The present invention relates to an expandable/ collapsible structure. In US-A-5024031, there is disclosed a radial expansion/retraction truss structure consisting of pairs of identical angulated rods connected together by a single hinge in "scissor" pairs. Such pairs are referred to herein as "simple angulated elements". When the structure is expanded or retracted, certain critical angles are constant which allows the overall geometry of the structure to remain constant as it expands or collapses. A number of different deployable structures are disclosed. However, owing to mobility requirements, there must be some clearance within each pivot joint, which makes the structure undesirably flexible, especially when the number of elements is large. Furthermore, whilst there is disclosed in US-A-5024031 a foldable "iris-type" dome structure having a circular plan, the structure consisting of concentric rings connected by scissor hinges, in fact only a limited number of shapes for the structures can actually be constructed from the simple angulated elements disclosed in US-A-5024031.

According to a first aspect of the present invention, there is provided an angulated element, the element comprising two pivotally interconnected angulated rods which embrace a constant angle as the element is folded and expanded, in which each rod has first and second portions, the length of the respective first portions of the rods being substantially the same, the length of the respective second portions of the rods being substantially the same, the angles between the first and second portions of the respective rods being different.

According to a second aspect of the present invention, there is provided an expandable/collapsible structure, comprising a generalised angulated element which embraces a constant angle as the generalised angulated element is

folded and expanded, the generalised angulated element comprising a plurality of elements each consisting of two pivotally interconnected angulated rods, wherein the ends of rods of each element are pivotally connected to respective ends of rods of an adjacent element such that each closed loop formed by such connection is a parallelogram, with any unconnected portions of rods of any terminal elements forming isosceles triangles. Thus, where there are terminal elements, isosceles triangles are formed at the ends of the structure.

According to a third aspect of the present invention, there is provided an angulated element, the element comprising two interconnected angulated rods which embrace a constant angle as the element is folded and expanded, in which each rod has first and second portions, the ratio of the lengths of the respective first portions of the rods being substantially equal to the ratio of the lengths of the respective second portions of the rods, the angles between the first and second portions of the respective rods being substantially the same.

According to a fourth aspect of the present invention, there is provided an expandable/collapsible structure, comprising a generalised angulated element which embraces a constant angle as the generalised angulated element is folded and expanded, the generalised angulated element comprising a plurality of elements each consisting of two pivotally interconnected angulated rods, wherein the ends of rods of each element are pivotally connected to respective ends of rods of an adjacent element such that each closed loop formed by such connection is a parallelogram, with any unconnected portions of rods of any terminal elements forming similar triangles. Thus, where there are terminal elements, similar triangles are formed at the ends of the structure. According to a fifth aspect of the present invention, there is provided an expandable/collapsible structure, the structure comprising a plurality of multi-angulated rods,

each rod being formed of at least three portions, the rods being pivotally connected to one another at at least one of the junctions between said portions such that one closed loop is a foldable base structure and any other closed loops are formed by parallelograms.

Structures of the fifth aspect of the present invention use multi-angulated rods, allowing a smaller number of parts and simpler pivot joints to be used whilst providing the same overall profile as disclosed in US-A-5024031. More general profiles can also be provided. The structure of the present invention retains the important features of the structure disclosed in US-A-5024031, but with the key advantages of higher stiffness and more general shapes allowing large and complex structures to be formed.

In one embodiment, for each multi-angulated rod, said portions are substantially the same length and the angles subtended at an origin by said portions of each rod are substantially the same with each successive portion of each rod being at an increasing distance from the origin.

The rods may be substantially identical.

All of the rods may be planar.

All of the rods may be non-planar.

Some of the rods may be planar and some may be non- planar.

In an alternative embodiment, for each multi-angulated rod, said portions are of substantially different length.

A three-dimensional structure may have a single layer the projection of which onto a plane is a structure according to the fifth aspect of the invention. Alternatively, it may have a base layer being a structure according to the fifth aspect of the invention, and a second layer being a structure according the fifth aspect of the invention, the projection of the second layer onto the base layer being substantially identical to said base layer.

The structures of all aspects of the present invention have many applications, including but not limited to portable shelters, roofs over stadia, swimming pools, and theatres, and garden furniture. Examples of the present invention will now be described with reference to the accompanying drawings, in which:

Fig. 1 shows an angulated element used for a Type I General Angulated Element (GAE, see below) ; Fig. 2 shows a general Type I GAE;

Fig. 3 shows an angulated element used for a Type II GAE;

Fig. 4 shows a general Type II GAE;

Fig. 5 is a diagram for explaining the geometry of a simple multi-angulated rod having the same kink angles;

Fig. 6 are plan views of a structure using multi- angulated rods;

Fig. 7 shows a further example of a structure using multi-angulated rods in which Figs. 7(a), (b) and (c) are respectively plan views of the structure in its fully deployed, partly deployed and fully collapsed configurations;

Fig. 8 shows a foldable closed loop structure which folds along a given polygon, the angulated element forming similar rhombuses;

Fig. 9 shows a foldable closed loop structure which folds along a given polygon, the angulated element forming similar parallelograms;

Fig. 10 shows two further examples of foldable closed loop structures which fold along a given shape;

Fig. 11 shows three foldable structures, Fig. 11(a) showing a base structure of several parallelograms,

Fig. 11(b) showing an extended structure with additional hinged bars, and Fig. 11(c) showing a rigidly connected structure formed by multi-angulated rods;

Fig. 12 shows a foldable elliptical structure formed by multi-angulated rods;

Fig. 13(a) to (c) are respectively a perspective view of the components used in a hinge of the structure, a perspective view of the assembled components, and a plan view of the components; and, Figs. 14(a) to (c) are respectively perspective views of an example of a double layer circular foldable structure in its fully deployed, partly deployed and fully collapsed configurations.

In the present description, reference will be made to a generalised angulated element (GAE) . Such an element is a set of interconnected angulated rods that form a chain of any number of parallelograms with either isosceles triangles (referred to herein as "Type I GAE") or similar triangles (referred to herein as "Type II GAE") at either end. As will be shown, a generalised angulated element embraces a constant angle as the element is folded or expanded. Separate proofs that the angles of embrace of Type I and Type II GAEs are given next.

GAE's without any parallelograms are considered first, for simplicity; it will be shown that the simple angulated element of US-A-5024031 is a special case of both Type I and Type II GAEs.

Type I GAE Before discussing the general Type I GAE, consider first the angulated element 10 shown in Fig. 1 formed of angulated rods 11,12, which has

AE=DE,BE=CE and in general ψ≠φ (1)

From Fig. 1, the sum of the internal angles in the quadrangle OGEF is 360° and, since ZOFE = ZOGE = 90° , the angle α can be expressed as α = 180 ^O -(ZAEF+/3+ZBEG) (2)

Because ΔADE and ΔBCE are isosceles triangles.

ZAEF=-£-&ϋ aanndd ZLBEΕGG=-±-±£ (3) 2 2

Hence, substituting Eq. 3 into Eq. 2

a ^* =180° - 'P* ^ ---constant ( 4 )

2 '

which shows that this element subtends a constant angle. Note that the simple element of US-A-5024031 is re-obtained, when φ = - .

A most interesting special case is obtained when either φ = 180°, or = 180°, which implies that one rod is angulated, while the other rod is straight.

More general Type I GAEs are made from two or more angulated elements. Fig. 2 shows an example with three elements 15,16,17, which satisfy the following conditions: (i) each closed loop is a parallelogram, i.e.

CE=BJ and EB=CJ,HJ=ΪP and ΪJ=HP (5)

(ii) ΔAED and ΔNPM are isosceles triangles, i.e.

Note that the structure shown in Fig. 2 can be regarded as being formed by "cutting" the element shown in Fig. 1 at the pivot point E and inserting parallelograms in between the triangles formed thereby.

Next, it will be shown that the angle α embraced by this element has constant magnitude. From Fig. 2, it can be seen that a = ZDON = α, + a ₂ + O ₃ (7) where a = 180 ° - (ZAEF+/3 ₁ +ZBEG) α ₂ = 180 ^O - ( ZBJK+0 ₂ +ZHJL) ( 8 ) α, = 180 ^O - ( ZHPQ+ 3,+ZMPR)

Condition (i) implies

ZBEC = ZBJC and ZHJI = ZHPI (9) and also

ZBEG+ZBJK = ZBEC and ZHJL+ZHPQ = ZHJI (10) Substituting Eqs. 8, 9 into Eq. 7 gives

α = 3 x 180 ^O - (ZAEF+^ ₁ +^ ₂ +/3 ₃ +ZMPR) ( 11 )

From condition ( ii) , we know that

Note that Eq. 9 can be rewritten as

^• /'1-01 ⁼ Φz'βz ^and 'Pz = ^3-^3 ⁽¹³⁾ and adding up these two equations gives

Adding φ + ₃ to both sides of Eq. 14 and tidying up gives φ β = ∑φ-∑φ+φ _z -β ₃ (15) where

Substituting Eq. 15 into Eq. 12, and the result into Eq. 11 gives

i.e. the angle of embrace for a Type I GAE is a constant.

Type II GAE

Consider first the angulated element 20 shown in Fig. 3 formed of angulated rods 21,22, which has

=== and ψ=φ (18)

DE BE

To show that the angle of embrace α is constant in this case, it is noted that Eq. 2 is still valid. Because ΔAED and ΔBEC are similar

ZBEG = ZDEF (19) and, substituting Eq. 19 into Eq. 2 a = 180°-ø (20)

Note that the simple element of US-A-5024031 is re-obtained when

A more general Type II GAE with three angulated elements 25,26,27 is shown in Fig. 4. This element satisfies the following conditions:

(i) each closed loop is a parallelogram (ii) the triangles on the sides, ΔAED and ΔNPM, are similar, i.e.

Note that the structure shown in Fig. 4 can be regarded as being formed by "cutting" the element shown in Fig. 3 at the pivot point E and inserting parallelograms in between the triangles formed thereby.

To show that the angle a is constant, we note that Eq. 11 is also valid for Type II elements. Also, because of condition (ii)

ZAEF = ZNPR (23) and hence Eq. 11 is equivalent to

It is interesting to note that, since

ZAED = φ _Λ - jt3, and ZMPN = φ _z - β _z (25) and since these angles are equal, Eq. 13, which is also valid for Type II elements, is equivalent to

∑ Ψ=∑ Φ (26)

which shows that the sum of the kink angles of the two sets of angulated rods that make up a Type II GAE is constant. For angulated elements consisting of two angulated rods only, Eq. 26 becomes φ = φ, which agrees with Eq. 18.

Multi-Angulated Rods

Next, it will be shown that in circular foldable structures made from identical, symmetric angulated elements, contiguous angulated rods can be connected rigidly to one another, to form multi-angulated rods. Consider two identical angulated rods 30,31 (A ₀ to A ₂ and A ₂ to A ₄ ) of semi-length I , lying in neighbouring sectors subtending equal angles α as shown in Fig. 5. No three of the nodes A^A^A^A _j lie in a straight line. Let node A ₂ be the connection point of the two elements 30,31. It will be shown that the angle between the two rods, ZA,A ₂ A ₃ , has constant magnitude.

Considering the first angulated rod 30, which lies between the lines OP ₀ and 0P ₂ , the distance of hinge A ₂ from point 0 is

A ₂ C, _gcos(ZA ₁ A ₂ C ₁ ) _(2?) sinα/2 sinα/2

where

ΔA.AJJC, ^• * ^** 90 ° -ZA ₂ A ₁ C ₁ = ( 28 )

Because

ZA ₂ A ₁ 0=ZS ₁ A ₁ 0+ZA ₂ A ₁ S ₁ =ZS ₁ A ₁ 0+-^^ (29)

Eq . 28 becomes

ZA ₁ A ₂ C ₁ = ZS ₁ A ₁ 0-o/ 2 ( 30)

Substituting Eq . 30 into Eq . 27

Also

ZA ₁ A ₂ O=180 ^o - -ZA ₂ A ₁ O=90° -ZS ₁ A ₁ O ( 32 )

Considering the second angulated rod 31 , the distance of hinge A ₂ from point O is where

ZA ₃ A ₂ C ₃ = ZS ₃ A ₂ C ₃ +α 2 = ZS ₃ A ₃ 0+α/2 ( 34 )

__ A ₂ C ₃ _ cos A ₃ A ₂ C ₃ ) ₍₃₃₎

² Sinα/2 sinα/2

Hence

Jcos (ZS,A,0+α/2)

OA ₂ = (35) sinα/2

Comparing Eq. 31 with Eq. 35,

ZS^O-α/2 = ZS ₃ A ₃ 0+α/2 (36) Also

ZA ₃ A ₂ 0 = ZA ₃ A ₂ C ₃ +90°-α/2 = 90°+ZS ₃ A ₃ O (37)

The angle between the two angulated elements can be calculated from Eq. 32 and Eqs. 36-37

ZA^A _J = ZA^O+ZA^O = lSC+ S _j A _j O-ZS^O = 180°-α = constant (38)

This proof can be extended to any number of contiguous rods of equal semi-length £ , provided that they are at an increasing distance from the centre: when they start to turn back towards the centre, the angle Zs,-A,-0 becomes negative and hence the above proof is no longer valid. Subject to this condition, the rods can be rigidly linked together to form a multi-angulated rod with a kink angle of 180°-α, Eq. 38. Fig. 6(a) shows a circular foldable structure 40 containing 48 five-segment multi-angulated rods 41. This structure has

Fig. 6(b) shows that modest shape changes can be made by varying the number of segments in some rods. Fig. 7 shows photographs of a model structure 50 with

360° x2 o ₍₄₀₎

24

whose 24 identical multi-angulated rods 51 each has a kink angle of 30° and consists of three segments of length I =

100mm. The fully deployed and fully folded or collapsed configurations of this model are shown in Fig. 7(a) and Fig. 7(c), respectively. Note that the rods 51 cannot fully overlap because of the physical size of the joints.

FOLDABLE STRUCTURES OF GENERAL SHAPE It might be expected that two-dimensional foldable structures with many different shapes might be made by a straightforward extension of the ideas introduced above. Indeed, an obvious way of doing this would be to divide any given boundary shape into straight segments and circular arcs, and then assemble together straight-edged, trellis-type structures of suitable length, and simple angulated elements as disclosed in US-A-5024031 with an appropriate angle of embrace. Unfortunately, a rod structure of this type is not foldable. The problem is that, although it is possible to vary the semi-length of the simple angulated elements that make up a circular sector, so that the hinges connecting this sector to its neighbouring trellis-type structure are equally or proportionally spaced in the radial direction, this can be done only for a particular configuration. The scissor hinges do not remain equally spaced when the configuration is varied. Hence, a circular sector cannot be connected to a structure consisting of straight rods, whose scissor hinges are always equally, or proportionally spaced.

To obtain the layout of a two-dimensional foldable structure with a boundary of prescribed shape one must begin by finding a foldable base structure, i.e. a structure consisting of angulated rods whose hinges lie on the prescribed boundary. Once a suitable layout for the base structure has been selected, extra members can be connected to it by means of scissor hinges, until the required shape and overall dimensions are obtained. It will be shown that such a structure is foldable and, subject to certain conditions, it remains foldable if

contiguous rods are firmly connected into multi-angulated rods.

Finding a base structure that meets all the shape and folding requirements of a given application is the key to a successful overall solution. The method will be explained by describing the procedure which has been followed for a series of representative examples. All of the examples are of the same basic type, continuous loop structures with a central hole of variable size. Such structures are suitable for foldable roofs for, e.g. sports stadia and tennis courts. Open loop structures are subject to fewer restrictions, and hence much easier to configure using any combination of GAEs.

Fig. 8 illustrates a simple technique as disclosed in US-A-5024031 to construct a single-loop foldable rod structure of any shape. Fig. 8(a) shows an illustrative, general polygon 60 which may be constructed from a series of simple angulated elements 61., to 61 ₆ as disclosed in US-A-5024031 whose internal hinges coincide with the vertices of the polygon. The semi-length of each angulated rod 62,-, 62,-' is equal to half the length of each respective side of the polygon 60 and the two rods 62 _jf 62/ in each respective element 61 _j form equal kink angles, which are equal to the corresponding internal angle of the polygon 60. Hence, in the fully folded configuration as shown in Fig. 8(b), the elements 61 _i overlap with the sides of the polygon 60. Note that half of the angulated rods 62,-, 62/ are hidden by the other rods 62,-, 62/. In general, of course, these angulated elements 61. are not symmetric and hence a radial mismatch develops as the structure is folded. However, the overall mismatch adds up to zero as shown in Fig. 8(c) because in this case the angulated elements form a chain of similar rhombuses whose diagonals are reduced in length by proportional amounts and also remain at constant angles during folding.

Fig. 9 shows a more general type of closed loop structure 70, whose internal hinges also coincide with the

vertices of the polygon 60 of Fig. 8(a). Here, the angulated rods 72 , , 72/ making up each element 71 _f are no longer identical, but still have a kink angle equal to the internal angle of the polygon 60 and form a chain of similar parallelograms 73. This property implies that the loop structure 70 is foldable, because the sides of the polygon vary by proportional amounts and hence no geometric mismatch builds up during folding. Note that the angulated elements used in this solution are simple Type II GAEs, i.e. without any parallelograms as in Fig. 3.

In addition to the above solutions for base structures forming closed loops of any shape, greater freedom is available in the case of loops with one or more axes of symmetry. Basically, any GAE can be used to form the basic repeating unit and since, by symmetry, all units behave in the same way, geometric compatibility in all configurations is automatically satisfied. Fig. 10 shows two loop structures whose innermost hinges lie on a rectangle with rounded corners. The base structure shown in Fig. 10(a) consists of identical rhombuses, and hence there is no need to invoke symmetry to prove that this structure is foldable. The base structure shown in Fig. 10(b), however, is based on a symmetric arrangement of GAEs of Type I or Type II. This can be seen by means of the central line of symmetry 85 which divides two opposite rhombuses 80,81 into similar isosceles triangles. Note that in the base structure shown as an example in Fig. 10, the quadrilaterals 80,81 around the line of symmetry need not be parallelograms, but may be general symmetrical quadrilaterals.

Any base structure can be extended by the addition of a pair of rods of any length, connected to one another and to the base structure by scissor hinges. The resulting structure will be foldable, like the original base structure. Repeating the same argument it can be shown that any number of pairs of rods connected by hinges to the base structure will leave its mobility unchanged.

Fig. 11(a) shows a general, small part of a rod structure consisting of angulated elements. Additional members are connected to its outer hinges as shown in Fig. 11(b), such that the quadrangles A ₂ A ₃ B ₁ B ₂ , etc. are parallelograms. This extended structure is foldable because all additional members are free to rotate with respect to the base structure but, in fact, no relative rotation between consecutive rods occurs as the structure is folded, i.e. ZA ₁ A ₂ A ₃ , ZB,,B ₂ B ₃ , etc. remain constant. Consider, for example, ZA ₁ A ₂ A ₃ . Because A.,A ₂ and A ₂ A ₃ remain parallel to B ₀ B, and B^B _Z , respectively,

ZA,A ₂ A ₃ = ZB _Q B^ = constant (41) since ZB ₀ B ₁ B ₂ is the kink angle of an angulated rod, which is fixed. In conclusion, this foldable structure can be made from multi-angulated rods similar to those described above as shown in Fig. 11(c) , but note that the kink angles along these multi-angulated rods are not the same. The same procedure is valid for all other closed loop base structures discussed in this section, as for any open loop base structure. Fig. 12 shows a foldable structure whose internal boundary has an elliptical shape which has not been achievable previously.

The two-dimensional solutions derived above easily extend to curved structures, such as domes, by projecting any two-dimensional solution onto a surface with the required shape. Thus, each multi-angulated rod can be curved out of plane. Of course, all connectors between multi-angulated rods should be perpendicular to the plane of projection.

The folding angle may be restricted if the rods are not allowed to overlap during folding. This problem can be solved by a proper design of the connections. An example of a suitable connector between two rods is shown in Figs. 13(a) to (c) . In the drawings, 90 is one of the rods whilst the other rod is in two parts 91,92. One part 92 of the other rod has a circular cross-section cylindrical post

93 of height H and the other part 91 of the other rod has a cap 94 which can be securely fitted onto the post 93. At the pivot point of the first rod 90 there is provided an open ring 95 of height h which is less than the height H of the post 93. The post 93 is inserted into the ring 95 and the cap 94 fitted over the part of the post 93 which projects through the ring 95. The parts 91,92 of the other rod can then be rigidly fixed to each other by some suitable means such as screws 96 so that the parts 91,92 of the other rod effectively act as one long rod. As the height H of the post 93 is greater than the height h of the ring 95, the two rods can rotate with respect to each other. Of course, other connectors will be suitable for allowing the rods to rotate with respect to each other. Fig. 14 shows a double layer model structure whose curved top layer is connected to the flat bottom layer by long bolts. The bottom layer is identical to the model shown in Fig. 7 and the orthogonal projection of the top layer onto the plane of the bottom layer is also identical to it. This model folds until the outer rods overlap fully, and thus demonstrates that the interference between rods connected to the same hinge has been successfully eliminated. Note that bracing elements could be added between the upper and lower cords, to increase the stiffness of the structure, if desired.

In summary, a general method for the design of two-dimensional foldable structures has been introduced. The new method extends and generalises the familiar trellis-type structures, based on a tiling of parallelograms whose sides are collinear, to structures based on any tiling of parallelograms. It has been shown that a rod structure of this type is (i) foldable and (ii) can be made from multi-angulated, rigid rods connected by scissor hinges. This result affords much greater freedom in the range of shapes that can be achieved, and of boundary conditions that can be met. This approach can be easily extended to three-dimensional dome structures.

Also, a family of elements for foldable structures has been introduced. These consist of angulated rods connected by scissor hinges. It has been shown that any element bounded by either isosceles triangles or similar triangles, with any number or parallelograms in between, maintains a constant angle of embrace.

Finally, a method for the design of structures consisting of multi-angulated rods that fold along their perimeter has been described and there is practically no limit to the range of perimeter shapes that can be achieved.