Volume Optimisation of Multi-stable Origami Bellows for Deployable Space Habitats

Origami bellows are formed by folding flat sheets into closed cylindrical structures along predefined creases. As the bellows unfold, the volume of the origami structure will change significantly, offering potential for use as inflatable deployable structures. This paper presents a geometric study of the volume of multi-stable Miura-ori and Kresling bellows, focusing on their application as deployable space habitats. Such habitats would be compactly stowed during launch, before expanding once in orbit. The internal volume ratio between different deployed states is investigated across the geometric design space. As a case study, the SpaceX Falcon 9 payload fairing is chosen for the transportation of space habitats. The stowed volume and effective deployed volume of the origami space habitats are calculated to enable comparison with conventional habitat designs. Optimal designs for the deployment of Miura-ori and Kresling patterned tubular space habitats are obtained using particle swarm optimisation (PSO) techniques. Configurations with significant volume expansion can be found in both patterns, with the Miura-ori patterns achieving higher volume expansion due to their additional radial deployment. A multi-objective PSO (MOPSO) is adopted to identify trade-offs between volumetric deployment and radial expansion ratios for the Miura-ori pattern.


Introduction
Origami is a traditional Japanese paper art that forms flat sheets into three-dimensional objects by folding along predefined creases. Due to its stowability, deployability, and programmable stiffness, origami has been applied in many fields ranging from the micro-/nanometre scale to the metre scale [1], such as nano-photonics [2] and inflatable shelters [3].
Origami bellows [4] are created by folding planar origami patterns into closed hollow cylindrical structures; these include the Miura-ori [5], Kresling [6], and waterbomb [7] patterns. Such bellows are shown to have rich mechanical properties including multi-stability [5,8,9], self-locking behaviour [10,11], and tunable stiffness [12][13][14]. As such, origami bellows have been applied in fields such as robotics [15,16], meta-materials [17,18], binary switch [19], and B Mengzhu Yang mengzhu.yang@bristol.ac.uk 1 Department of Aerospace Engineering, University of Bristol, Bristol BS8 1TR, UK self-folding structures [20][21][22]. Origami bellows are promising candidates for self-folding inflatable structures since the structure itself serves as the inflatable chamber. Inflation actuation utilises the pressure of compressed air or fluid to trigger the deformation of origami. For instance, Li et al. [23] introduced fluid-driven origami-inspired artificial muscles that can be programmed to achieve multi-axial motions including contraction, bending, and torsion. Melancon et al. [3] designed several bistable origami units that can be deployed through a single fluidic pressure input and combined these units to build functional structures at the metre scale. Melancon et al. [24] also used pressure to trigger a snap-through transition of a multi-stable Kresling. The inflatable origami may experience a significant volume change during the deployment process. Zhang et al. [25] approximated the volume of Kresling by assuming that the cross section of the oblique polygonal prism is constant. However, the actual volume change of inflatable origami has rarely been studied.
Inflatable origami structures have been proposed for aerospace applications due to their high deployment ratio, low system complexity and simple deployment mechanism [26]. The structure can be manufactured on Earth, folded into a compact configuration for launch and expanded to a larger volume once in orbit. Of particular interest is the use of origami concepts in deployable space habitats. Space habitats are where humans live in outer space and thus, need to withstand extreme space environments (e.g., radiation, meteoroids, and orbital debris). Deployable structures could significantly increase the amount of usable volume of space station modules [27] or habitats on the lunar surface. For example, the Bigelow Expandable Activity Module (BEAM) [28] has been successfully launched and expanded once docked to the International Space Station (ISS). Origami concepts such as the Miura-ori [29], accordion [27], Kresling, and Tachi-Miura polyhedron (TMP) fold patterns [30] have been considered for deployable habitats. For instance, the LUNARK habitat mission created a lunar habitat based on the Miura-ori pattern with a rigid carbon fibre shell, which can reach a volume expansion of 750% [29]. Morgan et al. [27] introduced a habitat using a modified accordion fold pattern accounting for material thickness, which has an 85% increase in length and volume between the stowed and expanded configurations. Yasuda et al. [30] designed volumetric origami cells made of TMP that can exhibit in-situ transition between flat-foldable and load-bearing states without modifying their predefined crease patterns or hitting the kinematically singular configuration. However, it is difficult to fairly compare the origami habitats to conventional habitat modules since they are irregular-shaped. Moreover, optimal design of origamibased space habitats has rarely been investigated.
In this work, we focus on two widely applied origami bellow patterns-Kresling and Miura-ori-and evaluate their potential as deployable space habitats. The remainder of the paper is organised as follows. Section 2 describes the crease patterns and internal volume calculation for bistable Miuraori and tristable Kresling bellows. A study on the volume ratio between different stable states is presented. To compare the performance of origami habitats to conventional habitat modules, the definition and calculation of effective volume of origami space habitats are introduced in Sect. 3. Next, optimisation methods are used to find patterns with high volume deployment ratio, whilst trading off against radial expansion. The findings of this paper are concluded in Sect. 4.

Miura-ori
The crease pattern and three-dimensional model of two-layer Miura-ori bellows are illustrated in Fig. 1. The pattern can be described by sector angles φ 1 and φ 2 , vertex spacing d 1 and d 2 , radius R, layer height H , number of unit cells around the circumference n and number of layers along the length of the bellows m. A single unit cell consists of two adjacent facets, as highlighted in Fig. 1a. For Miura-ori bellows, φ 1 and φ 2 satisfy: and Following Reid et al. [5], to form closed bellows, a cylindrical closure condition is introduced: where θ 1 and θ 2 are angles defined in Fig. 1b. Next, the angle ψ can be calculated as follows: which represents the folded state of the bellows, with ψ 0 representing a flat-folded configuration. The parameter d 2 and the layer height H are geometrically restricted to [5]: Thus, the geometric design parameters are given by φ 1 , φ 2 , R, d 1 , H , m, and n, whilst the remaining parameters can be calculated. The 3D geometry such as the positions of the vertices can be calculated using the SO(3) rotation matrices method proposed by Reid et al. [5]. Moreover, when the vertex spacing d 2 0, the Miura-ori pattern reduces to the Kresling pattern.

Bistability
To satisfy Eqs. (4) and (5), φ 1 and φ 2 (or φ 1 and φ 2 ) should be in the region bounded by [5]: The design space of the bellows can therefore be divided into three regions [5]: (i) invalid parameters, (ii) monostable region, and (iii) bistable region (including the flat-foldable line), as illustrated in Fig. 1d. Note that this definition of multi-stability does not depend on strain energy considerations, but rather geometric compatibility. Crucially, for the deployment between stable states, the origami structure must deform through bending and stretching of the facets-this is not modelled in this work. Here, we exploit the multi-stability of the origami bellows to provide compatible configurations for both the stowed and deployed configurations.

Volume Calculation
The volume of Miura-ori bellows can be calculated using the Cartesian coordinate positions of the vertices of a single unit in 3D space (as shown in Fig. 2) and then, multiplying by mn. Each single unit can be divided into six tetrahedra, Fig. 2. Sharing the same height and base area, Thus, the total volume can be expressed as follows: Fig. 2 Geometry and volume calculation of Miura-ori bellows. a 3D view of the geometry of a single unit cell in one-layer Miura-ori bellows, where A 0 B 0 is the cylinder axis of the bellows parallel to the z-axis. b The volume of a unit cell represented by the combination of six tetrahedra where B i B j is the vector from B i to B j , and B i A j is the vector from B i to A j , as highlighted in Fig. 2.

Kresling
The Kresling pattern can be derived straightforwardly from the Miura-ori pattern by setting the vertex spacing d 2 equal to zero. The prediction of multi-stability only depends on sector angles φ 1 and φ 2 , and thus, the design space is the same for normal Kresling and Miura-ori patterns (the left-hand region of Fig. 1d). Moreover, inspired by the tristable conical Kresling [31] and inflatable multi-stable Kresling [24], here, we extend the concept of tristable Kresling bellows.

Tristability
A normal Kresling pattern folds along the long diagonals (crease pattern 1, see Fig. 3a) of each unit cell. If the geometric parameters are within the bistable region [5], it will therefore have two stable states (see Fig. 3b). Hereafter, we denote the more compressed state as S 1 and the more deployed state as S 2 , with corresponding fold angles ψ 1 < ψ 2 . Melancon et al. [24] introduced an additional crease along the short diagonal of a panel of a normal Kresling; during deployment, the panels initially remain planar until they suddenly snap radially outward, folding along the short diagonals, and the structure assumes a new stable state. Inspired by this idea, we introduce additional creases along the short diagonals for all panels (crease pattern 2, see Fig. 3a). If the structure can fulfil the geometric compatibility (Eqs. 3-5) when folding along crease pattern 2, the Kresling bellows will enter the third stable state (S 3 ), as shown in Fig. 3c. The third stable state involves an additional relative rotation of the cross sections and axial extension compared to the second stable state (S 2 ). According to geometric constraints: and thus, sector angles φ 1 and φ 2 of crease pattern 2 can be expressed as follows: where φ 1 ∈ π 2 , π . For a compatible solution, φ 1 and φ 2 should satisfy the geometric constraints indicated in the righthand region of Fig. 1d [5]. When φ 1 ∈ π 2 , (n−1)π n , the structures will always find solutions for state S 3 ; when φ 1 ∈ (n−1)π n , π , sector angles should satisfy: For all Kresling patterns within the bistable region, it is found that a third compatible configuration S 3 exists.

Volume Calculation
In its folded configuration (see Fig. 4a), the geometry of the Kresling pattern can be described by height h 0 , radius r , and The red dashed lines along the long diagonals indicate the primary pattern (pattern 1); the blue dashed lines along the short diagonals form the secondary crease (pattern 2) for the third stable state; φ 1 and φ 2 are sector angles of crease pattern 1, whilst φ 1 and φ 2 are sector angles of crease pattern 2. b 3D geometry for stable states S 1 and S 2 when folding along crease pattern 1 (n 4, φ 1 77 • , φ 2 35 • ). The thick red line is the active crease whilst the thin blue line represents an inactive crease (i.e., no folding). c Corresponding 3D geometry at stable state S 3 when folding along crease pattern 2 (φ 1 134 • , φ 2 77 • ). The thick blue line is the active crease whilst the thin red line represents an inactive crease  27)) are highlighted. Note that the viewing perspective is adjusted for each stable state. a 3D geometry of Kresling bellows in a stable state; the geometry can be described by initial height h 0 , radius r , and initial rotation angle θ 0 . b Seven constituent tetrahedra of a unit cell when O 1 penetrates tetrahedron B 1 B 2 A 0 A 1 when folding along crease pattern 1 (state S 1 ). c Combination of three tetrahedra when O 1 does not penetrate tetrahedron B 1 B 2 A 0 A 1 when folding along crease pattern 1 (state S 2 ). d Combination of three tetrahedra when folding along crease pattern 2 (state S 3 ) initial rotation angle θ 0 : Thus, the vertex coordinates can be expressed as follows: where i 1, 2, · · · , n. The midpoint O 0 of axis A 0 and B 0 is given as follows: And the intersection point O i of crease pattern 1 and crease pattern 2 when folding along crease pattern 1 is calculated as follows: for i 1, 2, · · · , n. Since the Kresling pattern can be folded along either crease pattern 1 or crease pattern 2, it can be either a concave (Fig. 3b) or convex (Fig. 3c) when folded; therefore, the volume calculation method for S 3 is different from that for S 1 and S 2 . What is more, if the structure is highly compressed at S 1 or S 2 , it will lead to another volume calculation method. Taking a typical tristable Kresling (n 5, φ 1 74 • ,φ 2 41 • ) as an example: at S 1 , it is highly compressed (Fig. 4b), and the smallest repetitive unit can be regarded as a combination of seven tetrahedra; at S 2 , it deploys, and each unit can be partitioned into three tetrahedra (Fig. 4c); at S 3 , the short diagonals (e.g., B 1 A 2 ) pop out radially (Fig. 4d), the structure becomes convex, and the unit cell is composed of three tetrahedra. The volume of the Kresling bellows at different stable states can then be calculated as follows: First, consider the two different possibilities for the volume calculation when folding along crease pattern 1 to form states S 1 and S 2 : repetitive unit can be divided into the seven tetrahedra the total volume can be calculated as follows: Substituting Eq. (18) into the above expression: Fig. 4c. Similarly, the total volume can be expressed as follows: and Miura-ori bellows (H /R 2/3) at different stable states in the design space, with minimum stowed angle ψ min 10 • , 20 • , and 30 • indicated. V 1 , V 2 , V 3 represent the internal volumes at stable states S 1 , S 2 , and S 3 , respectively. All volume ratios are greater than one, indicating the volume increases for different stable states during the deployment. With the increase in ψ min , the feasible design space becomes smaller at boundary 1. a Kresling V 2 /V 1 , the ratio increases near boundary 1 with the increase in φ 1 . b Kresling V 3 /V 1 , the ratio in the region above boundary 1 is much greater than the region below it; the ratio increases along boundary 1 with increasing φ 1 . c Kresling V 3 /V 2 , the ratio is relatively large at boundary 2. d Miura-ori V 2 /V 1 , the ratio is largest along boundary 1 and for φ 1 π/2 Next, to calculate the volume at stable state S 3 , each unit can be divided into the three tetrahedra Fig. 4d. Thus, the total volume is calculated as follows: Equations (26) and (27) have the same first item since they both contain two tetrahedra whose base is A 0 A 1 A 2 or B 0 B 1 B 2 and has a height of h 0 . Figure 5 shows the volume change of Kresling and Miuraori bellows between different stable states across the feasible geometric design space [5]. In practical applications, the bellows cannot be folded flat due to the thickness of the material; therefore, we set a minimum stowed angle ψ min as a simple way to account for the panel thickness. Setting a minimum stowed angle will slightly reduce the feasible design space.

Volume Change of Multi-stable Origami Bellows
For the Kresling pattern, Fig. 5a and b indicates that both V 2 /V 1 and V 3 /V 1 increase near boundary 1 (g 1 (φ 1 , n) when ψ > ψ min ) with increasing φ 1 . Figure 5b also shows that V 3 /V 1 is comparatively small in the region below boundary 1. Moreover, the volume change between state S 3 and state S 2 (V 3 /V 2 ) is overall not significant (Fig. 5c), but the ratio is relatively large at boundary 2 ( f (φ 1 , n) when ψ > ψ min ). Similar to Kresling, for Miura-ori, V 2 /V 1 is the largest along boundary 1 and φ 1 π/2, as depicted in Fig. 5d. It is worth noting that increasing ψ min reduces the feasible design space at boundary 1.
It can be observed from Fig. 5 that all volume ratios (V 2 /V 1 , V 3 /V 1 and V 3 /V 2 ) are greater than one, suggesting that the bellows' internal volume increases monotonically during the deployment. This finding makes Kresling and Miura-ori bellows suitable for inflatable structures and space habitats deployed through pressurisation. In the next section, we shall explore the application of these origami bellows for deployable space habitats, which will require additional calculations to determine the effective internal volume as well as the volume required to transport the stowed bellows.

Optimisation of Inflatable Origami Bellows for Space Habitats
Cylindrical origami such as Kresling and Miura-ori is suitable candidates for self-folding inflatable structures since the structure itself serves as the inflatable chamber. A potential application of such inflatable cylindrical origami structures is as deployable space habitats, which are stowed compactly during launch before expanding in orbit to offer additional working and living space.
In this section, we study and optimise the deployment ratio (expanded/stowed volume) and radial expansion ratio for cylindrical origami structures. First, we briefly introduce the background and design constraints for the proposed origami space habitats. Next, we outline the concepts of stowed volume and effective deployed volume of the origami bellows, which enable fair comparison with conventional habitat designs. Finally, crease patterns are sought to maximise the deployment ratio of origami space habitats using particle swarm optimisation (PSO). Lastly, multi-objective PSO (MOPSO) is adopted to balance the trade-off between deployment ratio and radial expansion ratio for bellows based on the Miura-ori pattern.

Design Background
The dimensions of the rocket fairing used to transport the habitat to space are an important factor in the selection of an origami pattern and geometry; specifically, the aspect ratio determines the number of layers m that can be stacked. Here, we take the SpaceX Falcon 9 [32] as an illustrative example; its payload fairing measures approximately 4.5 m in diameter and 6.7 m in height [33]. Allowing for some margin, we take radius R fairing 2.2 m and height H fairing 6.5 m as upper limits for our stowed module dimensions, which defines the available volume V f of the fairing. Note that the aspect ratio H fairing /R fairing rather than the actual dimensions matters, since the geometry of the origami bellows can be simply scaled. Moreover, this preliminary study is based purely on geometric considerations; the thickness of the module walls is considered by setting a minimum stowed angle ψ min . Other considerations such as materials selection, specific design of the bulkhead, the connection between the bulkhead, and the rest of the module as well as the internal support structure of the module are not involved here.

Effective Volume Calculation
The internal and external geometry of origami bellows are irregular-shaped (as depicted in Figs. 2 and 4) compared to conventional habitat modules. To account for this and enable fair comparisons, we define a stowed volume V s and an effective deployed volume V d for the origami bellows. Stowed volume V s : the volume of the smallest cylinder that completely encloses the bellows in the stowed configuration S 1 . For both Kresling and Miura-ori bellows (Fig. 6a-i  and b-i), V s can be calculated as follows: where H s is the stowed height of a single layer, R s is the radius of the circumscribed circle, which can be calculated by measuring the distance between A 0 and the projection of A 1 on the x y plane. Effective deployed volume V d : the volume of an unobstructed prism contained by the innermost fold lines of the bellows at deployed configurations S 2 or S 3 (see the blue shaded areas in Fig. 6).
For Kresling at S 2 , as shown in Fig. 6a-ii: where H d is the deployed height of a single layer, C 1 is the intersection point of the projections of line A 1 B 2 and line A 2 B 3 on the x y plane. For Kresling at S 3 , as shown in Fig. 6a-iii: where C 1 is the intersection point of the projections of line where C 1 is the intersection point of the projections of A 3 A 4 and B 2 B 3 on the x y plane, P M1 where C 1 is the intersection point of the projections of A 2 B 2 and A 4 B 4 on the x y plane.

Definition of Optimisation Problem
The aim of the optimisation is to maximise the effective deployed volume of the structure that can be stowed in the payload fairing. Therefore, the ratio of the effective deployed volume (V d ) over the fairing volume (V f ),V d /V f , is chosen as the objective function. For Miura-ori patterns, the deployed volume is calculated at state S 2 , whereas for the Kresling bellows, we utilise the volume at state S 3 . This is motivated by the fact that V 3 is always greater than V 2 (see Fig. 5c), and that the axial stiffness at state S 3 was found to be up to several order of magnitude greater than at state S 2 and state S 1 for conical Kresling bellows [31]. Geometric parameters φ 1 , φ 2 , m, and n are selected as design variables, along with d 1 /R for the Miura-ori pattern. The upper limit for φ 1 and φ 2 is set to be 85 • to preclude structures that are too close to normal cylinders. The payload fairing height determines the integer number of layers m in the stowed configuration. Manufacturing considerations place limits on the maximum number of layers (m 8) and circumferential unit cells (4 ≤ n ≤ 8). Larger values for m and n result in an increased number of panels, folds, and vertices, which will increase the manufacturing and assembly complexity of the origami bellows. The radius of the circumscribed cylinder R s in the stowed configuration is set equal to the fairing radius R fairing . A minimum dihedral angle ψ min 20 • is used to account for the thickness of the panels in the stowed configuration, which is a constraint for the optimisation problem. Moreover, the fold pattern geometry should also lie in the feasible design region bounded by Eqs. (7)- (9). The constraints are handled using the static penalty function method [34], with the static penalty factor p selected as 10 5 here.
The optimisation design problem can thus be formulated as follows: For the Miura-ori pattern, the structure will experience radial expansion during the deployment, which will bring challenges for the design of the bulkhead section at the ends of the structure (e.g., maintaining air tightness and ensuring smooth deployment). Therefore, the radial expansion (R d /R s ) should also be considered, leading to a multiobjective optimisation problem: Note that for multi-objective optimisation, the deployed volume is necessarily calculated at state S 2 , even if the design parameters approach those of a Kresling pattern.

Optimisation Algorithm
The PSO [35] and MOPSO [36] algorithms are utilised for single-objective and multi-objective optimisation, respectively. Details of the algorithms are provided in Appendix A. The algorithm parameters are selected according to [35,36] and adjusted for the optimisation problem listed in Table  1.

Single-Objective Optimisation
The geometric parameters of the optimal designs are listed in Table 2, and the corresponding stowed and deployed configurations are presented in Fig. 7. The Convergence histories are illustrated in Appendix B. For the Kresling pattern, the achieved volume ratio V d /V f 2.56, whilst maintaining a constant cross sectional area. The increase in volume is therefore due to the change in height from 0.81 m per layer in the stowed state to 2.37 m in the deployed configuration. In the case of the Miura-ori pattern, in addition to the increase in height, the structure also shows a significant radial expansion during the deployment (R d /R s 1.84). Thus, the Miura-ori pattern can achieve much more significant volumetric expansion (V d /V f 8.31). It is interesting to observe that the optimal designs of these two patterns have the same number of layers, and the same stowed and deployed heights. Furthermore, both patterns utilise the maximum number of unit cells n 8. This can be explained by the exploration of the influence of n, which shows a trend of increasing volume with increasing n (see Appendix C). Lastly, the values for φ 1 and φ 2 are close to the flat-foldable region (g 1 (φ 1 , n) φ 1 − π/n), minimising the height of the stowed configuration. Relaxing constraints on the panel thickness and number of unit cells could further increase the effective volume. Appendix D shows the optimal results for different values of n and ψ min .

Multi-Objective Optimisation
The Pareto front for the multi-objective optimisation is shown in Fig. 8a. It is not continuous since the design parameters m and n are integers, and the objective function is highly nonlinear. Evidently, the two objectives −V d /V f and R d /R s are in strong competition with each other. The optimal design is normally chosen from the Pareto-optimal set according to the designers' requirements, such as assigning weights for objectives.
The results of the corresponding single-objective optimisation (SOO) should lie at the extremes of the Pareto-optimal set. The top-left solution (pointed out in Fig. 8a) on the Pareto front is also the optimal result for Miura-ori obtained in Sect. 3.5.1. The bottom-right solution should be the optimal result for Miura-ori with the objective of minimising R d /R s . The ratio R d /R s will reach its minimum (R d /R s 1) when the Miura-ori reduces to the Kresling pattern. Thus, the bottom-right solution should be the SOO result for minimising −V d /V f for the Kresling pattern that sets S 2 as the deployed state (solid purple square, −V d /V f −2.14, R d /R s 1.00). However, the optimiser fails to precisely find that SOO result. This can be explained by the fact that the optimiser will easily fall into local minima when it approaches any configuration that is close to the Kresling pattern (d 1 /R 2π/n, d 2 0), since for every Kresling pattern, R d /R s 1. Nonetheless, the optimiser has nominally found the single-objective solutions for both ends of the design space and provided a Pareto front to enable the selection of designs that balance the objectives. Here, to limit the radial expansion ratio and ensure a reliable result, we select an optimal design with a radial expansion ratio R d /R s ≤ 1.5,  Fig. 8a. The geometric parameters of the optimal design are listed in Table 2, and the stowed and deployed configurations of the optimal design are depicted in Fig. 8b.

Conclusions
In this paper, we investigate the volume of multi-stable origami bellows in their compatible configurations. The full geometric design space of bistable Miura-ori and tristable Kresling bellows is explored. The internal volume of the structures is shown to increase between different stable states during the deployment procedure, meaning that these bellows can be utilised as inflatable self-folding structures.
The application as a deployable space habitat is explored; the cylindrical origami structure would be stowed during launch in one stable configuration before expanding to an extended configuration in orbit. In order to enable fair comparison with conventional habitats, we define the stowed volume and effective deployed volume of the origami habitat. The dimensions of the SpaceX Falcon 9 payload fairing are used as a case study. Using a particle swarm optimisation (PSO) method, patterns are found that maximise the volume deployment ratio (effective deployed volume over stowed volume) of the origami habitat. The results show that the Kresling pattern can achieve a volume expansion ratio of 2.56, whilst the Miura-ori pattern can reach a ratio of 8.31, offering significant volume increases compared to a conventional cylindrical habitat. However, the radial expansion of Fig. 8 Optimal result of MOPSO for Miura-ori pattern bellows, trading off the change in deployed volume against the change in deployed radius. a Pareto-optimal front. The two objectives strongly compete with each other. The top-left solution on the Pareto front is also the result of the single-objective optimisation (SOO) for the Miura-ori pattern; the solid purple square represents the SOO result for minimising −V d /V f for the Kresling pattern, setting S 2 as the deployed state. An optimal design is selected by setting the radial expansion ratio R d /R s ≤ 1.5, as represented by a red dashed line and marked by an asterisk. b The stowed and deployed configurations of the selected optimal design (n 8, m the Miura-ori bellows will bring significant challenges for the design of the bulkhead section at the ends of the structure. Multi-Objective PSO is therefore used to balance the volume and radial expansion, allowing a final design to be selected from the Pareto front. The current study is a strictly geometric analysis of origami bellows for space habitats. Panel thickness (for thermal insulation, radiation protection, debris shielding, etc.) has not been considered, and practical designs would require careful consideration of the folds and vertices to enable compact stowage whilst maintaining air tightness. Furthermore, the structural mechanics has not yet been considered either for modelling the deployment (which requires panel bending and stretching) or the performance of the final deployed structure. Considering the various complexities in design and fabrication, it is at present difficult to assess whether origami-inspired space habitats are a worthwhile pursuit. Nonetheless, the present study shows that such deployable origami structures could offer a significant increase in habitat volume compared to conventional habitats for the same fairing dimensions. Moreover, the proposed multi-objective optimisation framework can be extended to include additional considerations for the design of optimal patterns for origami space habitats and other engineering origami applications.
R 40 mm, ψ > ψ min 10 • ), and Miura-ori bellows (m 1, φ 1 ∈ (0 • , 89 • ), φ 2 ∈ (0 • , 89 • ), R 60 mm, H 60 mm, ψ > ψ min 10 • ) at different stable states, showing an overall increase in the effective volume with the increase in n. Note that the colour bars indicate that base-10 logarithms of the volume ratios to effectively show a larger range. a Kresling, with the increase in n, the bellows more closely approximate smooth cylinders, leading to the increase in effective volume.
It can be observed from Fig. 10a and c that for both Kresling and Miura-ori bellows, V 1 /V s increases along the line closest to the flat-foldable line (g 1 (φ 1 , n) φ 1 − π/n), and when φ 2 hits the upper bound, V 1 /V s reaches the maximum value.
For Kresling, V d /V 3 shows a distinct overall increase when n ranges from 5 to 7. However, the range of V d /V 3 is narrow (from 0.83 to 0.95), indicating that Kresling has an overall ideal effective volume at the most deployed state (S 3 ). The ratio is relatively large around the upper boundary of the bistable region f (φ 1 , n) arctan((tanφ 1 (1 − sin(π/n))) / (1 + sin(π/n))). In this region, the angle ψ (here ψ > π/2) is relatively small, resulting in structures that are close to the corresponding right prism, and eventually leading to a big effective volume. Similarly, for Miura-ori, when φ 2 increases, ψ will be closer to π/2 and thus, has a big effective volume.

Appendix D: Sensitivity to Design Variables' Constraints
The influence of the domain of design variables on optimal results is briefly studied in this section. In the main paper, manufacturing considerations set the minimum stowed angle at ψ min 20 • and the number of circumferential unit cells at n max 8. Here, we explore the influence of ψ min and n max on optimal designs. Table 3 shows the optimal results when ψ min 10 • , 20 • and 30 • , whilst maintaining n max 8. With the increase in ψ min , the objective function value decreases since the feasible design space becomes smaller, as shown in Fig. 5. The Kresling pattern is more sensitive to the change of ψ min than the Miura-ori pattern. Table 4 shows the optimal results when n max 8, 9, and 10, while keeping ψ min 20 • . The table suggests that increasing n max is an efficient method to obtain a large deployment ratio, especially for the Miuraori pattern.

Appendix E: Software for Geometry and Volume Calculation for Miura-ori and Kresling Bellows
We have developed a MATLAB code [39] to generate the geometry and calculate the volume for multi-stable Miuraori and Kresling bellows. The interface of the software is shown in Fig. 11. Users choose a crease pattern (Kresling or Miura-ori) and then, input design parameters (n, φ 1 , φ 2 , R, m and d 1 /R for the Miura-ori pattern). The software will output the geometry at each feasible stable state and calculate the ratio of height change (H d S 2 /H s , H d S 3 /H s ), radial expansion (R d S 2 /R s , R d S 3 /R s ), internal volume change (V 2 /V 1 , V 3 /V 1 ), and effective volume change The geometry of the bellows in the stable states can be exported into the fold format. Fig. 11 Interface of the MATLAB code to generate the geometry and calculate the volume for multi-stable Miura-ori and Kresling bellows