Geometric Design Methodology for Deployable Self-Locking Semicylindrical Structures

: Due to their unique bistable characteristics, deployable self-locking structures are suitable for many engineering ﬁ elds. Without changing the geometrical composition, such structures can be unfolded and locked solely by the elastic deformation of materials. However, their further applications are hampered by the lack of simple and systematic geometric design methodologies that consider arbitrary structural curvature pro ﬁ les. This paper proposes such a methodology for double-layer semicylindrical grid structures to simplify their cumbersome geometric design. The proposed methodology takes joint sizes into account to ensure that the design results can be applied to actual projects without further adjustments. By introducing symmetry into the structural units (SUs) and selecting reasonable geometric parameters that describe the structural side elevation pro ﬁ le, a concise set of simultaneous nonlinear geometric constraint equations is established, the solution of which provides the geometric parameter values of the grid shape. On this basis, the remaining geometric parameter values, i.e., the geometric parameter values of the inner scissor-like elements (SLEs) of each SU, can be achieved from the equations derived from general SLEs. Design examples and the assembled physical grid structure indicate the feasibility and wide applicability of the proposed geometric design methodology.


Introduction
Since ancient times, the concepts of deployability and foldability have dramatically enhanced people's lives.Naturally, such concepts have also been introduced into the design of modern structures and mechanisms.From the mobile exhibition pavilion, mobile theater [1], and deployable swimming pool cover [2] to the retractable roof (e.g., [3,4]) and space truss (e.g., [5,6]), the applications of deployable structures have experienced remarkable development.In conjunction with their applications, theoretical analyses of these concepts have made enormous progress.Many studies have been conducted to summarize the geometrical principles of developable structural systems [7][8][9][10], to discuss their structural capabilities under completely expanded configurations bearing loads [11][12][13], and to probe the inherent regularities of their intermediate configurations during mechanical motions [14][15][16][17].In addition, the utilization of new materials also provides deployable structures with unique characteristics that enable them to operate in special environments [18,19], thereby facilitating an extensive imagination space for the further development of deployable structural forms.
Deployable structures have been the subject of increasing attention as a result of their many virtues, such as their industrial prefabrication, their reusability, and their rapid construction time [20].As one type of deployable scissor structure (e.g., [21,22]) composed of scissor-like elements (SLEs) (Figure 1), the deployable self-locking structure can support its own weight and bear a certain external load without imposing any geometric constraints when it is fully deployed.It is also bistable, that is, it is free of stress when either completely folded or completely expanded, which constitutes the basis for its geometric design [23,24].With these characteristics, the self-locking structure possesses much more excellent properties than traditional structures, such as faster unfolding and higher reliability.Therefore, the self-locking structure, which was invented by Zeigler [25], has been applied in a wide variety of fields.Without changing the geometrical composition, such structure can be unfolded solely by the bending elastic deformation of components [26].Among the many configurations of self-locking deployable grid structures, the semicylindrical shell is the most widely used structural style for temporary buildings (Figure 2).The double-layer semicylindrical grid structure is generally composed of hexahedral structural units (SUs, Figure 3), and each independent SU is characterized by the abovementioned integral grid structure.To further enhance the self-stabilizing capability of the entire grid after deployment, the studies focusing on SU configuration improvements are still in progress [27].Correspondingly, some progress has been made in the research on more refined mechanical models [16,22,28].Furthermore, in the direction of lightweighting semicylindrical grid structures, the reciprocal linkages proposed by Pérez-Valcárcel et al. [29,30] are expected to be a promising research direction.The spatial configurations of all deployable grid structures have a vital influence on their structural performance; therefore, the geometric design of the spatial configuration is critical [31][32][33].To reduce the design workload for grid structures with array characteristics or with complex spatial configurations, the geometries of repeatable parts and detachable simple parts can be separately designed first.Then, the final grid structure can be obtained by sequentially assembling the divided parts.Of course, this method of first designing each part separately and then assembling the whole structure also provides a way for effectively diversifying the types of deployable structures [34].In particular, the member lengths of the SLEs in each assembled structure must meet the foldability conditions of the overall grid structure to ensure that the structure can fold smoothly (Figure 1).In actual engineering endeavors, most scissor structures are usually flat plates (e.g., [35,36]) or shell surfaces with a constant curvature profile (e.g., [2]).Unfortunately, further applications of semicylindrical deployable grid structures are hindered by the absence of a geometric design methodology that permits an arbitrary curvature profile.To solve this problem, researchers have developed their systematic geometric design methodologies by assuming some approximate geometric constraints or by not fully considering the hub sizes [37].However, these methodologies require a large number of geometric parameters and the solution of complicated nonlinear equations, and thus are not very convenient for design purposes or for engineering applications.
Therefore, this paper proposes a new methodology for self-locking deployable semicylindrical structures.Compared to previous work (e.g., [37]), this approach does not require the introduction of the algebraic equation as a geometric constraint on the side elevation profile of the shell surface.There is also no need to assume approximate dimensions of the components, which thereby introduce additional geometrical incompatibilities, which would result in the grid structure failing to reach a minimum volume after folding.Rather, the proposed method first establishes the geometric constraint equations used to describe the side elevation profile by selecting appropriate length and angle parameters and then obtains the geometric parameter values of the shape.Combined with the equations derived from general SLEs, the geometric parameter values for the inner SLEs of each SU are obtained, after which the structural geometric design can finally be completed.Obviously, this approach considers the inner and outer SLEs of the SU separately, and thus, fewer coupling equations are required, thereby improving the calculation efficiency.
Moreover, the previous methods integrate the geometrical relationships of adjacent inner and outer SLEs into a single unfolded plan view, which is strictly only applicable to the computation of regular prismatic units in some special structures (e.g., [16,23]).In contrast, the method proposed in this paper considers the universal geometrical relationships of pairs of bars in SUs in three-dimensional space, making it more widely applicable.In addition, the introduction of symmetry not only benefits the design process but also facilitates the assembly of the structure.Therefore, the proposed methodology establishes the SU as a configuration with symmetrical characteristics and takes the joint sizes accurately into account so that the design results can be applied to actual engineering.The final examples and the model produced herein indicate the feasibility and broad applicability of the geometric design methodology for double-layer semicylindrical grid structures.

Construction of the Semicylindrical Grid Structure Configuration
The derivation presented in this section is based on the assumption that the grid joints are ideal joints, which are spatially articulated nodes with a size of zero.The construction of the double-layer semicylindrical grid configuration is illustrated in Figure 4.This derivation will also provide the basis for the mathematical description of the geometric configuration throughout this paper.An auxiliary global right-handed coordinate system is first established for convenience.Then, the projection curve of the structural outer layer surface with arbitrary curvature is plotted in the XOZ-plane, as shown in Figure 4a.The axis of symmetry of the surface is the Z-axis.One point of intersection between this curve and the X-axis is A1.S denotes the structural span.Similarly, the inner layer projection curve is plotted with identical curvature.The corresponding intersection point is denoted as a1, as shown in Figure 4b.The parameter t represents the structural thickness, which is the distance between A1 and a1.This pair of curves translates into a distance of W along the Y-axis (Figure 4c).Accordingly, the structural upper layer surface is obtained, where B1 and b1 are the corresponding points after translation.The inner layer surface is formed by reducing the length of a1b1 symmetrically (Figure 4d), and d denotes the reduced length of the two sides.Combined with the lengths of the bars that will be derived in the following sections, the peripheral SLEs of the structure are inserted into the anticipated SU shapes (Figure 4e).Finally, the inner SLEs are inserted into the corresponding SUs (Figure 4f).This step also marks the completion of the construction of the structural configuration.

Selection of Geometric Parameters
Based on whether an independent SU exists at the top of the grid, two types of configurations for the semicylindrical grid structure are given to facilitate the geometric design, as shown in Figure 5.In the geometric design, parameters are needed to establish a mathematical expression that describes the structural spatial configuration, and then a system of equations consisting of multiple expressions can be used to solve for the values of these parameters.However, there is a large number of geometric parameters suitable for establishing the above equations; therefore, to meet the needs of building functions and ensure the convenience of subsequent calculations, these parameters need to be reasonably selected.Obviously, for the purpose of effectively guaranteeing the spatial size of the structure, the parameters describing the geometric characteristics of the overall structure, including the net height, the lateral span, the facade length and the structural thickness, are suitable parameters for the geometric equations.In addition, to describe the changes in the curvature of the outer structural surface, angle variables reflecting the upper surfaces of the SUs should be selected as geometric parameters.Accordingly, the projected lengths of all SUs onto the side elevation profile of the structure can be selected as other parameters.Allowing for the symmetry of a semicylinder, only one half-span arch is considered.The number of SUs constituting the half-span structure is n.Each geometric parameter is marked in Figure 5. Ai and Bi (i = 1, 2,..., n + 1) represent the lateral vertexes of the unit top rectangles in the upper layer, while Ai' and Bi' (i = 1, 2,..., n + 1) represent the vertexes of the bottom rectangles in the inner layer, refer to the example in Figure 6.ai is the corresponding projection of Ai'.Li (i = 1, 2,..., n) is the transverse length of the i-th unit.Hi (i = 1, 2,..., n) is the height of Ai+1.W denotes the overall width of the structure.In each SU (e.g., Figure 6), there are two planes of symmetry: the xoz-plane and the yoz-plane.In addition, the xoz-plane is parallel to the XOZ-plane in the global coordinate system.The xoy-plane coincides with the bottom surface of the SU Ai'Bi'Bi+1'Ai+1' and is parallel to the upper surface AiBiBi+1Ai+1.For convenience, the variable representing the angle between the line AiAi+1 and the perpendicular line of Aiai in the global XOZ-plane is defined as θi (i = 1, 2,..., n).

Mathematical Descriptions of the Structural Profile
Known quantities should be presupposed based on the unique shape requirements.Assume that Hi, S, W, t, and d are known, for instance, and assume that Li and θi are the unknown quantities.According to the geometric constraints on the side elevation profile of the structure as well as the symmetry of each planar trapezoid Aiaiai+1Ai+1, the relationship of θi can be described as: where α is the selection coefficient.If structural configuration 1 is being referenced, α = 1; otherwise, α = 2.Because the SU has symmetrical characteristics and n ≥ 1, each angle parameter should meet the following constraint: Furthermore, all the joint heights in the two configurations can be represented in a unified form: For the present structural model, the height of each joint should vary between the following limits: ( ) where L' will be derived below to indicate the member length of the outer SLEs.The structural span can be denoted as follows: where β is the selection coefficient.If structural configuration 1 is being referenced, β = 1; otherwise, β = 0.In addition, S should be selected within a certain range:

Geometric Constraints on the Member Lengths of the SU
Consider the first SU separately (Figure 6).M0 and M1 are the pivots of the unit lateral trapezoids A1A1'B1'B1 and A1A1'A2'A2, respectively.Four equations exist between the member lengths simultaneously: The subscript refers to the line between two nodes.With the following foldability constraint: the following can be derived: In other words, the members in the outer SLEs have the same length.This conclusion is also applicable to the other units.Since the SUs are connected by the same side face, the members in the outer SLEs of the overall structure also have the same length.In the A1A1'B1'B1 plane: In the A1a1a2A2 plane, the explicit expression for the unit upper side length is as follows: Similar conclusions can be extrapolated for the other units: This equation also reveals the corresponding relationship between Li and θi in one SU.

Geometric Constraint Equations Describing the Structural Profile
From the derivation in the previous subsection, all the geometric parameters that determine the side elevation profile of the half-span structure can be expressed by the parameter Li and the parameter θi (i = 1, 2,..., n).Obviously, the final set of geometric constraint equations should have 2n unknown parameters, and 2n equations are needed for these unknown quantities.
First, the overall configuration of the double-layer semicylindrical grid structure requires that Equation (1) must be established.Second, the structural characteristics of each SU also require that Equation ( 12) must be established, where the number of equations is n.Finally, since the rise-span ratio has a great influence on the arch carrying capacity after the structure is fully deployed, Equation ( 5) for the structural span and the equation for the top joint height, that is, the equation when s = n in Equation (3), should be satisfied.At this point, n-3 equations are still required, and the corresponding number of equations should be selected in Equation ( 3) according to the needs of building functions to obtain the unique solution to the equations.From the above, the required number of equations is n ≥ 3. When n < 3, one of the structural spans and the top joint height is preferred as the control parameter.Due to the nonlinear characteristics of these equations, a numerical algorithm such as the quasi-Newton method can be used to solve them.

Pivot Endpoint Positions on the SLEs
For the geometric design of a grid structure comprising SLEs, in addition to the required length of each member, it is also necessary to obtain the pivot endpoint positions on the SLEs.As shown in the previous sections, concise expressions can be obtained by introducing symmetrical properties into the structural configuration.Similar effects can be achieved for the expressions used to obtain the pivot endpoint positions.The general formation of a hexahedral SU is illustrated in Figure 7. Oi is the upper terminal node of the inner SLE on the z-axis, and Oi' is the lower terminal node.M0 and Mi denote the pivots of AiAi'Bi'Bi and AiAi'Ai+1'Ai+1, respectively.Ni is the pivot of the inner SLE consisting of the members AiOi' and Ai'Oi.The variable w representing the unit width is equal to the structural width W. Due to the symmetrical nature of the SU, the number of unknown SLEs has changed from eight pairs to three pairs (marked with different colored bold lines in Figure 7).

Outer SLEs of the SU
Consider the lateral face Ai'AiBiBi' separately (Figure 8), where M0' is the projection point of pivot M0 onto the side Ai'Bi'.The distances between Ai and M0, M0 and Bi', and Ai' and Bi' are denoted as L01', L02', and L0', respectively.Accordingly, the following equations can be obtained: where 0 2 = − L ' w d .Then, consider the lateral face AiAi'Ai+1'Ai+1 (Figure 9), where Mi is the pivot.The distance between Ai and Mi is Li1', and the remaining half is Li2'.The intersection angle between AiAi+1 and the vertical line of Aiai is θi in the plane Aiaiai+1Ai+1.The following equations can be obtained: ( )

Inner SLEs of the SU
Generally, the two bars of one inner SLE within the SU are located on different planes.In this situation, the common perpendicular line between the axes of these two members can be used as the axis of the pivot, and their intersection points serve as the two endpoints of the pivot.To facilitate the geometric design of the structure, it is necessary to derive an explicit expression for the pivot endpoint coordinates; the theoretical derivation of this expression will be given below.
According to the above description, the straight lines along which the axis l1 and the axis l2 are located can be expressed as: , ( ) The points M and N are on these lines, as shown in the following: When j = 1, γ represents the letter m, and the corresponding case represents the letter n.Further derivations can be conducted for the above equation: ( ) ( ) Combined with Equations ( 17) and ( 20), the following can be obtained: The above equation can be abbreviated as follows: The solutions of these two linear equations are as follows: where the variables are listed below: The coordinates (m1, m2, m3) T and (n1, n2, n3) T can be obtained by providing the endpoint coordinates of axis l1 and axis l2.Since the outer contour of the SU has been determined in the previous section, the coordinates of one node per axis are known, that is, only the coordinates of the internal endpoints Oi and Oi' of the inner SLE are unknown.However, from the construction of the structural configuration, the x-coordinate and ycoordinate values of the two nodes are both zero, and thus, the z-coordinate values become the only two geometric unknowns.By establishing a value for one z-coordinate and the foldability constraint, the other coordinate value will also become known.It should be noted that the preset value has a great influence on the structural self-locking ability.At this point, the foldability constraint of the inner SLE should be expressed as follows: In this way, the problem of solving the member lengths of the inner SLEs and the pivot endpoint positions eventually reduces to the problem of solving the geometric parameter value (i.e., the z-coordinate value) inside an SU.To that end, a numerical solution can be obtained using a simple iterative method.

Geometric Design Methodology Considering Joint Sizes
The concepts and construction of the geometric design methodology for a half-span structure have been elaborated in the previous sections.However, the proposed methodology considering ideal joints can be used to preliminarily determine only whether the proposed set of design parameter values can obtain the required spatial configuration because the approach presented herein does not consider the joint dimensions.Thus, the results cannot be applied to an actual project.In this section, a disc-shaped hub joint commonly used in deployable grid structures is taken as an example to discuss the methodology considering joint sizes.Allowing for ideal joint conditions, the side elevation profile can be drawn using the geometric parameters of the overall structure.On this basis, the joint sizes will be considered in the following by regularly inserting the hub joints into the outer skeleton composed of ideal joints.In the proposed method, the hub center coincides with the ideal joint, and the ideal joint at this time becomes the intersection of a series of lines depicting the structural contour shape.
The hub joint is a discoid part with an equivalent radius r for connecting different rods, and the calculation model is shown in Figure 11.The spatial coordinate system still conforms to the right-hand rule, and the coordinate origin is at the centroid of the hub joint.The lines connecting the coordinate origin to the points Jk (k = 1, 2,…, 8) arranged counterclockwise along the circumference divide the disc into eight equal parts, where Jk represents the joint point between the rod end and the hub joint.

Spatial Configuration of SUs Considering Joint Dimensions
As mentioned earlier, for the preset geometric parameters that describe the side elevation profile of the structure, there are no differences between the structural model considering ideal joints and the one considering joint sizes.However, accounting for the sizes of joints will substantially change the member lengths, further resulting in changes in the unknown geometric parameters describing the grid structure profile and the unknown geometric parameter of the inner SLE of each SU.
Allowing for the symmetry of the SU and the convenience of assembly, the plane of each hub joint is supposed to be perpendicular to the hexahedral edge; for example, the joint at Ai is perpendicular to the intersection between the plane AiAi'Bi'Bi and the plane AiAi'Ai+1'Ai+1, the edge AiAi'.The SU model considering joint sizes is shown in Figure 12.The two inner hub joints are parallel to the xoy-plane, and the z-axis passes through their circle centers.A detailed description of the spatial transformation of the outer hub joints will be given in the following discussion.In addition, the selected geometric parameters describing the shape of the SU, including w, Li, and t, are identical to those describing the model considering ideal joints.Additionally, due to the symmetrical properties of the model, the three pairs of unknown SLEs are marked with different colored boldlines.The construction of the SU model comprises the following steps: 1.The original relative positions between the joint coordinate systems and the unit coordinate system are illustrated in Figure 13a.The ideal joint positions are replaced by the circle centers.The three axes of the coordinate system of one hub joint are all parallel to the corresponding axes of the SU. 2. Rotate the joints at points Ai, Ai', Bi', and Bi about their y'-axes through an angle θi, and rotate the others in the peripheral SLEs through an angle -θi. 3. Rotate these joints again about their x'-axes.The joints at the points Ai, Ai', Ai+1', and Ai+1 are rotated through an angle -φ, and the others in the peripheral SLEs are rotated through an angle φ.Accordingly, based on the model construction process, the new unit configuration is obtained (Figure 13b) and arctan( / ) ϕ = d t (Figure 9).

Outer SLEs Considering Joint Sizes
Based on the spatial transformation steps described above, joint point J7 on the hub joints Ai and Ai' as well as joint point J3 on the hub joints Bi and Bi' are still in the plane Ai'AiBiBi'.The new isosceles trapezoid consisting of these joint points is shown in Figure 14.M0' is still the projection of M0 onto the short side.But, the length of the short edge becomes L0 J '.And M0 partitions the peripheral rod into two segments L01' and L02'.Based on the side length without considering the joint sizes, a new side length can be obtained by reducing the length by 2Δw.The total length of each peripheral member can be derived as follows: The lengths of the two segments are expressed as follows: ( ) where In the other plane AiAi'Ai+1'Ai+1, the total length of each rod is also L' because of the foldability constraint and the symmetry of the structure (Figure 15).Similarly, after a series of transformations, point J5 on the hub joints Ai and Ai' and point J1 on the hub joints Ai+1 and Ai+1' are still on one plane.The side is reduced by 2∆Li.The longer side length is Li J , and the shorter side length is Li J '.The pivot Mi partitions the peripheral rod into two lengths Li1' and Li2'.Therefore, the following conclusions can be drawn: where cosθ Δ = i i L r .In addition, the distance between the hub centers of the two hub joints Ai and Ai+1 can be expressed as follows: For the overall structure considering joint sizes, when searching for the values of the unknown parameters Li and θi on the basis of the given geometric parameters t, r, d, W, S, and Hi, the geometric constraint Equation (12) needs to be replaced with Equation (35), and the simultaneous equations mentioned in Section 2.5 need to be solved.

Inner SLEs Considering Joint Sizes
For the inner SLEs, the spatial translation method presented above is not readily applicable for directly obtaining the rod length; therefore, the problem can be solved by calculating the position vector after the coordinate transformation of the rod endpoint.A brief introduction to the spatial transformation formula of the coordinate system is provided below: The elements of this matrix are the direction cosines between the axes of the two coordinate systems.For example, r12 is the direction cosine between the x-axis of A and the y-axis of B. In addition, A B R can be further written in an expanded form: where Cm represents the intermediate stage of each small step change from B to A. Therefore, the rotation matrix of each hub joint can be expressed according to the abovementioned transformation steps: where Rot ' is the rotation matrix rotating only about the x'-axis, and Rot ' is the rotation matrix rotating only about the y'-axis.
Using the abovementioned spatial transformation theory, the spatial coordinates of the joint points Jk between the inner SLE members and the outer hub joints of the SU can be obtained.
In addition, for the convenience of the structural design, the coordinate vectors of the joint points on the two inner hub joints of the SU are listed below: where zi1 and zi2 are the z-coordinate values of the upper and lower inner hub joints, respectively, of the SU labeled i.When zis is zero, o Jk ' p  is the position vector of joint point Jk in the local coordinate system of the hub joint.Furthermore, Equations ( 20) and ( 23)-( 27) can also be used to calculate the geometric parameters of the inner SLEs.

Design Examples of General Grid Structures
According to the design methodology proposed in this paper, the geometric design of an inner SLE is carried out separately after the design of the grid shape is completed, and the design should be based on the overall structural shape.In addition, for the geometric design of a general scissor structure that lacks a self-locking capability, it is fundamental to obtain the geometric parameters that determine the overall structural shape.Therefore, the design parameter values of the four sets of grid shapes that indicate the universal applicability of the proposed methodology to general double-layer deployable scissor structures are given first in Table 1.The outlines of the four grid structures are plotted in Figure 16.Configuration a is a simple geometric form with an arbitrary curvature profile, whereas configuration b is a more common geometric form with a constant curvature profile.Configurations c and d represent geometric forms with generalized arbitrary curvature profiles, and the half spans of these two configurations can be assembled into a new structural configuration when the two configurations have the same side dimensions and conform to the foldability condition of the assembled structure.The diversity of structural forms can be immensely enriched by a similar assembly method.

Design Example of One Self-Locking Grid Structure
For a self-locking deployable grid structure, the geometric design of the inner SLE determines the self-locking ability of the structure and whether it can be smoothly deployed.Configuration a is selected as the design basis of the self-locking grid structure considering joint sizes for the convenience of both the calculation and the assembly, and all the geometric parameter values are listed in Table 2.For the inner SLEs of each SU, AiOi' and Ai'Oi are selected to obtain the corresponding geometric parameter values.Moreover, to obtain a consistent table, the position vectors of the endpoints are listed separately in Table 3.

Physical Self-Locking Grid Structure
According to the results, a designed self-locking grid portion is subdivided into three components with array characteristics that are sequentially assembled into a complete physical grid structure.The members of this grid structure are made of hollow fiberglass tubes, the elastic modulus is 3.0 × 10 4 MPa, and the density is 1.89 × 10 −3 g/mm 3 ; the cross section of the pipe has an outer diameter of 17.5 mm and a wall thickness of 1.6 mm.The physical self-locking grid structure can be smoothly deployed, as shown in Figure 17, thereby verifying the feasibility of the design methodology.
In addition, the hubs are made of polyhexamethylene adipamide, featuring an elastic modulus of elasticity of 3.8 × 10 3 MPa and a density of 1.15 × 10 −3 g/mm 3 .Certainly, the joints can be manufactured using any material that offers enough rigidity and suitable strength.Previous studies have shown that the choice of joint material has minimal effect on the self-locking capability of the structure presented in this paper [26,31].In this study, the primary objective of selecting hub materials is to ensure that the structure has stable and reliable operational performance.Furthermore, with regard to further optimization of the hub structure design, the joint failure typically does not occur during the deployment process according to the relevant research we are carrying out.But there is a risk of destruction under ultimate load conditions after the structure is fully expanded.In conclusion, the hubs in the grid structure can meet the requirements for mechanism movement.
The output of a mature product requires a complete design process.For example, it has been found in practice that maintaining the self-locking performance index RF3 b within a certain range for each SU ensures smoother unfolding and folding of the entire structure [31].The example presented in this section only represents an interim result of a geometric design of the overall grid structure with suitable self-locking capability.

Figure 4 .
Figure 4. Construction of the double-layer semicylindrical grid model with ideal joints.(a) Projection of outer shell; (b) Projection of inner shell; (c) Two shell layers and outer layer surface; (d) Inner layer surface and SLEs in the lateral face; (e) Insert into other outer SLEs; (f) Insert into inner SLEs.

Figure 5 .
Figure 5. Two types of configurations for the semicylindrical grid structure.(a) Configuration 1: an independent structural unit exists at the top of the grid; (b) Configuration 2: two structural units exist at the top of the grid.

Figure 6 .
Figure 6.The first unit, at the bottom of the entire grid structure.

Figure 7 .
Figure 7. General configuration of one unit with ideal joints.
endpoint positions on the outer SLEs are expressed by the four equations shown above.

Figure 11 .
Figure 11.Real hub joint and corresponding geometric model.

Figure 12 .
Figure 12.General SU model with joint sizes.The hub joint center coincides with the vertex of the hexahedron, and each hub is supposed to be perpendicular to the hexahedral edge.

Figure 13 .
Figure 13.The construction of the SU model with considering hub joint sizes.(a) Original relative positions; (b) Final relative positions.

Figure 14 .
Figure 14.Outer SLE in the lateral face AiAi'Bi'Bi with considering joint sizes.

Figure 16 .
Figure 16.Four structural shapes of structure in side elevation.

Table 1 .
The parameter values of deployable grid structure with four structural configurations.

Table 2 .
The design results of a self-locking deployable grid structure with configuration a considering joint sizes.