Single Degree-of-Freedom Rigidly Foldable Cut Origami Flashers

Transcription

1 Robert J. Lang Lang Origai, Alao, CA 9507 e-ail: Sencer Magleby Deartent of Mechanical Engineering, Brigha Young University, Provo, UT 860 e-ail: Larry Howell Deartent of Mechanical Engineering, Brigha Young University, Provo, UT 860 e-ail: Single Degree-of-Freedo Rigidly Foldable Cut Origai Flashers We resent the design for a faily of deloyable structures based on the origai flasher, which are rigidly foldable, i.e., foldable with revolute joints at the creases and lanar rigid faces. By aroriate choice of sector angles and introduction of a cut, a gle degree-of-freedo (DOF) echanis is obtained. These structures ay be used to realize highly coact deloyable echaniss. [DOI: 0.5/.030] Introduction Many atterns fro the world of origai have alication in the world of engineering, articularly in the area of deloyable structures. Whenever there is a need for a echanis to transfor between a large, flat, sheetlike state (the deloyed state) and a uch saller state ( stowed ), origai-based echaniss can rovide efficient solutions. One origai echanis that has received considerable attention over the years is the attern called a flasher, which was introduced and exlored in the origai world by Paler and Shafer []; they develoed their concet fro the twist-fold fors of Kawasaki (see, e.g., Ref. []). However, the concet had a long existence outside of the origai world. As noted by Guest and Pellegrino (see Ref. [3] and references therein), several authors have discovered and exlored siilar structures ce the early 960s and it is regularly rediscovered. Nojia [] showed a variety of siilar fors with varying degrees of helicity and rotational syetry. For uroses of this work, we will refer to all of these atterns that are rotationally syetric, roughly flat in the deloyed state, roughly cylindrical in the stowed state, and deloy in a siral attern, as (generalized) flashers. Most work has focused on the use of the flasher echanis in concert with ebranes, or at least, structures that have distributed flexibility [5,6]. While any flashers have been deonstrated fro relatively stiff aterials, the basic flasher echanis and, to the best of our knowledge, all versions roosed and deonstrated to date are not rigidly foldable; they cannot be folded continuously fro the stowed to the deloyed state with rigid anels and ure revolute joints. To get around this roble, several alternatives have been roosed and/or deonstrated that work with rigid anels. Guest and Pellegrino roosed a structure coosed of searate anels joined by struts [7]. Zirbel et al. deonstrated a rototye solar array ug a flasher structure with rigid anels and finite-width ebrane hinges between the anels, in which the flexible ebrane hinges rovide the necessary additional coliance needed for deloyent [8], albeit at the cost of introducing otentially undesirable additional DOF into the otion. It is also otentially ossible to add DOF by triangulating soe fraction, erhas all, of the quadrilateral anels to allow soe aount of flexing across their diagonals. However, whether such an aroach guarantees full reachability across the range of otion and the effects on the DOF of the echanis reain oen questions for that aroach. Manuscrit received June 9, 05; final anuscrit received October 8, 05; ublished online March 7, 06. Assoc. Editor: Mary Frecker. An additional challenge with ug idealized flasher atterns in real-world engineering is the roble of thickness: idealized atterns assue zero (or negligible) thickness, but in real-world alications, the thickness of each anel is usually non-negligible. Finite thickness atters in two ways. First, it affects etric foldability: offsets and dislaceents of hinges fro their idealized zero-thickness ositions can affect the echanics of folding or even revent folding by turning a flexible echanis into a locked structure. Second, it affects injectivity, or self-intersection: the layers of a thick structure can collide with each other even if the corresonding zero-thickness odel does not self-intersect. Fortunately, recent work by Tachi [9] and Edonson et al. [0] deonstrated effective techniques for adating zero-thickness structures with non-negligible thickness anels, and such techniques are alicable here. Self-intersection aside, the roble of etric foldability reains. In articular, for alications with rigid anels, it would be desirable to have a folding attern that is rigidly foldable. Even better, it would be desirable for the folding otion to have a gle DOF, so that there is one and only one ath between the stowed and deloyed states. In this aer, we roose, describe, and analyze a eber of the flasher faily of echaniss that eets both criteria: it is rigidly foldable with lanar anels and ure revolute joints and transfors continuously fro a fully flat state to a coact, cylindrical configuration with a gle DOF. Like another well-known deloyable structure, the Miura-ori [], it is overconstrained according to the Kutzbach criteria, but by careful choice of angles in the design, we can realize a gle DOF in the deloyent otion; however, it is necessary to introduce a cut into the echanis. Furtherore, there is a tuning araeter for the sacing between layers in the stowed state, so that nonzero-thickness anels of varying thicknesses ay be accoodated in the folding attern. Throughout this work, we assue a zero-thickness odel, but note that the thickness-accoodating techniques of Tachi and/or Edonson ay be alied to any of the echaniss described to realize a full thick-anel echanis while reserving the gle-dof deloyent otion. Preliinaries Figures and show ileentations of the flasher structure by Scheel [] and Paler and Shafer [] that illustrate the fundaental structure. There is a central lanar region (henceforth, the central olygon) surrounded by a series of ountain and valley folds that eanate roughly radially, but are offset soewhat fro being center-directed. These iages show curved and bent facets, but it is ossible to create olyhedral (lanar facet) versions of the flasher [3,8]. Journal of Mechaniss and Robotics Coyright VC 06 by ASME JUNE 06, Vol. 8 / Downloaded Fro: htt://asedigitalcollection.ase.org/ on 03/07/06 Ters of Use: htt://

2 Fig. Scheel s wind-u ebrane. Left: nearly oen. Right: starting to close. Fro Ref. []. Figure 3 shows three versions of a olyhedral flasher with a square central olygon and fourfold rotational syetry constructed according to the algorith of Ref. [8]. There are four distinct tyes of folds in a flasher, as illustrated in Fig.. The diagonal folds eanate fro the corners of the central olygon and, in the folded for, roagate helically around the axis of syetry. Next, there are reverse folds, which roagate axially around the structure, each fold foring a siral that lies (nearly) in the sae lane. Both reverse and diagonal folds are shar folds, being folded nearly to 6. Next, there are the bend folds, which are (nearly) axis-arallel folds at the corners of the central olygon; they are the folds used for the layers to wra around the central olygon. And last, there are the central olygon folds, which aear to be continuations of the reverse folds but, unlike the latter, have a fold angle of about =, rather than 6. For a flasher to be rigidly foldable with a gle DOF, the fold angles around each vertex ust flex continuously in such a way that the fold angles along each fold are coatible when they eet u at every vertex. In general with flashers, this is not ossible; for all of the atterns shown to date, at the very least, facets ust bend along their diagonals. A general flasher attern consists of degree-6, degree-5, and degree- vertices, with the last usually being the ost nuerous. Degree- vertices individually have a gle DOF in their otion, and it will be these vertices that are the key to realizing gle-dof otion for a flasher. We now review briefly the iortant and relevant roerties of degree- vertices. 3 Degree- Vertices Figure 5 shows a generic degree- vertex, with four sector angles a i ; i ¼ ; ;, and four dihedral angles, c i ; i ¼ ; ;, with c i ½; Š. For ountain folds, c i < 0; for valley folds, c i > 0. If jc i j¼, the fold is fully folded; ifjc i jð0; 6Þ,itis artially folded; and if c i ¼ 0itisunfolded. If all four creases are artially or fully folded, then there ust be three ountains and one valley or three valleys and one ountain [3]. If we look at alternating (not consecutive) folds around Fig. 3 A olyhedral flasher with fourfold rotational syetry. To row: an ideal olyhedral flasher (left: crease attern and right: folded for). Middle row: the sae structure, odified to sread the layers to accoodate nonzero thickness. Botto row: the sae structure, but with additional reverse folds added to reduce the height. Note that the crease atterns and folded fors are shown at different scales; the diaeter of the folded for is aroxiately the diaeter of the central olygon of the crease attern in each case. the vertex, two will be of the sae tye and the other two will be of oosite tye. We call the two folds of the sae tye (c and c in Fig. 5) the ajor folds (or ajor air) of the vertex. The two folds of the oosite tye (c and c 3 in Fig. 5) are the inor folds of the vertex. We also recognize two secial cases: if a þ a ¼ a þ a 3 ¼, the vertex is straight-ajor, because the ajor folds are collinear. Siilarly, if a þ a ¼ a 3 þ a ¼, the vertex is straight-inor. Fig. The Paler Shafer origai flasher. Left: nearly fully deloyed. Right: stowed. Fro Ref. []. Fig. The four distinct tyes of fold in a flasher. Here, we use the origai convention of drawing ountain folds as solid lines, valleys as dashed, with different tones for the four failies of fold / Vol. 8, JUNE 06 Transactions of the ASME Downloaded Fro: htt://asedigitalcollection.ase.org/ on 03/07/06 Ters of Use: htt://

3 Straight-inor vertices are erfectly well behaved, but straightajor vertices are a secial case; it is not ossible for all four folds of a straight-ajor vertex to be artially folded at the sae tie. For a straight-ajor vertex to fold, first, c and c ust fold fro 0 to 6, and only then can c and c 3 fold (and then only if a ¼ a and a ¼ a 3 ). For non-straight-ajor vertices, if one fold angle is chosen, the other three are fully deterined fro trigonoetric relationshis between the sector angles and the fold angles. For the generic case, these relationshis can be rather colex (as shown in the Aendix). However, for a flat-foldable vertex one that can be ressed flat with all layers in a coon lane the relations collase to sile fors. First, as is well known fro Kawasaki s theore [], a degree- vertex is flat-foldable if and only if a þ a 3 ¼ a þ a ¼ () For a flat-foldable vertex, the ajor fold angles are equal [5] c ¼ c () and the inor fold angles are equal and oosite [5] c ¼c 3 (3) The relationshi between adjacent fold angles has been described by several authors [3,5,6]; a articularly sile and useful exression (derived in the Aendix) is tan c tan tan ¼ c tan c tan ¼ c tan c 3 tan ¼ c c tan c 3 ¼ ð a þ a Þ ð a a Þ Equation () also gives soe justification for the naes ajor and inor ; a consequence of Eq. () is that jc ; jjc ;3 j (5) with strict inequality at all artially folded angles and equality only at the flat (fully unfolded and fully folded) states. So, away fro the flat state, ajor folds are always larger than inor folds at a flat-foldable degree- vertex. We define the ratio in Eq. () as the fold angle ultilier l for the vertex. In general, l > : The fold angle ultilier is a easure of the geoetric advantage between a ajor and inor fold of the vertex. If we denote either ajor fold angle by c þ and either inor fold angle by c, we have that () Fig. 5 for. A degree- vertex. Left: crease attern. Right: folded ore broadly, for a esh whose interior vertices are all flatfoldable degree- vertices, if we can find a artially folded state involving all folds for a gle fold angle, then it is guaranteed that the entire attern folds soothly over the full range of fold angles fro fully flat to fully folded, with a gle DOF. This roerty is extreely owerful and useful, and we can eloy it when we are seeking gle-dof folding echaniss. If we can construct a fold attern consisting of flat-foldable degree- vertices and find a gle consistent artially folded state, then we have a gle-dof echanis. In rincile, such a attern could fold fro flat unfolded to flat fully folded. In ractice, selfintersection ay liit the full range of otion, but even if we do not need full flat-foldability, we can still use this technique to achieve gle-dof echaniss. And, we will now do this with flashers. Sile Flashers. Fold Angle Relations. We now turn our attention back to the flasher. For silicity, we will consider first a flasher that has no reverse folds at all, such as the one illustrated in Fig. 6. There are only diagonal, bend, and central olygon folds. This choice ensures that all of the interior vertices of the attern are degree-. We will first look at the constraints on the angles in the attern, assuing -fold rotational syetry on the ositions of the fold lines, -fold rotational syetry on the agnitudes of the fold angles, and =-fold rotational syetry on the sign of the fold angles (ountain/valley assignent). We also assue (at least initially) that is even and. The central olygon is a regular -gon, and so the interior angles at its corners are ð =Þ. We take d to be the angle between the incident diagonal fold and the side of the central olygon. In Fig. 6, we have ¼ and the central olygon is a square. Moving out along the diagonal folds, we have a sequence of degree- vertices. Denote the two angles to left and right of the dc li þ dc ¼ 6l; li þ ¼ 6=l; and dc c! dc dc þ dc ð =l; l Þat angles in between (6) c!0 There is a rearkable roerty ilicit in Eqs. () (), noted by Tachi [6]. For any two angles c i ; c j at a flat-foldable degree- vertex, tan ð=þc i = tan ð=þc j ¼ constant, indeendent of the state of foldedness. This roerty extends to any esh of degree- vertices: if c i and c j are connected by a ath containing only nonstraight-ajor flat-foldable degree- vertices, their half-angletangents are roortion by soe fixed value that deends on the sector angles around all of the vertices along the ath. Because of this constant of roortionality, it is guaranteed that c i, c j, and all folds in between can fold soothly all the way fro flat to fully folded, at least, if we ignore all vertices outside the ath. And, Fig. 6 A silified flasher, containing only diagonal, bend, and central olygon folds. Left: the full crease attern. Right: a close-u, with labeled sector angles. Journal of Mechaniss and Robotics JUNE 06, Vol. 8 / Downloaded Fro: htt://asedigitalcollection.ase.org/ on 03/07/06 Ters of Use: htt://

4 diagonal fold as a and b, resectively. If we are going to attet to create a gle-dof echanis ug the roerty described in Sec. 3, then this vertex ust be flat-foldable, with oosite angles suing to. This allows us colete the sector angle assignent around this vertex and, as well, around all of the other degree- vertices. Since, according to Eqs. () and (3), the ajor and inor folds around each of the degree- vertices are equal, this eans that the fold angles of all of the diagonal folds are equal to one another in agnitude, and the fold angles of all of the bend folds are equal to one another in agnitude. We denote the agnitude of the diagonal fold angles by c diag and the agnitude of the bend fold angles by c bend. All of the degree- vertices are siilar to one another (siilar in the geoetric sense), and because of that, the diagonal and bend fold angles at each degree- vertex are related to each other by Eq. (). So, u to now, the sector angles and fold angles are consistent with one another at every degree- vertex. Since each degree- vertex is a gle-dof echanis, the entire array of degree- vertices (aart fro those of the central olygon) ust be, itself, a gle-dof echanis if, that is, it is not locked by other interactions. We have not yet considered the degree- vertices around the central olygon. If we force each central olygon vertex to be develoable (its sector angles su to ), so that the crease attern is a flat sheet, that condition allows us solve for a a ¼ (7) While a is given by Eq. (7), angles b and d ay be chosen indeendently. But there is not colete freedo to choose. Looking at one of the vertices along the diagonal fold, if we choose b ¼ a, then the ajor folds at that vertex becoe collinear, resulting in the straight-ajor condition. As noted above, in a straight-ajor vertex, the ajor and inor folds becoe uncouled; such a vertex ust fold entirely fro flat to fully folded along the ajor crease before the inor crease can fold at all (if even ossible). So b ¼ a is forbidden if we want a gle-dof echanis where all the folds haen together. It is convenient to introduce a new angle, e, as the deviation fro straightness of the ajor folds, as illustrated in Fig. 6. We then have b ¼ a þ e (8) And because we now know the sector angles around these vertices, we can coute the fold angle ultilier between the ajor folds (diagonal folds) and inor folds (bend folds) at each vertex. We denote this fold angle ultilier by l db. It is given by l db ¼ cos a þ e a (9) If we choose any bend fold angle c bend, then every other bend fold has the fold angle 6c bend, with the sign deending on its ountain/valley assignent, and every diagonal fold has the fold angle 6c diag ¼ 6 tan l db tan c bend (0) with, again, the sign deterined fro the ountain/valley assignent.. Consistency in the Middle. Now we consider fold angles around the central olygon, and here a roble arises. Consider the black-dotted central olygon vertex in Fig. 6. It is clearly not flat-foldable, because oosite angles su to ð= þ =Þ, not. It is still a gle-dof echanis, and so if we choose a generic value of the diagonal fold angle, c diag, we can coute the fold angles of the two incident central olygon folds (which will, in general, be different fro one another in agnitude, as well as sign). Let us denote the dihedral angle of the valley fold of the central olygon by c c;v and that of the ountain fold by c c;. Ug the general exressions for oosite and adjacent dihedral angles fro the Aendix and the angles fro Fig. 6, we can solve for both c c;v and c c; in ters of c diag. The exressions are both rather large, and we oit the for brevity, but the iortant thing is this: they are quite different. The roble is that if we ove to the next central olygon vertex and coute the fold angles for the two incident central olygon folds, we will get the sae two values, but with oosite sign. Consistency fro one vertex to the next therefore requires that c c; ¼c c;v () and this is not the case; they have fundaentally different functional deendence uon c diag, no atter what values of a and d ight be chosen. So it is not ossible to find an assignent of fold angles around the central olygon consistent with the gle-dof echanis surrounding it. Well, then: How about if we sily cut out the central olygon entirely, so that its edges now becoe edges of the fold attern? Then, there would no longer be a consistency condition on the fold angles around each vertex of the central olygon because there are no central olygon fold angles to contend with. But there is still a consistency condition to consider. In fact, it just got ore colicated: by adding a hole, consistency ust be satisfied in both rotation angle (3DOF) and translation (three ore DOF). By design, all six conditions are satisfied at the unfolded and fully folded state. However, for a gle-dof echanis, both ust be satisfied across the full range of otion. If we cut out the central olygon, at each of its vertices, we have three fixed sector angles and two fold angles whose values are linked by the gle-dof echanis, and so we can solve for the angle between two adjacent sides of the central olygon what would have been the interior angle of the central olygon. We coute this angle by aking use of 3D rotation atrices. Define the usual rotation atrices about the x, y, and z axes as R x ð/ 0 cos / / A 0 0 / cos / cos / 0 / R y ð/ 0 0 A () 0 / 0 cos / cos / / 0 R z ð/ / cos / 0 A 0 0 Left-ultilying a vector by atrix R x ð/ Þ rotates the vector through angle / about the global x-axis, siilarly for R y ð/ Þ and R z ð/ Þ. If we have a local coordinate syste defined by a 3 3 atrix, then we can rotate that coordinate syste about its own local axes by right-ultilying by the transose of these sae atrices. We set u a coordinate syste centered on the central olygon vertex whose local x-axis runs along c c; and whose local x y lane contains folds c c; and c bend and describe this coordinate syste by soe atrix I. Then, we can find the direction vector for fold c c;v by successively rotating the coordinate syste about the local z-axis for each sector angle and about the local x-axis for each dihedral angle as we work our way around the vertex. The transfored coordinate syste is thus given by I 0 ¼ I R T z ð a d Þ RT x ð c bendþ ð aþ RT x ð c diagþ R T z ðdþ (3) R T z If the first coonent of I was the direction vector of fold c c;, then the first coonent of I 0 should be the direction vector of / Vol. 8, JUNE 06 Transactions of the ASME Downloaded Fro: htt://asedigitalcollection.ase.org/ on 03/07/06 Ters of Use: htt://

5 fold c c;v, and their dot roduct ust be the coe of the angle between the twofold in 3D. We denote that angle by a 3D. Its angle coe is cosa 3D ¼ cosa cosdcosða þ dþþ dða þ dþcosc bend cosc diag þða þ dþ dc bend c diag acosdcosc bend þadcosða þ dþcosc diag () Unfortunately, this quantity varies as the rest of the echanis changes its folded state (i.e., as c bend and c diag vary, which they do together). In the flat, unfolded state, the edges of the central olygon for a closed olygon. But, it turns out, for nearly all other artially folded states, the corner angles change, and so the central olygon no longer closes u. That eans we fail to satisfy translational consistency going around the central olygon. Perhas, a different choice of angles in the initial design would allow consistency, soe agic cobination? Unfortunately not, a 3D varies across the otion for every nonzero value of e. Perhas, instead, we could reove ore aterial around the center? Again, no. Consistency failure for a gle loo around the center eans that any loo that encloses the central olygon would fail in the sae way (just with uch greater algebraic colexity). And so, there is no consistent assignent of sector and fold angles that akes this attern an isoetric echanis. We are alost there: we can achieve consistency, isoetry, and gle- DOF otion at every other interior vertex, but the fold attern fails when we require consistency going around the loo of the central olygon. The solution, therefore, is to break the loo; we cut the attern fro the outside edge in to the center, so that there is no longer a loo condition around the central olygon to be satisfied. The cut ath can be anywhere fro the outside in, but for silicity, we will cut along one of the diagonal folds, as illustrated in Fig. 7 and then reove the central olygon. And that resolves the conflict! We now have a attern consisting entirely of degree- vertices in which the fold angles and sector angles at each interior vertex are utually consistent at every folded state, fro unfolded to fully flat (or as close to fully flat as we can get without self-intersection). We note that the idea of cutting a flasher fro edge to center is not without recedent. Tibbalds et al. introduced the notion of cutting ultile anels of a disklike for aart in order to roduce a gle-dof echanis [7] at the cost of introducing ultile sets of struts. If we create a gle-dof flasher, though, then we only need to create a gle cut, and we end u with a gle connected gle-dof echanis..3 Vertex Coordinates. So what does this attern look like when folded? For that, we need to coute the vertex coordinates in D and 3D. We denote each vertex of the crease attern by i;j, where i ¼ 0; indexes the vertices heading out along the diagonal folds fro the central olygon, and j ¼ 0; ; denotes the rotational osition around the origin, as illustrated in Fig. 8. Rotational indices wra around : i; ¼ i;0, and so forth. We assue for silicity that 0;0 ¼ ð; 0Þ. Define the D rotational atrix R ðþ k Then, we have that 0 cos k k k cos k C A (5) i;j ¼ R ðþ j i;0 (6) We further define angles / i;j and h i;j as the absolute angles (easured in a global coordinate syste as a rotation fro the x- axis) of the fold lines eanating fro i;j, as illustrated in Fig. 8. Fro consideration of the angles in Fig. 6, we have that h i;j ¼ ð d aþþ ie þ j= (7) / i;j ¼ ðdþþ ie þ j= (8) Now, we can coute the osition of i;j for i > 0 as the intersection of lines eanating fro lower-i vertices. We introduce the vector-valued function uðþ n ðcos n; nþ (9) the atrix deterinant detðx; yþ x y x y ¼ x y y x (0) and the line intersection function LINEINTða ; d ; a ; d Þ that returns the intersection between two lines eanating fro oints a and a with direction vectors d and d, given by detðða a Þ; d Þ LINEINTða ; d ; a ; d Þ ¼ a þ d detðd ; d Þ () Fig. 7 Cut lines on the sile flasher. We cut out the central olygon and cut in fro the edges along a diagonal fold. Fig. 8 Vertex and angle indexing in the sile flasher Journal of Mechaniss and Robotics JUNE 06, Vol. 8 / Downloaded Fro: htt://asedigitalcollection.ase.org/ on 03/07/06 Ters of Use: htt://

6 Then, iþ;0 ¼ LINEINT i;0 ; uðh i;0 Þ; i; ; u / i; () This relation, lus Eq. (6), allows us to recursively coute all of the oints of the crease attern, for as far out as we wish to go. An oen question is what to do for the outer boundary of the attern. For silicity, we have chosen to sily close the attern by connecting the oints n;j for soe value n. What about the folded for? Since we know the crease attern, and we know the angle of all of the folds, we can coute the folded for by rotating the facets of the crease attern relative to their neighbors about the known fold angles of their shared creases. This is done efficiently by constructing a sanning tree on the facets linked by their adjacency, then traverg the tree and coog rotations along the way. Ug these forulas, we have couted reresentative exales of gle-dof flasher echaniss. An exale is shown in Fig. 9 for araeters ¼, n ¼ 3, e ¼ 3 deg; d ¼ 3:5 deg, and c bend taking on various values fro 0 (fully flat) to (nearly) fully folded. The 3D lots verify the iossibility of achieving a closed central olygon. As can be seen in the first few iages, the central olygon oens u quite widely, then (in the iddle iage) it begins to curl u on itself. The axiu value of c bend in the figure was chosen to be the oint at which the central olygon closes back u on itself and, not incidentally, the originally cut edges of the ring eet u with each other once ore. It is ossible to take the attern all the way to a bend fold angle of c bend ¼ 80 deg, i.e., fully flatly folded, at which oint the entire attern lies in a coon lane. However, to get there, the anels of the attern ust intersect each other in nuerous laces and ways. No ractical alication would take this structure all the way to flatly folded (at least none that we can envision at resent). The stowed state for this structure stos when the central olygon recloses and the anels have soe sall dihedral angle between the. Such a odel ay be used in a thickening algorith, such as that of Tachi [9], which requires slightly angularly searated anels. There are two noticeably different otions in going fro the flat to the curled-u state. First, for sall c bend, ost of the otion haens on the diagonal folds, and as they fold u, the flat disk fors into a slightly curved vertical stack of layers. There then coes a oint where the diagonal folds are ostly folded, and the reaining otion coes fro the bend folds curling the layers around until they eet u again. So we can equate cos a 3D fro Eq. () and cos a 0 fro Eq. () and solve for the bend angle (or equivalently, the diagonal angle) that satisfies the equality. To kee the algebra tractable, we introduce the Weirstrauss substitution x tan c bend which gives rise to the following silifying substitutions: c bend ¼ x þ x x cos c bend ¼ þ x c diag ¼ ð l dbxþ þ ðl db xþ cos c diag ¼ ð l dbxþ þ ðl db xþ (5) (6) Substituting these into Eq. () along with taking d ¼ d lanar ¼ = e=; l db ¼ cos a þ e= a, and a ¼ = = gives cos a 3D ¼ x ðx þ Þ ðþþ e x cosðþ e þ cos x þ ðx þ x þ ÞcosðÞ e ðx þ Þ x cos e þ x cosðþþ e (7) This angle coe is araeterized on the variable x, which is the transfored version of c bend. We would like to know the value of x (and thus, by extension, c bend ) that akes this value equal. The Central Region. In general, the vertices of the central olygon are nonlanar in the artially folded state. However, we have found that a articular value of d gives rise to a central olygon that is lanar through the full range of otion: d lanar ¼ e (3) We chose d ¼ d lanar in Fig. 9. As noted already, as the echanis roceeds away fro the flattened state, the central olygon oens u, and then it recloses. The fully stowed state would be that where the central olygon has closed to its original state and the edges of the cut have coe back together. We would like to know what bend angle c bend gives rise to this state: this would define the full range of useful otion of the echanis. Recall that Eq. () gave the coe of the angle between folds c c; and c c;v, which was a 3D. The echanis has reclosed when that angle takes on the value of the interior angle of the original lanar central olygon, whose angle coe is cos a 0 ¼ cos ¼cos () Fig. 9 A gle-dof flasher for various values of c bend.fro uer left to lower right: c bend 5 0deg; 3deg; 5deg; 0 deg; 0 deg; 30 deg; 0 deg; 65 deg; and 87 deg. Note that the scale varies fro one subfigure to the next / Vol. 8, JUNE 06 Transactions of the ASME Downloaded Fro: htt://asedigitalcollection.ase.org/ on 03/07/06 Ters of Use: htt://

7 Fig. 0 A view fro the to of the flasher of Fig. 9 for c bend 5 80 deg, showing collisions with the inner layers that of the lanar central olygon. Setting the two values equal and solving for x gives two solutions: first, x ¼ 0, which is the unfolded state. The second solution is the desired fully folded state. We find sec x ¼ ffiffi l db cos e cos l db l db cos e e þ cos e cos = (8) which sets the axiu bend angle to be c bend;ax ¼ tan sec e s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 e 5 e e þ Oðe 3 Þ (9) As we fold the attern beyond c bend;ax, the attern selfintersects. There is also a sall aount of self-intersection even before closure. This can be seen in Fig. 0, which is a view looking down the barrel of the tubular for just before the central olygon closes. So, this is otentially a roble with ractical alications of this structure. However, this collision haens because of where we chose our cut, which results in one ortion of the attern wraing around another as we aroach the stowed state. In the stowed state, the cut edges realign with one another. We can therefore begin with the curled-u state and ake our cut soewhere else in a way that revents this wra-around issue. Two ossible alternate locations for a cut are illustrated in Fig.. Judicious choice of cut line can otentially eliinate selfintersection throughout the full range of otion, fro stowed to Fig. Crease attern with the addition of a air of reverse folds along one of the diagonals fully deloyed. We will show one such choice and its effects in Sec. 6. Another issue getting in the way of ractical alication is the fact that the stowed for is long and tubular, which arises fro the fact that this is the silest ossible flasher structure. That length can be reduced by choog a larger rotational order, but that strategy gives sall slivers of triangles near the inner ri, which are undesirable in alications. Scheel [] and subsequent investigators show that one can reduce the height of a flasher by incororating what in the origai world are called reverse folds into the attern; such were included in the constructive algoriths of Guest and Pellegrino [3] and Zirbel et al. [8]. We can incororate such folds into this flasher while reserving the gle-dof otion, as we show now. 5 Reduced-Height Flashers We now consider adding a air of reverse folds that eanates fro soe oint along one of the diagonal folds, as illustrated in Fig.. We denote by g the angle that the reverse fold akes relative to the diagonal fold. In order to avoid disturbing the gle-dof echanis that already exists, we will assue that the diagonal and bend folds are unchanged, excet for the addition of vertices and selective inversion of the fold tye (sign change of the fold angle). The nae reverse fold coes fro the origai world; such a set of folds inverts the arity of all of the diagonal and bend folds that lie within the V of the reverse fold. What freedo do we have in the choice of g? If we are to leave the agnitude of the fold angles unchanged along the diagonal and bend folds, then all of the new vertices ust be flat-foldable. That eans that both sides of the base of the V (where the reverse folds hit the diagonal) ust ake the sae angle g with the diagonal fold, as shown in Fig.. It also eans that each of the vertices where a reverse fold eets a bend fold ust be geoetrically siilar to one another, as well as flat-foldable. Considering what haens at two successive vertices along a reverse fold reveals that there is only one ossible set of sector angles at those vertices that ake the all geoetrically siilar, naely, sector angles of ð= e=þ and ð= þ e=þ (two of each), as illustrated in Fig.. With the four sector angles at each reverse/bend vertex assigned, the value of g is fully defined and can be worked out fro the interior angles of the shaded triangle in Fig.. We find that g ¼ a e ¼ e ¼ d lanar (30) Fig. A to view of the flasher with 5, n 5 3, 5 3 deg; d 5 3:5 deg, and c bend 5 87 deg. Lines A and B indicate ossible alternate cut lines. This is a nice result; it tells us that if we choose d ¼ d lanar and lace the ti of the reverse fold V along the first diagonal fold, the left side of the V will be arallel to the edge of the central Journal of Mechaniss and Robotics JUNE 06, Vol. 8 / Downloaded Fro: htt://asedigitalcollection.ase.org/ on 03/07/06 Ters of Use: htt://

8 olygon. If we lace the ti of the reverse fold V on a vertex of the central olygon, then the left side will be coincident with the edge of the central olygon. There is one thing still to check, however. We have three tyes of folds with distinct fold angle agnitudes: diagonal (c diag ), bend (c bend ), and now reverse (c rvrs ). We also have three tyes of interior vertices that enforce roortionality between the half-angletangents of the fold angles at each vertex. Each vertex can be labeled by the two tyes of fold incident on the vertex: diagonal/ bend (which we have already et, characterized by fold angle ultilier l db ), and now two new ones: reverse/bend, which will be characterized by a fold angle ultilier l rb, and reverse/diagonal, with a fold angle ultilier l rd. The shaded triangle in Fig. has one of each tye of vertex. There is a self-consistency condition that ust be satisfied going around this triangle. In articular, we ust have l rb ¼ l rd l db (3) For a general triangle of three flat-foldable vertices, this relationshi is not guaranteed to hold. What about this articular case? The value of l db was given by Eq. (9). For the other two, we find that l rb ¼ csc e l rd ¼ csc a þ e (3) (33) Substituting these into Eq. (3) reveals that the latter is satisfied for all values of a and e. So, no atter what sile flasher we start with, we can add one or ore reverse folds anywhere along the diagonal folds, and the resulting attern is guaranteed to be a gle-dof echanis. Equation (9) gave the value of c bend at the axially folded (stowed) state, c bend;ax. We can substitute that value back into Eq. (0), to find c diag at the axially folded state s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 c diag;ax ¼ tan csc e e 5 (3) And in the sae vein, the reverse fold angle, c rvrs, at the axially folded state will be Fig. A gle-dof reverse-folded flasher for various values of c bend. Fro uer left to lower right: c bend 5 0 deg; 3 deg; 5 deg; 0 deg; 0 deg; 30 deg; 0 deg; 65 deg; and 87 deg. c rvrs;ax ¼ tan csc e sec e sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 e 5 (35) The sacing between successive reverse folds can be chosen arbitrarily. The farther aart successive reverse folds are laced, the taller the resulting echanis. When lacing the reverse folds, one can think of each reverse fold as sliding along the diagonal, creating triangular and quadrilateral anels along each diagonal; each reverse fold breaks a diagonal fold into two segents whose lengths deend uon the osition of the ti of the V of the reverse fold. There is a secial case, where the ti of the V of the reverse fold coincides with an existing vertex along the diagonal. This choice gives rise to a articularly elegant crease attern consisting of triangular facets along the diagonals and near-rectangular anels everywhere else. It also creates degree-6 vertices along the diagonal, each created by the effective erging of two degree- vertices. This erging otentially increases the nuber of DOF of the echanis an issue we will coe back to resently but it does not alter the consistency between the values of c bend, c diag, and c rvrs given above. We now coute D and 3D reresentations of this structure. We introduce trily subscrited oints for the vertices of the reverse folds, as illustrated in Fig. 3. Fig. 3 A ortion of the crease attern of a reverse-folded flasher, with reverse folds eanating fro each of the diagonal vertices Fig. 5 A view fro the to of the flasher of Fig. for c bend 5 80 deg, showing collisions with the inner layers / Vol. 8, JUNE 06 Transactions of the ASME Downloaded Fro: htt://asedigitalcollection.ase.org/ on 03/07/06 Ters of Use: htt://

9 Fig. 8 A rigidly foldable hexagonal flasher near the endoints of the otion. Left: cbend 5 0:5 deg. Right: cbend 5 56 deg (different scale). ri;j;k ¼ hi;j g þ ke Fig. 6 A rigidly foldable flasher for various values of cbend. Fro uer left to lower right: cbend 5 0 deg; 3 deg; 5 deg; 0 deg; 0 deg; 30 deg; 0 deg; 65 deg; and 87 deg. In each rotational section, we define ri;j;k for vertices on the left side of the diagonal fold and si;j;k for vertices on the right side (as viewed fro the central olygon). For vertices on the diagonal folds, we define both ri;j;0 ¼ si;j;0 i;j (36) where fi;j g are the original vertices of the attern as defined in Sec.. As we ove out fro ri;j;0 along a reverse fold, the k index of ri;j;k increents each tie we hit a bend fold, and siilarly with si;j;k. We note that this gives ultile naes to the sae oint; not only does ri;j;0 ¼ si;j;0 but also ri;j; ¼ ri;j ;0 and s0;j;k ¼ r0;j ;kþ. This requires a bit of care in the bookkeeing of distinct vertices, but otherwise causes no robles. We further define qi;j;k as the absolute angle of the reverse fold eanating outward fro ri;j;k, and ri;j;k as the absolute angle of the reverse fold eanating outward fro si;j;k (like hi;j and /i;j, easured as a rotation relative to the x-axis in a global coordinate syste). With these definitions, the vertex coordinates and angles are as follows: qi;j;k ¼ hi;j þ g þ ke ri;j;kþ ¼ LINEINT ri;j;k ; uðqi;j;k Þ; iþk;jþ ; u /iþk;jþ (39) si;j;kþ ¼ LINEINT si;j;k ; uðri;j;k Þ; iþkþ;j ; u /iþkþ;j (0) where hi;j and /i;j were given by Eq. (7). The outerost vertices are a secial case, and their osition deends on how we choose to finish the attern. For the tubular flasher, we sily connected oints fn;j g for soe n. Doing that with this reverse-folded case will slice soe of the outerost anels, giving traezoidal and/or triangular facets. For silicity and elegance, we have chosen to terinate the attern along what would be bend folds, which gives roughly rectangular anels for all anels not along the diagonal folds. We have also chosen the cut line along one of the reverse folds (secifically, along the s0;0;k chain of folds), rather than along a diagonal, as in the revious exale. Figure shows this new flasher design fro unfolded to fully folded for the sae araeters as Fig. 9, with ¼, n ¼ 3, and e ¼ 3 deg. Once again, we have a rigidly foldable gle-dof echanis. (37) Fig. 7 A rigidly foldable flasher near the endoints of the otion. Left: cbend 5 deg. Right: cbend 5 85 deg (different scale). Journal of Mechaniss and Robotics (38) Fig. 9 A flasher with reverse folds only along the fs0;j;k g and fr0;j;k g lines with fivefold rotational syetry. To left: crease attern. To right: to view of the folded for. Botto left: ersective view of the flasher with equally saced cut lanes. Botto right: side view of sae. JUNE 06, Vol. 8 / Downloaded Fro: htt://asedigitalcollection.ase.org/ on 03/07/06 Ters of Use: htt://

10 6 Layer Collision Avoidance There is still the question of layer collisions. For the siulation of Fig., we have cut the crease attern along a reverse fold, rather than along a diagonal fold as in the siler flasher. This cut still gives a layer collision as we aroach the folded state, as can be seen in Fig. 5. The resence of layer collisions deends uon the osition and orientation of the cut. A straight cut in the flat state either along a diagonal fold, as in the sile flasher, or along a reverse fold, as in the revious exale assues a siral for in the 3D state, and it is the overla of that siral that gives rise to the collision in the nearly stowed state. We can avoid such a collision by choog a cut that is nearly straight in the stowed state, as was noted earlier in Fig.. Such a cut will give rise to a siral cut in the flat state, but with aroriate choice of cut, can give rise to a collision-free otion as the fold angle aroaches c bend;ax. In Fig. 6, we show a different cut osition with otherwise the sae araeters as Fig.. As before, this attern oves rigidly between the deloyed (flat) and stowed (cylindrical) shae. By aking the cut follow along a bend, we get the cut edges to ore cleanly line u in the stowed state. As c! c bend;ax, the cut edges realign and for a erfect butt joint. However, this articular introduces a very slight layer intersection near the deloyed state. Figure 7 shows to views in the nearly deloyed and nearly stowed states. This collision is slight and only results in a slight overla; colete eliination of the intersection could be achieved with a slight reoval of aterial fro a few of the anels. Contributing factors to the resence of collisions in the nearstowed state are the shar corners in the bend that arises fro low rotational order. In higher-rotational order, the residual overla is reduced. Figure 8 shows a hexagonal flasher with the sae cut attern; in this attern, there is still a very slight overla in the near-stowed state, but it is extreely sall, and only a tiny aount of aterial would need to be reoved to eliinate the overla. It sees likely that an aroriately chosen ath for the cut could entirely eliinate layer collisions; we leave that exloration as a toic for future work. reaining facets will also be claed to the gle-dof otion, which haens in this echanis. Another otential drawback of locating reverse fold tis at existing bend vertices along the diagonal is that it gives each siral ring a slightly different height, with inner rings being shorter than outer ones. (This behavior is ost clearly visible in the last subfigure of Fig..) This results in a less than otial acking in the stowed state, ce the sace above the shorter rings is essentially wasted. It would be ost efficient if each anel fully filled the vertical slice of sace allocated to it by its vertical rojection. As we have already noted, the ti of any given V of reverse folds ay be ositioned anywhere along the diagonal fold while aintaining the kineatics of the echanis, and if no reverse fold tis coincide with other vertices along the diagonal, then all of the vertices will be of degree- and all interior vertices and their incident facets will collectively exhibit gle-dof otion. The freedo to lace reverse folds anywhere along the diagonals allows us to ake all of the siral rings have exactly the sae height in the stowed state, and that height can be indeendently selected as a design variable in the construction of the crease attern. We constructed both of the flashers resented thus far by selecting angles that enforce gle-dof rigid foldability at all degree- vertices. It turns out that this choice results in a fortuitous behavior: the vertices of each near-linear chain of reverse folds lie in a coon lane throughout the full range of otion, and ce the central olygon itself is lanar for g ¼ d lanar ¼ = e=, that eans that any grous of vertices lie in a coon lane throughout the full range of otion. In articular, the sets of vertices fs 0;j;k g and fr 0;j;k g all lie in the sae lane as the central olygon throughout the full range of otion. If we construct a flasher with those as the only reverse folds, we obtain a structure like that shown in Fig. 9, in which the botto of the flasher lies in a horizontal lane, and each of the nearvertical facets akes the sae angle with that lane, an angle of c bend =. We could then envision cutting this conical structure in the stowed state with a series of horizontal lanes saced evenly in z; the intersection of each lane with the basic flasher thereby defines a set of reverse folds in the crease attern, and the folded for that incororates those reverse folds would be the result of 7 Constant-Height Flashers The choice of initiating the reverse folds at existing vertices has a certain elegance and cleanliness to it, and it akes all anels either quadrilaterals or triangles. This choice has an iortant side effect: it creates several degree-6 vertices. The nuber of DOF of a degree-n vertex is n 3, so a degree-6 vertex has three, not one, DOF. This could, in rincile, give the echanis extra DOF. However, those degree-6 vertices do not exist in isolation. The facets surrounding each degree-6 vertex are theselves connected to degree- vertices, and those vertices are constrained to gle- DOF aths in hase sace. If enough facets surrounding a highdegree vertex are constrained to gle-dof aths, then the Fig. 0 A constant-height flasher for 5 6, n r 5 9, e 5 3 deg, and h 5.8. Left: crease attern. Right: folded for. In this flasher, the cut runs along the fs 0;0;k g chain of creases. Fig. A wood veneer lainate fabricated constant-height flasher for 5 6, n r 5 9, e 5 3 deg, and h 5.8. To left: stowed state, to view. To right: stowed state, side view. Botto left: artially folded. Botto right: deloyed / Vol. 8, JUNE 06 Transactions of the ASME Downloaded Fro: htt://asedigitalcollection.ase.org/ on 03/07/06 Ters of Use: htt://

11 successive reflections of the folded for in the stack of lanes, as illustrated in Fig. 9. Suose we start at oint 0;0 ¼ ð; 0Þ and travel outward along the diagonal fold a distance d, where d is easured along the chain of diagonal folds (aking bends at each of the oints f i;0 g). Then in the fully folded (stowed) for, i.e., when c bend ¼ c bend;ax, the height z of the oint at distance d is given by s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi zd ð Þ ¼ d sec e e () This can be turned around to deterine the ositions of the oints of the V of each reverse fold as a function of the desired target height of the flasher. If h is the desired height, then each successive reverse fold along the jth diagonal should be initiated at a distance s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e d k ¼ h cos csc csc e () fro the oint 0;j for k ¼ ; ; n k, with the reverse folds and vertices constructed along the diagonals and bend folds in the sae way we constructed oints fs 0;j;k g and fr 0;j;k g. Now, the nuber of vertical divisions n k is decouled fro the nuber of anels n r in a gle siral. An exale of such a constant-height flasher with hexagonal rotational syetry is shown in Fig. 0. Choog a constant height for all the siral reverse folds gives the ost efficient ossible stowed for by aking full use of the vertical airsace above or below each tilted anel. In addition, dislacing the reverse folds fro the existing vertices along the diagonals, as in Fig. 0, ensures that all of the interior vertices are degree- vertices, aking it easier to obtain a gle-dof echanis even after adding a radial cut. 8 Exale Fabrication To verify the analysis resented above, we have fabricated several sall test ieces, ug laser-scored wood veneer lainate for ease of fabrication. The lainate used was Sanfoot wall covering, anufactured by Hokusan Cororation, which consists of a olyer-infused wood veneer sandwiched with layers of foil and olyer, with a total thickness of 0.3. We then laser-scored creases by scoring through the uer wood olyer layers, ug the foil layer as a blocking layer to revent cutting through the lower olyer layers, thereby creating flexible olyer hinges fro the layers below the foil. The wood and olyer lainate creates stiff, nonstretchable/nonwrinklable facets. The scored lines create flexible hinges with little excess lay in the hinges owing to the narrow width of the laser cuts (0. ), and the overall thinness of the aterial eans that we can get close to zero-thickness theoretical behavior without having to ileent ore colex thickness-coensating fabricational techniques (e.g., Refs. [9] and [0]). We tyically laser-scored colete (uncut) flasher atterns, then anually introduced the radial cut by cutting the attern with a shar knife. This allowed us to quickly exaine the effects of different radial cut atterns without having to build the rograatically into the araeterized odel that generated the crease attern. Figure shows hotograhs of one such flasher, ug the attern fro Fig. 0, in the flat (deloyed), artially folded, and fully stowed states. Manual aniulation of these test odels verifies the gle- DOF otion of the theoretical behavior, which is the case for arbitrarily stiff anels and arbitrarily free hinges. Real aterials, of course, are not erfectly stiff and/or free, and in the thin wood veneer odels, soe residual flexing of anels gives rise to soe additional bending odes, in uch the sae way that a gle- DOF Miura-ori ossesses twisting odes that rely on flexing of the anels. With suitable choice of fabrication technologies, such arasitic odes can undoubtedly be iniized. However, there is an inherent roble in echaniss based on flat-foldable degree- vertices such as this one (and the venerable Miura-ori [], aong others). In general, the greatest stacking efficiencies arise when the sector angles are close to 90 deg. Exactly 90 deg is a gular condition for which the ajor and inor fold angles becoe decouled. As the vertex angles ove away fro 90 deg, though, there grow offsets between the stacked layers, lowering the acking efficiency of the stowed state. Thus, the ost efficient structures will generally include any vertices with sector angles close to, but not quite, 90 deg. There is an inherent tradeoff, though. As one gets closer to the 90 deg gular condition, the geoetric advantage between the ajor and inor fold angles becoes huge and variable across the range of otion. As noted in Eq. (6), it varies between l and =l across the range of otion, and as we aroach the gular condition, l diverges. So, close to the gular state, at soe ositions, soe of the angles are very weakly couled to the rest of the echanis and so can readily flex with sall deforations of, e.g., anel bending or lay in the hinges. As one oves away fro the gular state, ajor and inor angles are ore strongly couled, but offsets between layers are greater, and stacking efficiency is lessened. In our structure, Fig. shows that the araeter e is a easure of how far one is fro the gular state for the axial/bend and reverse/bend vertices. For close acking, we would want e to be sall, but that will result in large geoetric advantages between the ajor and inor crosg creases. The way this relationshi layed out in the fabricated sales was that while in theory, the slightly olygonal curvature of the axial creases should have been sufficient to force gle-dof otion, in ractice, they felt floy due to the weak geoetric couling between the crossed creases. Evaluation of the tradeoff between stacking efficiency and coliance, ediated by the angle araeter e, requires consideration of any ore factors: residual flexibility of anels, lay in hinges, and fabricational inaccuracies, to nae a few. Exloration of such tradeoffs is soething we defer to the future. We note, however, that soe secial roerties of the constant-height flashers suggest several strategies for adding stiffness in ways that reduce or revent arasitic flexing odes, as we will ention in our conclusions. 9 Conclusions and Discussion In conclusion, we have resented a faily of flasher echaniss that ossess rigid foldability, thereby aking the suitable for the ileentation of deloyable echaniss with rigid hinged anels. Many of these echaniss exhibit gle-dof otion, which is another desirable trait. We resented constructive algoriths for the D crease atterns and 3D folded fors and the relevant fold angles of all creases. We resented secific exales with fourfold and sixfold rotational syetry, but the analysis is fully araeterized on the rotational order, so that other rotational orders ay be readily siilarly constructed. While it is ossible to design all of the vertices of a flasher to exhibit kineatic gle-dof otion coatible with the dynaic fold angles of all other interior vertices, it is not ossible to achieve such consistency around the central olygon as well. Consistency ay be restored by cutting the attern radially and reoving the central olygon. The result is a true gle-dof kineatic echanis, however, at the cost of breaking the radial syetry and otentially colicating its ileentation. A coon usage configuration for a flasher-based deloyable is to wra the echanis around a central ayload in the stowed state (e.g., Refs. [3] and [8]), and then deloy it to a disk around the ayload in the deloyed state. The cut flasher still suorts Journal of Mechaniss and Robotics JUNE 06, Vol. 8 / Downloaded Fro: htt://asedigitalcollection.ase.org/ on 03/07/06 Ters of Use: htt://

12 this ode of deloyent, as the stowed and deloyed states are identical to those of an uncut flasher. However, the cut flasher would only be solidly fixed along a gle edge of the central olygon, as the other edges will ove away fro the central hub, then coe back into contact over the course of the deloyent otion. This would entail greater echanical colexity, but would allow such flashers to be used in alications where the thickness and stiffness of the anels ust be accoodated in design, e.g., solar arrays. We seculate that, instead of introducing a radial cut, a gle- DOF echanis ay be achievable by further triangulation of soe or all of the quadrilateral anels with further hinges. Indeed, it is clear fro observation that flashers created fro flexible aterials and in hinges incororate anel flexing as art of their deloyent otion. Whether such triangulation can be done just enough to create a gle DOF is an interesting question and a otential avenue for further exloration. We also showed how to construct constant-height variants of gle-dof rigidly foldable flashers. The constant-height roerty rovides a ore efficient acking of facet anels than flashers in which the reverse fold vertices are aligned with bend vertices; it also allows one to avoid degree-6 vertices that could conceivably give rise to unwanted flexible otions. The constant-height variants also enable another intriguing ossibility. As noted above, in both the flat and stowed fors, all of the vertices of the uer reverse folds lie in the sae horizontal lane, and all of the vertices of the botto reverse folds, including the vertices of the central olygon, also lie in the sae horizontal lane. It is readily shown that the colanarity of these sets of vertices holds throughout the full range of otion fro flat to fully stowed. This roerty, in turn, suggests several interesting avenues for building uon constant-height flashers to create ore functional or robust echaniss. For exale, one could connect adjacent sets of anels with scissor-joints to add stiffness against out-of-lane otions, thereby circuventing the unwanted coliance of near-gular vertices noted in Sec. 8. We note, too, that the lanarity of the to and botto surfaces across the range of otion also suorts the concet of stacking flashers to create cellular fors that have overall rotational deloyent otions, in analogy with the translational deloyent of Tachi-Miura cellular fors [8]. More broadly, we have shown an aroach for constructing large-scale gle-dof echaniss by aking use of the unique roerties of flat-foldable degree- vertices secifically, the roortionality relationshi of Eq. (). This behavior was noted by Tachi [6], who deonstrated flat-foldable generalized Miuraori. As shown here, this roerty can be used to construct large gle-dof networks that rovide useful functionality even when the echanis is never folded all the way (or even close) to the fully flat state; individual vertex flat-foldability is used sily as a eans to attain constancy of tanð=þc i = tanð=þc j for every air fi; jg of vertices in the network. We exect that this roerty can be used to construct any ore colex, gle-dof, origai-based echaniss in the future. We build off of the work of Huffan [3], following his aroach and ug several of his results. We consider the trace of the vertex on the Gaussian shere, as shown in Fig. (analogous to Fig. 3 of Ref. [3]). The Gaussian shere is a dual-sace reresentation of the vertex and is the ath traced out on a unit shere by the surface noral vector as it travels around the vertex. The flat sectors of the vertex, being lanar, a to oints on the trace. We denote by fa i g the oints on the trace, which corresond to sectors with sector angles fa i g. Since the noral vector swees out an arc at each fold, the dihedral angles (fold angles) of the vertex a to arcs on the Gaussian shere. As noted by Huffan, the trace of a develoable degree- vertex always takes the general for of a (not necessarily syetric) bowtie, and the solid angles subtended by the two triangles of the bow are equal (this follows directly fro develoability of the vertex). Huffan derived relations between oosite dihedral angles for a general vertex (Eqs. (a) (c) in Ref. [3]). Ug our notation, they are c and ¼ a 3 a a a c c ¼ a a 3 c a a 3 (A) (A) For a flat-foldable vertex (a þ a 3 ¼ and a þ a ¼ ) with crease assignent as shown in Fig., these silify to c ¼ c ; c ¼c 3 (A3) Huffan also derived a relationshi between adjacent dihedral angles (Eq. (3) in Ref. [3]); here, we derive one that is soewhat siler. We consider the four triangles on the Gaussian shere ða ; a ; a Þ; ða ; a 3 ; a Þ; ða ; a ; a 3 Þ; and ða ; a ; a 3 Þ each coosed of two of the saller lettered triangles in Fig.. Because the first two of the four share the triangle F and the two halves of the bow-tie of the trace have equal area, the first air of triangles has equal area, as do the second air area ða ; a ; a Þ ¼ area ð a ; a 3 ; a Þ ¼ E þ F (A) area ða ; a ; a 3 Þ ¼ area ð a ; a ; a 3 Þ ¼ E þ G (A5) Ug a angent forula for triangle areas fro Ref. [3] (see unarked equation receding Eq. () in Ref. [3]), we can establish an equality for each air of triangles Acknowledgent This aer is based on the work suorted by the National Science Foundation and the Air Force Office of Scientific Research through NSF Grant No. EFRI-ODISSEI-07 at the Brigha Young University. Aendix Sector-Dihedral Relations Here, we derive new forulas relating the dihedral angles of a general degree- vertex to the values of the surrounding sector angles. A ortion of this derivation is also resented in Ref. [9]. Fig. Scheatic of a degree- vertex. (a) The vertex ebedded in a unit shere. Dashed lines are valley folds, and dotted lines are ountain folds. (b) The trace of the vertex on the Gaussian shere. Since c 3 is a ountain fold, its sign is negative / Vol. 8, JUNE 06 Transactions of the ASME Downloaded Fro: htt://asedigitalcollection.ase.org/ on 03/07/06 Ters of Use: htt://