EP3596278B1

EP3596278B1 - A foldable scissor module for doubly curved scissor grids

Info

Publication number: EP3596278B1
Application number: EP18711919.3A
Authority: EP
Inventors: Kelvin ROOVERS; Niels DE TEMMERMAN
Original assignee: Vrije Universiteit Brussel VUB
Current assignee: Vrije Universiteit Brussel VUB
Priority date: 2017-03-16
Filing date: 2018-03-15
Publication date: 2023-08-30
Anticipated expiration: 2038-03-15
Also published as: EP3596278A1; WO2018167247A1

Description

Field of the invention

The invention is related to scissor modules and grids built from scissor units, each unit consisting of a set of two rods connected at an intermediate rotation point. The invention is in particular related to modules and grids formed of polar scissor units.

Background of the invention

A deployable scissor grid consists of an articulated network of bars that has the ability to rapidly and reversibly transform in shape and in volume, allowing it to answer to changing needs. Most scissor grids transform between a compact bundle of bars and an expanded functional configuration, particularly useful for applications with a mobile or temporary character. As such they can be prefabricated in factory conditions, after which they are compactly transported and quickly erected on site with low skill and labour force requirements. Afterwards, they can rapidly return to their compact form either to be relocated and reused or to be stowed for later use. A different group of scissor grids comprises shell-like structures that deploy towards their perimeter. These are more useful for example as retractable roofs.
The basic building block of any scissor linkage is the scissor unit, also commonly known as a scissor-like element (SLE), a pantograph or a duplet. A scissor unit is a planar composition of two bars crosswise interconnected by a revolute joint, referred to as the intermediate hinge point. This joint allows a relative rotation of the bars about an axis orthogonal to the unit plane, which contains the two bars. The distance between the intermediate hinge point and an end point of a scissor unit is called a semi-length. Different types of scissor units exist, which vary in proportions and shape. The three main scissor unit types are the translational, the polar and the angulated scissor units. They can easily be distinguished through their unit lines, which are the imaginary lines connecting the upper and lower end point at both sides of the scissor unit. Each unit type comes in a basic and a generalized form, which in general allow to respectively generate simple and more complex shapes. In addition, a regular and irregular form can each time be discerned, which tend to give rise to grids with regular and irregular patterns respectively. Regular units are characterized either by rods that together with the unit lines describe a pair of congruent triangles or by parallel chords (i.e. the line connecting the upper nodes and the line connecting the lower nodes).
Translational scissor units consist of a pair of straight bars and have parallel unit lines throughout deployment. Polar scissor units consist of a pair of straight bars as well, but are characterized by unit lines intersecting at an angle that varies during deployment. The line connecting the intersection point and the intermediate hinge point is called the radial line. In the basic form of polar unit, this radial line is perpendicular to the transverse bisector line of the bars.
Scissor linkages are formed by linking multiple scissor units at their end nodes using revolute joints in two- or three-dimensional configurations. Most basic are the linear or curvilinear linkages formed by connecting a number of scissor units side by side, often within a plane. An open chain of scissor units in which the dihedral angles between adjacent unit planes are fixed forms a mechanism with one degree of freedom. Its deployment range will be determined by the scissor unit that first reaches one of its outer deployment stages. To optimize this range, the following constraint for any linkage of two scissor units with straight bars as shown in Figure 1a was presented by F. Escrig in 'Expandable Space Structures', Int Journal of Space Structures, 1(2), 79-91 : $a'_{i} + b'_{i} = a_{i + 1} + b_{i+1}$
This equation is also known as the 'deployability constraint'. Despite its name, it isn't always necessary nor sufficient for deployment. It ensures that all units in the linkage simultaneously reach their most compact state (when angles φ equal π) and the scissor linkage is theoretically reduced to a single line. As illustrated in Fig. 1b, this constraint can be graphically represented by an ellipse with focal points coinciding with the common end points of a pair of interconnected scissor units. As the sum of the distances to the focal points is the same for any point of the ellipse, both intermediate hinge points must be located on the same ellipse. Consequently, any planar linkage of translational or polar units complying with the deployability constraint contains a chain of tangent ellipses, where the tangent points coincide with the intermediate hinge points of the scissor units. In three dimensions the graphical representation of the deployability constraint for a linkage of two scissor units becomes a prolate spheroid (i.e. an ellipse revolved about its major axis). Hence, three-dimensional scissor grids complying to the deployability constraint contain networks of mutually tangent prolate spheroids, referred to as prolate spheroid packing. The use of these networks in the design of scissor grids is illustrated for example in "Generalized geometry of foldable scissor grids based on polar units", Roovers, K. and De Temmerman, N., Proceedings of the IASS 2016 Symposium - Spatial Structures in the 21st Century, Tokyo, 2016.
Under specific constraints the two ends of a (curvi)linear linkage of scissor units can be interconnected in order to create a single closed loop of units, called a scissor module. In general a scissor module will describe a spatial, polyhedral shape. In some cases this module can take up the shape of a planar closed ring. A module of polar units might be planar in a single deployment stage, but will be spatial in all other stages due to the varying angle between their unit lines. Forming a closed spatial module requires at least three scissor units.
K. Atake was the first to present a foldable scissor module consisting solely of polar scissor units for which the unit lines are not concurrent. This is described for example in US Patent 5,761,871 and in "ATAKE's structure - new variations of the scissors technique", In Brebbia, C. A. and Escrig, F., editors, Mobile and Rapidly Assembled Structures III (2000), pages 143-154. WIT Press. Atake's modules comprise an even number of regular basic polar units which alternate in orientation. Either opposing units or all units in these modules are identical. In addition, all rods in these modules have the same total length. As such, the unit lines of a module describe antiprisms. Since the unit lines in these modules are not concurrent, the dihedral angles α (see Fig. 3) between the unit planes of adjacent scissor units are fixed for each deployment stage and the modules generally are geometrically rigid.
The document 'Designing for uncertainty', Rosenberg, Master thesis, Dept. of Architecture, Massachusetts Institute of Technology, 2009 discloses in Figure 34 a number of foldable modules consisting of polar scissor units, wherein these modules comprise four parallel unit lines. This is realised by using so-called 'double scissor pairs', consisting of a planar linkage of two polar scissor units having two parallel unit lines. Such a double scissor pair behaves like a translational scissor unit. The modules formed of such double scissor pairs shown in Figure 34 of this document do not have fixed internal dihedral angles between two adjacent scissor units on either side of one of the parallel unit lines, for a given deployment stage (i.e. a given set of deployment angles of the scissor units making up the module). In each deployment stage, the scissor units on either side of the parallel unit lines can still rotate relative to each other.
Scissor grids are formed by combining multiple scissor modules. The manner in which they are combined leads to distinct spatial configurations. A first of these configurations is known as a single-layer grid (SLG). An SLG is formed by stacking multiple scissor modules consisting of the same amount of scissor units. Two modules are stacked by connecting the upper nodes of one module to the lower nodes of the other module. Hence, the scissor units lie in the base surface described by the grid. A double-layer grid (DLG) is formed by tessellating prismatic scissor modules along a base surface, as described for example in document "Evaluation of Deployable Structures for Space Enclosures", Hanaor and Levy, International Journal of Space Structures (2001), 16(4):211-229. As such, each module forms a cell in the scissor grid and units at a common edge of two cells are shared by two modules. The upper and lower chords of the scissor units describe two distinct layers.
The deployment process of a scissor grid generally knows three distinct stages : two outer stages (usually the functional ones, for example the stowed and the expanded configuration) and a transitional stage (during deployment from one outer stage to another). The deployment behaviour of scissor grids is usually described through its geometric compatibility in each of these three stages. A scissor grid is said to be geometrically compatible if all members in the grid fit together without deformations or tolerances. In contrast, it is incompatible when this geometric fit does not exist and some or all members have to elastically deform in order to achieve a certain stage of deployment.
Foldable scissor grids are geometrically compatible throughout all stages of deployment. The scissor grid therefore acts as a pure mechanism with a smooth and easily controllable deployment behaviour. Once erected, the mechanism should be locked in order to become a loadbearing structure. For this purpose, additional members are often added to the scissor grid, such as struts or cables. Anchoring multiple nodes of a scissor grid to the ground can equally be a strategy to lock its degrees of freedom. The process of manually locking a scissor grid is sometimes seen as a disadvantageous, time-consuming act, especially for large structures requiring many locking devices that might in addition be difficult to access. Clever design choices can nevertheless speed up this process. Evidently, since foldable scissor grids are compatible throughout deployment, intermediate deployment stages can also become functional, e.g. in a partially deployed cover to obtain partial shading.
Bistable scissor grids are geometrically compatible in the outer deployment stages and incompatible throughout or during a part of the transitional stage. Consequently, the deployment of a bistable grid requires an additional amount of effort to sufficiently deform its members and overcome these incompatibilities, after which it snaps into the next compatible configuration. This way, it becomes possible to create self-locking scissor grids, which, once deployed, remain standing under their own self-weight without external locking devices.

Brief description of the figures

Numerical references 1 through 8 are reserved in the appended drawings for numbering the scissor units within a module, and are therefore re-used in several of the drawings. Primed numbers (1', 2' etc) are used when identical units are applied in the same module : for example unit 1 and unit 1' are identical.

Figures 1a and 1b illustrate the deployability constraint of a linkage formed of scissor units.
Figures 2a to 2d illustrate the various types of polar scissor units.
Figure 3 shows three scissor modules according to the above-described prior art by Atake.
Figure 4 shows a scissor module according to a first embodiment of the invention, consisting of 4 polar scissor units.
In Figures 4, 5, 6a,6b, 10 and 11, the images from left to right and top to bottom represent: a 3D view of the scissor module, a view of the unrolled linkage, a top view, a front view and a side view.
Figure 5 shows a scissor module according to the first embodiment, consisting of 5 polar scissor units.
Figures 6a and 6b show a scissor module according to the first embodiment, consisting of 6 polar scissor units.
Figure 7 shows a scissor module according to a second embodiment of the invention, consisting of 4 polar scissor units.
Figure 8 shows a scissor module according to the second embodiment, consisting of 8 polar scissor units.
Figure 9 shows a scissor unit according to the second embodiment consisting of 6 units.
Figures 10 and 11 show scissor units according to a third embodiment of the invention, which combines the constraints of the first and second embodiments.
Fig. 12 illustrates the addition of joint lines with a uniform length in both layers added to a compatible module of polar units: (a) unrolled module with dimensionless joints; (b) unrolled module with joints; (c) three deployment stages in top and side view.
Fig. 13 illustrates existing concepts for DLGs consisting of all identical P_RB units combining modules with concurrent and non-concurrent unit lines (top view and three deployment stages in 3D view): (a) foldable flat DLG with trihexagonal pattern; (b) pyramid; (c) dome with truncated icosahedron pattern.
Fig. 14 illustrates that (a) a single module according to the first and/or the second embodiment of the invention is foldable, (b) as well as an open chain of such modules, (c) while closed loops of modules generally result in a bistable assembly.
Fig. 15 shows an example of a rotational surface, which is generated by rotating a curve about an axis.
Fig. 16 shows a simulation of the deployment process of three bistable DLGs based on rotational surfaces comprising four-unit modules according to the first embodiment.
Fig. 17 shows a scissor grid based on a catenoid in three deployment stages comprising four-unit modules according to the first embodiment: (a) 3D view and top view of a single row of modules that is foldable and symmetrical about a plane; (b) when linking multiple rows, maximum one row of modules can retain its desired symmetry during deployment while all other are incompatible. (b) shows a 3D view, top view, front view, and side view of a grid in various deployment stages.
Fig. 18 shows examples of bistable DLGs with negative double curvature making use of 4-unit modules according to the second embodiment, based on: (a) a hypar, (b) a helicoid and (c) a Möbius strip with three half-twists. From top to bottom : 3D view, top view, front view, side view.
Fig. 19 shows a DLG of P_IG units based on the helicoid comprising 4-unit modules according to the second embodiment: (a) principal curvature line network; (b) geometric support structure optimized to have torsion-free faces and comply to the geometric constraints imposed by the modules; (c) prolate spheroid packing on the support structure; (d) deployment simulation of the bistable scissor grid. Parts a, b and c show a front view. Part d shows a front view, top view and side view at various deployment stages.

Summary of the invention

The invention is related to a scissor module and to scissor grids in accordance with the appended claims. The invention is thus firstly related to a closed loop foldable scissor module according to independent claim 1 comprising at least four polar scissor units characterized in that :

At least one of the polar scissor units consists of rods that are not identical in terms of their length and/or their semilengths, i.e. at least one unit is not a regular basic polar (P_RB) unit,
The unit lines of the polar units are not all concurrent.

According to the present invention, the internal dihedral angles of the module are fixed for any deployment stage.
According to a first embodiment, the module is symmetrical with respect to at least one symmetry plane (P).
According to the first embodiment, the module may consist of

a first and second scissor unit, said first and second units being P_RB units, preferably these P_RB units are mutually non-identical units,
a third and fourth scissor unit, said third and fourth unit being mutually identical non-P_RB units,

the third and fourth unit are interconnecting the first and second unit,
the module is symmetrical about one symmetry plane (P) and the P_RB units are orthogonal to the plane of symmetry and have their radial line in the plane of symmetry.

A module according to the first embodiment may consist of an uneven number of scissor units, wherein at least one scissor unit is a P_RB unit, the remainder of the units consisting of pairs of mutually identical non-P_RB units.
Alternatively, a module according to the first embodiment may consist of an even number of scissor units higher than or equal to 6, formed by pairs of mutually identical non-P_RB units.
According to another alternative of the first embodiment, the module may consist of an even number of scissor units higher than or equal to 6, and comprise an even number of P_RB units, the remainder of the units consisting of pairs of mutually identical non-P_RB units.
According to a second embodiment, all scissor units consist of two rods of equal length, and wherein the opposing or the adjacent semi-lengths at the connection of two units are equal. The constraints of the second embodiment may be combined with those of the first or other embodiments.
A module according to the second embodiment may consist of an even number of scissor units. A module that combines the constraints of the first and second embodiment may consist of an uneven number of scissor units, an uneven sub-number of which being P_RB units, the remainder of the units consisting of pairs of mutually identical non-P_RB units.
The invention is furthermore related to a scissor module according to the invention, wherein the joints between scissor units are defined by joint lines placed between the spaced-apart units of the module.
The invention is also related to a double layer scissor grid comprising one or more modules according to the invention.
A scissor grid according to the invention may be based on a rotational surface and formed of modules according to the first embodiment, wherein the meridian curves of the rotational surface are populated by identical planar linkages of irregular generalized polar units and wherein the parallel circles of the rotational surface are populated by regular basic polar units that are identical within the same circle.
According to another embodiment, a scissor grid according to the invention may be formed of modules according to the second embodiment, and wherein the grid is based on a surface with double negative curvature.

Detailed description of the invention

Before describing the modules and grids according to the invention in more detail, the various types of polar scissor units are described. A distinction is made between basic polar units and generalized polar units. Both types have an irregular variant and a regular variant. The following abbreviations will be used hereafter

P_RB unit: regular basic polar scissor unit
P_IB unit: irregular basic polar scissor unit
P_RG unit: regular generalized polar scissor unit
P_IG unit: irregular generalized polar scissor unit

Each of the units has four semi-lengths, measured between the intermediate hinge point and the end points of the rods, where they are joined to the rods of other scissor units. The sizes of these semi-lengths, denoted by a, a', b and b', do not vary during deployment. Figures 2a and 2b respectively show the structure and geometrical parameters of a P_IB unit and a P_RB unit. Each unit comprises two rods 10 and 20 with respective semi-lengths a and a' above the transverse bisector line TT' and b and b' below the transverse bisector line. It is seen that a P_RB unit is a P_IB unit wherein a = a' and b = b'. A P_IB unit is fully determined by three size parameters plus one parameter to set the deployment stage. If one defines the three semi-lengths a, b and a' then for a certain deployment angle φ, all other parameters shown in the drawings can be found as a function of these four input parameters.
Figures 2c and 2d respectively show the structure and geometrical parameters of a P_IG unit and a P_RG unit. A P_RG unit is a P_IG unit wherein b = ka' and b' = ka with k a proportionality factor. Generalized polar units require one additional size parameter in order to be fully determined. The P_IG unit is the most general polar scissor unit that can be obtained by interconnecting two straight bars at an intermediate point using a revolute joint. The unit is fully determined by four size parameters plus one parameter to set the deployment stage. For example, by defining all four semi-lengths a, b, a' and b' and a deployment angle φ, all other parameters shown in the drawings can be found. As seen in the drawings, the rotation axis of the rods relative to each other is fixed with respect to each of the two rods.
Three examples of modules developed by Atake mentioned in the introductory section are illustrated in Figure 3. The images on the left show a 3D view of the module. The images on the right show the corresponding units of the module unrolled in a plane. In the first two examples, all the units are identical P_RB units. In the third example, the module consists of four P_RB units, with each pair of opposing units (one pair marked '1', the other '2') being identical.
As stated above, the present invention proposes a foldable scissor module formed of polar scissor units that is different from the Atake-approach : a foldable scissor module comprising at least four polar scissor units characterized in that :

At least one of the polar scissor units consists of rods that are not identical, meaning that the two rods differ in total length and/or semi-lengths,
The unit lines of the polar units are not all concurrent.

_RB

_IB

_RG

_IG

_RB

The theoretical line model of a scissor module refers to a module as illustrated in the majority of the appended drawings, wherein the bars of the module are represented as lines interconnected by dimensionless joints, i.e. a point with no dimensions that interconnects two articulated lines. In reality, the bars and the joint have a 2-or 3 dimensional structure, for example embodied by the addition of joint lines as described further in this text. In the presence of such 3-dimensional joints, the modules according to the majority of embodiments according to the invention remain foldable in the above-defined sense, i.e. geometrically compatible in every deployment stage. Some embodiments may however show small incompatibilities and result in slightly bistable modules, as will be illustrated for embodiments including joint lines. These embodiments are nevertheless foldable in the context of this description, given that the theoretical line model corresponding to the modules in question is effectively compatible throughout the three deployment stages.
According to the present invention, the internal dihedral angles α of a module according to the invention are fixed for any deployment stage. The internal dihedral angles are the angles between the unit planes of two adjacent scissor units of the module. One dihedral angle α is illustrated in Figure 3 for a module according to Atake. The fact that the internal dihedral angles α of a module are fixed for any deployment stage means that when a module is regarded in a given deployment stage wherein all the scissor units are deployed at a given angle, only one set of dihedral angles is possible. In other words, the module has exactly one geometrically compatible form for each deployment stage.
Three embodiments of a scissor module according to the invention are described hereafter: a first embodiment makes use of reflection symmetry to ensure a foldable scissor module. In its simplest form illustrated in Figure 4, it consists of four polar scissor units: two P_RB units 2 and 3 that are orthogonal to and have their radial line (line r in Figures 1 and 2) located in the plane of symmetry P, and two identical polar units 1 and 1' that interconnect the first two units. The units 1 and 1' are polar scissor units consisting of rods that are not identical, in the meaning as defined above. By being symmetrical about at least one plane, these modules can equally consist of more than 4 polar scissor units. To meet the symmetry condition, modules according to the first embodiment comprising an odd number of units contain at least one P_RB unit, with the other units consisting of rods that are not identical. This is illustrated in Figure 5 which shows a module according to the first embodiment comprising five polar units, one P_RB unit 3 and two pairs of identical polar scissor units 2/2' and 3/3', each of these four units consisting of two non-identical rods.
Modules according to the first embodiment formed of an even number of units of 6 or higher do not require P_RB units, as illustrated in the embodiment shown in Figure 6a. Nevertheless, modules according to the first embodiment having an even number of units of 6 or higher can still contain P_RB units. This is illustrated in Figure 6b which shows a module according to the first embodiment comprising six polar units, two of which are P_RB units (units 1 and 4), and further comprising two pairs of identical non-P_RB units 2/2' and 3/3' each of these four units consisting of two non-identical rods.
The feature that characterizes a module according to the first embodiment is that the module is symmetrical about at least one plane. Depending on the number of units n of which a module consists, this puts a number of further requirements on the choice of the polar unit types, as illustrated above : when n is uneven, at least one P_RB unit is required. The non-P_RB units in a module according to the first embodiment are applied in pairs of identical units.
The second embodiment combines two constraints in order to obtain a foldable module: firstly, all scissor units consist of two bars of equal length (a_i + b'_i = b_i + a'_i ), and secondly, either the opposing semi-lengths at the connection of two units are equal (a'_i = b _i+1 and b'_i = a _i+1, Fig. 7) or the adjacent semi-lengths at the connection of two units are equal (a'_i = a _i+1 and b'_i = b _i+1), thus embracing two congruent triangles which unrolled in a plane form a parallelogram or kite-shape respectively. Figures 8 and 9 show further examples of modules according to the second embodiment. The module of Figure 8 consists of six polar units wherein all opposing semi-lengths at the connection of two units are equal, i.e. all the pairs of congruent triangles form parallelograms. The module of Figure 9 consists of six units. Between units 2 and 3 as well as between units 6 and 1, the congruent triangles form a kite shape, between all other pairs, the congruent triangles form parallelograms. If no additional constraints are applied to a module according to the second embodiment, this module necessarily has an even number of polar units.
A third embodiment combines the constraints of the first and second embodiment. Figure 10 shows an example of such a module, comprising two P_RB units 1 and 4 and two pairs of identical polar (non-P_RB) scissor units 2/2' and 3/3', each of these four units consisting of two non-identical rods. The module is symmetric about the plane P (embodiment 1), while the constraints according to embodiment 2 are also satisfied. As such it becomes possible to form modules belonging to the second embodiment that consist of an uneven amount of scissor units as illustrated in Fig. 11, where a symmetric module is shown having one P_RB unit 3 and two pairs 1/1' and 2/2' of identical non-P_RB units. More generally, a module that combines the constraints of embodiments 1 and 2 may consist of an uneven number of units, formed of an uneven sub-number of P_RB units combined with pairs of identical non-P_RB units.
If for one deployment stage a geometrically compatible closed loop of scissor units with dimensionless joints can be found in accordance with the invention, for example according to the first, second or third embodiment, then the resulting scissor module is foldable. The design of the actual dimensions of the units requires the solving of a set of equations. The way in which these equations can be determined and solved is well known to the skilled person. The compatibility of the modules according to the invention has been empirically verified through parametric models and simulations of their deployment.
As stated above, the theoretical line model of a scissor module is based on the presence of dimensionless joints between two interconnected bars of the units included in the module. To manufacture a physical module and grid, its members have to be given a tangible volume without damaging the kinematic behaviour of the module/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 pair of bars. 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 theoretical 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. These joint lines can for example be added orthogonally to the unit lines, while lying within the scissor unit plane, as illustrated in Figures 12a and 12b for a module according to the second embodiment. Figure 12a shows the theoretical module, i.e. with dimensionless joints. Joint lines 11 and 12 that have the same length at the upper and lower nodes of the linkage are added to the module. When adding joint lines to a compatible module with dimensionless joints, compatibility of the deployed configuration can be maintained by dimensioning the joint lines proportionally to the lines running from the mid-point M between two bar ends orthogonally to the unit lines, as shown in Fig. 12 a and b : each joint line comprises two articulated parts which are respectively proportional in length to the corresponding orthogonal line, by a proportionality factor c. For example, joint lines with length c.m₂' and c.ms in Figure 12b are proportional to orthogonal lines m₂' and m₃ drawn in the theoretical model of Figure 12a. The proportionality factor c is free to choose. The joint lines c.m₂' and c.m_s are connected to their respective bars 30 and 31 by 2-dimensional joints J1 and J2, i.e. joints allowing one rotational degree of freedom about an axis orthogonal to the scissor unit's plane. These joints J1 and J2 ensure that the joint lines stay in the plane of the scissor unit to which they are connected. The joint lines of lengths c.m₂' and c.ms are interconnected by a 2-dimensional joint J3, allowing one rotational degree of freedom about the unit line 32 between the units comprising bars 30 and 31 respectively, ensuring that the joint lines remain orthogonal to the unit line 32.
The joints lines can be brought into practice by any suitable design, not necessarily by a pair of additional rods with the indicated lengths (such as c.m₂' and c.ms). Thanks to symmetry, the addition of joint lines for example in the manner as described above, allows geometrically compatible solutions for modules of the first embodiment. In modules according to the second embodiment, small incompatibilities 15 may arise during deployment (Fig. 12c) when joint lines are introduced. Thanks to the constraints according to the second embodiment, the module will again be compatible in the compact stage, as all unit lines will then theoretically become parallel. The module hence becomes slightly bistable. As stated above, the presence of these incompatibilities does not exclude the module of Figure 12 from the scope of the appended claims. The module is still regarded as foldable in the present context, given that the theoretical line module to which the joint lines are added is geometrically compatible throughout the deployment stages.
In any of the embodiments according to the invention, reinforcement members, such as bars, membranes or cables may be added to the modules in a deployed state and for example oriented along the unit lines or along the diagonals of a module. The presence of such reinforcement bars do not change the fact that scissor modules are employed in accordance with the invention.
Double-layer grids (DLGs) assembled from modules of polar scissor units for which the unit lines aren't all concurrent are subjected to strict constraints, as the internal dihedral angles α of a module are fixed for any deployment stage. To obtain a foldable DLG, these angles should be compatible at any node of the grid during all stages of deployment. As illustrated in the above-named references by K. Atake, a solution was found to generate foldable two-way scissor grids with single curvature making use of P_RB units for which the total lengths of all bars are equal.
By combining non-concurrent modules with concurrent modules, Atake additionally proposed a foldable grid based on a flat trihexagonal pattern (Fig. 13a). Furthermore he designed several spatial grids with a bistable deployment, e.g. a pyramidal structure based on the flat trihexagonal grid or a dome with truncated icosahedron pattern as shown in Fig. 13b,c. In this figure they have been made foldable by cutting the grid open during deployment.
The above-mentioned existing proposals by Atake all consist of P_RB units that have the same bar lengths throughout the grid. In contrast, the modules according to the invention give rise to various new design possibilities with a greater variety in curvature. When interconnecting multiple of these scissor modules to form open chains of modules, the resulting assembly remains foldable, as illustrated in Fig. 14b. However, when introducing closed loops of modules in these assemblies, they generally become bistable (Fig. 14c).
Two examples of scissor grids that can be generated from the modules according to the invention will be given in the following paragraphs. A first example is related to two-way DLGs based on rotational surfaces (Fig. 15-16). These scissor grids make use of the symmetrical four-unit module of Fig. 4 (the simplest form of the first embodiment). A rotational surface is generated by rotating a curve about a central axis 16 (Fig. 15). To translate this surface into a scissor grid, scissor units are arranged along the meridian curves 17 (i.e. the intersection curves of a rotational surface and the planes comprising its axis of rotation) and the parallel circles 18 (i.e. the intersection curves of a rotational surface and the planes orthogonal to its axis of rotation) of the rotational surface. The meridian curves are populated by identical planar linkages of P_IG (i.e. irregular generalized polar) units, while the parallel circles are populated by P_RB units that are identical within the same circle. Hence, the resulting scissor grid inherits the high degree of symmetry found in these rotational surfaces. The planar P_IG linkage allows a great geometric freedom. A single row of modules is foldable, as the modules have the freedom to adapt to the required angular deformations (Fig. 17a). In assemblies of multiple rows, angular incompatibilities arise at the internal grid nodes during deployment (Fig. 17b). To ensure compatibility in the compact stage and obtain a bistable scissor grid, the deployability constraint as defined in the introductory section (formula (1), with reference to Fig. 1a) must therefore be applied.
The second example makes use of the four-unit module of Fig. 7 (the simplest form of the second embodiment) to generate scissor grids based on surfaces with negative double curvature, for example the hypar, Möbius strip or helicoid (Figs. 18-19). In a first step, the smooth base surface is discretized using its network of principal curvature lines (Fig. 19a). This network is useful to obtain a mesh with a torsion-free geometric support structure, as described in "Geometric modelling with conical meshes and developable surfaces", Liu et al., 2006, ACM Transactions on Graphics, 25(3), p. 681-689. The geometric support structure of a mesh consists of the quadrangular faces formed by each mesh edge and the mesh normals at its ends (Fig. 19b). If the faces of this support structure are torsion free (i.e. planar), then they can be populated by planar scissor units to create double-layer scissor grids. In addition, the mesh must be optimized such that the geometric constraints as described by the second embodiment are met, which automatically ensures that the deployability constraint (formula (1)) is applied, as illustrated by the network of mutually tangent prolate spheroids in Fig. 19c. This network of prolate spheroids is a 3-dimensional extension of the tangent ellipse-representation illustrated in Figure 1b. An individual module is foldable, but again assemblies comprising closed loops of modules are bistable due to angular incompatibilities at the internal nodes (Fig. 19d).
The major advantages of the new foldable modules according to the invention is that they generate deployable scissor grids with 'freeform' double curvature using straight bars (existing concepts using straight bars mostly have no, single or spherical double curvature). Thanks to these straight bars, the resulting scissor grids can deploy towards compact bundles (in contrast to existing concepts for (freeform) doubly curved scissor grids using kinked rods, leading to less compact scissor grids). In addition, these new scissor grids display a bistable deployment behaviour using four-unit modules, which are more flexible than stiffer three-unit modules (mostly used in existing concepts for bistable scissor grids), which gives them a higher ability to cope with the incompatibilities during deployment.

Claims

A closed loop foldable scissor module comprising at least four polar scissor units (1, 1', 2, 2', 3, 3', 4, 5, 6, 7, 8) each unit consisting of a set of two rods (10, 20) connected by a revolute joint at an intermediate hinge point allowing a relative rotation of the two rods about an axis orthogonal to a unit plane comprising the two rods, wherein
• at least one of the polar scissor units consists of rods that are not identical in terms of their length and/or their semi-lengths (a, a', b, b'), wherein the semi-length is measured between the intermediate hinge point and an end point of the rod joined to a rod of another one of the polar scissor units, i.e. at least one unit is not a regular basic polar (P_RB) unit, wherein a regular basic polar unit is defined as a polar scissor unit which is fully determined by two size parameters and one parameter setting a deployment stage and in which corresponding semi-lengths of the two rods (a, a', b, b') have equal length,

• the unit lines (32) of the polar scissor units, which are the imaginary lines connecting the upper and lower end point of both sides of each of the polar scissor units, are not all concurrent,

• for each deployment stage, the foldable scissor module has a single geometrically compatible form with one corresponding set of internal dihedral angles between the unit planes of adjacent scissor units of the module.
The scissor module according to claim 1, wherein the module is symmetrical with respect to at least one symmetry plane (P).
The scissor module according to claim 2, consisting of
• a first and second scissor unit, said first and second units being P_RB units,

• a third and fourth scissor unit, said third and fourth unit being mutually identical non-P_RB units,
and wherein :
• the third and fourth unit are interconnecting the first and second unit,

• the module is symmetrical about one symmetry plane (P) and the P_RB units are orthogonal to the plane of symmetry and have their radial line in the plane of symmetry.
The scissor module according to claim 2, consisting of an uneven number of scissor units, and wherein at least one scissor unit is a P_RB unit, the remainder of the units consisting of pairs of mutually identical non-P_RB units.
The scissor module according to claim 2, consisting of an even number of scissor units higher than or equal to 6, formed by pairs of mutually identical non-PRB units.
The scissor module according to claim 2, consisting of an even number of scissor units higher than or equal to 6, and comprising an even number of P_RB units, the remainder of the units consisting of pairs of mutually identical non-P_RB units.
The scissor module according to claim 1 or 2, wherein all scissor units consist of two rods of equal length, and wherein the opposing or the adjacent semi-lengths at the connection of two units are equal.
The scissor module according to claim 7, consisting of an even number of scissor units.
The scissor module according to claim 7, wherein the module is symmetrical with respect to at least one symmetry plane (P), and wherein the module consists of an uneven number of scissor units, an uneven sub-number of which being P_RB units, the remainder of the units consisting of pairs of mutually identical non-P_RB units.
A scissor module according to any one of the preceding claims, wherein the joints between scissor units are defined by joint lines (11,12) placed between the spaced-apart units of the module.
A double layer scissor grid comprising one or more modules according to any one of the preceding claims.
The scissor grid according to claim 11, based on a rotational surface and formed of modules according to claim 1, wherein the meridian curves of the rotational surface are populated by identical planar linkages of irregular generalized polar units and wherein the parallel circles of the rotational surface are populated by regular basic polar units that are identical within the same circle.
The scissor grid according to claim 11 and formed of modules according to claim 6, wherein the grid is based on a surface with double negative curvature.