CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] The present application claims the benefit of U.S. Provisional Patent Application Nos. 63/315,047, filed on February 28, 2022, and 63/315,072, filed on February 28, 2022, the disclosures of each of which are hereby incorporated by reference herein in their entirety.

BACKGROUND

[0002] Deployable shelters are used for a variety of applications, including disaster relief and housing. The ability to transport and rapidly erect such shelters is an important design consideration. To date, available technologies suffer from a variety of deficits including lack of desired structural integrity, excessive complexity of articulating members, difficulty manufacturing components, limited configuration options, an inability to accommodate thick walls, and an inability to fold flat into a compact shape when not in use.

SUMMARY

[0003] Deployable shelters are used for a variety of applications, including disaster relief and temporary housing. The ability to transport and rapidly erect such shelters is an important design consideration. Foldable physical structures inspired by geometric characteristics of origami are disclosed herein. Such structures are foldable. However, unlike origami, embodiments of physical structures disclosed herein are flat when folded and then expand when unfolded. Expanded structures at least partially enclose a volume. The volume can be sufficiently large to be habitable, such that a structure can be used as a shelter. To exhibit this expansion during unfolding, a physical structure may include a plurality of walls connected together, for example with one or more hinges. More complex folds and complex structure shapes can be achieved by using wall(s) that include more than one panel. A wall itself may be foldable by folding its multiple panels together. In some embodiments, two panels form a hinge line at their connecting boundary (e.g., edges) such that the panels can be folded (and unfolded) by rotation about the hinge line.

[0004] Complex structures that include multiple walls and panels present a particular challenge to foldability. Certain combinations of shape and dimension would cause one or more panels to impinge on one or more other panels during folding or unfolding (presuming the panels are non-deformable). A structure that can fold and unfold between a flat-folded state and an expanded state without impinging on itself is kinematically compatible. Physical structures having kinematic compatibility are disclosed herein. Different panel and wall designs (e.g., including different shapes, numbers, and relative sizes) are disclosed that together produce physical structures having kinematic compatibility. Connectivity of walls and panels may be a factor considered in achieving kinematic compatibility.

[0005] Additionally, the walls and panels with sufficient thickness (e g., for use as a shelter) present additional challenges to achieving kinematic compatibility. Unlike folding of thin sheets, as in traditional origami, partial and complete folding of panels with finite (non-zero) thickness requires careful consideration of each hinge line relative to the inner and outer surfaces of the connected panels. Non-zero thickness panels require clearance when rotating to not impinge at edge(s). Different techniques that can be used to avoid such impinging are disclosed, including offset crease technique, offset panel technique, and hinge shift technique. Such techniques may be combined in a particular physical structure.

[0006] Deployable shelters are used for a variety of applications, including disaster relief and temporary housing. The ability to transport and rapidly expand such shelters is an important design consideration. Results presented herein show that through careful modeling and analysis of compound folding patterns, it is possible to arrive at origami-inspired foldable physical structures with sufficiently thick walls that can be moved as a single-degree-of-freedom system. That is, force applied to a single panel may be sufficient to cause the entire physical structure to move to its expanded state. Such force may be applied by pushing or pulling (e.g., lifting) on the single panel. In some embodiments, single degree-of-freedom systems are preferable to simplify physical structure expansion to a usable state. In some embodiments, habitable physical structures moveable with a single degree of freedom can be folded into a flat-folded state or unfolded into an expanded state in no more than 5 minutes (e.g., no more than 2 minutes).

Within certain examples of a single degree-of-freedom structure, required expansion load is found to vary during the expansion process and depends significantly on the location of the applied load. The results obtained from analytical analysis, and confirmed by simulation, can inform design decisions affecting size, geometric shape, and expansion load requirements for physical structures (e.g., shelters) disclosed herein. Mechanical assistance components, such as torsion springs, may be used to assist in unfolding to an expanded state, thereby reducing the load that need be applied to unfold a physical structure.

[0007] Locking mechanisms that stabilize a physical structure in an expanded state may also be used. Such locking mechanisms can prevent a structure from collapsing when unfolded (expanded). Because, in some embodiments, a physical structure may move with a single degree of freedom, a small number locking mechanisms (e.g., one locking mechanism) may be sufficient to prevent collapse towards a flat-folded state.

[0008] In some aspects, the present disclosure is directed to a physical structure having kinematic compatibility, the structure including walls connected together such that the walls are moveable between a flat-folded state and an expanded state. Each of at least two of the walls may include two or more panels. The walls may at least partially enclose a volume when in the expanded state

[0009] In certain embodiments, the structure includes one or more actuators, wherein the walls are moveable between the folded state and the expanded state by the one or more actuators. In certain embodiments, the one or more actuators is a single actuator. In certain embodiments, the one or more actuators includes two actuators disposed at different locations in the structure (e.g., corresponding to different hinge lines) (e.g., to apply force to different panels).

[0010] In certain embodiments, the structure is moveable between the folded state and the expanded state by moving with a single degree of freedom (e.g., by applying an expansion force to only one panel of one of the walls at a time). In certain embodiments, the structure includes a roof and moving with the single degree of freedom does not close the roof. In certain embodiments, the structure includes a roof that is one of the walls and moving with the single degree of freedom closes the roof.

[0011] In certain embodiments, the walls include a front wall and an opposing back wall. The front wall of the walls may include panels disposed and connected such that the panels fold and unfold away from the back wall when moving between the folded state and the expanded state. In certain embodiments, the panels are disposed and connected to have a fold with a vertex having a degree of four or more. In certain embodiments, the back wall includes only one panel. In certain embodiments, at least one of the walls includes panels that are disposed and connected so that the wall folds with both one or more mountain folds and one or more valley folds. [0012] In certain embodiments, the at least two walls includes a first wall and an opposing and identically foldable second wall (e.g., side walls). In certain embodiments, the first wall has a top edge and includes a hinge line along which the first wall folds that intersects the top edge at a non-corner location. In certain embodiments, the first wall has a bottom edge and a front edge adjacent to the bottom edge and the hinge line also intersects a corner of the bottom edge away from the front edge and the non-corner location is at least 0.2*H away from the front edge, where H is a height of the first wall. In certain embodiments, the two or more panels of the first wall includes a trapezoidal panel and a triangular panel. In certain embodiments, the trapezoidal panel has two parallel edges and a shorter one of the two parallel edges has a length of at least 0.2*H, where H is a height of the first wall. In certain embodiments, the first wall includes a triangular panel that forms less than half of the first wall. In certain embodiments, the at least two walls further includes a third wall that is adjacent to the first wall and to the second wall and the third wall includes three panels.

[0013] In certain embodiments, the two or more panels are connected with an offset crease method, an offset panel technique, a hinge shift technique, or a combination thereof. [0014] In certain embodiments, the two or more panels includes a panel having a non- uniform thickness (e.g., partial cutout) that corresponds to a different one of the two or more panels (e.g., such that the two or more panels are nested in the flat-folded state). In certain embodiments, the panels are non-deformable.

[0015] In certain embodiments, at least one of the two or more panels (e.g., at least one panel of each of the at least two walls) includes a tab extending along a hinge line and another panel (e.g., of the two or more panels) is connected to the at least one of the two or more panels at the tab.

[0016] In certain embodiments, the structure includes a roof. The roof may include one or more panels. In certain embodiments, the roof is one of the walls. In certain embodiments, the roof is a flat roof, such as an extended height roof; a gable roof; or a shed roof.

[0017] In certain embodiments, the walls include a front wall and at least some of (e.g., each of) the walls except the front wall is an extended height wall.

[0018] In certain embodiments, the walls include a back wall and at least some of (e.g., each of) the walls except the back wall includes two or more panels. [0019] In certain embodiments, the structure includes a floor. In certain embodiments, the floor includes a single panel. In certain embodiments, the floor is one of the walls.

[0020] In certain embodiments, the walls are connected together by hinges each of which connects adjacent ones of the walls. In certain embodiments, at least one (e.g., each) adjacent pair of the two or more panels is connected by one or more hinges. In certain embodiments, the structure includes one or more molded hinges each formed at least partially in one of the walls. [0021] In certain embodiments, the structure includes locking mechanisms disposed to lock the two or more panels of each of the at least two panels in position in the expanded state. In certain embodiments, the structure includes one or more locking mechanisms disposed to lock the walls in the expanded state.

[0022] In certain embodiments, the at least two walls includes two adjacent walls that are not parallel in the expanded state In certain embodiments, for at least one pair of adjacent ones of the walls, the adjacent ones are not directly connected. For example, a back wall of the walls may not be directly connected to a floor. In certain embodiments, one wall of the pair of adjacent walls moves (e g., horizontally and vertically) and rotates when the physical structure moves from the flat-folded state to the expanded state. In certain embodiments, the panels do not deform between the flat-folded state and the expanded state. In certain embodiments, the walls are disposed and constructed to have two symmetric comer folds corresponding to one of the walls (e.g., the back wall). In certain embodiments, the walls are disposed and connected to have at least one water bomb fold.

[0023] In certain embodiments, the walls include at least one window, at least one door, or at least one window and at least one door. In certain embodiments, at least one of the walls includes at least a portion of a window, at least a portion of a door, or at least a portion of a window and at least a portion of a door. In certain embodiments, at least one of the two or more panels for at least one of the at least two walls includes at least a portion of a window and/or at least a portion of a door.

[0024] In certain embodiments, the structure includes one or more mechanical assistance components (e.g., one or more active components and/or one or more passive components) (e.g., torsion springs). Each of the one or more mechanical assistance components may be disposed at a hinge line. At least one of the one or more mechanical assistance components may be disposed between two of the walls. At least one of the one or more mechanical assistance components may be disposed two of the two or more panels of one of the at least two walls.

[0025] In certain embodiments, the walls (e.g., each of the walls) include a rigid frame and soft shell. In certain embodiments, the walls (e.g., each of the walls) include a hard shell. In certain embodiments, the walls (e.g., each of the walls) include a sandwich composite. In certain embodiments, the walls (e.g., each of the walls) have been constructed using additive manufacturing.

[0026] In certain embodiments, the structure is wider than the structure is tall. In certain embodiments, the volume is at least 10 m ³ (e.g., at least 15 m ³ , at least 20 m ³ , at least 30 m ³ , at least 40 m ³ , at least 50 m ³ , at least 60 m ³ , or at least 100 m ³ ). In certain embodiments, the structure is habitable. In certain embodiments, the structure is a shelter. In certain embodiments, the structure entirely encloses the volume [e.g., in combination with a floor surface (e g., bare earth or a floor included in the physical structure)]. In certain embodiments, the walls include at least (e.g., and no more than) four walls. In certain embodiments, at least two (e.g., at least three) of the walls have a polygonal shape having at least four (e.g., at least 5) sides. In certain embodiments, the physical structure is a rectangular structure or a hexagonal structure. In certain embodiments, the physical structure is modular (e.g., has at least one open side to facilitate connection to another modular physical structure). In certain embodiments, each of the two or more panels has a thickness of at least 1 mm (e.g., at least 2 mm, at least 4 mm, at least 5 mm, at least 6 mm, at least 8 mm, at least 1 cm, at least 1.5 cm, or at least 2 cm) and no more than 50 cm (e.g., no more than 40 cm, no more than 30 cm, no more than 20 cm, no more than 15 cm, no more than 10 cm, no more than 8 cm, no more than 6 cm, no more than 5 cm, no more than 4 cm, no more than 2 cm, or no more than 1 cm). In certain embodiments, the physical structure includes seals disposed along edges of the walls and of the two or more panels such that the physical structure is sealed in the expanded state.

[0027] In certain embodiments, the physical structure has a lateral extent in the flat- folded state that is less than a total area of the walls (e.g., including any roof and/or floor) of the physical structure in the expanded state (e.g., the lateral extent is no more than 75%, no more than 50%, no more than 33%, no more than 25%, or no more than 20% of the total area). In certain embodiments, the physical structure has a lateral extent in the flat-folded state that is no more than an area of a largest wall of the physical structure in the expanded state. [0028] In some aspects, the present disclosure is directed to a method of unfolding a physical structure, the method including moving connected walls from a flat-folded state to an expanded state. At least two of the walls may include panels that are folded in the flat-folded state and unfolded in the expanded state. The walls may at least partially enclose a volume in the expanded state.

[0029] In certain embodiments, moving the connected walls to the expanded state includes moving with a single degree of freedom. In certain embodiments, moving with the single degree of freedom includes applying an expansion force to (e.g., pushing) only one of the panels at a time [e.g., at least partially with one or more mechanical assistance components (e.g., actuator(s))].

[0030] In certain embodiments, moving the walls into the expanded state includes applying an expansion force to the walls in a first location along a first direction for a first period of time and then applying an expansion force to the walls in a second location along a second direction for a second period of time after the first period of time. In certain embodiments, moving the walls into the expanded state includes applying the expansion force in the first location along the first direction until a torque threshold is met or exceeded and then applying the expansion force in the second location along the second direction. In certain embodiments, moving the walls into the expanded state includes applying an expansion force to a first panel of one of the walls for a first period of time and then applying an expansion force to a different second panel of one of the walls for a second period of time after the first period of time. In certain embodiments, an end of the first period of time and a beginning of the second period of time are determined based on a torque applied to the system. In certain embodiments, moving the walls into the expanded state includes applying the expansion force to the first panel until a torque threshold is met or exceeded and then applying the expansion force to the second panel.

[0031] In certain embodiments, moving the walls to the expanded state includes actuating one or more actuators. In certain embodiments, the one or more actuators is a single actuator. In certain embodiments, the one or more actuators is a plurality of actuators. In certain embodiments, moving the connected walls includes actuating only one of the plurality of actuators at a time. In certain embodiments, the method includes changing which actuator of the plurality of actuators is actuating based on a torque applied to the structure. In certain embodiments, changing which actuator of the plurality of actuators is actuating when a threshold torque is met or exceeded.

[0032] In certain embodiments, the structure is included in (e.g., is) a habitable shelter when the structure is in the expanded state.

[0033] In certain embodiments, the method includes folding the walls from the expanded state to the flat-folded state [e.g., by applying a folding force to the walls (e.g., with one or more mechanical assistance components (e.g., actuator(s)))]. In certain embodiments, the physical structure includes one or more torsion springs and moving the connected walls from the flat- folded state to the expanded state includes providing force from the one or more torsion springs. [0034] In certain embodiments, moving the connected walls from the flat-folded state to the expanded state includes translating (e.g., horizontally and vertically) and rotating at least one of the walls (e g., only one of the walls). In certain embodiments, each of the at least one of the walls includes only a single panel. In certain embodiments, translating and rotating each of the at least one of the walls includes unfolding along two symmetric comer folds for the wall.

[0035] In some aspects, the present disclosure is directed to a thick-walled physical structure. The structure may include connected panels. The panels may be movable between a flat unfolded state and a folded state. The panels may at least partially (e.g., entirely) enclose a volume in the folded state.

[0036] In some embodiments, at least some of the panels fold inward into the volume when the structure is moved from the flat unfolded state to the folded state (e.g., wherein the at least some of the panels are disposed at corners of one or more walls of the structure in the folded state).

[0037] In some embodiments, for at least one (e.g., at least two) pairs of adjacent ones of the panels, the adjacent ones of the panels are connected together only at ends of abutted edges of the adjacent ones of the panels.

[0038] In some embodiments, for each pair of adjacent ones of the panels, the adjacent ones of the panels are connected together along at least a central portion of abutted edges of the adjacent ones of the panels (e.g., entirely along abutted edges of the adjacent ones of the panels) [e.g., with one or more hinges (e.g., comprising fabric)].

[0039] In some embodiments, the structure comprises one or more energy storage hinges that each connect adjacent ones of the panels. Tn some embodiments, each of the one or more energy storage hinges comprises an elastomeric material (e.g., that is in an energetic state when the structure is in the folded state) (e.g., that is adhesively or mechanically fastened to the adjacent ones of the panels). In some embodiments, the elastomeric material is a barrier material (e.g., to water).

[0040] In some embodiments, the thick- walled structure is a single degree-of-freedom system.

[0041] In some embodiments, the panels comprise roof panels (e.g., of a pyramid hip roof). In some embodiments, at least two of the roof panels comprise a hole for a pin (e.g., a quick-release pin).

[0042] In some embodiments, the structure comprises a pin (e.g., a quick-release pin) that is insertable (e.g., inserted) into ones of the panels to maintain the panels in the folded state.

[0043] In some embodiments, each of at least two of the panels comprises a respective hole disposed in the panel shaped to accommodate a pin (e.g., a quick-release pin).

[0044] In some embodiments, the structure has kinematic compatibility.

[0045] In some embodiments, at least some of the panels comprise a chamfered edge

(e.g., wherein the at least some of the panels that fold inward into the volume comprise a chamfered edge).

[0046] In some embodiments, each of the panels has a thickness of at least 1 mm (e.g., at least 2 mm, at least 4 mm, at least 5 mm, at least 6 mm, at least 8 mm, at least 1 cm, at least 1.5 cm, or at least 2 cm) and no more than 50 cm (e.g., no more than 40 cm, no more than 30 cm, no more than 20 cm, no more than 15 cm, no more than 10 cm, no more than 8 cm, no more than 6 cm, no more than 5 cm, no more than 4 cm, no more than 2 cm, or no more than 1 cm).

[0047] In some embodiments, the physical structure has a blooming construction (e.g., an angled blooming construction or a blooming 90 degree construction). In some embodiments, the physical structure has a chamfered construction.

[0048] In some embodiments, the panels comprise a panel that rotates about an axis when moving between the unfolded state and the folded state and the panel comprises a hole for a pin and the hole is not aligned with the axis. In some embodiments, each of at least two of the panels comprises a hole disposed therethrough, wherein the at least two of the panels rotate about different axes when moving between the unfolded state and the folded state and the holes of the at least two of the panels are disposed such that a common pin can be inserted through the holes. In some embodiments, each of two or more of the panels comprise a hole sized and shaped to have a pin inserted therethrough. In some embodiments, the hole is disposed at an angle relative to a thickness dimension of the panel. In some embodiments, the hole is disposed such that when a pin is inserted therethrough the pin and the panel are not orthogonal.

[0049] In some aspects, the present disclosure is directed to a method of erecting a physical structure. The method may include using a winch to move the structure between an unfolded state and folded state or may include using a winch to move the structure between a folded state and an expanded state.

[0050] Any two or more of the features described in this specification, including in this summary section, may be combined to form implementations not specifically explicitly described in this specification.

BRIEF DESCRIPTION OF THE DRAWINGS

[0051] Drawings are presented herein for illustration purposes, not for limitation. The foregoing and other objects, aspects, features, and advantages of the disclosure will become more apparent and may be better understood by referring to the following description taken in conjunction with the accompanying drawings, in which:

[0052] FIG. 1 illustrates a basic three-sided single-vertex crease pattern;

[0053] FIG. 2 illustrates crease patterns to create a baseline model of a thin-walled shelter with the roof removed;

[0054] FIG. 3 illustrates visual confirmation of kinematic incompatibility in the baseline thin-walled model of FIG. 2 where panel (a) shows a partially folded state before the point of kinematic incompatibility is reached and panel (b) shows a partially folded state after the point of kinematic incompatibility is reached;

[0055] FIG. 4 illustrates a kinematically compatible physical structure having thin walls in various stages of folding/unfolding, according to illustrative embodiments of the present disclosure;

[0056] FIG. 5 is a side cut view of a partially folded structure of FIG. 4 illustrating kinematic compatibility, according to illustrative embodiments of the present disclosure; [0057] FIG. 6 illustrates variations in Yb and Y as a function of angle 0i in a thin-walled shelter according to FIG. 4 for (a) L/= 0, (b) L = 0.1, and (c) L = 0.3, according to illustrative embodiments of the present disclosure;

[0058] FIG. 7 illustrates a kinematically compatible physical structure having thin walls and a degree-4 vertex crease pattern in the front wall, according to illustrative embodiments of the present disclosure;

[0059] FIG. 8 illustrates (a) a side cut view and (b) a front cut view of panels affecting kinematic compatibility in a structure according to FIG. 7;

[0060] FIG. 9 illustrates variations in Yb and Y as a function of angle 3i in a thin-walled shelter according to FIG. 7 for (a) L/= 0, (b) L = 0.25, and (c) L = 0.4, according to illustrative embodiments of the present disclosure;

[0061] FIG. 10 illustrates simple folding models using (a) thin-walled panels, (b) thickwalled panels with an interior hinge, and (c) nesting thick-walled panels with an exterior hinge;

[0062] FIG. 11 illustrates panel intrusion (highlighted by circles) for panels in a wall that can occur for walls with finite (non-zero) thickness;

[0063] FIG. 12A illustrates folding sequence of a thick- walled single-vertex comer using (a) offset crease technique, (b) offset panel technique, and (c) alternate offset panel technique, according to illustrative embodiments of the present disclosure;

[0064] FIG. 12B illustrates a detail of a hinge shift technique, according to illustrative embodiments of the present disclosure;

[0065] FIG. 12C illustrates a detail of an offset panel technique, according to illustrative embodiments of the present disclosure;

[0066] FIG. 13 illustrates a kinematically compatible physical structure having thick walls without a roof (top row) and with a roof (bottom row) in various states from flat folded to expanded, according to illustrative embodiments of the present disclosure;

[0067] FIG. 14 illustrates a kinematically compatible physical structure having thick walls and a degree-4 vertex crease pattern in the front wall, according to illustrative embodiments of the present disclosure;

[0068] FIG. 15 illustrates dimensions of thick-walled structures for (a) a structure according to FIG. 13 and (b) a structure according to FIG. 14; [0069] FIG. 16A illustrates different styles of roofs that can be included on kinematically compatible physical structures including a (a) flat roof, (b) gable roof, and (c) shed roof, according to illustrative embodiments of the present disclosure;

[0070] FIG. 16B illustrates a possible extension section location for side panels of a physical structure in a flat-folded state, according to illustrative embodiments of the present disclosure;

[0071] FIG. 16C illustrates a physical structure having kinematic compatibility with extended height in an expanded state, according to illustrative embodiments of the present disclosure;

[0072] FIG. 16D illustrates a physical structure having kinematic compatibility with extended height and a flat roof in an expanded state, according to illustrative embodiments of the present disclosure;

[0073] FIG. 16E illustrates a physical structure having kinematic compatibility with a gable roof in an expanded state, according to illustrative embodiments of the present disclosure;

[0074] FIG. 16F illustrates a physical structure having kinematic compatibility with a shed roof in an expanded state, according to illustrative embodiments of the present disclosure;

[0075] FIG. 16G illustrates panel and wall thicknesses and arrangements to facilitate flatfolding, according to illustrative embodiments of the present disclosure;

[0076] FIG. 17 illustrates a thin- walled single-vertex fold model in partially folded state, according to illustrative embodiments of the present disclosure;

[0077] FIG. 18 illustrates variation of applied expansion torque about hinge line 1 and hinge line 2 as a function of 6i for the thin-walled single-vertex fold model shown in FIG. 17, according to illustrative embodiments of the present disclosure;

[0078] FIG. 19 illustrates a thick-walled shelter including a panel numbering referenced in FIGs. 20-25, according to illustrative embodiments of the present disclosure;

[0079] FIG. 20 is a geometric detail for calculating the center of gravity (CG) of panel 1 in the thick- walled shelter of FIG. 19, according to illustrative embodiments of the present disclosure;

[0080] FIG. 21 is a geometric detail for calculating the center of gravity (CG) of panel 2 in the thick- walled shelter of FIG. 19, according to illustrative embodiments of the present disclosure; [0081] FIG. 22 is a geometric detail for calculating the center of gravity (CG) of panel 3 in the thick- walled shelter of FIG. 19, according to illustrative embodiments of the present disclosure;

[0082] FIG. 23 is a geometric detail for calculating the center of gravity (CG) of panel 4 in the thick- walled shelter of FIG. 19, according to illustrative embodiments of the present disclosure;

[0083] FIG. 24 is a geometric detail for calculating the center of gravity (CG) of panel 5 in the thick- walled shelter of FIG. 19, according to illustrative embodiments of the present disclosure;

[0084] FIG. 25 is a plot of the variation of torque applied to panels 1, 3, and 5 as a function of angle 6i in the thick-walled shelter of FIG. 19, according to illustrative embodiments of the present disclosure;

[0085] FIG. 26 illustrates a structure having thick walls and a degree-4 vertex crease pattern in the front wall, according to illustrative embodiments of the present disclosure;

[0086] FIG. 27 is a plot of the variation of torque applied to panels 1, 3, and 5 as a function of angle 6i in the thick-walled shelter of FIG. 26, according to illustrative embodiments of the present disclosure;

[0087] FIG. 28 is a plot of the variation of torque applied to panels 1, 3, and 5 as a function of angle 6i in the thick-walled shelter of FIG. 19 that has a roof, according to illustrative embodiments of the present disclosure;

[0088] FIG. 29 is an example of a hinge that may be used in a physical structure, according to illustrative embodiments of the present disclosure;

[0089] FIGs. 30A-30B illustrate a locking mechanism that may be used to secure a physical structure in an expanded state and its possible placements, according to illustrative embodiments of the present disclosure;

[0090] FIG. 30C illustrates draw latches as a locking mechanism for a physical structure, according to illustrative embodiments of the present disclosure;

[0091] FIG. 30D illustrates straps as a locking mechanism for a physical structure, according to illustrative embodiments of the present disclosure;

[0092] FIGs. 31 A-C illustrate torsion springs that may be used to assist expansion of a physical structure, according to illustrative embodiments of the present disclosure; [0093] FIG. 32 illustrates a kinematically compatible structure highlighting locations where torsion spring(s) (or actuator(s)) can be disposed, according to illustrative embodiments of the present disclosure;

[0094] FIGs. 33A-D are annotated photographs of different views of a prototype physical structure having kinematic compatibility that includes molded hinges and embedded torsion springs, according to illustrative embodiments of the present disclosure;

[0095] FIGs. 34-50 illustrate alternative physical structures having kinematic compatibility, according to illustrative embodiments of the present disclosure;

[0096] FIG. 51 illustrates different relative dimensions for a physical structure that exhibit kinematic compatibility, according to illustrative embodiments of the present disclosure;

[0097] FIG. 52 illustrates an example of a sandwich composite construction, according to illustrative embodiments of the present disclosure;

[0098] FIGs. 53A-B illustrate physical structures that can be stored and/or transported in a conventional shipping container, according to illustrative embodiments of the present disclosure;

[0099] FIG. 54 illustrates a physical structure having a flat unfolded state that uses a bottom hinged construction, according to illustrative embodiments of the present disclosure;

[0100] FIG. 55 illustrates a physical structure having a flat unfolded state that uses a blooming 90 degree construction, according to illustrative embodiments of the present disclosure;

[0101] FIG. 56 illustrates a physical structure having a flat unfolded state that uses a angled blooming construction, according to illustrative embodiments of the present disclosure;

[0102] FIG. 57 illustrates a physical structure having a flat unfolded state that uses a topchamfered construction, according to illustrative embodiments of the present disclosure;

[0103] FIGs. 58A-B illustrate various close-up views of a bottom hinged construction, in this case having side-folding internal parts, according to illustrative embodiments of the present disclosure;

[0104] FIG. 59 illustrates individual panels of a physical structure having a flat unfolded state before being assembled, according to illustrate embodiments of the present disclosure;

[0105] FIG. 60 is a rendering of walls of a physical structure having a flat unfolded state after assembly, according to illustrate embodiments of the present disclosure; [0106] FIG. 61 illustrates three views of a foam board model with top-chamfered construction where the left and center views are of opposite sides of a flat unfolded state and the right view is of the model in a folded state, according to illustrative embodiments of the present disclosure;

[0107] FIG. 62 is a visual table illustrating various hinge designs and their compatibility with blooming constructions and top-chamfered constructions of physical structures having flat unfolded states, according to illustrative embodiments of the present disclosure;

[0108] FIG. 63 is a rendering of a physical structure having a flat unfolded state made using an angled blooming construction, according to illustrative embodiments of the present disclosure;

[0109] FIGs. 64A-C illustrate views of a table-top prototype of a physical structure having a flat unfolded state that uses a blooming 90 degree construction, according to illustrative embodiments of the present disclosure;

[0110] FIGs. 65A-B illustrate views of a table-top prototype of a physical structure having a flat unfolded state that uses a top-chamfered construction and remains folded using a pin and holes, according to illustrative embodiments of the present disclosure;

[0111] FIG. 66 illustrates elastomer hinges usable in physical structures, according to illustrative embodiments of the present disclosure;

[0112] FIG. 67 illustrates elastomer hinges usable in physical structures, according to illustrative embodiments of the present disclosure;

[0113] FIG. 68 illustrates stored energy (e.g., taped and mechanically fastened elastomer) hinges, according to illustrative embodiments of the present disclosure;

[0114] FIG. 69 illustrates a discrete torsion spring, according to illustrative embodiments of the present disclosure;

[0115] FIG. 70 illustrates a full-scale (life-size) prototype of a physical structure having a flat unfolded state that includes stored energy hinges, in this case continuous elastomer hinges, according to illustrative embodiments of the present disclosure;

[0116] FIG. 71 illustrates a full-scale (life-size) prototype of a physical structure having a flat unfolded state that includes stored energy hinges, in this case continuous elastomer and mechanically fastened hinges, according to illustrative embodiments of the present disclosure; [0117] FIG. 72 illustrates a prototype of a portion of a physical structure having a flat unfolded state that includes stored energy hinges, in this case continuous elastomer hinges, according to illustrative embodiments of the present disclosure;

[0118] FIG. 73 illustrates a full-scale (life-size) prototype of a physical structure having a flat unfolded state with a blooming 90 degree construction, in this case continuous elastomer and mechanically fastened hinges, according to illustrative embodiments of the present disclosure; and

[0119] FIGs. 74A-B illustrates views of a full-scale (life-size) prototype of a physical structure having a flat unfolded state with a blooming 90 degree construction, in this case continuous elastomer and mechanically fastened hinges, according to illustrative embodiments of the present disclosure.

[0120] Figures are not necessarily drawn or shown to scale. As reflected in the Parts List included herein, like parts labels between figures refer to similar (e.g., identical) parts. For ease of interpreting figures, not all parts are labeled in not all figures. For example, where a physical structure includes many walls or panels or hinge lines, a subset (e.g., none or more) may be labeled in any given figure. The structure and function of unlabeled parts will be readily apparent to those of ordinary skill in the art based on the parts that are labeled.

DETAILED DESCRIPTION OF CERTAIN EMBODIMENTS

[0121] In this application, unless otherwise clear from context or otherwise explicitly stated, (i) the term “a” may be understood to mean “at least one”; (ii) the term “or” may be understood to mean “and/or”; (iii) the terms “comprising” and “including” may be understood to encompass itemized components or steps whether presented by themselves or together with one or more additional components or steps; (iv) the terms “about” and “approximately” may be understood to permit standard variation as would be understood by those of ordinary skill in the relevant art; and (v) where ranges are provided, endpoints are included. In certain embodiments, the term "approximately" or "about" refers to a range of values that fall within 25%, 20%, 19%, 18%, 17%, 16%, 15%, 14%, 13%, 12%, 11%, 10%, 9%, 8%, 7%, 6%, 5%, 4%, 3%, 2%, 1%, or less in either direction (greater than or less than) of the stated reference value unless otherwise stated or otherwise evident from the context (except where such number would exceed 100% of a possible value). [0122] It is contemplated that systems, devices, methods, and processes of the disclosure encompass variations and adaptations developed using information from the embodiments described herein. Adaptation and/or modification of the systems, devices, methods, and processes described herein may be performed by those of ordinary skill in the relevant art.

[0123] Throughout the description, where articles, devices, and systems are described as having, including, or comprising specific components, or where processes and methods are described as having, including, or comprising specific steps, it is contemplated that, additionally, there are articles, devices, and systems according to certain embodiments of the present disclosure that consist essentially of, or consist of, the recited components, and that there are processes and methods according to certain embodiments of the present disclosure that consist essentially of, or consist of, the recited processing steps.

[0124] It should be understood that the order of steps or order for performing certain action is immaterial so long as operability is not lost. Moreover, two or more steps or actions may be conducted simultaneously.

[0125] As is understood by those skilled in the art, the terms “over”, “under”, “above”, “below”, “beneath”, and “on” are relative terms and can be interchanged in reference to different orientations of the layers, elements, and substrates included in the present disclosure. For example, a first layer on a second layer, in some embodiments means a first layer directly on and in contact with a second layer. In other embodiments, a first layer on a second layer can include another layer there between. Moreover, those of skill in the art will appreciate that the terms “back,” “front,” and “side” (e.g., in reference to walls) are relative terms that may depend on orientation of a viewer relative to a physical structure. That is, what is referred to as a front wall in this description may be considered a side wall in another context (e.g., based on where a physical structure is located when in use and/or where a viewer is standing when observing the structure). Generally, unless otherwise clear from context, for consistency and simplicity of explanation, in reference to the figures presented herewith, the largest wall (in area) that is closest to the viewer in the figure is referred to as a “front wall”; its opposite wall is referred to as a “back wall”; and walls that connect the front wall and the back wall are referred to as “side walls.”

[0126] Headers are provided for the convenience of the reader and are not intended to be limiting with respect to the claimed subject matter. [0127] The present disclosure encompasses a recognition that certain origami-inspired principles may be useful in the design of any of a variety of structures. The ancient art of paper folding or origami has inspired many modem applications in engineering that rely on folding patterns that allow objects (e.g., paper) to transition from one geometric configuration to another through a kinematically compatible transformation process. Prior applications of origami -based or origami -inspired concepts have ranged from reconfigurable miniature robots to space structures and certain foldable shelters that can be folded into a usable structure. Previously known foldable shelters include accordion style folding patterns that facilitate transformation from one configuration to another configuration (by folding and unfolding the accordion).

[0128] Origami patterns and origami-inspired structures are governed by specific requirements associated with the kinematics of folding. Varying the location of fold lines and vertices can create a multitude of patterns that fold into different shapes (and sizes). Previously known collapsible structures are usually found in the form of linkages or scissor like elements that allow for large expansion relative to the packaged state. Crease patterns often use known rules of origami such as Kawasaki’s theorem that requires an even number of lines connecting to each inner vertex and alternating angles summing to 180 degrees. As such, the present disclosure provides a solution to a long existing problem in providing habitable structures that can be easily assembled for use in, for example, humanitarian, military, or temporary applications through the application of origami principles as disclosed herein.

[0129] Disclosed herein are, inter alia, foldable physical structures. A physical structure includes one or more walls. Each wall may include one panel or may include multiple panels. The walls are connected together, for example by one or more hinges, such that they are movable between a flat-folded state and an expanded state. In the expanded state, walls at least partially enclose a volume. The volume may be sufficiently large to be habitable, such that the physical structure can act as a shelter, for example for humanitarian or military purposes. Panels are generally flat and non-deformable to promote flat-folding. Panels may have non-uniform thickness (e.g., partial cutouts corresponding to portions of other panels) to allow for nesting to achieve flat folding. While certain physical structures incorporate principles of origami into their design, physical structures disclosed in the present application differ from traditional origami in that disclosed physical structures are flat when folded and occupy volume when unfolded. That the physical structures occupy volume when unfolded results from the connections between walls. By designing physical structures to have certain geometric relationships between the panels and walls, kinematic compatibility can be achieved whereby folding and unfolding occurs without the structure impinging on itself (which may prevent (un)folding or otherwise require panels to be deformable). A particularly useful feature of certain physical structures having kinematic compatibility disclosed herein is that they operate with a single degree of freedom. A single degree of freedom for one or more walls (e.g., a physical structure including the one or more walls) refers to the wall(s) having a structure where force applied to one panel results in a compliant movement of all others in the wall(s). For example, a wall may include three panels (or more) connected together such that applying force to one panel in the wall to fold or unfold results in the other two panels (or more) moving to fold or unfold, respectively. In some embodiments, a group of panels [e.g., together included in (e.g., forming) one or more walls] have a single degree of freedom in that the panels are connected such that movement of one of the panels (e.g., due to applied force) causes movement of all of the panels in the group. In some embodiments, a physical structure is a single degree-of-freedom system such that applying force to one panel results in compliant movement of all panels in the physical structure.

[0130] A wall may include more than one panel. Panels may be connected at their edges such that, in an expanded state, they collectively form a flat wall. Walls and panels are said to “fold.” Folding occurs along hinge lines. Folding along a hinge line may be facilitated by one or more hinges disposed along the hinge line. The hinges provide freedom of rotation. No other freedom of movement may be intentionally provided by hinge(s) other than freedom of rotation (e.g., some natural, insignificant deflection may occur due to, for example, panel or wall weight). Hinge lines will naturally occur at locations where edges of adjacent panels abut. In some cases, edges of adjacent walls that abut are connected together and, in some cases, edges of adjacent walls that abut are not connected together. In some embodiments, including ones where a water bomb fold is included (as discussed further below), some adjacent walls are connected (e.g., a roof and a side wall) and others are not (e.g., the side wall and a front wall). Thus, walls are connected together to facilitate movement between a flat-folded state and expanded state but it is not necessary that amongst these connected walls each pair of adjacent walls be connected.

[0131] The following description provides illustrative embodiments of physical structures in addition to details of components, such as locking mechanisms and mechanical assistance components that may be used in physical structures. Locking mechanisms may be used to ensure a physical structure remains in an expanded state thereby “rigidifying” the structure (and can be unlocked for folding). Mechanical assistance components may be used to reduce the amount of force that need be applied to move a physical structure from a flat-folded state to an expanded state.

Thin-Walled Models

[0132] Thin-walled models can be used for an initial feasibility assessment of the origami-inspired structure (e.g., shelter) design concepts, as basis for development of the thickwalled physical structures presented later, or as final configurations for particular uses and embodiments.

[0133] FIG. 1 shows a basic three-sided single-vertex crease pattern consists of two 45° triangular panels on one side and two 90° rectangular panels on the other sides. The singlevertex crease pattern (vertex 1) is repeated by placing two additional 45° triangular panels opposite to the back wall resulting in vertex 2 and then mirrored to create a full rectangular box structure as shown in FIG. 2. A roof, for example consisting of a single rectangular panel (not shown), could fold behind the back wall with the top edge of the back wall serving as a hinge line location.

[0134] FIG. 2 shows a baseline thin-walled model with width and height of H and length of 2H (referred to subsequently as the “Baseline Model”). The term thin signifies an essentially zero thickness value, where each panel is represented as a two-dimensional surface. With the exception of the roof panel, the remaining members of the Baseline Model are connected in such a way that they behave as a single-degree-of-freedom system. The front wall of the Baseline Model includes three panels; the side walls include two panels; and the back wall includes one panel.

[0135] Kinematic compatibility refers to a physical structure’s ability to move between a flat-folded state and an expanded state without interference or intrusion of one panel into another (either on the same or different walls). Kinematic compatibility can be determined based on the presumption that panels must remain flat during the folding/expanding process with all the fold or hinge lines remaining straight (a presumption of no deformation or distortion). FIG. 3 shows the absence of kinematic compatibility in the Baseline Model as manifested by intrusion of the front wall panels into the back wall in the transition process. [0136] The intrusion problem can be resolved by considering any one of several possible configuration modifications to the Baseline Model. FIG. 4 shows a thin-walled physical structure 10 (referred to subsequently, for both thin- and thick- walled versions, as “Model 1”) with the same three-triangular panel arrangement for the front wall 20 (which includes panel P3 in FIG. 5) as in the Baseline Model, but with each side wall 20 divided into a triangular panel 22 (panel P l in FIG. 5) and a trapezoidal panel 22 (panel P2 in FIG. 5). The front wall 20 is connected to the side walls 20 and the side walls 20 are each connected to the back wall 20. No roof is shown, for clarity. The structure’s width is longer than its height (E+H as shown), and it folds (along hinge lines 30) into a larger rectangular footprint than the baseline model. The width extension, E, can be written as a function of the overall height, H through a length factor, Lf such that E = LfH. The physical structure 10 is shown moving (expanding) from a flat-folded state (left) to an expanded state (right).

[0137] To maintain kinematic compatibility, the vertical distance, Yb from the floor (base) to the potential intersection point on the back wall, shown as Ci in FIG. 5, must be greater than the vertical distance, Y from the floor to the top fold vertex of the front wall, shown as C2 in FIG. 5, for all values of angle 0i, which is the angle between the back wall and the floor. The Y and Yb values of the two points of interest (Ci, C2) are plotted in FIG. 6 as a function of back- wall angle 0i for various L/values. For L = 0, which corresponds to the baseline model in FIG. 2, the top fold vertex of the front wall (C2) has intruded into the back wall because Y > Yb for 0 < 9i < 72°. Only when L > ~0.3, the panel intrusion is eliminated and kinematic compatibility is achieved by keeping Y < Yb for 0 < 61 < 90° as shown in panel c of FIG. 6. This Lf value applies to thin-wall models.

[0138] Another example of a physical structure 10 having kinematic compatibility is shown in FIG. 7, in this case using a degree-4 vertex crease pattern in the front wall 20 (referred to subsequently, for both thin- and thick-walled versions, as “Model 2”). A degree-4 vertex is formed when four creases intersect at a single point. The main distinguishing feature in Model 2 relative to the Baseline Model is the addition of the small folding elements (SFE) in the front wall 20, which are described further subsequently. The structure has a single rectangular-panel floor 26 and back wall 20. The side walls 20 consist of two right-triangle panels 22 with 45° fold lines. The front wall 20 of the structure 10 folds inside of the structure 10 as the panels 22 of the side and back walls 20 fold over it. No roof panel 22 is shown for clarity. The physical structure 10 is shown in an expanded state (top left) and partially folded state (bottom left); no flat-folded state is shown. Model 2 has a benefit that it can more easily achieve a width to height ratio of 1. [0139] If the SFE is designed to fold away from the back panel, there would be no intrusion during the folding/unfolding process (that would occur in the Baseline Model with the SFE) and kinematic compatibility is ensured. A common degree-4 vertex crease pattern and a close-up view of the SFE are shown in FIG. 7. The folds are labeled as either mountain or valley fold. A mountain fold is created when two joined panels form a convex fold relative to the reference plane whereas a valley fold is developed when a concave fold is formed. According to the Maekawa theorem, the absolute value of the difference between the numbers of mountain and valley folds at a vertex must be two. The valley folds along the vertical and horizontal boundaries of the SFE cause the triangular panels to fold away from the back panel to prevent intrusion. Because the fold is a degree-4 vertex, the Kawasaki’s theorem requires i + 63 = 62 + 04 = 180° for the pattern to be locally flat-foldable. Therefore, in at least some embodiments, panels in a wall are disposed and connected so that the wall folds with both one or more mountain folds and one or more valley folds, for example as shown top and bottom right in FIG. 7.

[0140] Desirable (e.g., minimum) dimensions of the SFE, which includes three right triangles with two 45° angles, can be determined as a function of overall height of the physical structure and a length factor, L (= LfH which is shown in FIG. 7. Using the geometric properties shown in FIG. 8, the potential intersection point is characterized by point Ci at height Yb on the back wall and point C2 at height Yj on the front wall. Panels Pl through P4 are consistent with those in FIG. 7 (top left) with panel P2 in FIG. 8 hidden from the view in FIG. 7. [0141] To maintain kinematic compatibility, Yb must always be greater than Yj = Yi + Y for all values of i. The values of / and Yb are plotted in FIG. 9 as a function of i for various //values. The plot for Lf= 0 corresponds to the Baseline Model with no SFE. At Lf = 1/3, the front panel comes in contact with the back panel, and Lf = 0.4 results in //shorter than Yb for all values of 0i ranging from 0 to 90 degrees, thus, providing the desired kinematic compatibility. It is also evident from FIG. 7 that an //value greater than 0.5 would cause the SFE to protrude beyond the intended collapsed footprint of the physical structure. While such protrusion does not inhibit flat folding, it may nonetheless be undesirable for storage and transport of a physical structure. A value of 1.0 would eliminate P3 and reflect the entire front wall now folding away from the back wall. In some embodiments, these considerations would set an effective upper bound on /./ at 0.5.

Thick-Walled Physical Structures

[0142] The strict mathematics that describe origami patterns and folding procedures assume zero thickness panels. Since each panel edge is essentially a straight line, it coincides precisely with the only possible location of a fold or hinge line with an adjacent panel.

[0143] When used in structural applications, panels have some finite (non-zero) thickness (e.g., that is enough to provide sufficient structural rigidity). Due consideration must then be given to flat-folding of the panels. Thickening the panels adds complexity to origami-inspired structure (e.g., shelter) models. For example, since inner and outer surfaces of a thick panel are at some finite distance apart, a hinge location in a thickness direction along edge(s) becomes a design decision. To help with this discussion, a simple folding model forthick walls is shown in FIG. 10, including two options to accommodate flat-foldability in thick- walled panels. Panel a of FIG. 10 provides a thin-wall comparison; panel b shows a hinge 32 on an interior surface; panel c shows a hinge 32 on an exterior surface. As discussed further subsequently, a hinge shift technique may be used with thick-walled panels, where a combination of interior surface hinge(s) and exterior surface hinge(s) is used to prevent panels from impinging. FIG. 12B shows an example of such a technique.

[0144] If thin origami connections are used as default in a thick-walled system, it is possible to encounter panel intrusion and/or interference that prevents kinematic compatibility. That is, panel edges can conflict with each other at some point during folding/unfolding between a flat-folded state and expanded state (such that either state may not be able to be fully achieved). The basic three-dimensional single-vertex crease pattern shown previously in FIG. 1 is shown with thick-walled panels in FIG. 11. As highlighted, panel intrusion occurs along the diagonal fold line, and it is prevented only in the fully unfolded (expanded) state (top row).

[0145] A variety of folding techniques may be used with thick wall physical structures. In an offset crease technique, the original fold between two connected panels is widened and replaced by two parallel hinge lines to facilitate flat-foldability. Details about offset crease techniques can be found in the journal article: Ku, J.S. and Demaine, E.D., “Folding Flat Crease Patterns with Thick Materials,” Journal of Mechanisms and Robotics 8(3): 031003, 2016. Tn an offset panel technique, thickness of some panels is modified locally to allow folding through panel nesting. Details about offset panel techniques can be found in the journal article: Edmondson, B.J., Lang, R.J., Magleby, S.P., and Howell, L.L. “An Offset Panel Technique for Thick Rigidly Foldable Origami,” Proceedings of the ASME 2014 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, Buffalo, NY, Aug 17-20, 2014. In a hinge shift technique, position of the hinge line is shifted from one panel surface to another in an alternating fashion. Details about hinge shift techniques can be found in the journal article: Lang, R.J., Tolman, K.A., Crampton, E.B., Magleby, S.P., and Howell, L.L. “A Review of Thickness-Accommodation Techniques in Origami-Inspired Engineering,” Applied Mechanics Reviews, 70(1), 010805-010805-010820, 2018. A combination of these techniques may be utilized within a given physical structure, for example an offset panel technique and a hinge shift technique may be used in combination to account for thickness differences between panels or portions thereof. Manufacturability considerations may factor into which technique(s) are used for a given physical structure.

[0146] Extending the offset crease technique to a thick-walled single-vertex crease requires the addition of spacers as shown in panel (a) of FIG. 12A. Because of increased panel thickness, the two side panels 22 must have added spacers 23 to preserve flat-folding of the panels. Using the offset panel technique, it would be possible to keep each hinge line at the same location as in the thin origami model. However, as the name implies, it would require changing panel thickness in certain locations to create the necessary offsets for accommodating panelnesting and flat-foldability as shown in panel (b) of FIG. 12 A. The thickness variation is prominent near the angled fold line as shown by the local doubling of the panel thickness to create step-like panels that can nest together when folded. The number of panels is still kept at a minimum while maintaining kinematic compatibility between panels (e.g., within a single wall). Another local thickness allocation design that maintains uniform thickness in each triangular panel is shown in panel (c) of FIG. 12A. A single-vertex crease is still maintained and stationary tabs 24 to allow complete folding of the system are incorporated in design of some panels. A tab 24 may extend from an end of a panel 22 (e.g., at a right angle). An alternate offset panel technique (in which some, but not all, panels 22 include tabs 24 and variable panel thickness to promote flat folding) can be combined with a hinge shift technique (in which hinges alternate between interior and exterior surfaces of panels) to create kinematically compliant thick-walled physical structures (e.g., shelters), for example according to Model 1 or Model 2.

[0147] FIG. 12B provides another illustration of a hinge shift technique used in a physical structure in accordance with embodiments of the present disclosure. (Note that back wall 20 is connected to side walls 20 but not connected to floor 26). Looking from top to bottom along the right side wall 20, a first panel 22 is connected to the panel 22 that is the back wall 20 with a hinge 32 on a first (e g., interior) surface of the panel 22. A second hinge 32 connects the first panel 22 to a second panel 22 where the second hinge 32 is disposed on a second surface of the first panel 22 (opposing the first surface, e.g., an exterior surface) and a first (e.g., interior) surface of the second panel 22. A third hinge 32 connects the panel 22 that is the floor 26 to the second panel 22 where the third hinge 32 is disposed on a second surface of the second panel 22 that is opposite the first surface of the second panel 22 (e.g., an exterior surface). A dashed line connects the hinges 32, which highlights the alternating (“shift”) in hinge location. FIG. 12C provides a close up illustration of an offset panel technique where a tab protruding from one panel 22 defines a hinge line 30 for a second panel 22 to rotate about.

[0148] In a thick-walled version of Model 1 illustrated in FIG. 13, the hinge lines 30 are placed on the inner and outer edges of each side wall panel 22. The corner vertex fold behaves the same way as the thin-walled version discussed previously, but now the back wall 20 is forced (in this example) to have a vertical and horizontal displacement in addition to rotation throughout the folding process, indicating that the back fold does not mirror zero-thickness kinematics. Thus, the back panel 22 cannot be directly connected (e.g., have a fixed hinged connection) to the floor 26 and requires two symmetric comer folds (since it is made of a non-deformable single panel). In some embodiments, a physical structure includes two identical corner folds that include a wall (e g., a back wall). The thick-walled Model 1 also has added tabs 24 designed into the floor panel 22 to preserve flat folding in addition to double thickness side panels for nesting. The additional hinge 32 connecting the front wall 20 to the floor panel improves stability and may include an inherent seal in the hinged connection. A roof 28 is attached to the inner top edge of the back panel and is comprised of two panels 22 joined at one hinge line 30. The size of each roof panel 22 is dependent on the extension of the trapezoidal panel 22 in the side wall 20 and the height of the structure. The top row of FIG. 13 shows a thick- walled version of Model 1 without the roof 28, while the bottom row of FIG. 13 shows the same version with the roof 28. [0149] A thick-walled version of Model 2 is shown in FIG. 14. The physical structure 10 has additional complexity due to the SFE in the front wall. Similar to thickening the corner vertex, an alternate offset panel technique is also used to create a thick SFE. The corner vertex fold behaves the same way as the thin-walled version discussed previously, but now the back wall 20 is forced (in this example) to have a vertical displacement in addition to rotation throughout the folding process, indicating that the back fold does not mirror zero-thickness kinematics. Thus, the back panel 22 cannot be directly connected (e.g., have a fixed hinged connection) to the floor 26 and requires two symmetric corner folds (since it is made of a non- deformable single panel). The thick-walled Model 2 also has added tabs 24 designed into the floor panel 22 to preserve flat folding in addition to double thickness side panels for nesting. The additional hinge 32 connecting the front wall 20 to the floor panel improves stability and may include an inherent seal in the hinged connection.

[0150] Thick-walled versions of Model 1 and Model 2 illustrated in FIGs. 13-14 are shown with rather large (exaggerated) thickness relative to the other dimensions to mainly highlight the change from thin to thick origami. Because, in certain embodiments, structures are intended for use as temporary shelters and exposed to various environmental and loading conditions, representative sizes must be examined to ensure designs meet both structural and deployment requirements.

[0151] Thick-walled versions of Model 1 and Model 2 with length of 3.61 m (142 in) and 4.42 m (174 in), respectively, are shown in FIG. 15 (panel (a) and panel (b), respectively). In both cases, H = 2.286 m (90 in). Both structures can be scaled up or down in overall size by adjusting the height and kinematically compatible Lf value, with other dimensions changing in a consistent manner. The width and height dimensions in panel (a) of FIG. 15 together with changes in wall thickness accommodate kinematic compatibility. With the dimensions shown, approximately up to six structures (e.g., shelters) can be stored and transported inside an ISO IC container. It is noted that Lf= 0.23 in panel (a) of FIG. 15 while kinematic compatibility is maintained (whereas it would not have been maintained in a thin-walled model). However, the structure remains kinematically compatible due to thickness variation in combination with the use of the hinge shift technique which causes the back wall to translate vertically and/or horizontally somewhat as it rotates, adding clearance for the front wall to rotate without collision. An identical method was developed to check compliance for the thick-walled models that incorporated effects of panel thickness. Thus, lower Lf values (e.g., > 0.2) may still produce kinematically compatible structures with thick walls when hinge locations are appropriately selected. In some embodiments, kinematic compatibility is maintained for LF > 0.2 (e.g., > 0.23), for example due to choice of folding technique(s) (e.g., hinge shift, offset panel, and/or offset crease) to accommodate flat-folding.

[0152] Both the height and roof geometry can be modified with three examples shown in FIG. 16A. The first modification requires a length extension to increase the height of the side and back walls 20 while maintaining the front wall 20 at its original height. A narrow panel 22 in the roof 28 would fold over to cover the opening created by the difference in height of the front-wall panels 22. A gable roof 28 can also be used to increase the height-to-width ratio of a physical structure 10, as shown in panel (b) of FIG. 16A. This modification also requires a roof 28 with multiple panels 22 that folds to conform to the corresponding triangular top edge of the side walls 20. In addition to a gabled roof, a shed roof could be utilized to obtain similar benefits while reducing design complexity as shown in panel (c) of FIG. 16A. FIG. 16E illustrates an example of a physical structure 10 with a gable roof 28. FIG. 16F illustrates an example of a physical structure 10 with a shed roof 28.

[0153] In some embodiments, there is a small unused space in a flat folded state of a physical structure that can be used to extend the top of the side panels to create taller shelters. This space is highlighted in FIG. 16B. At least two height extension concepts are envisioned. The first concept can include a certain amount of width extension to increase shelter height. The flat-folded and expanded physical structure 10 is shown in FIG. 16C without a roof 28 and in FIG. 16D with a roof. The front wall 20 maintains its original height, meaning an extended roof 28 with an additional fold panel 22 may be used to fully enclose a volume (e.g., fully seal the structure). The roof 28 would fold over just as it does in the original roof concepts but would require an additional step of folding in the additional roof flap to cover the opening created by the difference in height of the panels 22 of the front wall 20 and the panels 22 of the side walls 20.

[0154] Another concept developed to address the height-to-width ratio is a gable roof implementation. FIG. 16E illustrates an example of a physical structure 10 with a gable roof 28. A triangular extension of the side panels 22 is added to support a gable roof 28. This concept also requires a roof 28 with multiple panels 22 that folds to conform to the triangular shape of the side walls 20. In addition to the angled roof, a shed roof could be utilized to maintain the same benefits, while reducing complexity. FIG. 16F illustrates an example of a physical structure 10 with a shed roof 28.

[0155] In some embodiments, a roof includes a plurality of panels that each rotate about one of at least two hinge lines to allow the folded roof to fit in the footprint of the entire flat- folded physical structure. That is, multiple panels may be needed in a roof to prevent the lateral extent of the roof from extending beyond the lateral extent (e.g., footprint) of the rest of the physical structure. The thickness of roof panels may need to be controlled to ensure flat folding, depending on how the rest of a physical structure folds. For example, FIG. 16G illustrates various folding schemes where the first two ensure flat folding and the third scheme does not. In these examples, because of the location of the hinge lines, a roof panel must have a thickness value that is equal to or larger than a back panel to ensure flat folding

Motion Analysis of Physical Structures Folding and Expanding

[0156] Origami-inspired structures represent a collection of flat (e.g., non-deformable) panels connected by multiple hinge lines. The folding process is assumed to occur slowly, thus, neglecting inertial effects and assuming a quasi-static loading condition. The only loads acting on a structure as it is unfolded to an expanded state are the weight of each homogeneous panel and the applied force or torque needed to maintain equilibrium. For the purpose of understanding folding/unfolding, all hinge lines are assumed to be frictionless. As the structure has articulating members, the method of virtual work is used to analyze it at different points during the unfolding (expansion) process. Thin-walled models were analyzed first due to their reduced complexity followed by thick-walled models.

[0157] FIG. 17 shows a thin-walled model considered with the vertical center of gravity (CG) location, z of each panel highlighted. While the base is held stationary, the side and back panels can move simultaneously as a single-degree-of-freedom system with 0i used as the only independent variable. Such movement may be facilitated by an actuator as discussed subsequently or may be accomplished by a pushing or pulling force applied by a person

[0158] Considering the applied torque T i about hinge line 1 along with the weight of each panel, the principle of virtual work for this system can be expressed as [0159] Since the base provides only one rotational degree of freedom to panels 1 and 3 at their corresponding hinge lines, the reaction loads along the hinge lines 1 and 2 are fixed in position and do no work in the virtual work analysis.

[0160] Using the geometric relationships in FIG. 17, the vertical CG position of each panel relative to the base or the reference plane as a function of angle 0i is found as

[0161] Differentiating Eqs. (2)-(4) with respect to 9i, substituting the results into Eq. (1) and simplifying the equation gives

Eq. (5) gives the value of torque required about hinge line 1 to keep the system in equilibrium for 0 < 9i < 90°.

[0162] To explore the effect of loading location on the required torque, hinge line 2 is selected. Following a similar procedure as above while maintaining 91 as the independent variable and where T 1 is identical to the expression given in Eq. (5).

[0163] Assuming the back wall’s height, H = 2.13 m (84 in), length = 2H, panel thickness ~ 0.0032 m (0.126 in), to mimic thin origami, and using the material density of PET (1420 kg/m ³ or 0.0513 lb/in ³ ), results in Z i = 41.36 kg (91 lb) and W2 = HG = 10.34 kg (22.75 lb). The applied expansion torque about hinge lines 1 and 2 were calculated using Eqs. (5) and (6) as a function of angle 9 and plotted as shown in FIG. 18.

[0164] As expected, maximum torque occurs at 9i = 0 when the model is in a fully flat- folded state. The most striking observation is that at 9i = 0, T2 is exactly twice the magnitude of T 1. During the unfolding process, the torque required to maintain equilibrium decreases with the rate of decrease in T2 being noticeably greater than that for T 1, with the two reaching the same value at i = 68.5°. Beyond the intersection point, it appears that less torque would be needed about hinge line 2 than hinge line 1 to bring the thin-walled single-vertex fold model to an expanded state (wherein 0i = 90°). It is also noteworthy that while T ₂ is zero at 0i = 90°, T 1 is over 135.6 N-m (1200 Ib-in). A close examination of Eq. (5) shows that while the first term on the right-hand side goes to zero at di = 90°, the second and third terms do not. To verify the analytical results, a motion simulation of the same model was performed using SolidWorks with the results matching those in FIG. 18 for all values of 0i. Therefore, in some embodiments of a single degree-of-freedom physical structure, reduced torque can be achieved when moving between a flat-folded state and an expanded state by applying force to a first panel for a first period of time before applying force to a second panel for a second period of time after the first period. Despite the switch, force need only be applied to one panel at a time to expand the physical structure as it is a single degree-of-freedom system.

[0165] Adapting the analysis to thick-walled physical structures affects the CG location of each thick panel. To keep the derivation general, the thinnest panel is given a thickness of t with the others adjusted to 2t or 3t to accommodate both kinematic compatibility and nesting for flat-folding of structures. Each CG location no longer aligns with the plane that defines the folding angles with the amount of offset depending on the thickness of each panel. A thickwalled version of Model 1 without a roof was analyzed using the panel numbering system shown in FIG. 19. Three possible unfolding conditions are considered with the external torque applied to rotate panel 1, 3, or 5.

[0166] Using the principle of virtual work and applying a torque to panel one gives ) ( )

[0167] The angle at the base of panel 1, 0i is the only independent variable (it is a single degree-of-freedom system). Because of the increased thickness, the CG location determination for some panels may involve several intermediate steps that are not shown here. However, the key figure used in each calculation is provided.

[0168] In addition to rotation at its base, panel 1 also experiences vertical and horizontal displacements that prevent it from being permanently attached to the base along its bottom edge as highlighted in FIG. 20.

[0169] The vertical CG position of panel 1 is found as

[0170] The vertical displacement z _t is due to the thickness of panel 2 whereas z _tr is due to the thickness of panel 3. Each term can be determined from certain views of the structure.

Comparison of Eq. (8) to Eq. (2) shows the elevated complexity as a result of increased panel thickness.

[0171] The vertical CG location for panel 2 is highlighted in FIG. 21, and it is calculated as

[0172] The additional vertical displacements z _t and z _tr also appear in Eq. (9). Because N2 is always normal to panel 1, a side view can be used to determine z _P 2, the vertical distance from the edge of panel 1 to the CG of panel 2.

[0173] Panel 3 has a complex geometry due to its trapezoidal shape and split thickness of t and 2t as shown in FIG. 22. The vertical CG location of panel 3, Z3 is found as The expressions for CG3 _y and CG3 _Z are obtained from geometric decomposition of panel 3 considering the variation in thickness as shown in FIG. 22. The movement of panel 3 is simply a rotation about the side of the base of the structure.

[0174] FIG. 23 shows various geometric attributes of panel 4 with its vertical CG location obtained as

[0175] The geometry of panel 5 is shown in FIG. 24, including the length extension W.

With a constant thickness of t, the vertical CG location of panel 5 is found as

[0176] The virtual work results for the thick- walled version of Model 1 are obtained from the solution of Eq. (7) for a single torque applied to panel 1, panel 3, or panel 5 as identified in FIG. 19. For the latter two cases, the T 1861 term in Eq. (7) is replaced with T3863 and T5865, respectively. Multiple SolidWorks models were developed for validating the analytical results through motion simulation. Each thick panel is modeled with a uniform density of 133.14 kg/m ³ (0.00481 lb/in ³ ) estimated from a foam core polyethylene terephthalate glycol (PETG) sandwich plate, with geometric dimensions: t = 0.0254 m (1 in), W = 0.914 m (36 in), E = 1.02 m (40 in), and H = 2.13 m (84 in).

[0177] The plots of torque as a function of 0i for three cases are presented in FIG. 25. The analytical results match those found by motion simulation in SolidWorks. Similar to the results for the thin- walled single-vertex fold model, it is evident that the magnitude of torque to unfold the thick-walled version of Model 1 from the flat-folded state (wherein i = 0) to the expanded state (wherein 61 = 90°) can vary substantially depending on its application location. It appears that for i < -67°, applying the torque to panel 1 would be most efficient. Beyond that point, the best location shifts to panel 3 and panel 5. Both the analytical and motion simulation results indicate a sharp increase in the applied torque T 1 to maintain equilibrium near 0i = 90°. In contrast, both T 3 and T5 reach their minimum values at i = 90°. Therefore, as previously stated, force may be applied, whether by actuator or manual pushing or pulling for example, to one panel for one period of time and to another panel at another period of time to minimize overall torque applied to the system.

[0178] Following a similar procedure as above in the thick-walled version of Model 2, shown in FIG. 26, and using the same density panels with I = 0.0254 m (1 in), W= E = 0 m, H = 2.21 m (87 in), and Lf= 0.368, yields the results shown in FIG. 27 for variation of applied torque as a function of angle 0i. The trends appear to be fairly similar to those seen in FIG. 25 with the applied torque T 5 being smaller in magnitude in FIG. 27.

[0179] In the version of Model 2 shown in FIG. 26, panel (a), first panel 22 (“(1)”) has a constant thickness, and second, third, fourth, and fifth panels 22 (“(2)”, “(3)”, “(4)”, “(5 )”) with non-uniform thicknesses. Note that there are two congruent second, third, and fourth panels 22 in panel (a) of FIG. 26 (one of the third panels is hidden and therefore not labelled). Panel (b) shows a side view corresponding to panel (a) that illustrates the non-uniform thicknesses as integral multiples of a baseline thickness (e.g., t or If). Panels (c) and (d) of FIG. 26 show how fourth and third panels 22 nest when folding from an expanded state to a flat-folded state.

[0180] The virtual work analysis was repeated for a thick-walled version of Model 1 with a single-panel roof. Prior to unfolding the entire structure to its expanded state, the roof panel is first unfolded to the front of the flat-folded shelter similar to that depicted in FIG. 13. The roof motion beyond that point, is divided into two stages: (1) the roof is in contact with and slides over the ground, and (2) the roof comes in contact with and slides along the side-wall panels as soon as it loses contact with the ground, as shown in FIG. 13. Both contact surfaces were assumed frictionless for simplicity of analysis. (Note that by first unfolding the roof separately, the structure including the roof becomes a version of a single degree-of-freedom structure as the roof will now move to its final position in the expanded state simultaneously with the other panels/walls.)

[0181] An example of a sandwich roof includes glass-reinforced PETG facesheets with PET foam core. The foam core density is approximately 75 kg/m ³ (4.68 lb/ft ³ ), whereas the PETG lamina density is roughly 1334.8 kg/m ³ (83.33 lb/ft ³ ). Each panel is assumed to have four plies on either side of the core with average ply thickness of 0.0003 m (0.012 in). The core thickness is either 0.019 m (0.75 in) or 0.044 m (1.75 in), where the larger thicknesses correspond to panel 2 and the bottom and side portion of panel 3. The overall shelter dimensions are approximately 2.13 m (7 ft) tall, 2.51 m (8.25 ft) wide, and 4.27 m (14 ft) long.

[0182] The applied expansion torque results from both the analytical solution and SolidWorks motion simulation are shown in FIG. 28 with excellent agreement. The visible jump at 0i = 33.4° represents the transition point between stages 1 and 2 previously mentioned. The result is predictable, because once the roof transitions to stage 2, the full weight of the roof panel is on the rest of the structure as the normal force due to contact with the ground goes to zero. Once again, it appears that for most values of i, panel 1 results in the smallest value of torque to lift the shelter to its upright position. Outlier points near i = 0 in each case from the SolidWorks simulations appear to be a numerical artifact which do not appear in the analytical results and can reasonably be disregarded. Roofs

[0183] A roof may be provided with or included in a physical structure. A physical structure with a roof disposed thereon may fully enclose a volume, for example in combination with a floor included in the physical structure or with bare ground (e.g., dirt or concrete) on which the physical structure is located. (Generally, a floor included in a physical structure would include only one panel but may include multiple panels, for example to promote compactness in a flat-folded state.) A roof may be a separate piece from a kinematically compatible physical structure (e.g., a flat sheet of material added to the top of a physical structure after it is expanded). Such a separate piece may be temporarily secured to the physical structure during use. A roof may also be part of a physical structure in the sense that it is connected to one or more wall(s) (e.g., by one or more hinges). A roof may therefore be considered a “top” wall in a physical structure. (A floor may therefore also act as a “bottom” wall in a physical structure.) As with other walls disclosed herein, a roof may include only one panel (e.g., like a back wall in certain embodiments) or may include multiple panels. Physical structures may be designed that include one of any number of roof styles. For example, a roof may be a flat roof, a shed roof, or a gable roof. Different roof structures can be employed with only minimal adjustments made to panels in walls, thereby allowing easy customizability for a physical structure to have a desired roof style without compromising flat-foldability.

[0184] FfGs. 16D-F show examples of physical structures 10 that include roofs 28. In FIG. 16D, the physical structure 10 includes a flat roof 28. However, the flat roof 28 has a second panel 22 that folds down to extend the height of front wall 20 while maintaining overall kinematic compatibility (adding the height to panels 22 included in front wall 20 instead would inhibit flat folding). FIG. 16E illustrates a gable roof 28. The gable roof 28 includes three panels 22 that rotate about three hinge lines 30 (one of which is where the roof 28 is connected to the back wall 20). In FIG. 16F, the roof 28 of the physical structure 10 is a shed roof that includes one panel 22 connected to the back wall 20.

[0185] In some embodiments, a roof is lifted by moving walls from a flat-folded state to an expanded state. FIGs. 16B-D, 16F, 35, 37A-37B, and 41-46 show examples of such movement. Such physical structures can be advantageous to simplifying installation of a roof (e.g., to fully enclose a volume) as compared to previous technologies which required separated lifting and installation of a roof. In some embodiments, an initial movement of one or more panels of a roof followed by movement (e.g., with a single degree of freedom) can result in a roof being properly disposed on a physical structure (e.g., by first unfolding a roof).

Hinges

[0186] Adjacent panels in a physical structure may be connected together by one or more hinges. Adjacent walls in a physical structure may be connected together by one or more hinges. In some embodiments, a hinge provides a connection (e.g., between adjacent panels). In some embodiments, a hinge restricts all but one degree of freedom: rotation about one axis. A hinge may be a butt hinge, barrel hinge, piano hinge, pivot hinge, overlay hinge, offset hinge, strap hinge, scissor hinge, spring hinge, flag hinge, flexible material hinge (e.g., a membrane material or single-sided adhesive sheet). A combination of such hinges may be used within a single physical structure. A hinge may be a molded hinge that is formed at least partially in a wall (e.g., in one or more panels thereof). For example, such as in a butt hinge, flag hinge, or piano hinge, a hinge may include a knuckle that is formed by a portion of a panel; a separate pin may be provided. In some embodiments, a molded hinge includes one or more knuckles formed from a first panel and one or more knuckles formed from a second panel and a pin. The first panel and the second panel may be on different walls. In some embodiments, a wall includes multiple panels where each pair of adjacent panels is connected by at least one hinge. In some embodiments, at least one pair of adjacent panels in a wall are not connected by any hinge. In some embodiments, the at least one pair of adjacent panels in the wall are not connected at all. [0187] In some embodiments, a first panel is connected to a second panel along a first edge of the first panel by a first hinge disposed at an interior surface of the first panel and the first panel is connected to a third panel along a second edge of the first panel (different from the first edge) by a second hinge disposed at an exterior surface of the first panel (e.g., in an offset hinge technique) (e.g., as shown in FIG. 12B). The first, second, and third panels may be panels in a same wall. The first panel and one of the second and the third panel may be panels in a same wall while the other of the second and the third panel is in a different wall (e.g., as shown in FIG. 12B).

[0188] In some embodiments, a physical structure includes one or more flexible hinges. A flexible hinge may include an elastomeric material. A flexible hinge may act as a seal (e.g., to water and/or air from outside the physical structure). Tn some embodiments, a flexible hinge runs along an entire length of a hinge line. In some embodiments, a flexible hinge runs along an entire length of an edge of a panel, for example along an entire length of both panels connected by the flexible hinge. Such flexible hinges may act as seals, for example when constructed with an impermeable (e.g., to water) material. In some embodiments, a physical structure is sealed when in an expanded state. A physical structure may include one or more seals (e.g., one or more gaskets). For example, each seal may be disposed along an entire length of an edge of a respective panel. In some embodiments, a (e.g., each) seal is at least partially compressed when a physical structure is in an expanded state. In some embodiments, a physical structure includes a seal for each hinge line. In some embodiments, a physical structure includes a seal for each edge of each panel (e.g., that runs along the edge of the panel). A seal may be shared for edges of adjacent panels that contact when a physical structure is in an expanded state (one seal that seals the contact between the edges of the panels, whether the panels are connected or not). Techniques described herein may facilitate incorporation of seals without including flat-folding or kinematic compatibility.

[0189] FIG. 29 illustrates an example of a molded hinge 32 that is a butt hinge. One panel 22 includes a single knuckle 34 for the molded hinge 32. Another panel 22 includes two knuckles 34. The molded hinge 32 also includes a pin 33 that is not a part of either panel 22. When the pin 33 is inserted through the three knuckles 34, the panels 22 are connected together are operable to rotate about hinge line 30. Such hinges may be used in physical structures 10 disclosed herein.

Locking Mechanisms

[0190] A structure may be sufficiently stable in an expanded state (e.g., due to resistance provided by one or more mechanical assistance components, as described further subsequently). In some embodiments, one or more locking mechanisms may be used to “rigidize” a structure when it/they are locked (engaged). Locking mechanism(s) may prevent partial or total collapse of a physical structure from an expanded state. Collapse may occur, absent the locking mechanism(s), due to an applied force on the physical structure, whether from the environment (e.g., wind) or artificially applied (e.g., by someone pushing or leaning on the structure). Numerous different locking mechanisms may be used including, but not limited to, bent sliding bars, latches (e.g., draw latches), cam locks, straps, buckles, clamps, clasps, or pins. Two or more different types of locking mechanisms may be used in a single physical structure. Generally, a locking mechanism has a locked position or state that prevents folding along one or more hinge lines (e.g., by spanning the hinge line). A locking mechanism may intersect a hinge line. A locking mechanism generally acts on two adjacent panels, whether within the same wall or between adjacent walls, but may span across more than two panels, for example a strap that acts across three panels within a wall. A locking mechanism need not necessarily act on two adjacent panels, for example may span to different, non-adjacent panels (e.g., on different walls). A locking mechanism may be, for example, a brace that braces panels (e.g., on different walls) to prevent undesired unfolding. A locking mechanism may be integral with a physical structure or may be an accessory part that is applied to (e.g., installed on) the structure, for example after expansion.

[0191] Where a physical structure has a single degree of freedom, a single locking mechanism may be sufficient to fully rigidize a physical structure. Nonetheless, even where a physical structure has a single degree of freedom, additional locking mechanism(s) may be provided, for example in order to provide redundancy, additional rigidity (e.g., to limit panel flex), multiple options for a person to secure the physical structure (e.g., if the structure is setup in a location that blocks one or more of the locking mechanisms). In some embodiments where a physical structure has multiple degrees-of-freedom, multiple locking mechanisms may be used (e.g., needed) to fully rigidize the structure. It is desirable for a locking mechanism to not impede folding and unfolding of a physical structure when it is in an unlocked (disengaged) position. For example, a locking mechanism may be intentionally designed to be thin, be disposed on an outer surface of a physical structure when the physical structure is in a flat-folded state, or recessed within (e.g., disposed within a recess) a panel of a physical structure so that it does not protrude therefrom. In some embodiments, a locking mechanism, or portion thereof, is applied to panel(s) after a physical structure is moved to an expanded state. For example, a strap or latch (e.g., draw latch) may be applied to attachment points on two adjacent panels after expansion. A portion of a locking mechanism may be formed from a panel (e.g., a molded portion of the panel).

[0192] An illustration of one possible rigi dizing method for physical structure 10 using locking mechanisms 40 is provided in FIGs. 30A-B. The locking mechanisms 40 are disposed at a subset of the hinge lines 30 in the physical structure 10 and are sufficient to prevent the physical structure from moving between an expanded state (shown in FIG. 30A) and a flat- folded state. Physical structure 10 includes roof 28, which may be secured by an internal locking mechanism (not shown) or may be freely moveable even when the physical structure 10 is rigidized by the locking mechanisms 40. FIG. 30B shows a series of steps to move one of the locking mechanisms 40 from an unlocked position to a locked position. When the one of the locking mechanisms 40 is unlocked, the hinge line intersection portion 42 does not intersect the hinge line 30 at which the locking mechanism 40 is disposed. When locked, the hinge line intersection portion 42 intersects the hinge line 30, thus preventing folding along the hinge line 30. The lock/unlock maintaining portion 44 prevents undesired movement between locked and unlocked position. To move from locked to unlocked (or vice versa), someone can rotate the bent sliding bar around its hinge line intersection portion 42 so that its lock/unlock maintaining portion 44 is not obstructed by recessed portions of panel(s) 22 and can slide up and down. Rotating back into a recessed portion of panel(s) 22 after sliding up or down fixes the locking mechanism 40 in the other position (locked or unlocked, depending on direction). FIGs. 30C- 30D show alternative locking mechanisms 40 of a draw latch (FIG. 30C) and a strap (FIG. 30D).

Mechanical Assistance Components

[0193] While origami is easily manipulable by hand, larger structures, especially those made with rigid sheets of material for panels, may require significant force to expand or fold. In some embodiments, a physical structure is moved between a flat-folded state and an expanded state (whether folding or expanding) by one or more persons. If the physical structure is sufficiently low weight (e.g., due to its small size) and/or any hinges/connections are sufficiently low friction, such movement may be trivial to make. For example, if a physical structure fits on a table, it is likely that a single individual can expand and fold the physical structure without trouble. Such motion may be trivial in some single degree of freedom structures and in some multiple degree-of-freedom structures. However, when a physical structure is heavier and/or larger, for example when dimensioned to be habitable, it may be difficult or even impossible for a single person (or multiple people) to fold or expand a physical structure. Accordingly, one or more mechanical assistance components may be provided that lower (or eliminate) the amount of force required to move a physical structure to an expanded state from a flat-folded state. [0194] In accordance with various embodiments, any of a variety of active or passive mechanical assistance components may be used. For example, an electronically controlled actuator may be used as an active mechanical assistance component or a torsion spring may be used as a passive mechanical assistance component. Hydraulic, electric, or pneumatic active or passive mechanical assistance components may be used. For example, a motor (e.g., that includes a gear box) may be used to provide rotation to assist in moving between a flat-folded state and an expanded state. Mechanical advantage could be created by incorporating a load reducing component with a mechanical assistance component (e.g., a long beam that acts as a lever arm). Passive mechanical assistance components are generally simpler and generally facilitate easier fabrication of physical structures. In some embodiments, a mechanical assistance component can be engaged or disengaged from a physical structure (e g., electronically or mechanically). A mechanical assistance component may be engaged in order to provide assistance in expansion into an expanded state and disengaged to not hinder folding into a flat-folded state. For example, in some embodiments, an electronically controlled actuator can be turned on or off or a torsion spring can be designed to be engaged or disengaged (or to be added or removed). In some embodiments, a mechanical assistance component is asymmetric so that it provides assistance during expansion but does not resist (or resists to a lesser extent) folding, even while remaining engaged. Similarly to locking mechanisms, mechanical assistance components may be arranged in recessed portions of panel(s) to facilitate flat folding. Moreover, a portion of one or more panel(s) may form a portion of a mechanical assistance component (e.g., attachment point(s) that another portion of the mechanical assistance system connects to). In some embodiments, two panels are connected together with one or more mechanical assistance component(s) (e.g., torsion spring(s) and/or actuator(s)).

[0195] In some embodiments, shelter concepts may include large connected hard-shell panels that fold on to each other when packaged for transportation. Consequently, the expansion of such shelters requires a varying amount of force depending on the weight of each panel. To bring the required expansion force to a manageable level, active and/or passive mechanical assist component(s) can be integrated within the physical structure. Such assistance can be provided by, for example, spring(s), like torsion springs, which serve a similar function in other applications such as lifting the hood of a car or a garage door. In some embodiments, the springs could be loaded when the shelter is in its flat-folded state (e.g., for transportation state), with the stored energy released when the structure is unpacked and moved to its expanded state. Torsional springs in specific folding regions can be considered as the passive energy storage device that take advantage of the weight of the structure when it is collapsed.

[0196] Torsion springs may be placed in locations to maximize their ability to store energy, while not impeding the motion of the shelter during expansion. Certain constraints may exist or be desirable to apply when configuring a physical structure. For example, total spring length, outer coil diameter relative to wire diameter, spring size(s) (e.g., coil or wire diameter) relative to panel size (e.g., thickness), fatigue factor of safety, factor of safety against yielding, or a combination thereof may be deliberately constrained in configuring a physical structure. In some embodiments, a total length of spring(s) on a hinge line does not exceed 50% (e.g., does not exceed 40%, does not exceed 30%, or does not exceed 20%) of a length of the hinge line (e g , of a side length of a panel corresponding to the hinge line). In some embodiments, an outer coil diameter of a torsion spring is no less than three times (e.g., no less than four times, no less than five times, or no less than ten times) a wire diameter of a wire from which the torsion spring is formed. In some embodiments, an outer coil diameter of a torsion spring is no more than a thickness of a panel in a physical structure. For example, an outer coil diameter of a torsion spring disposed at a hinge line may be no more than a thickness of a smallest of the two panels that are connected at the hinge line. In some embodiments where a panel has a variable thickness, a smallest of the two panels refers to the panel with the thinnest local thickness where the torsion spring is connected. In some embodiments, a fatigue factor of safety of a torsion spring is at least 1.25 (e.g., at least 1.5 or at least 2). For example, a fatigue factor of safety against bending of a torsion spring may be at least 1.25 (e.g., at least 1.5 or at least 2). In some embodiments, a factor of safety against yielding of a torsion spring is at least 1.25 (e.g., at least 1.5 or at least 2). Having a sufficient factor of safety (e.g., against bending or against yielding) for torsion spring(s) in a physical structure may ensure reusability of the physical structure as it is able to withstand repeated expanding and flat folding. Stiffness depends on wire diameter to the fourth power (see subsequent Eq. 16) and therefore wire size is a particularly important constraint to consider when designing a torsion spring.

[0197] An analytical model used to determine the expansion loads for thin and thick shelter designs can be expanded to include the benefits of incorporating torsion springs. The virtual work analysis requires the inclusion of contributions by torsion springs. The two-panel (clamshell) model is used to introduce the formulation for a torsion spring along a single hinge line as shown in FIG. 31 A. The torque created on hinge line 1, Ts, is simply the spring torsional stiffness k (Ib-in/deg) multiplied by the deflection of the spring, 0 (Ts = k0). The virtual work of a spring is simply the torque about the hinge line multiplied by the virtual angular displacement 66 or Uk=Ts(80).

[0198] Therefore, the virtual work of a torsion spring of stiffness kl located at hinge line 1 is calculated as

[0199] Note that the term 0FI signifies the free angle of the spring on hinge line 1. The free angle is the angle that the two legs of the torsion springs make with one another when no load is applied. An example of several common free angles are illustrated in FIG. 3 IB.

[0200] The torsional stiffness of the spring can be determined from its geometric and material properties. The wire diameter, outer coil diameter, number of coils, modulus of elasticity, free angle, and leg length all contribute to the torsional stiffness of the spring in the pertinent equations.

[0201] The number of equivalent coils can be expressed in terms of each leg length (LI and L2) and the mean coil diameter (D), which is simply the outer coil diameter ( _o ) minus the wire diameter (d).

[0202] The number of active coils must be determined from N _e and the number of coils in the body Nb).

[0203] Finally, the torsion stiffness k is found using N _a , d, D, and the modulus of elasticity of the spring material (Mod). The torsional stiffness for torsion springs of round wire in terms of units of torque per revolution is shown in Eq. 16.

[0204] The constant in the denominator is increased from 10.2 to 10.8 to account for friction between the coils, in accordance with suggestions in the literature. Different geometric and material properties can be chosen to yield different spring stiffness values. This equation does not, however, determine if a certain set of parameters will yield a spring that can be manufactured or one that does not yield or fail in fatigue, which may be an important consideration for certain use cases of physical structures.

[0205] To further detail the addition of mechanical advantage in the analysis of a system of folding panels, a torsion spring is placed along hinge line 1 of the thin four-panel model as shown in FIG. 17.

[0206] With 9i as the driver angle (the angle considered for motion in a single degree-of- freedom system that results in expansion of the physical structure), the virtual work for the model in FIG. 17 is formulated as

[0207] Expressed in terms of the virtual displacement 80i and simplified, Eq. (17) becomes

[0208] The nontrivial equation for the torque becomes:

[0209] The analytical model can be used to derive a relationship between expansion torque and angle 0i where torque decreases with increasing angle, and agrees with simulation results obtained with SolidWorks.

[0210] If a set of one or more torsion springs is instead placed along hinge line 2 (torsion springs not shown in FIG. 17), the expansion load requirements will be different. Changing the location of the torsion spring will affect the load requirements. The governing virtual work equation including the contribution of the spring is given as

[0211] The only change desired is the location of the spring, so the torsional stiffness k2 and 9F2 are assumed to be the same as the previous case.

[0212] Writing the equation in terms of the virtual displacement 801 and simplifying the result gives [0213] The virtual displacement term cannot go to zero, so the equation for the torque must be

[0214] From Eq. (21), it is obvious that not only the change from 0i to 02 will contribute to a change in required expansion torque, but also the derivative of 02 with respect to 0i. The analytical model can be used to derive a relationship between expansion torque and angle that agrees with simulation results obtained with SolidWorks.

[0215] Analyzing the required amount of torque needed to maintain equilibrium about hinge line 1 using these models shows that torque requirements can be reduced significantly when using a spring as compared to not using a spring - up to a factor of ~10 for small 0i angles. Therefore, mechanical assistance components, such as torsion springs, can significantly aid to moving between a flat-folded state and expanded state. In some embodiments, a physical structure as disclosed herein can be expanded in a period of no more than 5 minutes (e.g., no more than 2 minutes).

[0216] FIG. 31C illustrates examples of torsion springs 52 attached to panels 22. In both illustrations, the outer diameter of the torsion spring 52 is smaller than the smallest panel. In both illustrations, the torsion spring 52 connects two panels 22 at a hinge line 30. In the bottom illustration, the torsion spring 52 connects to one of the panels 22 at a tab 24 that extends from an end of the panel 22.

[0217] FIG. 32 illustrates a physical structure in order to highlight the effect that the placement of torsion springs can have on the expansion load requirements. Due to symmetry, six different hinge lines were investigated. The hinge lines are numbered according to the connecting panels with “B” signifying a connection to the floor (base). Results obtained using an analytical model based on the foregoing paragraphs, and verified with SolidWorks, show that for all applied torque locations, the highest reduction in required torque occurs when a torsion spring is placed along hinge line 23. The next most efficient hinge placement is along hinge line 45, meaning that the two 180-degree hinge lines are desirable locations to place torsion springs. The analysis performed contemplated only using torsion spring(s) along a single hinge line. A very significant reduction in expansion torque is possible if torsion springs are incorporated along multiple or all hinge lines. [0218] In some embodiments, one or more actuators may be incorporated into a physical structure to assist in expanding a structure from a flat-folded state to an expanded state. In some embodiments, no additional force need be applied beside actuation from actuator(s) in order to move a physical structure to its expanded state. An actuator may be passive (e.g., pneumatic) or active (e.g., electronically controlled). FIG. 17 shows an example of how an actuator 50 could be positioned. With reference to FIG. 32, an actuator could be incorporated at any of the highlighted hinge lines to assist in expanding the structure. In a single degree-of-freedom system, one actuator can be sufficient, and may be all that is desired, to achieve expansion. In some embodiments, more than one actuator is used, for example one for each of a subset of hinge lines. In some embodiments, multiple actuators are used so that applied torque can be minimized by changing from pushing with one actuator to pushing with another after some critical angle In some embodiments, an actuator is used to apply a force but is not permanently integrated into a physical structure (e.g., is removed after expansion and/or to allow for folding). In some embodiments, one or more winches may be used with a physical structure to assist in expanding a structure from a flat-folded state to an expanded state. That may involve incorporating one or more cables for attaching the winch(es) into the physical structure.

Additional Examples of Physical Structures Having Kinematic Compatibility

[0219] FIGs. 33A-51 show additional examples of physical structures in accordance with embodiments of the present disclosure. It will be appreciated by those of ordinary skill in the art that feature(s), such as, for example, (i) folding, including hinge line placement, (ii) panel number, shape, and arrangement within a wall, (iii) number, shape, and arrangement of walls, (iv) size, (v) folds, (vi) hinges, (vii) hinge presence and locations, and (viii) mechanical assistance component operability and location, from one or more of these examples can be combined with feature(s) from one or more others of these examples can be combined to form additional embodiments that are contemplated to be within the scope of the present disclosure. [0220] A 3D-printed prototype physical structure 10 has been created and is shown in FIGs. 33A-33D. The prototype incorporates a mechanical assist system that includes multiple mechanical assistance components. In this example prototype, the one or more mechanical assistance components are torsion springs 52. The molded hinges 32 disposed at the hinge lines 30 include pins 33 (aligned along the hinge lines 30) that pass through knuckles 34 that are formed by a portion of panels 22. In this case, additive manufacturing (3D printing) facilitates facile fabrication of knuckles 34 as a portion of panel 22 (such that at least a portion of hinges 32 are formed by a portion of panels 22). The hinges 32 appear large based on their design but those of ordinary skill in the art will appreciate that smaller relative hinge sizes can be used, whether by scaling down the hinges or using the same size hinges in a larger size physical structure. Only one hinge 32 is shown per hinge line 30 (which is disposed at abutting edges of adjacent panels 22). The 3D-printed version of Model 1 shown in FIGs. 33A-33D is shown at various stages of expansion in FIG. 33A and at different views in the expanded state in FIGs. 33B-D. Large hinges were used in the prototype for simplicity. It is possible to significantly decrease the size of hinges (e.g., knuckle(s) and/or pin(s)) relative to a size of the physical structure (e.g., width, length, and/or height dimension(s) and/or panel thickness(es)) from what is shown in the prototype, if desired.

[0221] In this example, the panels are all printed separately and connected together hinges 32 that include knuckles 34 and a pin 33. The fabrication of the printed physical structure shows the feasibility of the design and allows a better understanding of the movement of the structure. Note the addition of torsion springs 52 along various hinge lines 30. The torsion springs 52 are aligned perpendicular to hinges 32 but other relative orientations can be used (e.g., depending on the torsion spring 52 design used). The torsions springs 52 aid in the ease of expanding the structure and prohibit the structure from (partially or fully) collapsing once it is in the expanded state.

[0222] An illustrative physical structure 10 is shown in FIG. 34. This physical structure 10 eliminates the use of SFE by using one solid rectangular panel 22 in its place. The front wall 20 has three trapezoidal and one rectangular panels 22, with the rectangular panel 22 folded open to close the gap created where SFE was located in Model 2. Connecting this small rectangle also has potential to aid in rigidizing the structure once expanded. Note that no roof panel 22 is included for clarity of interpretation of the figure, but one could be included. Note also that the rectangular panel 22 on the front wall 20 introduces a second degree of freedom to this example. [0223] FIG. 35 illustrates an example of a physical structure 10 in accordance with embodiments of the present disclosure. The physical structure 10 is a rectangular shelter that folds into a rectangular footprint. This structure 10 has dimensions that are restricted to the width being at least twice the height, and the length being at least three times the height in order to maintain kinematic compatibility. The structure 10 has two rectangular end walls 20 and side walls 20 that consist of a trapezoid panel 22 and two 45-degree right triangle panels 22. The triangle panels 22 allow for collapse of the end walls 20. The roof 28 includes five rectangular panels 22 that fold on top of one another as the structure collapses. A method of unfolding the physical structure 10 from its flat-folded state (top left) to its expanded state (bottom right) includes steps of: (i) anchor down the physical structure; (ii) pull on the side rectangular walls 20; and (iii) once fully expanded, rigidize the hinge lines with locking mechanisms (not shown in FIG. 35). A mock-up the physical structure 10 shown in FIG. 35 made of foam board and tape is shown in FIG. 36. The ability to successfully fold and unfold the physical structure validates feasibility (e.g., kinematic compatibility) of the design.

[0224] FIGs. 37A-B show an alternate physical structure 10 that is similar to the structure shown in FIGs. 35-36 but with a roof 28 that includes only a single panel 22. FIG. 37A shows side views during expansion while FIG. 37B shows isometric type views during expansion. Because the top edges of the side rectangular walls 20 of the physical structure 10 are always at the same height, a roof panel that slides over the top with rails or slots could be utilized to enclose the structure (such a roof 28 is still connected to side walls 20 but just not with a hinge). A side view of the shelter can be seen in FIG. 37A, showing that the side rectangular panels 22 of side walls 20 maintain the same height as each other throughout expansion. This structure 10 has dimensions restricted to the width and length being at least twice that of the height of the structure 10 to maintain kinematic compatibility. The length and width, however, have no upper bound.

[0225] FIG. 38 illustrates an additional physical structure 10 in accordance with embodiments of the present disclosure. The physical structure 10 includes rectangular front and back walls 20 and square side walls 20. The physical structure 10 incorporates what is commonly referred to as a water bomb origami base pattern in the side walls 20. A similar “stretched” water bomb pattern is used on each rectangular wall 20 of the shelter. The structure folds flat into a rectangular footprint with a folding process that allows only one direction of motion of the roof 28, which is in the vertical direction. The physical structure includes a roof 28 that includes a single panel 22 and a floor 26 (not visible) that includes a single panel 22. The water bomb base pattern is a square pattern that can collapse into a triangle. The stretched side pattern folds into a trapezoid. A more detailed image of the water bomb folding pattern can be seen in FIG. 39. A physical model of this pattern as a multi-fold panel was fabricated using PET foam core and two plies of PETG.

[0226] Notably, the use of the water bomb fold requires that some edges of adjacent walls 20 be unconnected to allow folding (though the walls remain indirectly connected through other routes). For example, the vertical edges where adjacent perimeter walls 20 (e.g., a side wall 20 and the front wall 20 or a side wall 20 and the back wall 20) meet when the physical structure 10 is in the expanded state are not connected. Nonetheless, the walls 20 of the physical structure 10 are connected together, with side walls 20 being connected to the front and back walls 20 through respective connections to roof 28 (and floor 26, not shown). In other physical structures without water bomb folds, there is a connection for each pair of adjacent walls for at least some of the walls (e.g., wherein a back wall and a floor are not connected). Similar connectivity or non-connectivity may exist for panels within a given wall. Tn some embodiments, locking mechanism(s) are provided at non-connected edges to stabilize the walls or panels that meet there in the expanded state. For example, the vertical edges between a side wall 20 and the front or back wall 20 in the physical structure of FIG. 38 may be locked together in the expanded state (since no hinge exists to keep them together). Note that these vertical edges are not hinge lines. A physical structure that includes a water bomb fold move to an expanded state with a single degree of freedom or may require manipulation of more than one degree of freedom.

[0227] Because of the symmetry of the folding process, the physical structure 10 shown in FIG. 38 provides potential for modularity. The structure 10 could be created without the two square end panels, and successively joined to create an extended rectangular footprint of tailorable size. The structures 10 could then be expanded simultaneously as shown in FIG. 40. The process could also begin with expanding each structure individually, and combining them after being rigidized. End panels could then be attached to either end of the modular system to enclose the structure and potentially aid in rigidity. Alternatively, some physical structures 10 could be provided with a single side wall 20 to act as end units in the overall modular structure, in order to enclose it.

[0228] FIG. 41 illustrates a physical structure 10 that folds similarly to the physical structures shown in a modular arrangement in FIG. 38 but having a square footprint (e.g., cubic shape) instead of rectangular. This structure 10 would collapse into a square footprint. This physical structure 10 can also be combined with either similar or dissimilar structures with the same folding process. A modular system combining four of the physical structures of FIG. 41 can be seen in FIG. 42. Note that this modular system is arranged as a 4x1 row but other array dimensions can be achieved (MxN in size generally) and by selective positioning of walls 20, internal structure can be given such as rooms and hallways. Doors to provide entry and exit to rooms could be installed after expansion. In some embodiments, a physical structure includes one or more panels that are solid (e.g., made of sheet(s) of rigid material) and one or more panels that are frames (e.g., including rods of material defining the perimeter of the panel(s)). Such a combination would allow for a construction of a larger room from smaller physical structures where the frame panels (or walls made of frame panels) act as support structures for a roof for the room and/or facilitate easy expansion and flat-folding (e.g., using a single degree-of-freedom motion).

[0229] The water bomb pattern folds compactly, so it was used in multiple concept designs. FIG. 43 illustrates a hexagonal physical structure 10 that employs six water bomb patterns, one on each side of the hexagon. This physical structure 10 can serve as a central hub for a modular system of structures (e.g., shelters). In part attributable to the utilization of the water bomb pattern, this concept has great potential for modularization. If the dimensions of the hexagonal roof 28 and base 26 (not shown) are created such that sides are equivalent in magnitude as other physical structures (e.g., shown in FIGs. 38 and 41), the shelters could be connected to an end side of the hexagon. Each connection would remove one water bomb pattern, and instead have an open end that would house a connection to another physical structure. An example of a modular system that combines the physical structure of FIG. 43 with the physical structure of FIG. 38 can be seen in FIG. 44.

[0230] Designs that incorporate connected water bomb patterns have panels (e.g., on adjacent walls) that are in contact when expanded, but move away from one another as the structure collapses as shown in FIG. 45. This combination poses both a challenge in rigi dizing the structure and weather sealing. Proper seals (e.g., gaskets) and/or rigidizing methods would eliminate this concern. For example, gasket(s) may be included at adjacent edges of panels (e.g., on adjacent walls) that is/are sufficiently compressed (e.g., to prevent water and/or air penetration) when a physical structure is in an expanded state. Designs that incorporate this pattern may also have solid panel roofs, which is ideal for both structural integrity and waterproofing. The symmetry of the folds allows for this, and facilitates simple expansion. Simple extension of the structure vertically and potentially a push or pull on one edge of a triangle of the water bomb pattern would be sufficient to expanded the structure.

[0231] Thickening the panels adds complexity and introduces numerous challenges in origami-inspired design. For example, when successive water bomb panels of finite thickness are joined, the comers of the panels can intersect, thus interfering with the folding operation. An example of this can be seen in a CAD image of a physical structure shown in FIG. 46. With thick panels, the inner and outer surfaces are at some finite distance apart, giving rise to multiple options for hinge location along an edge in the thickness direction. For two thick panels, precisely locating a hinge line to coincide with opposite sides of panels facilitates a proper folding process. Hinges can be constructed and attached to appropriately align surfaces of adjacent panels to fold and expand unimpeded in a water bomb fold.

[0232] An example of a hinge shift technique was discussed above. In some embodiments, hinge lines are placed on inner and outer edges of each panel. In some embodiments, a comer vertex fold behaves the same way, but now the back panel is forced to raise and lower throughout the folding process, indicating that the back fold does not mirror zero-thickness kinematics. This means that the back panel cannot have a hinged connection to the base and requires two symmetric comer folds. The folding process for this model is shown in the illustrative physical structure 10 of FIG. 47. The angles of the two-wall model are maintained in the thick folding of a two-comer vertex thick fold due to the same folding procedure and parallel panel faces. The front of this fold remains in the same plane and contains a simple folding of one panel on one hinge line 30.

[0233] In order for the large triangle that includes the majority of the front wall to fold flat with the other panels, it must be connected to the triangles on either side of the front wall. This means that the front wall will fold lower than the level of the hinge line on the sides of the structure. Maintaining kinematic compatibility can be achieved by constructing the floor panel to have additional tabs to allow the large triangle to nest inside of it. Due to the thickness allocation to preserve flat-folding, some locations on the structure may have a local increase in panel thickness. The added tabs, local thickness increase, and nesting of the panels are shown in FIG. 48. [0234] In order to create a fully enclosed shelter, the opening on the front of the shelter created by the added tabs can be sealed. This can be accomplished by extending a front (e.g., triangular) panel to create an overhang that acts as a flap that seals the opening during the folding process. This extension and sealing process is shown in FIG. 49.

[0235] While certain physical structures disclosed herein are designed to be a single degree-of-freedom system, the roof panel may or may not be considered a part of this single degree-of-freedom system, in accordance with various embodiments. A roof can be integrated into a physical structure to fully seal and cover the structure. The roof can be attached to the inner top edge of the back panel and is comprised of two panels joined at one hinge line. The size of each roof panel can be dependent on, for example, the extension of the side trapezoid and the height of the structure. FIG. 50 shows an example of a physical structure that includes a roof. While the roof may initially need to be unfolded (e.g., manually), once unfolded the structure can be expanded with a single degree of freedom and therefore may be considered a structure that has a single degree of freedom. The expansion process may include: (i) anchoring down the physical structure, (ii) folding over the roof, (iii) locking one or more roof hinges, (iv) expanding the structure, and (v) rigidizing with locking mechanism(s).

[0236] FIG. 51 illustrates examples of kinematically compatible thin-walled structures 10. (Analogous thick- walled physical structures are contemplated, using appropriate thickness(es) and/or folding technique(s) (e.g., in accordance with FIG. 26 panel (a)).) Characteristic dimensions E, H, w, and L are labelled. A desired overall size and shape of the physical structure 10 can be achieved while maintaining kinematic compatibility by controlling these characteristic dimensions. For example, the top schematic on the right illustrates that kinematic compatibility can be maintained when L = 0 and w = 0 if E is sufficiently larger than 0. As shown in the middle schematic on the right, if E = 0 while w = 0, kinematic compatibility can be maintained if L is sufficiently larger than 0. As shown in the bottom schematic on the right, some combination of L > Q, E > 0 and w > 0 can be used to achieve/maintain kinematic compatibility. The precise relative values of these characteristic dimensions can be selected based on desired geometry (e.g., in a flat-folded and/or expanded state) and/or desired size or relative overall dimension (e.g., height to width, width to length, length to height). In these illustrated examples, a front wall 20 may have three panels 22 (e.g., top schematic) or more than three panels (e.g., at least 6 panels) (e.g., middle and bottom schematics). Side walls 20 include two panels 22 where one panel 22 is a triangle and one panel 22 is a trapezoid, though other shapes for such panels may be used. Back wall 20 includes only one panel 22.

Flat-Folded State

[0237] In some embodiments, panels are flat (e.g., planar). For example, FIGs. 12A-16F illustrate flat panels. In some embodiments, a wall includes panels that are in direct contact (e.g., stacked or side by side) in a flat-folded state. In some embodiments, panels from different walls are in direct contact (e.g., stacked or side by side) when in a flat-folded state. In some embodiments, at least a portion of one panel is nested in at least a portion of another panel in a flat-folded state. For example, FIGs. 12A (panel b), 12B, 19, and 26 illustrate examples of such arrangements. In some embodiments, a lateral extent (e.g., footprint) of a physical structure is no larger than an area of a wall (e g., a back wall, a roof, or a floor) or a floor when the physical structure is in a flat-folded state (e.g., if at least one of the walls includes a single panel). For example, FIGs. 13, 14, 16A-F, 19, 33A-35, 38, 39, and 47 illustrate some such arrangements. In some embodiments, a lateral extent of a physical structure is smaller than any wall of the physical structure (in an expanded state) when the physical structure is in a flat-folded state (e.g., if each wall includes multiple panels). In some embodiments, a flat-folded physical structure occupies a volume no more than 50% (e.g., no more than 40%, no more than 30%, no more than 20%, or no more than 10%) of a volume that the physical structure occupies in an expanded state. Therefore, in some embodiments, physical structures can be efficiently transported and then expanded once delivered to their location(s) of intended use. In some embodiments, a physical structure has a lateral extent in a flat-folded state that is less than a total area of the walls (e.g., including any roof and/or floor) of the physical structure in an expanded state. For example, the lateral extent may be no more than 75%, no more than 50%, no more than 33%, no more than 25%, or no more than 20% of the total area. In some embodiments, a physical structure has a lateral extent in a flat-folded state that is no more than an area of a largest wall (e.g., a front wall, a back wall, a side wall, a roof, or a floor) of the physical structure in an expanded state. In a flat-folded state, a physical structure is both flat and folded (not unfolded). In some embodiments, a physical structure in a flat-folded states has some thickness corresponding to a sum of thicknesses of multiple walls of the structure. [0238] It may be desirable to use fewer hinge lines where feasible given other constraints (e.g., dimension(s) of the enclosed volume in an expanded state). Reducing the number of folds in a physical structure (while maintaining other desired features, such as dimension(s)) can improve manufacturability. Additionally, increasing the number of hinge lines (e.g., by introducing a higher degree vertex or higher degree vertices) generally increases the degrees of freedom of that fold (e.g., as with the degree-4 vertices discussed previously), and may provide a more unpredictable, uncontrollable, and inconsistent movement between a flat-folded state and an expanded state. Kinematic compatibility and/or flat-folding may prohibit use of higher degree vertex/vertices in some cases.

Physical Structures Having a Flat Unfolded State

[0239] In some embodiments, a physical structure includes connected panels that are movable between a flat unfolded state and a folded state. The panels at least partially (e.g., entirely) enclose a volume in the folded state. Each of the panels may be included in a wall of a set of one or more walls (e.g., two or more walls). Each wall may, independently, include one panel or may include multiple panels. The walls are connected together, for example by one or more hinges, such that they are movable between a flat unfolded state and a folded state. In the folded state, walls at least partially enclose a volume. The volume may be sufficiently large to be habitable, such that the physical structure can act as a shelter, for example for humanitarian or military purposes. In this way, in some embodiments, a physical structure operates in an inverse manner to other physical structures disclosed herein.

[0240] A wall may include more than one panel. Panels may be connected at their edges such that, in a folded state, they collectively form a flat wall. Walls and panels are said to “fold.” Folding occurs along hinge lines. Folding along a hinge line may be facilitated by one or more hinges disposed along the hinge line. The hinges provide freedom of rotation. No other freedom of movement may be intentionally provided by hinge(s) other than freedom of rotation (e.g., some natural, insignificant deflection may occur due to, for example, panel or wall weight). In some embodiments, a hinge, such as a flexible material hinge, facilitates some minimal translation motion in addition to rotation. Hinge lines will naturally occur at locations where edges of adjacent connected panels abut. In some cases, edges of adjacent walls that abut are connected together and, in some cases, edges of adjacent walls that abut are not connected together. In some embodiments, including ones where a water bomb fold is included (as discussed further below), some adjacent walls are connected (e.g., a roof and a side wall) and others are not (e.g., the side wall and a front wall). Thus, walls are connected together to facilitate movement between a flat unfolded state and a folded state but it is not necessary that amongst these connected walls each pair of adjacent walls be connected.

[0241] The following description provides illustrative embodiments of physical structures in addition to details of components, such as locking mechanisms and mechanical assistance components that may be used in physical structures. Locking mechanisms may be used to ensure a physical structure remains in a folded state thereby “rigidifying” the structure (and can be unlocked for unfolding). Mechanical assistance components may be used to reduce the amount of force that need be applied to move a physical structure from a flat unfolded state to a folded state.

[0242] A variety of folding techniques may be used with thick wall physical structures. In an offset crease technique, the original fold between two connected panels is widened and replaced by two parallel hinge lines to facilitate flat unfolding. Details about offset crease techniques can be found in the journal article: Ku, J.S. and Demaine, E.D., “Folding Flat Crease Patterns with Thick Materials,” Journal of Mechanisms and Robotics, 8(3): 031003, 2016. In an offset panel technique, thickness of some panels is modified locally to allow folding through panel nesting. Details about offset panel techniques can be found in the journal article: Edmondson, B.J., Lang, R.J., Magleby, S.P., and Howell, L.L. “An Offset Panel Technique for Thick Rigidly Foldable Origami,” Proceedings of the ASME 2014 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, Buffalo, NY, Aug 17-20, 2014. In a hinge shift technique, position of the hinge line is shifted from one panel surface to another in an alternating fashion. Details about hinge shift techniques can be found in the journal article: Lang, R.J., Tolman, K.A., Crampton, E.B., Magleby, S.P., and Howell, L.L. “A Review of Thickness-Accommodation Techniques in Origami-Inspired Engineering,” Applied Mechanics Reviews, 70(1), 010805-010805-010820, 2018. A combination of these techniques may be utilized within a given physical structure, for example an offset panel technique and a hinge shift technique may be used in combination to account for thickness differences between panels or portions thereof. Manufacturability considerations may factor into which technique(s) are used for a given physical structure. [0243] Adjacent panels in a physical structure may be connected together by one or more hinges. Adjacent walls in a physical structure may be connected together by one or more hinges. In some embodiments, a hinge provides a connection (e.g., between adjacent panels). In some embodiments, a hinge restricts all but one degree of freedom: rotation about one axis. A hinge may be a butt hinge, barrel hinge, piano hinge, pivot hinge, overlay hinge, offset hinge, strap hinge, scissor hinge, spring hinge, flag hinge, flexible material hinge (e.g., a membrane material or single-sided adhesive sheet). A combination of such hinges may be used within a single physical structure. A hinge may be a molded hinge that is formed at least partially in a wall (e.g., in one or more panels thereof). For example, such as in a butt hinge, flag hinge, or piano hinge, a hinge may include a knuckle that is formed by a portion of a panel; a separate pin may be provided. In some embodiments, a molded hinge includes one or more knuckles formed from a first panel and one or more knuckles formed from a second panel and a pin The first panel and the second panel may be on different walls. In some embodiments, a wall includes multiple panels where each pair of adjacent panels is connected by at least one hinge. In some embodiments, at least one pair of adjacent panels in a wall are not connected by any hinge. In some embodiments, the at least one pair of adjacent panels in the wall are not connected at all. [0244] In some embodiments, a first panel is connected to a second panel along a first edge of the first panel by a first hinge disposed at an interior surface of the first panel and the first panel is connected to a third panel along a second edge of the first panel (different from the first edge) by a second hinge disposed at an exterior surface of the first panel (e.g., in an offset hinge technique). The first, second, and third panels may be panels in a same wall. The first panel and one of the second and the third panel may be panels in a same wall while the other of the second and the third panel is in a different wall.

[0245] In some embodiments, a physical structure includes one or more flexible hinges. A flexible hinge may include an elastomeric material. A flexible hinge may act as a seal (e.g., to water and/or air from outside the physical structure). In some embodiments, a flexible hinge runs along an entire length of a hinge line. In some embodiments, a flexible hinge runs along an entire length of an edge of a panel, for example along an entire length of both panels connected by the flexible hinge. Such flexible hinges may act as seals, for example when constructed with an impermeable (e.g., to water) material. In some embodiments, a physical structure is sealed when in a folded state. A physical structure may include one or more seals (e.g., one or more gaskets). For example, each seal may be disposed along an entire length of an edge of a respective panel. In some embodiments, a (e.g., each) seal is at least partially compressed when a physical structure is in a folded state. In some embodiments, a physical structure includes a seal for each hinge line. In some embodiments, a physical structure includes a seal for each edge of each panel (e.g., that runs along the edge of the panel). A seal may be shared for edges of adjacent panels that contact when a physical structure is in a folded state (one seal that seals the contact between the edges of the panels, whether the panels are connected or not). Techniques described herein may facilitate incorporation of seals without compromising flat unfolding. [0246] A structure may be sufficiently stable in a folded state (e.g., due to resistance provided by one or more mechanical assistance components, as described further subsequently). In some embodiments, one or more locking mechanisms may be used to “rigidize” a structure when it/they are locked (engaged). Locking mechanism(s) may prevent partial or total collapse of a physical structure from a folded state. Collapse may occur, absent the locking mechanism(s), due to an applied force on the physical structure, whether from the environment (e.g., wind) or artificially applied (e.g., by someone pushing or leaning on the structure). Numerous different locking mechanisms may be used including, but not limited to, bent sliding bars, latches (e.g., draw latches), cam locks, straps, buckles, clamps, clasps, or pins. Two or more different types of locking mechanisms may be used in a single physical structure. Generally, a locking mechanism has a locked position or state that prevents folding along one or more hinge lines (e.g., by spanning the hinge line). A locking mechanism may intersect a hinge line. A locking mechanism generally acts on two adjacent panels, whether within the same wall or between adjacent walls, but may span across more than two panels, for example a strap that acts across three panels within a wall. A locking mechanism need not necessarily act on two adjacent panels, for example may span to different, non-adjacent panels (e.g., on different walls). A locking mechanism may be, for example, a brace that braces panels (e.g., on different walls) to prevent undesired unfolding. A locking mechanism may be integral with a physical structure or may be an accessory part that is applied to (e.g., installed on) the structure, for example after erection.

[0247] Where a physical structure has a single degree of freedom, a single locking mechanism may be sufficient to fully rigidize a physical structure. Nonetheless, even where a physical structure has a single degree of freedom, additional locking mechanism(s) may be provided, for example in order to provide redundancy, additional rigidity (e.g., to limit panel flex), multiple options for a person to secure the physical structure (e.g., if the structure is setup in a location that blocks one or more of the locking mechanisms). In some embodiments where a physical structure has multiple degrees-of-freedom, multiple locking mechanisms may be used (e.g., needed) to fully rigidize the structure. It is desirable for a locking mechanism to not impede folding and unfolding of a physical structure when it is in an unlocked (disengaged) position. For example, a locking mechanism may be intentionally designed to be thin or recessed within (e.g., disposed within a recess) a panel of a physical structure so that it does not protrude therefrom. In some embodiments, a locking mechanism, or portion thereof, is applied to panel(s) after a physical structure is moved to a folded state. For example, a strap or latch (e.g., draw latch) may be applied to attachment points on two adjacent panels after expansion. A portion of a locking mechanism may be formed from a panel (e g., a molded portion of the panel).

[0248] In some embodiments, a physical structure includes one or more energy storage hinges. An energy storage hinge may include an elastomeric material. An elastomeric material may store energy during moving to a folded state that at least partially encloses a volume. Thus, an elastomeric material may have an energetic state (e.g., that occurs when a physical structure is in a folded state). The stored energy may then be released during unfolding to promote rapid and complete unfolding toward an unfolded state. An elastomeric material may be adhesively or mechanically fastened to adjacent panels in a physical structure. An energy storage hinge may be used in combination with another type of hinge (e.g., a piano hinge) to connect adjacent panels. A torsion spring may be used in place of, or in combination with, an energy storage hinge.

[0249] A locking mechanism, such as a pin (e.g., a quick-release pin), may be used (e.g., inserted into two or more holes in a physical structure) to maintain a physical structure in a folded state. For example, in some embodiments, upon removal of a quick-release pin, a physical structure may rapidly unfold to an unfolded state (e g., a flat unfolded state). Such a locking mechanism may be particularly useful where stored energy hinges and/or torsion springs are used that otherwise add additional unfolding force. A pin may be a quick-release pin, a rode, a bolt, or a screw, for example. A physical structure may include two or more holes for use with one or more pins. Holes may be disposed in any suitable location A pin may be of any suitable dimension. In some embodiments, a hole is disposed in a panel of a physical structure at an angle relative to a thickness dimension of the panel. In some embodiments, a hole is disposed in a panel of a physical structure such that when a pin is inserted therethrough the pin and the panel are not orthogonal. Using angled holes and/or non-orthogonal pin and panel pairs may allow for using simpler pin designs. For example, such arrangements may provide some friction or resistance to unfolding due to panel(s) needing to rotate to unfold and pin(s) preventing that rotation from occurring while inserted in holes. For example a simple rod of sufficient length could act as such a pin; there would be no need for any special pin ends to prevent the pin from unfolding. In some embodiments, a physical structure includes a panel that rotates about an axis during folding and unfolding and the panel includes a hole for a pin that is not aligned with the axis. In some embodiments, a physical structure includes holes, each disposed in a corresponding panel, where the corresponding panels for at least two of the holes rotate about different axes during folding and unfolding. Such an arrangement facilitates use of simple pin designs since the different rotation axes work against each other thereby inhibiting the physical structure from unfolding when the pin is inserted without needing any mechanism to keep the pin in place. A quick-release pin, or other locking pin, may be used nonetheless to provide a greater margin of safety.

[0250] In some embodiments, a physical structure includes one or more damping mechanisms. In some embodiments, connections between adjacent panels (e.g., at abutted edges) are waterproof and/or include a gasket. In some embodiments, fixed equipment is included within interior space (e.g., within the volume that is at least partially enclosed by panels) for quick startup upon erection. In some embodiments, a quick erecting shelter that is pre-attached to inside space for quick erection upon opening is included. In some embodiments, a structure includes one or more locking mechanisms (e.g., latching systems) for securing panels for transport in a folded or flat unfolded state. In some embodiments, a physical structure at least partially encloses a volume having a side with an area of at least 256 sq. ft., at least 500 sq. ft., at least 800 sq. ft., or at least 1000 sq. ft.

[0251] FIG. 54 illustrates an exemplary physical structure 10 having a flat unfolded state that uses a bottom hinged construction. Such a physical structure may be made with thick walls. For example, in some embodiments, a wall and/or panel thickness is 2 inches. Thicker and thinner thicknesses may also be used. Tn the figure, lines in red (lines extending orthogonally from the center of each of the four sides) are connected at ends only, not continuously. The red point specifies point of connection. The physical structure 10 may be erected into its folded state using one or more straps (e.g., slings), as shown. A winch may be used to raise the middle, vertical strap portion in order to erect the physical structure 10. The physical structure 10 is a single degree of freedom system, though panels 22 that form roof 28 may need to be separately moved into the folded position (enclosing a volume) depending on how the structure 22 is folded. Exemplary dimensions are shown though other dimensions (e.g., in other proportions) may be used.

[0252] This physical structure 10 is not kinematically compatible based on the principles of origami, specifically at the fold lines that are not connected. Because the structure is flat in an unfolded state, as opposed to flat in a folded state, the lack of kinematic compatibility does not necessary impair use of the structure. In some embodiments, a physical structure having a flat unfolded state is not kinematically compatible but the lack of kinematical compatibility is immaterial. In some embodiments, a physical structure having a flat unfolded state is not kinematically compatible but the extent of the compatibility is reduced (e.g., minimized) such that the structure is still sufficient to serve one or more intended purposes (e.g., as an emergency or temporary shelter or habitat). In certain applications, the lack of kinematic compatibility of a bottom hinged construction of a physical structure may make such a construction not preferred, for example as compared to blooming constructions and top-chamfered constructions, which will be subsequently described.

[0253] FIG. 55 illustrates an exemplary physical structure 10 having a flat unfolded state that uses a blooming 90 degree construction. Such a physical structure may be made with thick walls. For example, in some embodiments, a wall and/or panel thickness is 2 inches. Thicker and thinner thicknesses may also be used. In this example, all hinge lines are connected continuously along their length. This physical structure 10 is kinematically compatible based on the principles of origami. The physical structure 10 may be erected into its folded state using one or more straps, as shown. A winch may be used to raise the middle, vertical strap portion in order to erect the physical structure 10. The physical structure 10 is a single degree of freedom system. Exemplary dimensions are shown though other dimensions (e.g., in other proportions) may be used. [0254] FIG. 56 illustrates an exemplary physical structure 10 having a flat unfolded state that uses an angled blooming construction. Such a physical structure may be made with thick walls. For example, in some embodiments, a wall and/or panel thickness is 2 inches. Thicker and thinner thicknesses may also be used. In this example, all hinge lines are connected continuously along their length. This physical structure 10 is kinematically compatible based on the principles of origami. The physical structure 10 may be erected into its folded state using one or more straps, as shown. A winch may be used to raise the middle, vertical strap portion in order to erect the physical structure 10. The physical structure 10 is a single degree of freedom system. Exemplary dimensions are shown though other dimensions (e.g., in other proportions) may be used. FIG. 63 illustrates another exemplary physical structure 10 having a flat unfolded state that uses an angled blooming construction.

[0255] FIG. 57 illustrates an exemplary physical structure 10 having a flat unfolded state that uses an top-chamfered construction. (“Top -’’chamfered refers to the fact that the chamfers of physical structure 10 are oriented on a top surface of the physical structure 10 when in a flat unfolded state in order to allow the physical structure 10 to fold thereby at least partially enclosing a volume.) Such a physical structure may be made with thick walls. For example, in some embodiments, a wall and/or panel thickness is 2 inches. Thicker and thinner thicknesses may also be used. In this example, all hinge lines are connected continuously along their length. This physical structure 10 is kinematically compatible based on the principles of origami. The physical structure 10 may be erected into its folded state using one or more straps, as shown. A winch may be used to raise the middle, vertical strap portion in order to erect the physical structure 10. Exemplary dimensions are shown though other dimensions (e.g., in other proportions) may be used. Chamfered construction may be preferred for a physical structure having a flat unfolded state, for example, due to advantages in fabrication, such as the ability to use 3D printing and/or other rapid and light fabrication techniques. Gaskets may also be readily used with chamfered constructions. Chamfer location and design may be tailored to application. In some embodiments, reducing the number and/or length of chamfers may be desirable. The physical structure 10 is not a single degree of freedom system, which may make erection into its folded state more difficult. For example, one generally cannot simply push on one panel or wall to erect the structure 10 but could use strap(s) similarly to other systems that are single degree of freedom (e g., in FIGs. 54-56). [0256] FIGs. 58A-B illustrate various close-up views of a bottom hinged construction for a physical structure having a flat unfolded state, in this case having side-folding internal parts. FIG. 58A illustrates a single corner of a physical structure in four states of folding. FIG. 58B illustrates in more detail the lack of kinematic compatibility in this design where normal hinges are applied along the blue (angled) lines while the pair of adjacent panels that are darker are not connected (i.e., no connection along the red line). The lack of connection at that location allows the panels to slide past each other during folding to erect the structure. In some embodiments, some center space is still required to accommodate folding and unfolding. In some embodiments, there is an axis-point constraint, which may require, for example, a linear guide with a ball joint-type connection.

[0257] FIG. 59 illustrates individual panels of a physical structure having a flat unfolded state before being assembled. FIG. 60 is a rendering of an assembled of a physical structure 10 having a flat unfolded state. The physical structure 10 includes single panel walls 20, 22.

Certain panel 22 are not included in any wall 20 because they form internal structures in the folded state, for example the smallest triangular panels are not part of any wall. These non-wall panels 22 assist in making physical structure 10 into a single-degree-of-freedom system. Some panels 22 are part of roof 28. Physical structure 10 includes hinges 32 that hinge along hinge lines 30. (Note that representative ones of, but not all of, panels 22, walls 20, portions of roof 28, hinge lines 30, and hinges 32 are labelled in FIG. 60 for visual clarity.) FIG. 61 illustrates three views of a foam board model with top-chamfered construction. The left and center views are of opposite sides of a flat unfolded state. The right view is of the model in a folded state. [0258] FIG. 62 is a visual table illustrating various hinge designs and their compatibility with blooming constructions and top-chamfered constructions of physical structures having flat unfolded states. The leftmost column specifies where the hinge axis is located along the thick walls. The next column to the right illustrates exemplary schematics of hinging mechanisms using rigid hinges (illustrated by circular elements). The next column to the right illustrates exemplary schematics of potential flexible material hinges that also have a hinge axis located in a position indicated in the leftmost column. The two rightmost columns indicate whether the hinging mechanisms are feasible or not feasible with blooming constructions (second from rightmost column) and top-chamfered constructions (rightmost column). [0259] FIGs. 64A-C illustrate three views of a table-top prototype of an exemplary physical structure having a flat unfolded state that uses a blooming 90 degree construction. FIGs. 64A-B illustrate a whole (FIG. 64A) and partial, magnified (FIG. 64B) view of the prototype in its flat, unfolded state. FIG. 64B illustrates various hinges and hinge lines at one of the areas of the structure that will fold into the at least partially enclosed volume that is at least partially enclosed when the structure is erected into its folded state. FIG. 64C illustrates the structure erected into its folded state, where it fully encloses a volume (due to its roof). The physical structure may be secured in its folded state using one or more pins (e.g., quick-release pins). Panels of the physical structure may be connect with spring-loaded embedded hinges. In that way, when the pin(s) are removed, the structure may spontaneously revert back to its unfolded state. Thus, breakdown of the structure may occur faster than setup of the structure. [0260] FIGs. 65 A-B illustrate two views of a table-top prototype of an exemplary physical structure 10 having a flat unfolded state that uses flexible hinges. In this example, the prototype physical structure uses a top-chamfered construction. Pin 46, in this case a quickrelease pin, can be inserted into holes 48 in roof 28 of the physical structure 10. In this example, one pin and two holes are used, but in general any suitable number of pins may be used with any suitable number of holes. The exact number and/or placement of pins and/or holes may vary based on, for example, the number of walls, roof panels, and/or dimensions of a physical structure. The structure 10 is made of printed material bonded to canvas. The various panels 22, such as those defining single-panel walls 20, are defined by the printed material. The canvas not only acts as an outer shell for the physical structure 10 but also provides continuously connected hinges for the structure.

[0261] FIG. 66 illustrates views of hinges 32 that could be used in a physical structure. Some of the hinges 32 are rigid (barrel hinges) and others of the hinges 32 are elastomer hinges (exemplary stored energy hinges). FIG. 67 illustrates views of fabric hinges 32 that could be used in a physical structure. FIG. 68 illustrates views of taped and mechanically fastened elastomer hinges, which are examples of stored energy hinges. FIG. 69 illustrates a discrete torsion spring applied to a portion of a structure that is unfolded in an expanded state, but could be applied to a structure that is folded when erected. Such a torsion spring can provide additional unfolding force to a physical structure, for example to drive it from a folded state in which it at least partially encloses a volume toward a flat unfolded state. [0262] FIGs. 70-74B illustrate various prototypes of physical structures that have a flat unfolded state and a folded state in which they at least partially enclose a volume. FIG. 70 illustrates a life-size prototype of a physical structure that includes continuous elastomer hinges that are taped. FIG. 71 illustrates a life-size prototype of a physical structure that includes continuous elastomer hinges that are mechanically fastened. A magnified view of the mechanical fastening is also provided. FIG. 72 illustrates a portion of a prototype of a physical structure that includes piano hinges, fabric (canvas) hinges, and a hinge made of G-10 (also called garolite), which is a high-pressure fiberglass laminate. The panels are made of foam-core sandwich material. FIG. 73 illustrates views of an assembly process for a full-scale (life-size) prototype of a physical structure that has a flat unfolded state and uses flexible hinges with a blooming 90 degree construction. FIGs. 74A-B illustrates views of another full-scale (life-size) prototype of a physical structure that has a flat unfolded state and uses flexible hinges with a blooming 90 degree construction. FIG. 74A shows people placing the physical structure in its folded state while FIG. 74B shows the people placing the physical structure in its flat unfolded state.

[0263] Those of ordinary skill in the art will readily recognize that one or more features of embodiments of physical structures having a flat unfolded state and a folded state that at least partially encloses a volume may be used with, or appropriately adapted for use with, physical structures having a flat-folded state and an expanded state, and vice versa. Therefore, feature(s) disclosed in this section are not limited to use only with physical structures having a flat unfolded state and feature(s) disclosed in other sections are not limited to use with physical structures having a flat-folded state.

Materials and Size

[0264] Many materials are suitable for use in physical structures disclosed herein. In some embodiments, a panel or wall includes a polymer. For example, thermoplastics, such as, for example terephthalates (e.g., polyethylene terephthalate or polyethylene terephthalate glycol) or polyethylenes (e.g., HDPE, UHMWPE), may be used. Non-polymer materials such as, for example, a metal, composite, wood, cardboard, or ceramic may be used. Material selection may be guided by intended location and operational environment of the physical structure to be used. In addition to external loads (e g., wind, snow), moisture and temperature variations may also be considered in selecting material(s) for a physical structure. In some embodiments, a panel (e.g., forming all or part of a wall) includes a rigid shell or a soft shell on a rigid frame. Thus, in some embodiments a physical structure forms a frame (e.g., on or over which a soft shell, such as a canvas shell or waterproof shell, may be applied). Panels may be formed by additive manufacturing. In some embodiments, panels (e.g., in multiple walls) include (e.g., are lined with) a barrier material, such as a water barrier.

[0265] A sandwich composite (sometimes referred to as a “sandwich-structured composite”) may be used as a panel material. A core density of a sandwich composite may vary in a panel and/or from panel to panel within a physical structure. Variable core density may be used to minimize weight, maintain structural integrity, maintain thickness needed to ensure flatfolding, or a combination thereof. FIG. 52 illustrates a sandwich composite topology for a panel, the composite having regions of high density material and regions of low density material, in order to maintain structural integrity while reducing weight.

[0266] In some embodiments, a physical structure is wider than the structure is tall. A physical structure is moveable between a flat-folded state and an expanded state. In an expanded state, a physical structure at least partially (e.g., entirely) encloses a volume. For example, a physical structure in an expanded state may enclose at least 3 sides, at least 4 sides, at least 5 sides, or at least 6 sides of a volume. A physical structure may act in combination with a floor surface, such as bare ground or concrete, to at least partially (e.g., entirely) enclose a volume. In some embodiments, a volume at least partially enclosed by a physical structure is at least 10 m ³ (e.g., at least 15 m ³ , at least 20 m ³ , at least 30 m ³ , at least 40 m ³ , at least 50 m ³ , at least 60 m ³ , or at least 100 m ³ ). Smaller volumes may be at least partially enclosed for smaller structures.

[0267] In some embodiments, a physical structure has a height of at least 2 m (e.g., at least 3 m, at least 4 m, or at least 5 m) and no more than 20 m (e.g., no more than 15 m or no more than 10 m). Such physical structures are suitable as habitable structures as they are sufficiently tall for a human to stand upright in them. In some embodiments, a physical structure has at least one of a length and a width of no more than 12 m (e.g., no more than 6 m) such that it may fit in a standard shipping container when in a flat-folded state.

[0268] A panel may be solid, for example a solid sheet of material (e.g., a sandwich composite). In some embodiments, a panel has a thickness of at least 1 mm and no more than 50 cm. For example, in some embodiments, a panel has a thickness of at least 2 mm, at least 4 mm, at least 5 mm, at least 6 mm, at least 8 mm, at least 1 cm, at least 1.5 cm, or at least 2 cm. For example, in some embodiments, a panel has a thickness of no more than 40 cm, no more than 30 cm, no more than 20 cm, no more than 15 cm, no more than 10 cm, no more than 8 cm, no more than 6 cm, no more than 5 cm, no more than 4 cm, no more than 2 cm, or no more than 1 cm. [0269] In some embodiments, a panel has a non-uniform thickness. For example, a first portion of a panel may have a thickness that is less than a thickness of a second portion of the panel. A panel with non-uniform thickness may include portions with different thicknesses that are integral multiples (e.g., lx, 2x, 3x) of a baseline thickness. A first panel may have a non- uniform thickness that corresponds to a different panel (e.g., that is directly connected to the first panel along a hinge line), for example corresponds such that the first panel and the different panel are nested in a flat-folded state. Using integral multiple thicknesses for non-uniform thickness panels allows for simple designs that facilitate flat-folding (e.g., due to panel nesting), for example as shown in FIGs. 16G, 19, 22-23, 26, 48. In some embodiments, in a wall of a physical structure, a first panel has a thickness of at least 5 cm (e.g., at least 10 cm, at least 15 cm, or at least 20 cm) and no more than 50 cm and a second panel has a thickness of at least 1 mm and no more than 20 cm (e.g., no more than 15 cm, no more than 10 cm, or no more than 5 cm).

[0270] In some embodiments, a physical structure is habitable (e.g., for living and/or for work). In some embodiments, a physical structure is a shelter, for example a disaster relief shelter, a humanitarian aid shelter, a temporary occupancy shelter (e.g., for homeless persons), or a military shelter. Physical structures may be used in applications where compactness during transport is desirable. For example, when using a structure in space, the structure must first be transported to space. Physical structures disclosed herein can occupy significantly less space in a flat-folded state than in an expanded state and therefore are suitable for applications where reduced volume during transport is particularly desirable, for example due to hard limits on available volume. One such application is use in space. For example, a physical structure may an expandable solar array, habitable structure (e.g., living and/or working module) for astronauts, or case, housing, or container for use in performing experiments in space. Another application for some physical structures is as collapsible body armor (e.g., using flexible material hinge(s)). Other applications of physical structures disclosed herein include as camping huts (e.g., tents), ice fishing huts, collapsible barricades, aid stations, and temporary jobsite trailers (e g , for construction). One advantage of structures disclosed herein, including physical structures that are moveable between a flat unfolded state and a folded state and between a flat-folded state and an expanded state, is that they can be transported pre-assembled so that minimal time is needed to set them up.

[0271] In some embodiments, physical structures can be stored and transported in a standard size shipping container (e.g., intermodal container). In some embodiments, the shipping container is a physical structure or can be used as a shelter. FIGs. 53A-53B illustrate examples of physical structures disclosed herein that can be stored and transported in a shipping container. For example, seven physical structures (FIG. 53A) or six physical structures (FIG. 53B) can be stored and transported in a shipping container. Because the physical structures can be flat folded, packing and unpacking the shipping container is relatively simple. In some embodiments, evenly spaced dividers (e g., removable dividers) are disposed on an interior of a shipping container so that each physical structure, in its flat folded state, can be slotted into one of the spaces created by the dividers. In some embodiments, the dividers are removable so that the shipping container is usable, for example as a shelter, before repacking the physical structures that were previously stored therein. The physical structures 10 shown in FIGs. 53 A- 53B are an extended- width version of the physical structure 10 shown in FIG. 26 panel (a). In both examples, some panels have a variable thickness in order to accommodate nesting for flat folding. Some of the panels 22 in FIGs. 53A-53B are shown with lines indicating variable thickness (at the location where thickness changes from one values to another (e.g., from t to If) (as opposed to hinge lines 30, which are labelled).

Windows and Doors

[0272] A physical structure may include one or more windows. In some embodiments, a panel includes a window (e.g., positioned within an interior of the panel or adjacent to an edge of the panel). In some embodiments, a wall includes a window (e.g., spread over two or more panels). In some embodiments, a window wraps over two walls (e.g., is disposed at an edge where two walls meet in an expanded state). A window may be a hole in which a commercially available window (e.g., a double pane window) can be installed. Windows may be disposed on different walls of a physical structure. Multiple windows may be disposed on a single wall of a physical structure. A window may have any suitable perimeter shape, for example a rectangular, rounded (e.g., circular), or triangular shape.

[0273] A physical structure may include one or more doors. In some embodiments, a panel includes a door (e g., positioned adjacent to an edge of the panel). In some embodiments, a wall includes a door (e.g., spread over two or more panels). In some embodiments, a door wraps over two walls (e.g., is disposed at an edge where two walls meet in an expanded state). A door may be a hole in which a commercially available door can be installed. Doors may be disposed on different walls of a physical structure. Multiple doors may be disposed on a single wall of a physical structure. A door may have any suitable perimeter shape, for example a rectangular, rounded (e.g., circular), or triangular shape. In some embodiments, a physical structure includes one or more doors and one or more windows.

Utilities

[0274] Because physical structures according to embodiments of the present disclosure have thick walls, they can accommodate utilities. In some embodiments, a panel includes one or more embedded conduits that can be used for plumbing, electrical wire(s), data transmission line(s) (e.g., Cat 6 line(s) and/or fiber optic line(s)). In some embodiments, a panel includes one or more embedded ducts for HVAC. In some embodiments, a panel includes one or more embedded conduits and one or more embedded ducts. By embedding conduit(s) and/or duct(s) inside of panel(s), flat folding can be maintained. (If conduit(s) and/or duct(s) were disposed on interior and/or exterior sides of panel(s), a physical structure may not be able to achieve a flat- folded state.) In some embodiments, electrical wire(s) and/or data transmission line(s) can be disposed through conduit(s) and left in place when folding a physical structure into a flat-folded state. In some embodiments, electrical wire(s) and/or data transmission line(s) may be removed from conduit(s) prior to folding and inserted after a physical structure is moved into an expanded state. In some embodiments, conduit(s) in adjacent panels (e.g., disposed in a same wall or in adjacent walls) connect when a physical structure is in an expanded state. In this way, electrical line(s), data transmission line(s), plumbing, or a combination thereof can extend from any point in a physical structure to any other point (e.g., across an entire wall, across multiple walls, or from a first point that is on a first wall to a second point that is on a second wall). A physical structure may include one or more active heating and/or cooling systems, one or more passive heating and/or cooling systems, or at least one active heating and/or cooling system and at least one passive heating and/or cooling system. Such systems may improve habitability of a physical structure (e.g., for use in more extreme environments).

[0275] Certain embodiments of the present disclosure were described above. It is, however, expressly noted that the present disclosure is not limited to those embodiments, but rather the intention is that additions and modifications to what was expressly described in the present disclosure are also included within the scope of the disclosure. Moreover, it is to be understood that the features of the various embodiments described in the present disclosure were not mutually exclusive and can exist in various combinations and permutations, even if such combinations or permutations were not made express, without departing from the spirit and scope of the disclosure. The disclosure has been described in detail with particular reference to certain embodiments thereof, but it will be understood that variations and modifications can be effected within the spirit and scope of the claimed invention.

Parts List

10 kinematically compatible structure

20 wall / front wall / back wall / side wall

22 panel

23 spacer

24 tab

26 floor

28 roof

30 hinge line

32 hinge

33 pin

34 knuckle

40 locking mechanism

42 hinge line intersection portion 44 lock/unlock maintaining portion 46 pin / quick-release pin hole actuator torsion spring strap / sling