor and Levy, 2001). When linking scissor units to form scissor modules or grids, various geometric constraints need to be met in order to obtain a deployable assembly. A well-known example is the deployabil-

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Please cite this article as: K. Roovers, N. De Temmerman, Deployable scissor grids consisting of translational units, International Journal of Solids and Structures (2017), http://dx.doi.org/10.1016/j.ijsolstr.2017.05.015

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Fig. 5. Illustration of the deployability constraint.

Fig. 1. Singly curved deployable scissor grid in two deployment stages.

Fig. 2. (a) A translational, (b) polar and (c) angulated scissor unit in its simplest form.

Fig. 3. Examples of scissor modules, i.e. a closed loop of scissor units interconnected side by side.

Fig. 4. (a) Single-layer scissor grid based on Hoberman’s Iris dome consisting of angulated scissor units; (b) spherical double-layer scissor grid with rhomboid lamella pattern consisting of polar scissor units and based on Escrig (1985).

ity constraint, which was proposed by Escrig (1985) and is useful for linkages consisting of scissor units with straight bars. For the two scissor units shown in Fig. 5 which are linked at their ends with semi-lengths a, b, c and d (measured between the intermediate hinge point and an end point of a rod) this constraint states:

a+b=c+d

(1)

Hence the sum of semi-lengths coming together in any node of the scissor grid should be equal. It ensures that all scissor units in the linkage simultaneously reach their most compact state and thus the linkage is theoretically reduced to a single line. The kinematic behaviour of a scissor grid depends on the type of scissor unit used, the geometric constraints that have been applied and the configuration in which the units are assembled. This behaviour is described by the scissor grid’s geometric compatibility

during the two outer deployment stages (e.g. the expanded, functional stage and the compact, stowed configuration) and the transition stage. A scissor grid is geometrically compatible when all its members fit together without deformations. If geometric compatibility exists for all deployment stages, then the grid forms a pure mechanism with a smooth deployment and is referred to as being foldable. A foldable scissor grid needs to be locked once erected in order to obtain a rigid load-bearing structure. A contrasting behaviour is displayed by the so-called bistable scissor grids, which are compatible in the outer deployment stages and incompatible during deployment (Gantes and Konitopoulou, 2004). These incompatibilities need to be overcome by the actuation forces, inducing strains in the members and resulting in a snap-through deployment behaviour. Bistable grids have the benefit of being selflocking, removing the need for external locking devices after erection when subjected to small external loads. The non-linear effects that accompany this snap-through behaviour however complicate the design process (Gantes, 2001). Literature mentions a variety of scissor grid concepts. Each concept is formed from a different combination of scissor units using different sets of constraints to ensure a deployable assembly, each time giving rise to a different geometric potential. The geometric design of scissor grids can be straightforward for flat or singly curved shapes with a high degree of symmetry. On the other hand, once three-dimensional grids with double or freeform curvature are desired, the complexity of the design process quickly rises. The resulting low accessibility to generate and explore scissor grid geometry poses a barrier in their design process. However, scissor grids consisting of translational scissor units form an exception to this issue. Indeed, translational units allow constructing spatial double-layer scissor grids with a myriad of shapes and a foldable deployment behaviour, of which the design can be simplified to a set of two-dimensional problems, as all its upper and lower layer nodes are located on parallel lines. A part of the design potential of translational units has previously been demonstrated by Zanardo (1986), Escrig and Valcárcel (1993), Meurant (1993), Pellegrino and You (1993), Sánchez-Cuenca (1996), Langbecker and Albermani (20 0 0) and De Temmerman (20 07). Nevertheless, many more design options have remained unexplored. By for the first time setting up the global mathematical rules to generate scissor grids consisting of translational units, we have managed to uncover its full geometric potential. This paper presents this potential, covering the existing and a range of new scissor grid types. Section 2 discusses the basic geometric and kinematic concepts regarding translational scissor units and the grids that they form. Section 3 will afterwards present the different grid shapes and configurations possible in case the deployment relies entirely on the scissor motion and all other rotations are locked. The possible configurations when additionally allowing the dihedral angles between the planes of adjacent scissor units to vary are presented in Section 4. Section 5 completes this geometric and kinematic overview by presenting multiple methods, including one new method, to incorporate joints with tangible dimensions into these theoretical line models without compromising the kinematic behaviour of the grid.

Please cite this article as: K. Roovers, N. De Temmerman, Deployable scissor grids consisting of translational units, International Journal of Solids and Structures (2017), http://dx.doi.org/10.1016/j.ijsolstr.2017.05.015

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n  



 i + hi = 0 w

3

(7)

i=1

Since ti = ki−1 ti−1 we know that



ti =

i−1 



k j t1

(8)

j=1

and Eq. (6) can be rewritten as n 

ki = 1

(9)

i=1

 i are parallel to a common plane and all Furthermore, since all w hi are normal to this plane, we can split Eq. (7) into two separate equations: n 

hi = 0

(10)

 i = 0 w

(11)

i=1

Fig. 6. Different forms of translational scissor unit, all characterized by parallel unit lines.

n  i=1

2. Concepts regarding translational units and their linkages A translational scissor unit is characterized by unit lines which are parallel throughout deployment (Fig. 6). In its basic form it consists of two identical straight bars giving rise to a linear motion, or in three dimensional space to planar motion. Therefore they are called plane-translational units (Fig. 6, bottom). Its generalized form consists of two bars of proportional lengths through which it becomes possible to generate curved spatial scissor grids. Hence they are called curved-translational units (Fig. 6, top). For both cases we can distinguish a regular version (Fig. 6, right), where the intermediate hinge point is located in the middle of the bars, and an irregular version (Fig. 6, left), where this point is located eccentrically. The most general translational unit – i.e. the irregular curvedtranslational unit – can be defined by three size parameters l, e and k (where l and k are non-zero positive values and |e| < l) and a deployment parameter, e.g. the angle φ (which ranges between 0 and π ). If e = 0 the unit is plane-translational and if k = 1 the unit is regular. Using the sine and cosine rules we find

t =



2l 2 + 2e2 − 2(l 2 − e2 ) cos φ

(2)

 w = w =

l 2 − e2 (1 + k ) sin φ t

h = h 2el = (1 + k ) t

(3)

(4)

Translational units consist of straight bars and hence benefit from the deployability constraint, given by Eq. (1), to obtain a compactly deployable linkage. For a linkage of translational units (Fig. 7), Eq. (1) becomes

li = ki−1 li−1

(5)

In order to form a module of n translational scissor units, the following equalities must apply (Fig. 7):

kn tn = t1

(6)

Hence the constraints needed to close the spatial scissor module of translational units can be studied into separate two-dimensional planes. Eq. (10) can be studied in the scissor unit planes, or in the unrolled linkage (Fig. 7a). Since all h are parallel, it can be rewritten as n 

hi =

i=1

n  i=1



2ei li ( 1 + ki ) ti



=0

(12)

=0

(13)

or through Eq. (8) as n 



ei li ( 1 + ki ).

i=1

i−1  j=1

 1 kj

When applying the deployability constraint this is further reduced to n 

( ei ( 1 + ki ) ) = 0

(14)

i=1

Eq. (11) on the other hand can be studied in a plane P normal to the unit lines (Fig. 7c). The projection of the scissor module onto P becomes a polygon where each edge corresponds to a scissor unit i and has length wi . The interior angles α i of the polygon correspond to the angles between the scissor unit planes. If n = 3 these interior angles are determined by w1 , w2 and w3 for any state of deployment and can be calculated using the cosine rule (Fig. 8, left). If n > 3 a total of n − 3 interior angles α can be chosen independently of the scissor unit dimensions and deployment stage (Fig. 8, right). In fact, for any fixed state of deployment, the scissor module becomes an n-bar planar linkage with revolute joint axes formed by the unit lines. Therefore, the module obtains an additional n − 3 in-plane kinematic degrees of freedom on top of the scissor action. Scissor modules of translational units are foldable. Indeed, Eqs. (9) and (13), required for forming a compatible scissor module, consist of only size parameters, which remain invariant throughout deployment. Therefore, if we design a scissor module of translational scissor units such that these equations are met for one deployment stage, they will automatically be met for any other deployment stage. The third condition for compatibility given by Eq. (11) requires the possibility to form a closed polygon with edge lengths wi , which is always possible if the largest edge length

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Fig. 7. General linkage of n translational scissor units: (a) planar linkage; (b) the closed spatial module formed by the linkage if Eqs. (9)–(11) are met; (c) projection of the closed module onto a plane P normal to the unit lines.

Fig. 8. The spatial configuration of a three-unit module is fixed for any deployment stage, while a four-unit module has an additional degree of freedom.

doesn’t exceed all the others combined. Fig. 9 shows a practical design method that follows from this observation. Starting from any prism with an n-sided polygonal base, arbitrary crosses are drawn on each of the n faces such that their ends meet. The resulting

crosses will then describe a foldable module of translational scissor units. Its deployment range will generally be limited however, as the deployability constraint hasn’t been met. Similar to a single module, if we design a scissor grid consisting of translational units such that it is compatible in one deployment stage, it will generally be foldable. Its theoretical deployment range is determined by the scissor unit in the grid that first reaches one of its outer deployment stages (when φ = π or φ = 0). As mentioned before, the deployability constraint improves the deployment range by ensuring that all units in the linkage simultaneously reach their most compact stage (where φ = π ). However, there exist a number of limiting factors for the deployment range. Firstly, compatibility is only possible as long as Eq. (11) can be met for any closed loop of scissor units in the grid. For triangulated scissor grids this requires careful design, as will be shown in Section 4.1. For other grids we’ve noticed that this tends to be less of a problem as long as the interior angles α don’t approach π . Secondly, scissor units for which the projection lines onto P intersect must be avoided, as these units will collide when folding towards its compact state and consequently block the mechanism. This fact forms the largest limitation on the geometric freedom of scissor grids consisting of translational units. The unit lines of the doublelayer grid should intersect its base surface at maximum one point, thus eliminating the option to use a closed volume, like a sphere, as a base surface. A workaround however does exist, as shown in

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Fig. 9. (left) Drawing method of a foldable translational module of n units based on a prism with n-sided base and (right) the deployment of the resulting module.

prevented during deployment. These deformations can be studied in the projection of the grid onto a plane P normal to the unit lines. We’ve found two ways to ensure that the interior angles of the projected grid remain constant. In a first method the entire projected grid scales uniformly during deployment, thus keeping its shape unchanged, while in a second this shape changes but the interior angles still remain invariant. 3.1. Shape-invariant projected grid One way to fix all interior angles α i throughout deployment is by ensuring that all edge lengths in the projected grid scale proportionally, and thus that the projected grid only changes in scale, not in shape. For any two units i and j of the scissor grid this can be expressed as:

wi = constant wj

Fig. 10. Examples of single-layer scissor grids of translational units in different deployment stages.

Section 4.2, by connecting two grid halves at only one layer of nodes. Single-layer grids of translational units have limited possibilities, with the most useful configurations being linear or curved towers with tubular sections, as shown in Escrig and Valcárcel (1993) (Fig. 10). More interesting are the double-layer grids, of which a full overview will be given in the following sections. Foldable double-layer grids of translational units contain at least one kinematic degree of freedom given by the scissor action, with possibly additional in-plane mobility resulting from nontriangulated modules. Often this extra mobility is undesired, requiring additional bracing elements to stiffen the grid. Sometimes however, in-plane angular distortions of the grid are essential for obtaining a foldable deployment process. In these cases the joint hubs interconnecting multiple scissor units need to be designed such that they enable the additional rotations, potentially resulting in added design and manufacturing complexity and an increased sensitivity to errors. Therefore it becomes useful to specifically look for solutions that do not rely on angular grid distortions in order to be foldable. 3. Scissor grids of translational units without angular distortions In order to combine a foldable deployment process with joints that fix the angles between scissor planes and thus contribute to the rigidity of the grid, angular distortions of the grid need to be

(15)

Since size parameters l, e and k in Eq. (3) remain constant, Eq. (15) becomes

t j sin φi = ti sin φ j



l 2j + e2j − (l 2j − e2j ) cos φ j sin φi . = constant li2 + e2i − (li2 − e2i ) cos φi sin φ j

Eq. (16) is true if



ej ei = li lj

or

ej ei =− li lj

(16)

and

φi = φ j

(17)

Geometrically this means that the triangles formed by a pair of semi-lengths and the corresponding unit line must all be similar throughout the linkage. Consequently, these grids must always comply with the deployability constraint. Due to the similarity of triangles and Eqs. (3) and (4), we obtain

hj hi = wi wj

or

hj hi =− wi wj

(18)

Therefore, a three-unit module must consist of only planetranslational units (where h = 0), since in a three-unit module of curved-translational units (where h = 0), all three units will be coplanar. Indeed, if all values h differ from zero, one edge length w of the projected triangle will equal the sum of the other two edge lengths:

h1 + h2 + h3 = 0

⇒

w1 ± w2 ± w3 = 0

(19)

The simplest way to design a double-layer grid complying with Eq. (17) is to use plane-translational units, for any such grid adhering to the deployability constraint consists of a chain of all similar triangles. Escrig (1985) presented some regular two-way and three-way planar grids and Pellegrino and You (1993) and You and Pellegrino (1997a) used this method to create closed-loop expandable planar rings (Fig. 11). To access and easily explore the many

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Fig. 11. Planar scissor grids of plane-translational units during deployment: (a) regular two-way grid; (b) expandable ring based on Pellegrino and You (1993).

design options, we presented a general method to generate regular and irregular planar grids based on any planar circle packing in Roovers and De Temmerman (2015) (Fig. 12). You (20 0 0) proposed unconventional scissor components that at their ends embrace the same similar triangles and therefore have projected lengths that vary synchronously with the rest of the grid (Fig. 13). These components additionally give rise to a height difference in order to generate non-planar grids. As demonstrated before, non-flat modules of curvedtranslational units adhering to Eq. (17) must consist of at least four units. Particularly useful is the module of four identical regular units shown in Fig. 14(a). It was used by Escrig and Valcárcel (1993) to generate slanted planar or piecewise-planar grids (Fig. 15). Double curvature can also be generated using lamella patterns (Fig. 16a), as was originally demonstrated by Meurant (1993). New variations on these proposals can be generated by using non-identical irregular units (Figs. 14b and 16 b). This doesn’t affect the base surface described by the grid, but allows to locally vary the density of the grid and the sizes of the scissor bars. Of course this intervention comes at the cost of a decreased uniformity of components. These scissor grids with a shape-invariant projection display a particularly interesting kinematic feature. Since Eq. (17) states that the angle φ is equal for each unit in the grid, the units in the grid don’t only reach their most compact stage together (when φ = π ), but also their furthest deployed configuration (when φ = 0). The deployment range therefore has reached a maximum. In grids of plane-translational units the upper and lower layer of nodes will coincide when φ becomes zero. The bars become orthogonal to the unit lines and are therefore reduced to a single layer (Fig. 17a). During the expansion of grids of curved-translational units, the

Fig. 12. Design of a general scissor grid of plane-translational units based on a circle packing: (a) boundary surface for the circle packing; (b) circle packing; (c) resulting scissor grid in two deployment stages.

Fig. 13. Deployment of one of the unconventional scissor units proposed by You (20 0 0) to introduce height in planar DLGs of translational units without inducing angular distortions.

area of the projected grid reaches a maximum when the shortest bar in each unit is orthogonal to the unit lines. When further deploying the grid, the area of the projected grid will decrease again while the grid still increases in height until it is again reduced to a single line (Fig. 17b). This retraction in area following a first expansion is typical for all scissor grids consisting of curved-translational units, but its effect is greatest for this specific case. Of course the functional outer deployment stage will most likely not correspond to this theoretical outer deployment stage.

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Fig. 14. Modules with reflection symmetry consisting of four (a) regular or (b) irregular curved-translational units.

Fig. 17. Full deployment range of scissor grids with a shape-invariant plan for which each scissor unit is theoretically reduced to a single line in both outer deployment stages (side and top view).

3.2. Angle-invariant projected grid

Fig. 15. Pyramidal grids of curved-translational units.

Fig. 16. Scissor grids of curved-translational units with lamella rhomboid patterns: (a) grid of identical units proposed by Meurant (1993); (b) generalization using nonidentical irregular units.

Alternatively, the angles between the unit planes are invariant if the projected grid forms a tessellation of polygons of which each pair of opposing edges remains parallel and of the same length. Consequently, the grid must consist of even sided modules where opposing scissor units are identical. Odd sided modules can nevertheless still occur in these grids if they consist of units embracing all similar triangles (adhering to Eq. (17)). Triangular modules must therefore consist of three plane-translational units. Compliance with the deployability constraint is not required for this grid type, but is nevertheless recommended. Regular two-way grids based on this principle were previously studied by Zanardo (1986), Sánchez-Cuenca (1996) and Langbecker and Albermani (20 0 0). They proposed grids that are generated by translating two planar linkages of regular curved-translational units alongside each other, resulting in quadrangulated doublelayer grids based on translational surfaces (Fig. 18). It is a popular and easy method to obtain a myriad of shapes. In addition, it is possible to have the scissor units come together at right angles, making way for simple and uniform joint hubs. We propose two generalizations of this existing concept to greatly increase its potential. A first one can be made by using irregular units (where k = 1). Since all pairs of opposing units are identical, this results in grids with alternating structural thickness t (Fig. 19). Secondly and more interestingly, it is possible to generate grids using a myriad of other patterns. Depending on the pattern, two, three, four or even more independent directions of curvature can be distinguished, which can be assembled by tiling an equal amount of planar linkages with a uniform or alternating structural thickness t (Fig. 20). This generalization opens an abundance of new possibilities, ranging from the highly symmetrical shapes obtained by translating the same planar linkage along the different

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Fig. 18. Generating a two-way scissor grid by translating two planar linkages alongside each other: (a) input linkages; (b) spatial set up before translation; (c) resulting grid in several deployment stages.

Fig. 19. Saddle-shaped two-way grid with alternating structural thickness t.

curvature directions (Fig. 20a) to the amorphous shapes obtained by combining different curves (Fig. 20b and c); or from uniform joints that embrace all equal angles (Fig. 20a and c) to a warped grid in which not a single joint or angle is the same (Fig. 20b). Especially when combining various linkages with different curvature this can locally give a jagged or staggered effect to the resulting scissor grid. 4. Scissor grids of translational units with angular distortions By allowing the angles of the projected grid to vary during deployment, the design constraints become less strict and the amount of design options increases. Firstly, it becomes possible to generate curved double-layer grids with triangular modules of translational units. Also non-triangulated grids present more

Fig. 20. Various double-layer grids of translational units generated by tiling the scissor units of multiple planar linkages along a pattern: (a) hexagonal tiling; (b) rhombille tiling; (c) truncated square tiling.

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Fig. 21. General layout for a compatible triangulated scissor grid of translational units: top view and unrolled scissor units of two adjacent cells.

options, making use of the fact that any module of translational units complying with Eqs. (9), (11) and (12) is foldable. Interestingly, if these foldable scissor grids comply with the deployability constraint, they can be translated into bistable scissor grids by intentionally using joints that do not enable the required angular distortions. The joints are designed such that they ensure compatibility in the expanded configuration, while the deployability constraint ensures compatibility in the folded configuration, when all scissor bars are theoretically aligned. 4.1. Grids with a regular triangular pattern Triangular patterns have the advantage of being geometrically rigid, making them very useful in scissor grid design. In any cell of the pattern the interior angles α are fully determined by the scissor units and vary during deployment when using curvedtranslational units. These angles must be compatible (i.e. at each interior vertex add up to 2π ) in a scissor grid for any deployment stage in order for the grid to be foldable. A general solution for compatibility is shown in Fig. 21. The grid is formed by three scissor units with projected lengths w1 , w2 and w3 , which are repeated and scaled by factors k1 , k2 and k3 along the three grid directions. As such we obtain a scissor grid consisting of two sets of similar modules. All interior nodes of the grid are identical and the six α angles coming together at these nodes equal the six interior angles of the two triangular modules, which therefore always add up to 2π . To form the configuration shown in Fig. 21, Eq. (12) must be met for each module, which becomes

h1 + h2 + h3 = 0

k2 h1 +

h2 + h3 = 0 k1

(20)

(21)

When applying the deployability constraint, this gives

e1 ( 1 + k1 ) + e2 ( 1 + k2 ) + e3 ( 1 + k3 ) = 0

(22)

Fig. 22. Modules used for triangulated scissor grids of regular scissor units.

k2 e1 ( 1 + k1 ) +

e2 ( 1 + k2 ) + e3 ( 1 + k3 ) = 0 k1

(23)

A first solution to these equations is found when k1 = k2 = k3 = 1, thus if all units are regular and the structural thickness t is constant throughout the grid (Fig. 22). Then Eqs. (22) and (23) become

e1 + e2 + e3 = 0

(24)

Consequently, all modules in the grid become identical and the grid itself describes a plane (Fig. 23a). However, due to the congruency of the grid cells, the sum of any three consecutive angles α of an interior node equals π (Fig. 22). By keeping e1 constant throughout the grid and varying e2 and e3 between different rows of cells, it therefore is possible to generate singly curved doublelayer grids based on cylinder surfaces. If e1 = 0, then e2 = −e3 .

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Fig. 23. Triangulated scissor grids where (a) all modules are identical and (b) the inclination of the modules varies across the different rows to give rise to cylindrical curvature.

Fig. 25. (a) Triangulated conical scissor grid for which R = 10, nu = 10, k = 0.7 and t = 1; (b) triangulated conical scissor grid in which the rows of modules describe helical paths.

Fig. 24. Triangulated scissor grid based on one regular plane-translational and two identical irregular curved-translational units.

Thus the units along the first direction are plane-translational and the units along directions 2 and 3 form each others mirror image within a module (Figs. 22a and 23 b). This grid type was first proposed by De Temmerman (2007). The concept was later extended in Roovers and De Temmerman (2014) for the case where e1 = 0 (Fig. 22b), which gives the impression of a slanted grid. The latter paper additionally presents a graphical design method for this type of scissor grid. A second solution identically has k1 = 1 and e1 = 0, but now k2 = 1 (Fig. 24). Combined with Eq. (9), Eqs. (22) and (23) therefore become

e2 (1 + k2 ) = −e3 (1 + k3 )

⇒

e3 = −e2 k2

(25)

The resulting grid describes the surface of a right cone (Fig. 25a). Its modules scale down towards the apex of the cone at which their dimensions approach zero, therefore typically leading to a

Fig. 26. Scale model of a foldable conical scissor grid.

central oculus. The circular rows of modules can be designed to meet at their ends in order to describe a continuous closed surface. However, if e2 = 0 (i.e. for a non-flat grid), these closed loops have to be broken up again to enable the angular deformations required for deployment. If not, the grid will display a snap-through behaviour, which strongly increases with the steepness of the cone. As a consequence the grid can easily and firmly be locked by interconnecting its two ends, which can be accomplished using a simple hook or buckle (Fig. 26). Based on the partial projected grid shown in Fig. 27, this grid type can be designed by defining the following four geometric parameters: the number of units nu along

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Fig. 27. Layout of a conical grid with radius R and nu scissor units at the perimeter.

a ring, the radius R and structural thickness t at the base, and the scale factor k = k2 = 1/k3 . The dimensions of the scissor units can then be calculated as follows:

w1 = 2R sin

π

(26)

nu

w2 = w3 = R 1 + k2 − 2k cos

l1 = l2 =

π nu

1 2 w1 + t 2 2

(27) Fig. 28. Triangulated scissor grid forming a regular four-sided pyramid in three stages of deployment.

(28)

 e2 = t

w22 1 − 2 4 (1 + k ) (4l12 − t 2 )

(29)

A third solution is found when k1 = 1 and e1 = 0 (Fig. 21). In this case five design parameters completely determine the shape of the scissor grid. For example when choosing l1 , e1 , k1 , k2 and the parameter t to set the deployment stage, we find k3 , l2 and l3 through Eqs. (5) and (9), and e2 and e3 through Eqs. (22) and (23):

e2 =

−e1 k1 (1 + k1 )(1 − k2 ) (k1 − 1 )(1 + k2 )

e3 = −k1 k2

e1 ( 1 + k1 ) + e2 ( 1 + k2 ) 1 + k1 k2

(30)

(31)

Similar to the previous solution, the resulting grid also describes the surface of a right cone with unit dimensions that approach zero towards the apex. It differs in the fact that the modules are now arranged along helical curves, leading to non-planar boundary nodes. If well designed, the grid can still be closed at its ends once deployed in order to form a continuous grid (Fig. 25b). The remaining combination for values k1 and e1 , which is k1 = 1 and e1 = 0, simply yields a planar grid. Indeed, Eqs. (22) and (23) give

e2 (1 + k2 )(1 −

1 )=0 k1

(32)

which since k1 = 1 and k2 > 0 means that e2 = 0 and therefore also e3 = 0. A last triangulated scissor grid slightly differs from the pattern of Fig. 21. The planar grid shown in Fig. 23(a) can be assembled to generate three-, four- or five-sided pyramidal grids consisting of only two different scissor units (Fig. 28). This simple new proposal was inspired by the pyramid with hexagonal base shown in Escrig and Valcárcel (1993). By reducing the amount of sides and slicing open the grid during deployment to enable the required angular

deformations, the grid has been made compatible and its deployment range has drastically been improved. Again, linking the ends of the grids firmly locks it to obtain a rigid pyramidal structure. Considering the scissor module of Fig. 22(a), a pyramidal scissor grid with nf faces can be defined by choosing a structural thickness t and semi-length l. Its remaining size parameter can be calculated as:

t e=± 2



1−

1 2 4 sin nπ

(33)

f

4.2. Grids with other patterns By allowing angular grid distortions it has become possible to generate various singly curved triangulated scissor grids. For doubly curved triangulated grids, scissor unit types with intersecting unit lines will need to be used (e.g. as demonstrated in Roovers and De Temmerman (2015)). In case of double-layer grids consisting of modules of four or more translational scissor units even more design options exist. To our knowledge, none of these options have currently been mentioned before in other literature. Nevertheless, the possibilities are endless. Firstly, the scissor grids of Section 3 can be generalized to increase their freedom in shape. For example, scissor grids with lamella rhomboid patterns can now be modelled on various surfaces of revolution, useful for their high degree of symmetry (Fig. 29). Closed volumes can be obtained by attaching two halves at a single layer of nodes (Fig. 29b). In general almost any arbitrary non-triangulated pattern can be mapped on any arbitrary surface and translated into a scissor grid, ideally afterwards optimised to adhere to the deployability constraint (Fig. 30). If well designed, this scissor grid will be foldable. Points of attention are that the base surface should intersect any unit line at only one point, and that any closed loop of scissor units in any deployment stage must adhere to Eq. (11). This latter condition should be verified for each individual design, e.g. through simulation of the deployment in a digital design environment. Increasing the amount of units per module increases the ease at which certain geometries can be approximated. Of course, all the

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Fig. 30. Foldable double-layer scissor grids generated by mapping a rhombille pattern onto an arbitrary terrain and afterwards optimising the scissor units to match the deployability constraint.

Fig. 29. Foldable scissor grids with rhomboid lamella pattern based on surfaces of revolution: (a) dome-like grid with base curve and axis of revolution; (b) creating a closed volume by linking two identical halves at a single grid layer.

resulting additional degrees of freedom need to be locked once the functional deployment stage is obtained in order to ensure a rigid structure. 5. Joints in translational scissor grids In previous sections scissor rods have been represented by lines and joints by points. To manufacture these grids, its members have to be given a tangible volume without damaging the kinematic behaviour of the grid. At the intermediate hinge point of each scissor unit a simple pivot hinge (e.g. using a bolted joint) suffices to serve as a revolute joint interconnecting the rod pair. The joints at the end points of the scissor units on the other hand usually have to interconnect a spatial configuration of multiple scissor units and enable the correct rotational motion of each unit, generally about different rotation axes. One way to achieve this could be through high-tech ball joints. These would however drastically increase the manufacturing cost of the overall structure. Often a better solution is to incorporate joint lines in the line models of the scissor grid that introduce spacings between the different axes of rotation

and consequently serve as placeholders for volumetric joints with a simpler design. Three orthogonal directions for the joint lines can be distinguished that affect the grid differently: one direction parallel to the unit lines (i.e. parallel to h; Fig. 31b), one normal to the scissor  × h; Fig. 31c) and one parallel to unit plane Pu (i.e. parallel to w  ; Fig. 31d). the projected grid lines (i.e. parallel to w Similar to the condition of Eq. (7), the joint vectors j together  and h have to form a closed loop during any deployment with w stage in order to obtain a foldable scissor module or grid. When adding joint vectors that translate a scissor unit along the unit lines this poses no problem as the joint vector at the start point and end point of each rod add up to 0 (see Fig. 31b). Important is that both rods of a scissor unit are translated an equal amount, to ensure that the intermediate hinge points of both bars stay aligned along their common revolute axis. A translation of scissor bars normal to the scissor plane equally results in joint vectors cancelling each other out (Fig. 31c). In addition, since this translation is parallel to the revolute joint axes, the two rods of a scissor pair can be translated a different amount or even in opposite directions. Consequently, the joint lines can provide space for two straight volumetric bars to coexist side by side. Escrig (2012) used this principle for a scissor and joint arrangement similar to Fig. 32. Attention has to be given that this translation doesn’t cause crossing scissor bars to hinder each other during deployment, an issue that makes this joint type less fit in case the interior angle α between different scissor units is small (as is the case for high valency nodes found in for example triangulated scissor grids). It should additionally be noted that even though previous two translations do not affect the global geometry or kinematics of the scissor grid, they do introduce eccentricities which should be kept to a minimum from a structural point of view. Indeed, these eccentricities induce bending moments in the generally slender joint pins.

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Fig. 32. Possible joint and rod configuration when adding joint lines normal to the unit planes (based on Escrig (2012)): (a) top view of the line model; (b) detailed top view; (c) parallel view.

Especially for the double-layer scissor grids without in-plane angular distortions of Section 3, we wish to retain a distortion-free grid when adding joint lines. This is true if any two scissor units i and q with projected lengths that are proportional or identical during deployment are given joint vectors for which the sum of joint lengths j1 + j2 equally is proportional or identical. Thus for any unit i and q of the grid

if

Fig. 31. Three main directions along which joint lines can be added: (a) scissor unit in a linkage without joint lines; (b) joint lines parallel to the unit lines; (c) joint . lines normal to the unit plane Pu ; (d) joint lines parallel to w

A translation of the bars parallel to the projected lengths does influence the spatial configuration of the scissor module or grid and has to be considered with greater care (Figs. 31d and 33). In this case Eq. (11) becomes n   i=1



ji,1 + w  i + ji,2 = 0

(34)

wi = constant wq

then

ji,1 + ji,2 wi = = constant jq,1 + jq,2 wq

(35)

For the scissor grids with a shape-invariant projection of Section 3.1 this means that the joint vectors describe a grid that is similar in shape to the grid described by the projected lengths (Fig. 34). Pellegrino and You (1993) and You and Pellegrino (1997a) presented joint hubs for a foldable ring according to this principle. For the angle-invariant grids of Section 3.2, Eq. (35) can easily be achieved by letting the joint lines of sets of identical units also be identical, or even simpler by making al joint lines equal in length. A two-way scissor grid based on translational surfaces in which joint lines with equal lengths have been added is presented in Langbecker and Albermani (20 0 0). For the grids of Section 4 that already require angular distortions to deploy, the sizing constraints for the joint vectors are less strict. Most critical are the triangulated grids. Of course, to maintain compatibility in these grids the modules with a similar shape must remain similar. Thus the joint sizes for one module can be freely chosen, from which the other joint dimensions follow using the three scale factors k1 , k2 , and k3 . As the grid deploys and the interior angles distort, the joint lines stay confined in the scissor unit plane and rotate along with the scissor units. The angular distortions should therefore be enabled at the central nodes of the joint hubs.

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Fig. 34. Adding joint lines to a scissor grid with shape-invariant projection: (a) grid without joint lines; (b) grid with joint lines in multiple deployment stages. The sum of joint lengths corresponding to each scissor unit is proportional to the projected length w of that scissor unit.

Any scissor grid that was designed to be compatible (and thus to comply with Eq. (11)) will therefore also comply with Eq. (34) after adding these joint lines. In addition they do not alter the kinematic behaviour: grids designed to be free from angular distortions will remain so. As the joint lines are no longer confined to the scissor unit planes, this new method could improve the uniformity of the joint hubs by modelling the joint vectors on regular patterns. For example, for the triangulated scissor grid of Fig. 36 we could therefore model the joint hubs on a regular triangular pattern, making them all equal in size and equally distributed in direction. Since the joint vectors remain invariant, possible in-plane angular distortions will now require the scissor units to rotate about the end points of the joint hub instead of its centre point. An experimental model of such a joint is shown in Fig. 37 and Fig. 1 shows its use in a scissor grid. Kim et al. (2013) propose an alternative high-tech solution for a joint that allows the same motion. Fig. 33. Influence on the internal angles in a scissor module of adding joint lines  : (a) module without joint lines; (b) joint lines with a combined length parallel to w not proportional to w; (c) joint lines with a combined length proportional to w, therefore retaining the shape of the original projected grid in the current deployment stage.

As all unit lines in a scissor grid of translational units remain parallel throughout deployment, we have found an alternative design solution for the joint hubs. In this case, the joint vectors remain invariant throughout deployment, even if angular distortions occur in the scissor grid (Fig. 35). The joint vectors independently form fixed closed loops, meaning that n   i=1



ji,1 + ji,2 = 0

(36)

6. Conclusions This paper presented the main characteristics of scissor grids composed of translational units and set up the criteria to make these scissor grids foldable. Where past research targeted specific design solutions, we’ve managed to use these general principles to maximize the design potential of this scissor grid type. The resulting overview demonstrates that their design possibilities are vast. The first solutions presented in this paper are constrained by the fact that the angles between scissor units are fixed in order to enable simpler joint designs. They tend to lead to scissor grids with elementary shapes and a high degree of symmetry and uniformity. More complex shapes can be achieved as well, e.g. based on translational surfaces or, more general, on planar tilings. However, using this tiling principle with many directions of different curvature can locally result in jagged effects, which might be undesirable. Triangulated scissor grids also impose strict constraints on

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Fig. 36. Triangulated scissor grid with cylindrical curvature to which joint lines are added that form independent closed loops. The scissor units rotate about the ends of the joint hubs, and no longer about the central node of the joint hub.

Fig. 35. Module with joint lines that form independent closed loops in two stages of deployment.

the design freedom. A general solution for a compatible doublelayer grid consisting of three-unit modules was presented from which the individual design options could be extracted. Their geometry is restricted to singly curved shapes. Once angular distortions are allowed in non-triangulated scissor grids, a whole range of unexplored possibilities open up and the solutions truly become limitless. This work presented select examples to illustrate this geometric freedom. A great design freedom is equally encountered for the joint hubs. Translational scissor units can freely be translated within a scissor grid along the unit lines or along the direction of their revolute axes if this benefits the joint design. In the direction normal to the unit lines different methods exist as well to provide spacing between different axes of rotation. These can however change the geometry and dimensions of the scissor grid and thus should be considered early in the design process. Of course, despite the kinematic possibility to introduce all these eccentricities for the benefit of the joint design, they should always be kept to a minimum from a structural point of view.

Acknowledgements Funding: Research funded by a PhD grant of the Agency for Innovation by Science and Technology (IWT).

Fig. 37. Experimental joint hub for the triangulated grid of Fig. 36 based on fixed joint vectors in which four out of six joint pins can rotate in-plane: (left half) exposed central layer with joint vectors and revolute axes; (right half) joint hub with cap to keep the pins in place.

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Langbecker, T., Albermani, F., 20 0 0. Foldable positive and negative curvature structures - geometric design and structural response. J. Int. Assoc. Shell Spatial Struct. 41 (134), 147–161. Meurant, R., 1993. Circular space trusses. In: Parke, G., Howard, C. (Eds.), 4th International Conference on Space Structures. Thomas Telford, London, Surrey, pp. 2053–2062. Pellegrino, S., You, Z., 1993. Foldable ring structures. In: Parke, G., Howard, C. (Eds.), 4th International Conference on Space Structures. Thomas Telford, London, Surrey, pp. 783–792. doi:10.1680/ss4v1.19683.0084. Roovers, K., De Temmerman, N., 2014. Design of a Foldable Triangulated Scissor Grid for Single-Curvature Surfaces. In: 4th International Conference on Mobile, Adaptable and Rapidly Assembled Structures (MARAS2014). Oostende, pp. 195– 206. doi:10.2495/mar140161. Roovers, K., De Temmerman, N., 2015. Digital Design of Deployable Scissor Grids Based on Circle Packing. In: Proceedings of the IASS 2015 Symposium - Future Visions. Amsterdam, p. 12. Sánchez-Cuenca, L., 1996. Geometric models for expandable structures. In: Escrig, F., Brebbia, C. (Eds.), Mobile and Rapidly Assembled Structures II. Computational Mechanics Publications, Southampton, pp. 94–102. You, Z., 20 0 0. Deployable structure of curved profile for space antennas. ASCE J. Aerosp. Eng. 13 (4), 139–143. doi:10.1061/(ASCE)0893-1321(20 0 0)13:4(139). You, Z., Pellegrino, S., 1997. Cable-Stiffened pantographic deployable structures part 2: mesh reflector. AIAA J. 35 (8), 1348–1355. doi:10.2514/2.243. You, Z., Pellegrino, S., 1997. Foldable bar structures. Int. J. Solids Struct. 34 (15), 1825–1847. doi:10.1016/S0 020-7683(96)0 0125-4. Zanardo, A., 1986. Two-dimensional articulated systems developable on a single or double curvature surface. Meccanica 21 (2), 106–111. doi:10.1007/BF01560628.

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Dr. ir. arch. Kelvin Roovers (∗ 1989) is a member of the Research Lab for Architectural Engineering (AE-LAB) of the Vrije Universiteit Brussel (VUB), Belgium. He holds the degrees of Doctor (2017) and Master (2012) in Architectural Engineering Sciences. His main expertise entails deployable scissor grids and parametric design. His recently completed doctoral thesis entitled “Deployable Scissor Grids: Geometry and Kinematics” presents a comprehensive overview of the design potential of scissor grids.

Prof. dr. ir. arch. Niels De Temmerman (∗ 1977) is a member of the Research Lab for Architectural Engineering (AE-LAB) of the Vrije Universiteit Brussel, Belgium and is chair of the research group TRANSFORM Transformable Structures for Sustainable Development. Currently, he chairs the Structural Morphology Group (WG 15) of the IASS (International Association for Shell and Spatial Structures). He is an architectural engineer (2002) with a PhD on deployable structures (2007). His main expertise entails the design and analysis of transformable structures (deployable structures and kit-of-parts systems) for architectural applications.

Please cite this article as: K. Roovers, N. De Temmerman, Deployable scissor grids consisting of translational units, International Journal of Solids and Structures (2017), http://dx.doi.org/10.1016/j.ijsolstr.2017.05.015