Abstract:
Improved reversibly expandable structures are formed from novel loop assemblies comprising a plurality of links, each of said links having at least one center pivot joint and a plurality of end pivot joints, each of at least two of said plurality of end pivot joints proximate to the outer edge of said loop assembly and connected to another link; each of said plurality of links being connected to another one of said plurality of links by at least two end pivot joints thereby forming a link pair, said loop assembly comprising at least three link pairs, each of said at least three link pairs connected to at least two other link pairs through at least one of said end pivot joints; each of said at least three link pairs connected to a central piece that is central to the loop assembly, said central piece being rotatable around a central axis, wherein the rotation of the central piece reversibly expands said loop assembly.

Description:
REFERENCE TO RELATED APPLICATION 
   This application claims the filing date of provisional patent application, Ser. No. 60/267,240, filed Feb. 7, 2001. 

   BACKGROUND OF THE INVENTION 
   U.S. Pat. Nos. 4,942,700, 5,024,031, 6,082,056 and 6,219,974, hereby incorporated by reference as if fully disclosed herein, teach methods for constructing reversibly expandable truss-structures in a wide variety of shapes. The teachings therein have been used to build structures for diverse applications including architectural uses, public exhibits and unique folding toys. 
   Utilizing the teachings of these patents, self-supporting structures that maintain their overall shape as they expand or collapse in a synchronized manner may be constructed. A basic building block of such structures is a “loop-assembly” which consists of three or more scissor units (disclosed in the &#39;700 and &#39;031 patents) or polygon-link pairs (disclosed in the &#39;056 and &#39;974 patents), each consisting of a pair of links that are pinned together at pivots lying near the middle of each link. Such a loop assembly comprises a ring of interconnected links which can freely fold and unfold. However, at the center of such a ring, a space or void is opened up as the ring expands, resulting in lessened structural stability. 
   It is, therefore, desirable to provide additional stability and structural stability to such a loop assembly while retaining its ability to expand and contract. It is also desirable to provide a central location to provide a means to mechanically drive the entire assembly. 
   In accordance with the present invention a novel loop assembly is presented that incorporates an additional useful feature. I have discovered a way to provide a link-pair that lies at the center of the assembly. The middle pivot of this central link-pair is located at the center point of the assembly as a whole. Further, this pivot always maintains its location at the center of the loop assembly as it extends and retracts. 
   Loop assemblies having such central link-pairs are better stabilized and better self-supported than those without such a feature. The movement of structures built from such assemblies are better synchronized. Further, central link-pairs offer a conveniently placed point of control for folding structures. By simply introducing a rotary motion of one such link relative to its paired link, a force is translated in an even, symmetric fashion to the entire assembly, thereby opening and closing it. Thus, a motor may be conveniently attached to one central link and the motor shaft fixed to the paired central link to provide a well-placed, stabilized means to drive the entire assembly. 
   Further, in addition to such mechanical improvements, such central link-pairs lead to new functional applications, such as the construction of expanding wheels and spreading mechanisms. 
   SUMMARY OF THE INVENTION 
   In accordance with the present invention reversibly expandable structures are formed from loop assemblies comprising interconnected pairs of links which lie essentially on the surface of the structure or parallel to the plane of the surface of the structure. The links in the loop assembly have at least three pivot joints. At least some of the polygon links however, have more than three pivot joints. One of the pivot joints on each link is a pivot joint for connecting to another link to form a link pair. Each link also has at least one internal pivot joint and one perimeter pivot joint. The internal pivot joints are used for interconnecting adjacent link pairs to form the loop assembly. Loop links are additionally joined to a central piece located at the center of the loop assembly. The central piece can be a circular construction with pivot points to which the pivots on the loop links are joined. The rotation of the central piece through a plurality of degrees clockwise and counterclockwise, expands and contracts the entire loop assembly. The central piece can alternatively be constructed of scissor pairs which open and close, resulting in the expansion and contracting of the loop assembly. 
   Loop assemblies can be joined together and/or to other link pairs through the perimeter pivot joints to form structures. 
   In one preferred embodiment of the present invention link pairs may be connected to adjacent link pairs to form a loop assembly through hub elements that are connected at the respective internal pivot joints of the two link pairs. Similarly hubs elements can be used to connect loop assemblies together or loop assemblies to other link pairs through the perimeter pivot joints to form structures. In yet another embodiment of the present invention the pivot joints can be designed as living hinges as described more fully below. 
   Structures built in accordance with the subject invention have specific favorable properties, including: a) The ability to use highly rigid materials rather than bending or distortion of the mechanical links, allowing for a smooth and fluid unfolding process; b) The use of compact, structurally favorable and inexpensive joints in the form of simple pivots; c) Retaining the strength and stability of the structure during folding and unfolding since all movement in the structure is due to the actual deployment process, without floppiness in the structure; d) A wide range of geometries; e) Inexpensive manufacture of structures with flexible hinges that are formed continuously with the links themselves; f) Convenient assembly of structures of many different shapes through kits of the necessary parts; g) The ability to create a space-filling structure by arranging linkages in a three-dimensional matrix; h) Structures have additional stability and structural stability because of the central piece, while still retaining its ability to expand and contract; and i) Structures have a central location to provide a means to mechanically drive the entire assembly. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  shows a link  1  having three pivots. 
       FIG. 2  shows link  1  joined to link  2  by pivot 
       FIG. 3  links  1  and  2  are shown rotated about their common pivot  3  to a different relative position. 
       FIG. 4  links  1  and  2  are again shown in a different relative rotational position. 
       FIG. 5  shows a linkage consisting of four links which are joined in a loop by pivots. 
       FIG. 6  shows the linkage of  FIG. 5  in a different position. 
       FIG. 7  shows a linkage consisting of four links which are joined in a loop by pivots. 
       FIG. 8  shows the linkage of  FIG. 7  in a different position. 
       FIG. 9  shows a linkage consisting of six links joined in a loop arrangement via pivots. 
       FIG. 10  shows the linkage of  FIG. 9  in a different position. 
       FIG. 11  shows the linkage of  FIG. 9  in yet another different position. 
       FIG. 12  shows a link having four pivots that form an isosceles triangle and pivots that lie on a second triangle that is the mirror image of the first triangle formed. 
       FIG. 13  shows a linkage, which is an alternate embodiment of the invention. 
       FIG. 14  shows the linkage of  FIG. 13  drawn in a different position. 
       FIG. 15  shows a linkage consisting of five scissor pairs. 
       FIG. 16  shows the linkage of  FIG. 15  in a different position. 
       FIGS. 17 and 18  show the linkage of  FIG. 15  in perspective views. 
       FIG. 19  shows a linkage consisting of thirteen scissor pairs. 
       FIGS. 20 and 21  the linkage of  FIG. 19  in two different positions. 
       FIGS. 22 ,  23  and  24  show perspective views of the linkage of  FIG. 19  in different positions. 
       FIG. 25  shows a scissor pair comprised of two links which have a different profile than those shown in the previous drawings. 
       FIG. 26  shows a linkage comprised of twelve perimeter scissor pairs and one central scissor pairs, all of whose pivot locations are similarly distributed to the linkage in FIG.  19 . 
       FIG. 27  shows the linkage of  FIG. 26  in a partially expanded position. 
       FIG. 28  shows the linkage of  FIG. 26  in a fully expanded position, an embodiment of the invention as an expanding wheel. 
       FIGS. 29 ,  30  and  31  each show a perspective view of the linkage of FIG.  26 . 
       FIG. 32  shows an alternate embodiment of the invention consisting of two scissor pairs which form a four bar linkage. 
       FIG. 33  shows a four sided linkage consisting of four perimeter scissor-pairs and one central scissor pair. 
       FIGS. 34 and 35  show the linkage of  FIG. 33  in two different positions. 
       FIGS. 36 ,  37  and  38  show perspective views of the linkage of  FIG. 33  in different positions. 
       FIG. 39  shows an alternate embodiment of the invention consisting of two scissor pairs forming a four bar linkage. 
       FIG. 40  shows the linkage of  FIG. 39  in a folded position. 
       FIG. 41  shows a four-sided linkage in a folded position. 
       FIG. 42  shows the linkage of  FIG. 41  in an opened position. 
       FIGS. 43 and 44  show perspective views of the linkage of  FIG. 41  in two positions. 
       FIG. 45  shows an exploded view of the linkage of  FIG. 41  in an open position, with a motor shown ready to be attached to the central link. 
       FIG. 46  shows the linkage of  FIG. 41  in its assembled form, where the motor has been joined to the central link and the shaft is fixed to the central link. 
       FIG. 47  shows the linkage of  FIG. 41  in its closed position, the shaft having been rotated and driving the entire linkage. 
   

   DETAILED DESCRIPTION OF THE INVENTION 
     FIG. 1  shows a link  1  having three pivots  3 ,  4  and  5 . Lines connecting these three pivots form an isosceles triangle  13 , with pivots  4  and  5  lying on the base and pivot  3  at the apex. 
     FIG. 2  shows link  1  joined to link  2  by pivot  3 . Link  2  has an additional pivot  6 . Pivots  4 ,  5  and  6  are equidistant from pivot  3 . A line  7  is drawn through pivots  4  and  6 . A second line  8  is drawn through pivots  5  and  6 . 
   In  FIG. 3  links  1  and  2  are shown rotated about their common pivot  3  to a different relative position. A line  9  is drawn through pivots  4  and  6 . A second line  10  is drawn through pivots  5  and  6 . The angle formed between Lines  9  and  10  is identical to the angle formed between lines  7  and  8  as shown in FIG.  1 . 
   In  FIG. 4  links  1  and  2  are again shown in a different relative rotational position. The similarly drawn lines  11  and  12  again form an angle, which is identical to that formed in FIG.  1  and FIG.  2 . 
   In general, given two joined links, the first having three pivots whose center points form an isosceles triangle, the second having two pivots whose distance is identical to the sides of that triangle, which links are joined by the pivot at the apex of the first link, the angle formed by drawing lines that lie on the center points of the three un-joined pivots is constant and unchanging for any relative angle between the two links. 
   In  FIG. 5  a linkage  15  is shown consisting of four links  20 ,  22 ,  24  and  26  which are joined in a loop by pivots  21 ,  23 ,  25  and  27  respectively. The figure formed by connecting the center points of these four pivots is a parallelogram. Thus the linkage may be seen to be a parallel four-bar. 
   Link  20  has three pivots  27 ,  28  and  21  whose center points lie on the vertices of isosceles triangles. Likewise link  22  has three pivots  23 ,  29  and  21  which form an isosceles triangle, which is similar, but of a different size, than that triangle formed by link  20 . Line  30  is drawn through pivots  28  and  25 . Line  31  is drawn through pivots  29  and  25 . 
     FIG. 6  shows the linkage  15  in a different position. Lines  32  and  33  are drawn through pivots  28 ,  25  and  29 ,  25  respectively. The angle formed between lines  30  and  31  shown in  FIG. 5  is identical to the angle formed between lines  32  and  33  shown in FIG.  6 . 
   In general, given a parallel four-bar linkage, each link being joined to two neighboring links, where two of the links have an additional pivot each of which form an isosceles triangle with the other two pivots of that link, which two triangles thus formed are similar, the lines drawn between each of those additional pivots and the pivot connecting the two links opposite forms an angle which is constant and unchanging for any relative position of the linkage. 
   In  FIG. 7  a linkage  170  is shown consisting of four links  172 ,  174 ,  176  and  178  which are joined in a loop by pivots  173 ,  175 ,  177  and  179  respectively. The figure formed by connecting the center points of these four pivots is a rhomb. Thus the linkage may be seen to be a parallel four-bar with equal sides. 
   Link  172  has three pivots  179 ,  173  and  180  whose center points lie on the vertices of isosceles triangles. Likewise link  174  has three pivots  175 ,  181  and  173  which form an isosceles triangle which is whose sides are the same length as that triangle formed by link  172 , but whose base is of different length. Line  185  is drawn through pivots  180  and  177 . Line  186  is drawn through pivots  181  and  177 . 
     FIG. 8  shows the linkage  170  in a different position. Lines  187  and  188  are drawn through pivots  180 ,  177  and  181 ,  177  respectively. The angle formed between lines  187  and  188  shown in  FIG. 7  is identical to the angle formed between lines  186  and  185  shown in FIG.  8 . 
   In general, given a equal-sided parallel four-bar linkage, each link being joined to two neighboring links, where two of the links have an additional pivot each of which form an isosceles triangle with the other two pivots of that link, which two triangles thus formed have equal length sides, but bases of different lengths, the lines drawn between each of those additional pivots and the pivot connecting the two links opposite forms an angle which is constant and unchanging for any relative position of the linkage. 
   In  FIG. 9  a linkage  38  is shown consisting of six links  40 , 42 , 44 , 46 , 48  and  50  joined in a loop arrangement respectively via pivots  41 , 43 , 45 , 47 , 49  and  50 . Link  40  may be seen to have three pivots:  51 ,  41  and  55 . Pivots  51  and  41  lie towards the perimeter of the loop assembly, while pivot  55  lies towards the interior of the loop assembly. Thus pivots of each of these types shall be hereinafter referred to as perimeter pivots and interior pivots respectively. Additionally to interior pivot  55 , there are five other interior pivots  56 , 57 , 58 , 59  and  60 . 
   Linkage  38  is further comprised of two centrally located links  64  and  65 . Three interior pivots  55 , 57  and  59  respectively connect links  40 , 44  and  48  to central link  64 . Three other interior pivots  56 , 58  and  60  respectively connect links  42 ,  46  and  48  to central link  65 . Central links  64  and  65  are themselves attached by pivot  66 . 
   Thus linkage  38  may be seen to consist of a region of outer links and central links. The outer links have, in general, perimeter pivots, which serve to connect them into a loop arrangement, and interior pivots which server to connect the outer links to the central links. The central links are pivotally attached to each other via a central pivot. 
     FIG. 10  shows linkage  38  in a different position;  FIG. 11  shows linkage  38  in yet another different position. For each position of the linkage, central links  64  and  65  can provide a convenient and stable point to drive the linkage; simply by rotating these two links relative to each other, forces will be transmitted to the outer links in a symmetric fashion. 
     FIG. 12  shows a link  72  having four pivots  81 ,  82 ,  85  and  86 . Pivots  81 ,  82  and  86  form an isosceles triangle and pivots  85 ,  82  and  86  lie on a second triangle that is the mirror image of the first triangle formed. 
     FIG. 13  shows a linkage  70 , which is an alternate embodiment of the invention. Linkage  70  comprised of four links  72 ,  74 ,  76  and  78 , which are connected together by pivots  82 ,  87 ,  90  and  86  to form a parallel four-bar linkage. Like link  72 , link  74  has four pivots  82 ,  87 ,  84  and  85 , which lie on the vertices of mirrored isosceles triangles. Line  91  is drawn between pivot  81  and  85 . Line  92  is drawn between pivot  84  and  83 . The intersection of lines  91  and  92  is at the center point of pivot  90 . 
     FIG. 14  shows linkage  70  drawn in a different position. Line  93  passes through pivots  81  and  85 . Line  94  passes through pivots  84  and  83 . The intersection of lines  93  and  94  is again at the center point of pivot  90 , which lies opposite the two four-pivot links  72  and  74 . The angle formed between lines  93  and  94  is identical to that formed between line  91  and  92  in FIG.  13 . 
   In general, given an equal-sided parallel four-bar linkage where two of the links each have two pivots, and two other links each have four pivots which lie on the vertices of two mirrored isosceles triangles, and of those four pivots, two lie on the mirror line and are attached to neighboring links, and two are side pivots which remain unattached, two lines may be drawn each connecting between the side pivots of neighboring links, which two lines will form an angle that is constant and unchanging for any relative position of the linkage, and will always intersect that pivot which lies opposite the two four-pivot links. 
     FIG. 15  shows a linkage  100  consisting of five scissor pairs  120 ,  130 ,  140 ,  150  and  160 . Each Scissor pair is comprised of two links joined by a centrally located pivot. For example, scissor pair  120  is comprised of links  121  and  122  joined by pivot  103 , the others are similarly formed. 
   Based on their general position and function, scissor-pairs  120 ,  130 ,  140  and  150  shall be referred to as perimeter scissor-pairs, whereas  160  shall be referred to as a central scissor pair. 
   The links in scissor-pair  120  and  150  each have four pivots. They are joined both to their neighboring scissor-pairs— 130  and  140  respectively—and to the central scissor pair  160 . They are thus called centrally attached perimeter pairs. The links in scissor pairs  130  and  140  each have three pivots. They are joined only to their neighboring perimeter pairs, and are thus not centrally attached. A line drawn through side pivots  104  and  102  intersects central pivot  125 . Likewise, lines drawn through  106 , 108  and  110 , 112  and  114 , 116  and  118 , 119  respectively all intersect central pivot  125 . 
     FIG. 16  shows linkage  100  in a different position. The five lines drawn through side pivots  102 , 104  and  106 , 108  and  110 , 112  and  114 , 116  and  118 , 119  respectively all intersect central pivot  125 . 
     FIGS. 17 and 18  show linkage  100  in perspective views. 
     FIG. 19  shows a linkage  200  consisting of thirteen scissor pairs. There are twelve perimeter scissor-pairs forming a complete loop-assembly. Four perimeter pairs  205 ,  220 ,  235  and  250  are centrally attached. Eight perimeter pairs  210 ,  215 ,  225 ,  230 ,  240 , 245 ,  255 ,  260  are not centrally attached. The thirteenth scissor pair  265  is comprised of two links  266  and  267 , which are attached by center pivot  270 . Pair  265  is pivotally attached by a total of eight pivots to pairs  205 , 220 , 235  and  250 . 
     FIGS. 20 and 21  shows linkage  200  in two different positions. By rotating links  266  and  267  relative to one another, central scissor pair  265  may be seen to drive the entire assembly in a symmetric and stable fashion. Center pivot  270  remains at the geometric center of the entire assembly in all positions. 
     FIGS. 22 ,  23  and  24  show perspective views of linkage  200  in different positions. 
     FIG. 25  shows a scissor pair  302  comprised of two links  304  and  306 , which links have a different profile than those shown in the previous drawings. 
     FIG. 26  shows a linkage  300  comprised of twelve perimeter scissor pairs and one central scissor pairs, all of whose pivot locations are similarly distributed to linkage  200 . Due to the different profile of the individual links, the overall shape of the linkage is a circle. 
     FIG. 27  shows linkage  300  in a partially expanded position.  FIG. 28  shows linkage  300  in a fully expanded position. In this position the overall shape of the linkage is a circle. Thus linkage  300  shows an embodiment of the invention as an expanding wheel. 
     FIGS. 29 ,  30  and  31  each show a perspective view of the linkage  300 . 
     FIG. 32  shows an alternate embodiment of the invention consisting of two scissor pairs  310  and  320  which form a four bar linkage  311 . 
   In  FIG. 33  a four sided linkage  370  is shown consisting of four perimeter scissor-pairs  310 ,  330 ,  340  and  350  and one central scissor pair  360 .  FIGS. 34 and 35  show linkage  370  in two different positions. The center pivot of central scissor pair  360  always remains in the center of the linkage. 
     FIGS. 36 ,  37  and  38  show perspective views of linkage  370  in different positions. 
     FIG. 39  shows an alternate embodiment of the invention consisting of two scissor pairs  410  and  420  forming a four bar linkage  405 . The relative position of the pivots is identical to linkage  370  shown in  FIG. 32 , however links  412  and  414  each have an additional pivot, respectively  421  and  422 . Lines  433 ,  431  and  432  drawn between  421 ,  422  and  415 ,  417  and  416 ,  417  respectively may be seen to form a right triangle. 
     FIG. 40  shows linkage  405  in a folded position. The triangle formed by lines passing through  421 ,  422  and  415 ,  417  and  416 ,  417  respectively is similar to that formed in  FIG. 39 , but of different size. 
     FIG. 41  shows a four-sided linkage  500  in a folded position.  FIG. 42  shows linkage  500  in an opened position. Central scissor pair  505  may be seen to drive linkage  500  by a relative rotation between each of its links. 
     FIGS. 43 and 44  show perspective views of linkage  500  in two positions. 
     FIG. 45  shows an exploded view of linkage  500  in an open position, with motor  510  shown ready to be attached to central link  503 .  FIG. 46  shows  500  in its assembled form, where motor  510  has been joined to central link  503  and the shaft  511  being fixed to central link  504 . 
   In  FIG. 47  linkage  500  is shown in its closed position, the shaft  511  having been rotated and thereby driving the entire linkage. 
   It will be appreciated that the instant specification, drawings and claims set forth by way of illustration and not limitation, and that various modification and changes may be made without departing from the spirit and scope of the present invention.