Design of Hierarchical Structures for Synchronized Deformations

In this paper we propose a general method for creating a new type of hierarchical structures at any level in both 2D and 3D. A simple rule based on a rotate-and-mirror procedure is introduced to achieve multi-level hierarchies. These new hierarchical structures have remarkably few degrees of freedom compared to existing designs by other methods. More importantly, these structures exhibit synchronized motions during opening or closure, resulting in uniform and easily-controllable deformations. Furthermore, a simple analytical formula is found which can be used to avoid collision of units of the structure during the closing process. The novel design concept is verified by mathematical analyses, computational simulations and physical experiments.


Supplementary Note 1: Construction of planar hierarchical structures
The rotate-and-mirror method for creating 2D hierarchical structures consists of two steps. A detailed graphical illustration for constructing a level-3 hierarchy is given in Supplementary Fig. 1. Firstly, a square rigid unit (level-0) is rotated counter-clockwise about a selected vertex from the original coordinate system (X, Y) to a new system (x', y'), by an angle of ϕ. Then the rotated unit is mirrored about the vertical or horizontal line in (X, Y) system, which passes through one of the four vertices. Continuing the mirror procedure to the rigid unit creates a 3×3 assembly, i.e., level-1 hierarchy. This 3×3 assembly will be treated as an equivalent unit for constructing level-2 hierarchy after being rotated clockwise by an angle of ϕ. Continuing the rotate-and-mirror procedure will produce hierarchies of higher levels, as long as the rotational direction is counter-clockwise for assembly/unit from even levels and clockwise for assembly from odd levels. This simple rule guarantees that the assemblies are joined together by connecting adjacent units at the edges of the assemblies by two vertices.
The counter-clockwise rotation from coordinate system (X, Y) to a new system (x', y') can be realized by , when the hierarchy level of the building unit/assembly N =0, 2, 4, … [S1] while the clockwise rotation from (x', y') to (X, Y) can be realized by , when the hierarchy level of the building unit/assembly N = 1, 3, 5, … [S2] where the transformation matrix T is When ϕ=45, the structure is in its fully expanded state and It should be pointed out that the two types of construction assemblies (types 1 and 2) as mentioned in the paper could contain different numbers of units, resulting in a variety of hierarchical structures with different patterns. Theoretically, infinite levels of 2D hierarchies could be achieved through the simple rotate-and-mirror procedure. Practically, however, only limited levels are useful depending on specific applications and manufacturing constraints. A key reason for limited applications of high level hierarchies is the significantly increased degrees of freedom.

Supplementary Note 2: Degrees of freedom of hierarchical structures constructed by the rotate-and-mirror method
The degrees of freedom can be determined by geometrical and mathematical analyses on the void space of the hierarchies. Supplementary Fig. 2 shows void spaces of level-1 and level-2 hierarchies. For level-1 structure, the rotation of the entire structure is governed by a single variable θ due to geometric constraints as illustrated in Supplementary Fig. 2a. Angles for the level-1 assembly are shown in Supplementary Fig. 2b. The enclosed void area of level-2 structure is surrounded by four sub-level assemblies ( Supplementary Fig. 2c). Assuming the rotation of sub-level assemblies are independent of each other, it will have four unknown angles θ j (j=1 to 4). Assuming that there are four independent variables θ j (j=1 to 4) for the level-2 hierarchy in Supplementary Fig. 2c, the relationship between these four variables can be determined by considering the geometric constraints. The void space of the level-2 structure can be simplified to a polygon ABCD, whose horizontal and vertical projections of each edge, Δx and Δy, can be calculated by: . where a is the size of the rigid unit. Since the polygon ABCD is enclosed, we have . Hence, θ 1 =θ 3 and θ 2 =θ 4 . It shows that the two diagonal assemblies which enclose the void area are dependent on each other. Therefore, level-2 hierarchies constructed by the rotate-and-mirror method have two independent variables, i.e., F 2 =2. It should be noted that the degrees of freedom of level-2 hierarchy are independent of the number of assemblies in horizontal and vertical directions, which is fundamentally different from hierarchical structures constructed by the fractal cut method. For the level-3 hierarchical structure, similar void area can be considered to determine the degrees of freedom, as shown in Supplementary Fig. 3. Assuming that there are five independent variables in the structure, the horizontal and vertical projections of each edge of the polygon ABCD, Δx and Δy, can be calculated by: Since the polygon ABCD is enclosed, . It can be determined that θ 2 =θ 4 and θ 3 =θ 5 . Therefore, level-3 hierarchy created by the rotate-and-mirror method has three independent variables, i.e., F 3 =3. Supplementary Fig. 4 shows the fully expanded structures of level-2 ( Supplementary Fig. 4a) and level-3 ( Supplementary Fig. 4e) hierarchies as well as various deformation states. Note that rigid units of the same color belong to the same group, and each group would rotate synchronously. By using similar method, the degrees of freedom for level-4 and level-5 structures can be determined. For a level-i hierarchy, we may define i+2 characteristic angles θ j (j=1 to i+2). For example, we may assume that level-4 structure has six independent variables ( Supplementary Fig. 5), while level-5 structure has seven ( Supplementary Fig. 6). Mathematical calculations prove that both the two hierarchical structures have two variables dependent, respectively. Therefore, F 4 =4 and F 5 =5. Generally, the degree of freedom for level-N hierarchical structures created by using the rotate-and-mirror method shall follow F N =N.

Supplementary Note 3: Degrees of freedom of hierarchical structures constructed by the fractal cut method
Gruebler's equation (1) is the most commonly used for determining DOF or mobility of a planar structure.
where M is the mobility or degrees of freedom of a structure, L is the number of links or solid elements and J is the number of joints. Level-2 fractal cut hierarchy is investigated by using Gruebler's equation. Supplementary Fig. 7 shows the degrees of freedom for several fractal cut hierarchies. Although level-1 hierarchy has only one degree of freedom, the number of DOF for level-2 hierarchies from the fractal cut method varies, depending on the number of assemblies along the edges, m and n. In the simplest case, m=n=2, the DOF is five. In more general cases, the DOF is equal to 2m+2n+3. Motion simulations by Working Model reveal the possible independent variables of level-2 and level-3 hierarchies as shown in Supplementary Fig. 8. Different from the hierarchies created by the rotate-and-mirror method, the diagonal assemblies of the fractal cut designs may rotate independently.

Supplementary Note 3: Computational simulation
We apply Working Model in this study to simulate the deformation of 2D hierarchical structures of different levels. The degrees of freedom of level-2 and level-3 hierarchical structures are verified, as well as the deformation patterns. Supplementary Fig. 9 shows the motion of a level-2 hierarchy during a closing process. In this model, five blue assemblies represent the first degree of freedom (type 1 assembly) and four pink assemblies represent the second degree of freedom (type 2 assembly). The simulation is conducted in two steps. In the first step, motor 1 ( Supplementary Fig. 9a) applies a clockwise momentum to one the rigid units in one of the five blue assemblies. In the second step, motor 2 applies a momentum in the opposite direction (clockwise) to one of the rigid units in one of the four pink assemblies. A displacement constraint is imposed in a type 2 assembly during step one, and this constraint is deactivated in step two. One edge of a rigid vertex unit is fixed throughout the simulation. Supplementary Video 1 shows the whole closing procedure of this level-2 hierarchy, clearly demonstrating the synchronized motion of the assemblies of the same type. The motions of the two types of assemblies are independent of each other. Therefore the degrees of freedom for level-2 structure are two, F 2 =2.
We have also carried out simulations of hierarchical patterns extracted from the fractal cut method under identical boundary and "loading" conditions. Supplementary Video 2 show four independent degrees of freedom for the level-2 hierarchical structure, i.e. F 2 =4. If more rigid units are included in such a level-2 pattern, the degrees of freedom will increase rapidly, while the level-2 structure from the rotate-and-mirror method will always have two degrees of freedom no matter many rigid units are included in the structure.
A level-3 hierarchy from the rotate-and-mirror method containing four level-2 assemblies is simulated in a similar manner using Working Model (Fig. 6a-e). Supplementary Video 3 show the three independent variables in the closing process.
It should be pointed out that rigid units belonging to different assembly types could collide with each other during the closing process. To ensure a successful full closure, the mechanisms of collision are analytically investigated.
The distance between points A, Cʹ, Dʹ and B can be determined mathematically in the process of rotating different assemblies. In the case of level-3 hierarchical structure shown in Supplementary Fig. 10, there are three independent variables, θ 1 , θ 2 and θ 3 . To prevent the collision, the length of segments AB, ACʹ and BDʹ, D 1 , D 2 and D 3 must satisfy the condition D 1 ≥ D 2 + D 3 when rotating different types of assemblies. D 1 , D 2 and D 3 can be obtained as proper closing procedures are designed. As shown in Supplementary Fig. 11, the curved surface plotted from Eq. S5 defines a lower bound (ceiling) of θ 1 , above which collision would occur (Cases 4, 5 and 6). Therefore, to achieve fully closure without collision, one may simply select a route in the diagram from point A (fully open) to point B (fully closed) without crossing the ceiling. Supplementary Videos 4 and 5 show the closing processes of cases 1 to 6, respectively, which confirm the validity of the collision condition we have discovered.

Supplementary Note 5: Extension to 3D Hierarchies
The rotate-and-mirror method can be easily extended to constructing 3D hierarchical structures with low degrees of freedom and synchronized deformation. Similar to 2D hierarchies, the first step to construct a 3D hierarchical structure is to rotate a cubic rigid unit or a cubic assembly by using a transformation matrix T on even levels and T -1 on odd levels,