Foldable Hexagonal Structures Based on the Threefold-Symmetric Bricard Linkage

A number of large space structures are closed loops consisting of bars or trusses connected by joints with internal mobility so that the structures have deployable performance. Compared to the 2D linkage assemblies in designs, 3D linkages [1], especially 3D overconstrained linkages [2,3] with single mobility, are preferred to obtain the structure with fewer links and joints as well as easy and reliable actuation and control. They also have potentials in the design of platforms in parallel robots [4,5]. 3D overconstrained linkages have been developed since the first one, Sarrus linkage [6] invented in 1853. Then a number of 4R, 5R, and 6R overconstrained linkages were proposed and studied [7–11]. About the single-loop overconstrained linkage with the maximum number of bars, the Bricard linkage is the only 6R overconstrained linkage independent of other linkages, which contains three different types of mobile octahedra [12] and three additional types of mobile linkages [13]. Recently, some novel linkages [14–22] were derived, such as the mixed double-Goldberg 6R linkages by combining two Goldberg 5R linkages through a common link pair or the Bennett linkage [21], six classes of 6R linkages for circular translation by type synthesis based on 6R overconstrained mechanisms [22].

Multiple overconstrained linkages can be combined for realizing many novel forms of deployable structures with specific design objectives [23–29]. For instance, Chen et al. realized one-degrees-of-freedom (DOF) folding of waterbomb tessellation with thick panels by employing two kinds of Bricard linkages [23]. Wang and Kong constructed deployable polyhedron mechanisms by adopting Bricard linkages and other linkages [24]. Qi et al. constructed a class of deployable mechanisms by several plane-symmetric Bricard linkages [25]. Adopting the threefold-symmetric Bricard linkage, Zeng and Ehmann presented a series of parallel hybrid-loop manipulators [26], Shang et al. constructed a deployable robot [27], while Yang and Chen proposed a one-DOF spatial linkage to realize the polyhedral transformation between tetrahedron and truncated tetrahedron [28]. Xiu et al. constructed Fulleroid-like Archimedean mechanisms based on a Sarrus-like overconstrained eight-bar linkage [29].

To apply overconstrained linkages into practical applications, the cross-section of their actual bars should be considered in the design state. Based on Crawford and Hedgepeth's principle [30], Pellegrino et al. revealed a model of a 4R linkage with a square and a bundle with a square cross-section as the deployed and the folded configurations, respectively [31]. This linkage is, in fact, an alternative form of the Bennett linkage. Later, Chen and You studied the family of the alternative form of Bennett linkage symmetrically and proposed a general method to realize the deployable square frames with square cross-section bars [32,33]. Viquerat et al. provided one-DOF foldable rectangle structure based on plane-symmetric 6R linkages, which are also fabricated with square cross-section bars [34], followed by multi-DOF 8-bar and 10-bar foldable rings [35,36]. Gan and Pellegrino proposed a specific plane hexagonal frame that can be folded into a dense bundle with isosceles triangular cross-section bars [37], which is actually an alternative form of threefold-symmetric Bricard linkage [38]. Yet, a general method to design deployable hexagonal structures with isosceles triangular bars to realize the deployment between a plane and a bundle configurations is still an unexplored research study.

Therefore, the aim of this paper is to present all possible construction of foldable equilateral hexagons based on the threefold-symmetric Bricard linkage, which can be deployed between a planar hexagon and a hexagonal cross-section bundle with one DOF. The layout of the paper is as follows. In Sec. 2, the geometric relationship between the Bricard linkage and its alternative form is set up. Section 3 is devoted to constructing the physical model with equilateral triangle cross-section bars by studying the relationship between design parameters and kinematic variables. Discussions on physical blockage of the obtained hexagonal structures are given in Sec. 4. Conclusions end the paper in Sec. 5.

The corresponding kinematic curves for some twist angles are depicted in Fig. 2. When $α=π/3orα=2π/3$⁠, there are two particular configurations, a planar triangle and a compact bundle [38], which offer a folding element for constructing large-scale deployable structures. Here, we named the original form of the threefold-symmetric Bricard linkage in a form so that its links are perpendicular to their joints under the D-H notation [39]. Yet in many application cases, the foldable hexagonal structures are desired more often than the triangle one, which leads to the alternative form of this threefold-symmetric Bricard linkage, in which the physical links are no longer perpendicular to the joint axes.

Therefore, an alternative form of the threefold-symmetric Bricard linkage can be obtained when the pair of c and d is determined.

In the deployment process, when A, C, and E meet at a point while the points B, D, and F meet at another point simultaneously, the alternative form becomes a bundle, which is called the compactly folded configuration (abbr. the folded configuration), as shown in Fig. 3(a). When A, B, C, D, E, and F are on the same plane, the linkage is fully flattened to form an equilateral hexagon, which is called the fully deployed configuration (abbr. the deployed configuration), as shown in Fig. 3(b). Kinematic variables θ, φ are denoted as θ_f, φ_f and θ_d, φ_d at these two configurations, respectively.

Considering Eqs. 2(b) and (13), the relationship between θ_d and θ_f can be obtained, which is plotted in Fig. 4 for some twist angles α’s.

Further study shows that there will be a feasible pair of θ_d and θ_f for all possible α ∈ [44.4π/180, 135.6π/180] to obtain the alternative forms with deployed and folded configurations. (1) When 44.4π/180 ≤ α < 57.4π/180, the curve between θ_f and θ_d has two loops; (2) when 57.4π/180 ≤ α < π/3, some other paths exist between these two loops; (3) when π/3 ≤ α < π/2, the curve has two loops with the origin at their centers; and (4) when α = π/2, it becomes a single loop. Situations of α ∈ (π/2, 135.6π/180] are the same as those of α ∈ [44.4π/180, π/2) as the symmetric property of the linkage.

Furthermore, according to Eqs. (8) and (10), the pair of |c/l| and |d/l| is determined for each folded configuration, i.e., θ_f. For the above four situations, the relationship between the pair of |c/l| as well as |d/l| and θ_f is depicted in Fig. 5. For situation (1), there is no feasible θ_d when θ_f is close to zero according to Fig. 4(a), thus the absolute value of θ_f could not be too small although there may exist the pair of |c/l| and |d/l| as shown in Fig. 5(a). For situations (3) and (4), θ_f could not reach π and –π as there is no feasible θ_d according to Figs. 4(c) and 4(d), as well as the pair of |c/l| and |d/l| according to Figs. 5(c) and 5(d). Meanwhile, for all situations, when φ_f gets close to zero, |d/l| is too large to be adopted in practice.

To show the foldability, the fully deployed and the compactly folded configurations of a designed structure for each of the above situations are given in Fig. 6. All of these structures can be folded to a compact bundle, where a small value of twist angle α may generate physical interference, i.e., the deployed configuration of Fig. 6(a) is a dual-loop polygon.

Because six equilateral triangles surrounding one common vertex in a plane can form a regular hexagon with no gap, equilateral triangle cross-section bars are preferred to construct the hexagonal structure to obtain the minimum cross-section at the folded configuration. Taking sides of these equilateral triangles as the unit length, the physical structure can be constructed with three design parameters, angle λ, angle ω, and the bar length L, see Fig. 7. Due to the symmetry, $λ∈[0,π]$ and $ω∈[0,2π/3]$⁠.

Figure 8 shows a hexagonal structure at its deployed configuration, where A₀B₀C₀D₀E₀F₀ is the original Bricard linkage and ABCDEF is the alternative form of the linkage. Revolute joints are set along GA, QB, JC, SD, ME, and UF to connect six equilateral triangle cross-section bars AGP-BHQ (it is cut from a triangular prism by triangles AGP and BHQ, while AB, GH, and PQ are the three sides), BHQ-CJR, CJR-DKS, DKS-EMT, EMT-FNU, and FNU-AGP. It should be noticed that AP, GP, BH, HQ, etc. are common sides of adjacent bars just at the folded configuration, while ABCDEF, GHJKMN, and PQRSTU are three paralleled planes. The cartesian coordinate system O-xyz is established, where O is the center of GHJKMN, the z axis is the normal of the plane, the x axis is directing to point H, and the y axis is set by the right-hand rule. Meanwhile, one local frame is set at each R-joint (like Fig. 1). As OX⊥GH, triangle XYZ is the right section of bar AGP-BHQ, then set $XY¯=YZ¯=ZX¯=1$⁠. Projections of A, B, P, Q, Y, Z in the plane GHJKMN are denoted as A′, B′, P′, Q′, Y′, Z′, respectively.

For any possible α ∈ [44.4π/180, 135.6π/180], ω and λ are not one-to-one related. In many practical needs, a regular hexagon is expected at the deployed configuration, i.e., ω = π/3. In this case, from Eq. (18), the relationship between λ and α is plotted in Fig. 9(c).

For example, the deployment process of a structure with α = π/3, ω = π/3, and λ = 18.83π/180 is shown in Fig. 10. Hence, even with the same folded and deployed configurations, when the tilted angle of triangle bars λ is different, the corresponding twist of the original linkage α will be different, in turn, the relationship among kinematic variables, θ_f, φ_f and θ_d, φ_d, will also be different, which is set up as follows.

Thus, c and d can be solved by the above Eqs. (35) and (36). Then, θ_f and φ_f are obtained according to Eqs. (8) and (10). The relationships among θ_d (θ_f) and λ, ω are plotted in Fig. 11(a), while the relationships among φ_d (φ_f) versus λ, ω are plotted in Fig. 11(b).

In particular, for the regular hexagon, i.e., ω = π/3, the relationships between θ_d/θ_f, φ_d/φ_f and λ are shown in Figs. 12(a) and 12(b), respectively. The amounts of rotation angles, β = |θ_f − θ_d| for the group of joints A/C/E and β = |φ_f − φ_d| for the group of joints B/D/F, are plotted in Fig. 12(c). Intuitively, the greater the β, the greater the possibility of interference of the linkage.

These curves can be divided into three sections, in section II where π/3 < λ ≤ 2π/3, all six joints rotate more than π, while in section I or III, the rotation of three joints is larger than π and that of the other three is less. Only when λ = π/2, six joints rotate the same angles to move from a folded configuration to a deployed one or vice versa, such a physical model is shown in Figs. 13(a) and 13(b). In fact, the twist of the linkage is α = 78.46π/180, whose kinematic curve is plotted in Fig. 13(d) where the folded and deployed configurations are marked by F and D, respectively. It can be found that although the whole rotation of six joints is identical, over the deploying process, joints A/C/E and joints B/D/F rotate with different velocities, and it cannot realize the transformation physically as physical interference occurs, see Fig. 13(c). Therefore, the physical blockage cannot be detected in these linkages by simple observation, and it is essential to study the situation on the physical blockage.

In practice, even though the theoretical linkage can work efficiently with no bifurcation, there may be some physical blockages. There are two types of physical blockages, the first one is that just one set of three alternative joints of the structure meets at a configuration before reaching the compactly folded configuration.

Figure 14(a) shows an example of this type, in which points A, C, and E coincide, while points B, D, and F are still separated. The alternative situation, in which points B, D, and F meet and points A, C, and E are separated, will also render this type of blockage.

The second type is that just three pairs of bars of the structure reach the fully deployed configurations before reaching the fully deployed configuration. Figure 14(b) shows an example of this type, in which end surfaces of bar-pairs AB and BC, CD and DE, and EF and FA at joints B, C, and D meet, respectively, while end surfaces at joints A, C, and E do not reach the fully deployed configurations. Similarly, full deployments of bar-pairs BC and CD, DE and EF, and FA and AB before reaching the fully deployed configuration will also render this type of blockage.

As π/3 ≤ α ≤ 2π/3, the motion path of threefold-symmetric Bricard linkage is a closed one. According to the characteristics of the above illustrations, four configurations B₁ (θ_d, φ_b), B₂ (θ_b, φ_f), $B1′$ (θ_b, φ_d), and $B2′$ (θ_f, φ_b) are possible to generate physical bifurcations, where θ_b and φ_b represent the values of θ and φ at corresponding blockages, respectively. According to the features of these two types of physical blockages, the first type is related to the folded configuration, i.e., θ_b = θ_f, φ_b ≠ φ_f or θ_b ≠ θ_f, φ_b = φ_f, and the second one is related to the deployed configuration, i.e., θ_b = θ_d, φ_b ≠ φ_d or θ_b ≠ θ_d, φ_b = φ_d. Thus, B₂ and $B2′$ are the first type, while B₁ and $B1′$ are the second one.

Therefore, if there is a continual path between D and F, all of B₁, B₂, $B1′$⁠, and $B2′$ are not on this path, the structure can realize the transformation between the fully deployed and the compactly folded configurations physically; otherwise, there must be some physical blockages. It should be noticed that the choosing of motion path and direction is determined by physical parameters. There could be different results even for linkages with the same kinematic variables, such as linkages with twist $α=π/2−arctan2$ in Figs. 8 and 9 in Ref. [38], and their motion paths are depicted in Fig. 15. For the linkages with the same twist angle, both of their deployed configurations are at D, and their motion directions are opposite due to their physical geometries. The first linkage moves along path I and terminates at T due to the physical blockage, while the second one is able to finish the folding from D to F passing U and V with no physical blockage.

An example of physical blockages is shown in Fig. 16 as α = 7π/18 to show different cases, there are two continual paths between D and F. In Fig. 16(a), B₁, B₂, $B1′$⁠, and $B2′$ are all located on one of these paths, thus no blockage exists if the linkage works along the other path. In Fig. 16(b), B₁ is located on one of these paths and B₂, $B1′$⁠, and $B2′$ are located on the other one, thus one physical blockage point occurs at least. Meanwhile, B₁ and B₂ are located on one of these paths, $B1′$ and $B2′$ are located on the other one in Fig. 16(c), thus two physical blockage positions occur.

Meanwhile, the motion path of the threefold-symmetric Bricard linkage is two non-intersected branches rather than a closed one when α < π/3 or α > 2π/3. The fully deployed and the compactly folded configurations, D and F, must be on one of these branches at the same time, e.g., the structure as shown in Fig. 17 when α = 57.42π/180.

According to this approach, we can find that there must be some physical blockages in the previous linkage with ω = π/3 and λ = π/2, as shown in Fig. 13.

In this paper, we propose a method to design general deployable hexagonal structures based on alternative forms of threefold-symmetric Bricard linkages. The relationship between threefold-symmetric Bricard linkage and its alternative form is analyzed, which derives the conditions of kinematic variables for the alternative form to realize the desired deployment between the fully deployed and the compactly folded configurations, i.e., a regular hexagon and a bundle. To facilitate practical applications, regular triangular cross-section bars are considered, and the obtained conditions of kinematic variables are hence transformed into the relationship among design parameters. Furthermore, two types of physical blockages are studied, and a simple way is presented, in which the appearance of physical blockage can be judged conveniently.

The proposed method supplies a systematic way to design deployable hexagonal structures, where bars are with a regular triangular cross-section. Other types of bars, such as those with other cross-sections or polygonal faces, will be studied in the future. Meanwhile, tessellating some of these structures to generate novel deployable structures is also one of our future works.

F. Yang wishes to appreciate the financial supports from the Natural Science Foundation of Fujian Province, China (Grant No. 2019J01209; Funder ID: 10.13039/501100003392), and Fujian Provincial Department of Education (Grant No. TJ180027; Funder ID: 10.13039/501100003410). Y. Chen thanks the support of the National Natural Science Foundation of China (Grant Nos. 51825503 and 51721003; Funder ID: 10.13039/501100001809). Z. You wishes to acknowledge the support of Air Force Office of Scientific Research (Grant No. FA9550-16-1-0339; Funder ID: 10.13039/100000181). He was a visiting professor at Tianjin University during the course of this research.