Abstract:
A foldable unit comprised of linkages in the form of tubular prisms is provided. The unit may be additionally comprised of polygonal panels. Linkages and panels correspond to a central polyhedral design that may be specified from a wide variety of geometric arrangements.

Description:
This application is based on and claims priority of U.S. Provisional Patent Application Ser. No. 62/023,240 filed Jul. 11, 2014. 
    
    
     BACKGROUND OF THE INVENTION 
     Mechanisms convert forces into movement; structures resist forces to maintain stability. Deployable structures are, in effect, hybrids embodying both structural and mechanical behaviors. 
     In general, there are two types of deployable structures: those comprised of mechanical linkages, and those based on folding principles similar to origami shapes. Each type utilizes a different approach to material composition and fabrication techniques. Beyond considerations of physical construction, there are significant differences between each type in terms of their geometric organization and kinematic constraints. 
     What are the comparative advantages mechanical vs. origami-based structures?As a broad generalization, linkage based structures give the designer more freedom to select and optimize both mechanical performance and spatial configurations. There are fewer constraints in configuring link-joint arrangements of mechanical assemblies than in folding sheet-hinge arrangements. This freedom increases the ease in achieving both strength and controlled movement by incorporating structurally favorable shapes, optimized boundary conditions and kinematically redundant networks. 
     Nonetheless, origami-type deployable structures have significant advantages as well. One advantage is that surface and structure are, in effect, one entity, eliminating the need to add additional covering or cladding. A second benefit is that the cost of construction is economical. Manufacture can be efficiently performed by 2D sheet-based machinery such as laser-cutters or water-jets. Plates and hinges may be developed as material laminates and assembly may be minimized through simple folding processes. 
     This invention discloses a novel and unique class of folding mechanisms that combine benefits of both mechanical and origami approaches. Structures built according to this system have an origami-like folded construction. Yet at the same time, like many mechanically deployable structures, they are comprised of synchronized linkages that can be arranged in wide variety of structural compositions. 
     Structures built according to this invention are based on folding units that correspond to various polyhedral shapes. Units can be assembled into three dimensional lattice-like assemblies that maintain a high degree of synchronization and structural integrity during folding. 
     SUMMARY OF THE INVENTION 
     A foldable unit comprised of linkages in the form of tubular prisms is provided. The unit may be additionally comprised of polygonal panels. Linkages and panels correspond to a central polyhedral design that may be specified from a wide variety of geometric arrangements. 
     Also disclosed are means to connect foldable units together by fixed connections between linkages, thus making foldable assemblies. Such assemblies, which may aggregate as many units as desired, extend in either two or three dimensions. Assemblies thus formed retain the kinematic behavior of the unit itself. 
     Certain embodiments of units and assemblies under this invention are capable of being substantially flattened for ease of packing and transport. 
     Further disclosed are modular linkage elements that provide universal building blocks for creating a wide variety of foldable units and structural arrangements. 
     Accordingly, it is an object of the invention to provide foldable mechanisms or units which utilize both mechanical and origami approaches. 
     Another object of the invention is to provide a foldable mechanism or unit which incorporates synchronized linkages that can be arranged in numerous configurations. 
     A further object of the invention is to provide foldable mechanisms or units which correspond to various polyhedral shapes. 
     Other objects of the invention will be apparent from the following description. 
    
    
     
       BRIEF DESCRIPTION 
       For a fuller understanding of the invention, reference is made to the following drawings in which: 
         FIG. 1  shows four-sided linkage  52  which in the form of a tubular prism. 
         FIGS. 2 and 3  show perspective views of linkage  52  in successively folded positions. 
         FIGS. 4, 5 and 6  show plan views of linkage  52  in three successively folded positions. 
         FIG. 7  shows a three-sided prism  51 : a central polyhedral design to define a first folding unit. 
         FIG. 8  shows linkages  52 , 54 , 56 , and panels  57 , 58  in exploded view around polyhedral design  51 . 
         FIG. 9  shows folding unit  60  where linkages  52 , 54  and  56 , panels  57 ,  58  have been hingedly connected as shown. 
         FIGS. 10, 11 and 12  show folding unit  60  in successively folded positions. 
         FIG. 13  shows a six-sided prism  71 : a central polyhedral design to define a second folding unit. 
         FIG. 14  shows linkages  72 , 73 , 74 , 75 , 76 , 77  and panels  78 , 79  in exploded view around polyhedral design  71 . 
         FIG. 15  shows folding unit  70  where linkages  72 , 73 , 74 , 75 , 76 , 77  and panels  78 , 79  have been hingedly connected as shown. 
         FIGS. 16, 17 and 18  show folding unit  70  in successively folded positions. 
         FIG. 19  shows folding units  50 , 70 , 80  and  90  in an exploded perspective view. 
         FIG. 20  shows a folding assembly  100  where units  50 , 70 , 80  and  90  are joined together by fixed connections as shown. 
         FIGS. 21 and 22  show folding assembly  100  in two successive positions. 
         FIG. 23  shows folding units  50 ,  70 ,  80  and  90  in an exploded plan view. 
         FIGS. 24, 25 and 26  show plan views of folding assembly  100  in three successive positions. 
         FIG. 27  shows polyhedron  161  which is a central polyhedral design to define another folding unit under this invention. 
         FIG. 28  shows linkages  162 ,  163 , 164 , 165  and panels  166 , 167 , 168 , 169  in exploded view around polyhedron  161 . 
         FIG. 29  shows unit  170  where linkages  162 ,  163 , 164 , 165  and panels  166 , 167 , 168 , 169  have been hingedly joined together as shown. 
         FIGS. 30,31 and 32  show unit  170  in successively folded positions. 
         FIG. 33  shows ten units  175 , 180 , 185 , 190 , 195 , 200 , 205 , 210 , 215  and  220  in an exploded perspective view. 
         FIG. 34  shows a folding assembly  225  where the ten units are joined together by fixed connections as shown. 
         FIGS. 35,36 and 37  show assembly  225  in successively folded positions. 
         FIG. 38  shows a perspective view of six-sided linkage  250  which is in the form of a tubular prism. 
         FIGS. 39 and 40  show perspective views of linkage  250  in successively folded positions. 
         FIG. 41  shows polyhedron  241  which is a central polyhedral design to define another folding unit under this invention. 
         FIG. 42  shows six linkages  250 , 251 , 252 , 253 , 254 ,  255  and eight panels  260 , 264 ,  261 , 262 , 265 , 266 , 267 , 268  in exploded view around polyhedron  241 . 
         FIG. 43  shows folding unit  300  where the six linkages and eight panels have been hingedly connected as shown. 
         FIGS. 44,45 and 46  show folding unit  300  in successively folded positions. 
         FIG. 47  shows eight folding units  300 , 305 , 310 , 315 , 320 , 325 , 330  and  335  in an exploded perspective view. 
         FIG. 48  shows a folding assembly  350  where the eight units are joined together by fixed connections as shown. 
         FIGS. 49 and 50  show assembly  350  in further folded positions. 
         FIG. 51  shows polyhedron  400  which is a central polyhedral design to define another folding unit under this invention. 
       In  FIG. 52  shows linkages  412 , 414 , 416 , 418  and panels  420 , 422  in exploded view around polyhedron  400 . 
         FIG. 53  shows eight additional linkages  430 , 431 , 432 , 433 , 434 , 435 , 436  and  437  in exploded view around polyhedron  400 . 
         FIG. 54  shows folding unit  450  where the twelve linkages and two panels have been hingedly connected as shown. 
         FIGS. 55,56 and 57  show folding unit  450  in successively folded positions. 
         FIG. 58  shows six units  450 , 455 , 460 , 465 , 470  and  475  in an exploded perspective view. 
         FIG. 59  shows folding assembly  480  where the six units have been joined together by fixed connections as shown. 
         FIGS. 60,61 and 62  show assembly  480  in successively folded positions. 
         FIG. 63  shows six linkages  510 , 515 , 520 , 525 , 530  and  535  in exploded view around polyhedral design  501 . 
         FIG. 64  shows folding unit  600  where the six linkages have been hingedly connected. 
         FIGS. 65-71  show unit  600  in a series of positions. 
         FIG. 72  shows unit  600  in a flattened position. 
         FIG. 73  shows eight units  600 , 605 , 610 , 615 , 620 , 625 , 630  and  635  in an exploded perspective view. 
         FIG. 74  shows a folding assembly  650  where the eight units have been joined together by fixed connections as shown. 
         FIGS. 75-80  show assembly  650  in successively further folded positions. 
         FIG. 81  shows assembly  650  in a flattened position. 
         FIG. 82  shows an exploded view of a three layers  710 ,  711  and  712  that may be laminated together to form a linkage. 
         FIG. 83  shows a plan view of the assembled laminate  720 . 
         FIG. 84  shows a perspective view of laminate  720  in a flat configuration. 
         FIGS. 85, 86 and 87  show  720  in successive folded positions as it is formed into a linkage. 
         FIG. 88  shows three modular linkages  720 , 730  and  740  in exploded view. 
         FIG. 89  shows linkages  720 , 730  and  740  having been joined together by making fixed connections as shown. 
         FIGS. 90 and 91  show reconfigurable unit  750  in two positions. 
         FIG. 92  shows unit  800  which is comprised of six linkages  810 , 820 , 830 , 840 , 850  and  860  and three motors  815 ,  825  and  845 . 
         FIGS. 93,94,95 and 96  show unit  800  in various positions as controlled by motors  815 ,  825  and  845 . 
     
    
    
     DETAILED DESCRIPTION 
       FIG. 1  shows a perspective view of a linkage  52  which is in the form of a tubular prism and is comprised of four planar links  2 , 4 , 6  and  8  which are connected by hinges  12 , 14 , 16  and  18  in a closed loop. Hinges  12 , 14 , 16  and  18  are all parallel to one another defining an extruded direction. Link  2  has an end edge  22 . Likewise links  4 , 6  and  8  have end edges  24 , 26  and  28  respectively. Edges  22 , 24 , 26  and  28  which are non-parallel to the extruded direction, lie in a common plane. 
       FIGS. 2 and 3  show perspective views of linkage  52  in successively folded positions. Extruded hinges  12 , 14 , 16  and  18  remain parallel to one another, and end edges  22 , 24 , 26  and  28  continue to share a common plane.  FIGS. 4, 5 and 6  show plan views of linkage  52  in three successively folded positions corresponding to  FIGS. 1,2 and 3  respectively. Linkage  52  is essentially an extrusion of the polygons that are shown in  FIGS. 4, 5 and 6 . 
       FIG. 7  shows a three-sided prism  51  which is a central polyhedral design that will define a folding unit according to an embodiment of this invention. 
       FIG. 8  shows three linkages  52 ,  54  and  56  in exploded view around polyhedral design  51 . Also shown are two triangular panels  57  and  58 . 
       FIG. 9  shows folding unit  60  where linkage  52  has been hingedly joined to panel  58  by edge  24 . Linkage  52  has also been hingedly joined to linkage  54  by edge  26 ; to panel  57  by edge  28  and to linkage  56  by edge  22 . In a similar fashion linkages  54  and  56  have been joined to each other by one each of their end edges, to panel  57  by one each of their end edges, and to panel  58  by one each of their end edges. 
       FIGS. 10, 11 and 12  show folding unit  60  in successively folded positions. As panels  57  and  58  are shifted relative to one another, the extruded directions of linkages  52  and  56  are reoriented as they draw together. Simultaneously, planar links  2 ,  4 ,  6  and  8  change their angles relative to each other and take on a more flattened configuration. As the same time, linkage  54  swings downwards. 
       FIG. 13  shows a six-sided prism  71  which is a central polyhedral design that will define a folding unit according to a second embodiment of this invention. 
       FIG. 14  shows six linkages  72 , 73 , 74 , 75 , 76  and  77  in exploded view around polyhedral design  71 . Also shown are two hexagonal panels  78  and  79 . 
       FIG. 15  shows folding unit  70 . Linkage  72  is hingedly connected by its end edges to neighboring linkages  73  and  77  and panels  78  and  79 . Similarly linkages  73 , 74 , 75 , 76  and  77  are hingedly connected to neighboring linkages by two each of their end edges, and to panels  78  and  79  by their remaining end edges. 
       FIGS. 16, 17 and 18  show folding unit  70  in successively folded positions. As panels  78  and  79  are shifted relative to one another, there is a reorientation of the extruded directions of linkages  77 ,  72  and  73  as they draw together above panels  78  and  79 . Likewise the extruded directions of linkages  74 , 75  and  76  draw together below panels  78  and  79 . As linkages  72 , 73 , 74 , 75 , 76  and  77  reorient their direction, they take on a more or less flattened configuration to accommodate their new positions. 
       FIG. 19  shows folding units  50 ,  70 ,  80  and  90  in an exploded perspective view. Units  50  and  80  each have three linkages, and units  70  and  90  each have six linkages. 
       FIG. 20  shows a folding assembly  100 . Units  50  and  70  are joined together by fixed connections between the end edges of linkages  56  and  76 . Similarly units  70 , 80  and  80 , 90  and  90 , 50  are respectively joined to each other by their end edges  75 , 85  and  83 , 93  and  92 , 52 . 
     The joined linkages of units  50 ,  70 ,  80  and  90  form a closed loop. Joined linkage  75 , 85  has a parallel orientation to joined linkage  52 , 82 . Likewise joined linkage  56 , 76  has a parallel orientation to joined linkage  83 , 93 . 
       FIGS. 21 and 22  show folding assembly  100  in two successive positions. Joined linkage  75 , 85  maintains a parallel orientation to joined linkage  52 , 82 . Likewise joined linkage  56 , 76  is maintains a parallel orientation to joined linkage  83 , 93 . Folding assembly  100  is thus a parallel mechanism that is itself comprised of multiple linkages. 
       FIG. 23  shows folding units  50 ,  70 ,  80  and  90  in an exploded plan view. 
       FIGS. 24, 25 and 26  show plan views of folding assembly  100  in three successive positions. 
       FIG. 27  shows a polyhedron  161  comprised of four triangular faces and four rectangular faces defining the central polyhedral design of a folding unit according to another embodiment of this invention. In  FIG. 28  four linkages  162 ,  163 , 164  and  165  are shown in exploded view around polyhedral design  161 . Also shown are four triangular panels  166 , 167 , 168  and  169 . 
       FIG. 29  shows folding unit  170 . Linkage  162  is hingedly connected to linkage  163  by one end edge, and is further connected to panels  166 , 167  and  169  by its three other edges. Similarly linkages  163 , 164  and  165  are each hingedly connected to another linkage by one end edge, and connected to three triangular panels by their remaining edges respectively. 
       FIGS. 30,31 and 32  show folding unit  170  in successively folded positions. As panels  166 , 167 , 168  and  169  shift towards a flattened configuration, the extruded directions of linkages  162  and  164  draw closer together. Likewise, the extruded directions of linkages  163  and  165  draw closer together. As linkages  162 ,  163 , 164  and  165  reorient their direction, they take on a more or less flattened configuration to accommodate their new positions. 
       FIG. 33  shows folding units  175 , 180 , 185 , 190 , 195 , 200 , 205 , 210 , 215  and  220  in an exploded perspective view.  FIG. 34  shows a folding assembly  225 . Folding unit  180  is joined to units  175  and  185  by fixed connections between end edges of their respective neighboring linkages. Similarly, each folding unit is connected to other units by fixed connections between their end edges of neighboring linkages.  FIGS. 35, 36 and 37  show assembly  225  in successively further folded positions. 
       FIG. 38  shows a perspective view of a linkage  250  which is in the form of a tubular prism and is comprised of six planar links  230 , 231 , 232 , 233 , 234  and  235  which are connected by six parallel hinges in a closed loop. The upper edges of these six links lie in a common plane. 
       FIGS. 39 and 40  show perspective views of linkage  250  in successively folded positions. The upper edges of links  230 , 231 , 232 , 233 , 234  and  235  continue to lie in a common plane. 
       FIG. 41  shows polyhedron  241  which is comprised of six rectangular faces and eight hexagonal faces, defining the central polyhedral design of a folding unit according to another embodiment of this invention. 
     In  FIG. 42  six linkages  250 , 251 , 252 , 253 , 254  and  255 , each having six links, are shown in exploded view around polyhedral design  241 . Also shown are two hexagonal panels  260 , 264  as well as six rectangular panels  261 , 262 , 265 , 266 , 267  and  268 . 
       FIG. 43  shows folding unit  300 . Six sided linkage  250  is hingedly connected to linkages  254  and  255  by two of its end edges, and is further connected to panels  260 , 261 , 265  and  266  by its four end other edges. Similarly linkages  251 , 252 , 253 , 254  and  255  are each hingedly connected to two neighboring linkages by two edges each, and connected to four panels by their remaining edges respectively. 
       FIGS. 44, 45 and 46  show folding unit  300  in successively folded positions. Linkages  250 , 251 , 252 , 253 , 254  and  255  take on various shapes and, at the same time, reorient their extruded directions relative to each other. 
       FIG. 47  shows folding units  300 , 305 , 310 , 315 , 320 , 325 , 330  and  335  in an exploded perspective view.  FIG. 48  shows a folding assembly  350 . Folding unit  330  is joined to units  305 , 315  and  325  by fixed connections between end edges of their respective neighboring linkages. Similarly, each folding unit within assembly  350  is connected to other units by fixed connections between their end edges of neighboring linkages.  FIGS. 49 and 50  show assembly  350  in successively further folded positions. 
       FIG. 51  shows polyhedron  400  which is comprised of six rectangular faces and eight hexagonal faces defining the central polyhedral design of a folding unit according to another embodiment of this invention. While polyhedral design  400  has the same basic geometry as polyhedral design unit  241  shown in  FIG. 41 , the folding unit that is derived will utilize a different arrangement of links and panels. 
     In  FIG. 52  four linkages  412 , 414 , 416  and  418 , each having four links, are shown in exploded view around polyhedral design unit  400 . Also shown are two rectangular panels  420  and  422 . 
     In  FIG. 53  eight linkages  430 , 431 , 432 , 433 , 434 , 435 , 436  and  437 , each having six links, are shown in exploded view around polyhedral design  400 . Also shown are four-sided linkages  412 , 414 , 416  and  418  as well as panels  420  and  422 . 
       FIG. 54  shows folding unit  450 . Six sided linkage  430  is hingedly connected to linkages  431 , 433  and  436  by three of its end edges, is further connected to four-sided linkages  412  and  418  by two of its edges, and is further connected to panel  420  by its remaining edge. Similarly linkages  431 , 432 , 433 , 434 , 435 , 436  and  437  are each hingedly connected to three neighboring six-sided linkages, two neighboring four-sided linkages and to one panel respectively. 
       FIGS. 55,56 and 57  show folding unit  450  in successively folded positions. Linkages  430 , 431 , 432 , 433 , 434 , 435 , 436  and  437  take on various shapes and, at the same time, reorient their extruded directions relative to each other. 
       FIG. 58  shows folding units  450 , 455 , 460 , 465 , 470  and  475  in an exploded perspective view.  FIG. 59  shows folding assembly  480 . Unit  455  is joined to units  450 , 460 , 465  and  475  by fixed connections between end edges of their respective neighboring linkages. Similarly, each folding unit within assembly  480  is connected to other units by fixed connections between their end edges of neighboring linkages.  FIGS. 60,61 and 62  show assembly  480  in successively further folded positions. 
       FIG. 63  shows central polyhedral design  501  which is in the form of a cube. Six linkages  510 , 515 , 520 , 525 , 530  and  535  shown in exploded view around polyhedral design  501 . 
       FIG. 64  shows folding unit  600 . Linkage  515  is hingedly connected to linkages  510 , 520 , 525  and  535  by each of its four end edges. Likewise linkages  510 , 520 , 525 , 530  and  535  are all connected to four neighboring linkage by each of their four end edges. 
       FIG. 65  shows unit  600  in a second position where linkages  515  and  530  have been flattened relative to their configuration in  FIG. 64 , while linkages  510 , 520 , 525  and  535  remain in their original square configuration. 
       FIG. 66  shows unit  600  in a third position where linkages  520  and  535  are successively flattened relative to their configuration in  FIG. 64 , while linkages  510 , 515 , 525  and  530  remain in their original square configuration. 
       FIG. 67  shows unit  600  in a third position where linkages  510  and  525  are successively flattened relative to their configuration in  FIG. 64 , while linkages  515 , 520 , 530  and  535  remain in their original square configuration. 
     The various positions of unit  600  shown in  FIGS. 65-67  demonstrate that unit  600  has three degrees of freedom, wherein three angles may be positioned independently of one another. 
     In  FIG. 68  unit  600  is shown in an elongated position wherein linkages  510 , 515 , 520 , 525 , 530  and  535  all have identical angles between their respective links. 
       FIGS. 69 and 70  show two other positions of unit  600  wherein linkages  510 , 515 , 520 , 525 , 530  and  535  continue to have identical angles between their respective links. 
       FIG. 71  shows unit  600  in a further flattened position.  FIG. 72  shows unit  600  as it approaches a fully flattened position. 
       FIG. 73  shows folding units  600 , 605 , 610 , 615 , 620 , 625 , 630  and  635  in an exploded perspective view.  FIG. 74  shows a folding assembly  650 . Unit  635  is joined to units  600 ,  610  and  620  by fixed connections between end edges of their respective neighboring linkages. Similarly, each folding unit within assembly  650  is connected to other units by fixed connections between their end edges of neighboring linkages.  FIGS. 75 and 76  show assembly  650  in successively further folded positions. 
       FIGS. 77,78,79 and 80  show assembly  650  in a series of different configurations. 
       FIG. 81  shows assembly  650  as it approached a fully flattened position. 
       FIG. 82  shows an exploded view of a three layers  710 ,  711  and  712  that may be laminated together to form a linkage. Layers  710  and  712  are each comprised of a multiplicity of polygonal shapes that have been cut from a rigid sheet of material. Layer  711  is cut from a sheet of flexible material and will provide a hinges for the linkage. 
       FIG. 83  shows a plan view of the assembled laminate  720  which will be folded into a linkage. Laminate  720  is comprised of a multiplicity of polygonal regions. Regions  713 , 714 , 715  and  716  are each comprised three layers: two rigid layers sandwiching an interior flexible layer. Region  721  is comprised of a single flexible layer, thereby forming a hinge between  714  and  715 . Shaded regions  722  and  724  are comprised of a single rigid layer and will serve as pockets for later assembly. Regions  723  and  726  are comprised of comprised of a rigid layer and a flexible layer and are hingedly connected to regions  716  and  715  respectively and will serve as attachment flaps for later assembly. 
       FIG. 84  shows a perspective view of laminate  720  in a flat configuration.  FIGS. 85 and 86  show  720  in successive folded positions as it is formed into a linkage.  FIG. 87  shows  720  wherein pocket  722  and flap  723  have been fixedly joined together, forming a closed loop. Laminate linkage  720  forms a modular linkage that serves as a building block to create folding units under this invention. 
       FIG. 88  shows three modular linkages  720 , 730  and  740  in exploded view. Flap  747  is aligned with pocket  724 . Likewise flap  725  is aligned with pocket  746 . In a similar fashion flaps  727 , 736 , 734  and  745  are aligned with pockets  737 , 726 , 744  and  735  respectively. 
       FIG. 89  shows linkages  720 , 730  and  740  having been joined together by making a fixed connection between flaps and pockets of the respective linkages as shown. After being joined together, linkages  720 , 730  and  740  have a hinged connection between them. It should be noted that these hinge connections are not created by the assembly process—they pre-exist as are part of the modular linkages. This ensures ease of assembly and precise alignment. 
       FIGS. 90 and 91  show reconfigurable unit  750  in two positions where linkages  720 , 730  and  740  have been further joined to three additional modular linkages. 
       FIG. 92  shows reconfigurable unit  800  which is comprised of six linkages  810 , 820 , 830 , 840 , 850  and  860 . A motor  825  is attached to linkage  820 , and has an aligned orientation along the hinge that connects linkage  820  to  830 . The shaft of motor  825  is attached to linkage  830 . Motor  825  thus serves as a controllable actuator for the relative angular position between linkages  820  and  830 . In a similar fashion, motor  815  is attached to linkage  810  with its shaft attached to linkage  850 , and has an aligned orientation along the hinge connecting linkages  810  and  850 . Likewise, motor  845  is attached to linkage  840 , with its shaft attached to linkage  860 , and has an aligned orientation along the hinge connecting linkages  840  and  860 . Motors  815 ,  825  and  845  control three angles, thus providing actuation for the three degrees of freedom of reconfigurable unit  800 . 
       FIGS. 93,94,95 and 96  show unit  800  in various positions each specified by a unique combination of angles as controlled by motors  815 ,  825  and  845 . 
     It should be noted that unit  800  may be programmably positioned by controlling three angles within the unit, utilizing various means of actuation, such as pneumatic or hydraulic devices. Further, various different combinations of hinges may be selected for control still providing the ability to programmably position a particular folding unit. 
     While there has been described what are at present considered to be the preferred embodiments of the present invention, it will be apparent to those skilled in the art that the embodiments described herein are by way of illustration and not of limitation. Various modifications of the disclosed embodiments, as well as alternative embodiments of the invention, will become apparent to persons skilled in the art upon reference to the description of the invention. Therefore, it is to be understood that various changes and modifications may be made in the embodiments disclosed herein without departing from the true spirit and scope of the present invention, as set forth in the appended claims, and it is contemplated that the appended claims will cover any such modifications or embodiments.