Based on the unequal diagonal angles of the angulated element (AE), a reconfigurable dual-AE unit is designed by articulating two AEs symmetrically at nodes A and B, as shown in Fig. 1a. Parameters of the dual-angulated-element unit satisfy that AC = AC′ = BC = BC′ = CD = CE = C′F = C′G = L and DH = EI = FJ = GK = l, where ∠BCD = ∠ACE = ∠AC′G = ∠BC′F is denoted by γ; the angle required to rotate BC clockwise to BC′ is defined as θ, and the diameter of each node is e. The obtained reconfigurable dual-AE unit has two motion modes: scissor-like deployable mode and rotation mode. In scissor-like deployable mode, the dual-AE unit can deploy as a scissor-like element, and the distance between the two axes Lp is calculated in Eq. (1) and described in Fig. 3. 1Lp=2l+Lsin⁡θ2-sin⁡γ+θ2,2dp=L(1-cos⁡γ)+l.

When nodes A and B coincide with each other, the unit reaches its switching configuration; then the two angulated elements are equivalent to two rigid bars, and the dual AEs are in rotation mode. The dual-AE unit becomes two links connected by an R joint, which can make AE-2 rotate counterclockwise around AE-1. The link lengths of the unit are set as L, and the diameter of each node is e. In order to get a better folding effect and avoid linkage interference, the link angle γ of the AE is derived. 3γ=π-arcsin⁡e2L By connecting three dual-AE units with spatial RRR chains, a reconfigurable Bricard-like mechanism is constructed, as shown in Fig. 2c. In the RRR chain, the lengths of two links are equal; the axes of the adjacent R joints are perpendicular to each other; and the first and the third R joints are, respectively, parallel to the R joints of its articulated dual-AE unit. When nodes A and B coincide, R joints at nodes A and B will replace the virtual R joints to form the Bricard linkage, as shown in Fig. 2d.

The 3D model of the Bricard-like mechanism is shown in Fig. 2. There are two modes for the reconfigurable deployable mechanism including prism deployable mode and Bricard turnover mode. In the prism deployable mode, it can deploy from a bundle shape to a regular triangle loop on a plane. Then it goes into the switching configuration, in which both the prism deployable motion and the Bricard turnover motion can be realized. When three dual-AE units turn into the rotation mode by fixing six AEs each as a rigid link, the mechanism switches to the Bricard turnover mode which can perform the same infinite turning motion as a TFS Bricard linkage. To analyze the parameters of the model, the standard model proposed by Denavit and Hartenberg (2021) is used.

As shown in Fig. 2d, the coordinate systems of links of the Bricard linkage are established, where zi is the axis along the revolute axis of joint i, and xi is the axis along the common normal direction from axes zi to zi+1. When nodes A and B coincide with each other, each AE with a link of the RRR chain is fixed as a rigid link, and the six rigid links are connected by six R joints to form a TFS Bricard linkage. The geometric parameters of the Bricard linkage contained in the Bricard-like mechanism satisfy the conditions in Eqs. (4)–(7). And referring to Chen et al. (2005), its closure equations are derived in Eq. (8). Then the relationship curve between θ and φ of the Bricard-like mechanism in the Bricard turnover mode is depicted in Fig. 3. 4a1=a2=a3=a4=a5=a6=dp;5α1=α3=α5=π2α2=α4=α6=3π2;6d1=d2=d3=d4=d5=d6=0;7θ1′=θ3′=θ5′=π-φ,θ2′=θ4′=θ6′=π-θ;8cos⁡θ+cos⁡φ-cos⁡θcos⁡φ=0.

By connecting a number of identical Bricard-like mechanism units in the switching configuration, a multimode deployable arch is constructed. The serial number of the Bricard-like mechanism unit is denoted by (n), as shown in Fig. 4. In order to ensure the symmetry of the mechanism, it is suggested that n is an odd number. From the top view of the multimode arch, the revolute joints of the ith Bricard-like mechanism are denoted by J(2i-1), J(2i), J(2i+1), J(2i+2), J(2i+3), and J(2i+3)′; the revolute axes J(2i-1), J(2i+1), and J(2i+3) are directly out of the paper, while J(2i), J(2i+2), and J(2i+3)′ are parallel to the paper. Therefore, these revolute axes are simplified in Fig. 4, where a dot in a circle represents the direction of the axis pointing out of the paper, and where the dotted line represents the direction of the axis paralleling the paper.

Since the Bricard-like mechanism has 1 degree of freedom (refer to Chen et al. 2005), three identical dual AEs move synchronously. The multimode deployable arch is composed of a number of identical Bricard-like mechanisms, and adjacent Bricard-like mechanisms share their intermediate dual AEs; all of the Bricard-like mechanisms in the arch move synchronously with the first Bricard-like mechanism. Therefore, the multimode deployable arch has 1 degree of freedom as well.

As shown in Fig. 5, the coordinate systems of units (1)–(3) of the deployable arch are established, where zi is the axis along the revolute axis of joint i, and xi is the axis along the common normal direction from axes zi to zi+1. Since the multimode deployable arch is composed of a number of identical Bricard-like mechanisms and they are symmetrically arranged as in Fig. 4, the D–H parameters of the entire arch can be determined by the D–H parameters in Appendix B. As shown in Fig. 8, according to the properties of the three-symmetric Bricard mechanism, the vectors S1, S2, and S3 are guaranteed to intersect at a point, which is named O. Additionally, since S2 is the vector sum of S1 and S3 and since S4 is the vector sum of S3 and S5, then S2 and S4 will necessarily intersect at point O. Moreover, due to the non-collinearity of P2, P4, and O, they lie in the same plane. As shown in Fig. 5, the arch is constructed using identical Bricard-like mechanisms, and adjacent Bricard-like mechanisms share AEs. As a result, S4 and S6 also intersect at point O. Therefore, all the triangular envelope surfaces formed by the Bricard-like mechanisms in the arch are coplanar and intersect at point O.

In the scissor-like deployable mode, φ remains still at 60∘; the mechanism will fold and deploy as the value of θ varies. To avoid link interference, the range of θ in the scissor-like deployable mode is [3arcsin⁡(e/2L), π). Here, we use the length L, the width W, and the height H of the enveloping cuboid to represent the size of the mechanism in the scissor-like deployable mode; here we take 11 Bricard-like units as an example as shown in Fig. 6, and they are derived as follows. 9L=n+12sin⁡θ2+sin⁡θ2-arcsin⁡e2LL+l,10W=3sin⁡θ2+sin⁡θ2-arcsin⁡e2LL+l,11H=2Lcos⁡θ2-arcsin⁡e2L+e.

The volume Vs of the enveloping cuboid of the mechanism in scissor-like deployable mode is derived in Eq. (12). The lengths of links L, l, and e can be given different values. Suppose L = 15 mm, l = 6 mm, and e = 4 mm, then the mesh grids of Vs with respect to θ and n is drawn in Fig. 7. 12Vs=23nsin⁡θ2+sin⁡θ2-arcsin⁡e2LL+l2Lcos⁡θ2-arcsin⁡e2L+e

In arch deformable mode, the Bricard-like mechanism is in turnover mode; each dual AE mechanism with a link of the RRR chain is fixed as a rigid link with the link length of dp, and the six rigid links are connected by six R joints to form a TFS Bricard linkage, as shown in Fig. 8. Then, an area of the sector-like figure enclosed by an arch is derived, as shown in Fig. 5.

The origin of the coordinate system is denoted by Pi (i = 1, 2, …, 2n+3); the sector along Zi (i = 1, 3, 5, …, 2n+3) is denoted by Si. The following is an example of the first Bricard-like mechanism to show the process of calculating the area of the sector-like figure enclosed by an arch. After the first coordinate transformation, the position matrix of S2 can be obtained; then after the second coordinate transformation, the position matrix of S3 can be obtained. The direction of S2 can be obtained by adding S1 and S3. Similarly, the position and direction of S4 can be obtained, as shown in Eqs. (13)–(14). 13S1+S3=S2,14S3+S5=S4. The intersection point of S2 and S4 is denoted by O, and the angle between S2 and S4 is denoted by η; then an isosceles triangle P2OP4 can be obtained, and the area of the triangle P2OP4 can be obtained. 15cos⁡η=S2⋅S4S2⋅S4,16P2P4=2l+Lcos⁡arcsin⁡e2L+Lsin⁡12arccos⁡cos⁡θcos⁡θ-1,17tan⁡η2=P2P42h,18SΔP2OP4=12⋅P2P4⋅h. Since the deployable arch is composed of a number of identical Bricard-like mechanisms in turnover mode, and since the Bricard-like mechanism is symmetrically distributed, the area of the sector-like figure enclosed by the arch is denoted by S and can be obtained from Eqs. (13)–(19). 19S=nSΔP2OP4 The lengths of links L, l, and e can be given different values. Suppose L = 15 mm, l = 6 mm, and e = 4 mm, then the mesh grids of S with respect to θ and n are drawn in Fig. 9.

As shown in Fig. 10, the Bricard-like mechanisms of the arch are changed with the angle of θ, and the shape of the arch is changed accordingly; that is, when the angle of θ of the Bricard-like mechanisms become larger, the Bricard-like mechanisms in the arch will become closer. When the number of Bricard-like mechanisms is greater than or equal to 6, the special configuration (cube shape) appears at θ=π2. Besides, it is easy to see that the arch radius R is equal to the link length dp.

The multimode deployable arch discussed above can also be used as a construction unit to construct a center-driven multimode deployable mechanism. As shown in Fig. 11a, the arches are denoted by Aj​​​​​​​ (j = 1, 2, 3), and the joints in each arch can be denoted as Aj–Ji (i = 1, 2, etc.); by connecting revolute joints A1–J3 and A2–J1, A2–J3 and A3–J1, A3–J1 and A1–J1 separately, a new multimode deployable mechanism is obtained. Since the whole mechanism is composed of three identical arches, a TFS Bricard-like mechanism is constructed among these three arches. As a result, there are 3n+1 Bricard-like mechanisms connected with each other. When one of the Bricard-like mechanisms moves, the whole mechanism will move synchronously with this Bricard-like mechanism; in other words, the center-driven multimode deployable mechanism is a 1 degree-of-freedom mechanism.

Based on the above three-branch center-driven multimode deployable mechanism, revolute joint A4–J1 is added to A1–J3(A2–J1), and A4–J3 is added to A2–J5. Similarly, with the construction approach above, A5–J1 is added to A3–J1(A2–J3), A5–J3 is added to A3–J5, A6–J1 is added to A1–J1(A3–J3), and A6–J3 is added to A1–J5; thus, a six-branch central-driven multimode deployable mechanism is constructed. The specific construction method is shown in Fig. 11b.

Since the six-branch mechanism is composed on the basis of the three-branch mechanism, the three added arches and three-branch mechanism both share the common angulated elements, and the motion of each dual AE of the three-branch mechanism is synchronized, the six-branch mechanism and the three-branch mechanism both move synchronously; that is, the six-branch mechanism has 1 degree of freedom as well. In addition, we can change the size of the mechanism by increasing or decreasing the number of the Bricard-like mechanisms in each arch.

Furthermore, as shown in Fig. 12, on the basis of the multimode center-driven deployable mechanisms, new mechanisms can be obtained by adding shorter arches at different positions of the multimode center-driven deployable mechanisms. The newly obtained mechanisms still have two motion modes, namely, scissor-like deployable mode and spherical deformable mode, which can be switched through the transition configuration. Furthermore, since the newly added arches still share the common AEs, i.e., all the AEs of the whole mechanism still move synchronously, as a result, the multimode center-driven deployable mechanisms with arches still have only 1 degree of freedom, and the stiffness of the whole mechanism has been improved. However, in the case that the interference of the links is considered, the more arches added, the smaller deformation range of the mechanism under the spherical transformation mode.

In the scissor-like deployable mode, φ remains still at 60∘; the spherical mechanism will fold and deploy as the value of θ varies like the arch. To avoid link interference, the range of θ in the scissor-like deployable mode is [3arcsin⁡(e/2L), π) as well. Here, we use V3 and V6 of the enveloping cuboid to represent the size of the mechanism in the scissor-like deployable mode, respectively; the lengths of links L, l, and e can be given different values. Suppose L = 15 mm, l = 6 mm, and e = 4 mm, then the mesh grids of V3 and V6 with respect to θ and n are drawn in Fig. 13.

When the arch is in arch deformable mode, the length from the position of each revolute joint to the vertex of the sector-like figure is equal, and the multimode center-driven deployable mechanism is constructed by connecting three identical arches. The multimode center-driven deployable mechanism will envelop a spherical mechanism, the radius of the enveloping sphere will change as the value of θ varies, and the multimode center-driven deployable mechanism is in spherical deformable mode. We use the radius R of the enveloping sphere to represent the size of the mechanism, and it is calculated in Eq. (20), the lengths of links L, l, and e can be given different values. Suppose L = 15 mm, l = 6 mm, and e = 4 mm, then the relationship curve between θ and R of the spherical mechanism in the spherical deformable mode is described in Fig. 14. In addition, when θ=33π/36, θ=17π/18, and θ=35π/36, the states of the mechanism in spherical deformable mode are shown in Fig. 15. 20R=L1-cos⁡γ+ltan⁡θ2