**2. Soft Origami-Based Design**

An important design parameter of the link is the number of polyhedron faces [34], as they can limit the movement when folding if they are very close, like the case of four faces. To generate a symmetrical model, we decided to build a six face polyhedron as shown in Figure 2 representing every state of folding, from collapse (Figure 2a) to deployment (Figure 2d).

**Figure 2.** Cylinder polyhedron origami with one section (*h*1) and six faces (*n* = 6). (**a**) Top view collapsed state. (**b**–**d**) Folding state.

A structure having several Kresling layers (for example five) is able to generate a cylinder composed by the same number of sections. The versatility of this design allows each section to be compressed or extended independently, so the five-section cylinder can have a variety of possible lengths, where the maximum height is the extended height of all sections. This cylinder model meets the scalability parameter by its bending property and the number of sections mentioned above. Now, the challenge is to make the cylinder reconfigurable. To achieve this, a number of cylinders have been joined together. Each cylinder is renamed as *link*, regardless of its number of sections, and one layer will be renamed as *single link*. The link connection represents a greater challenge, because it requires a mechanism at each end of the link, as shown in Figure 3a, where the input *IN* and output *OUT* unions of the link are shown in blue and orange, respectively. This allows the link to know which end has been placed in the next link.

**Figure 3.** Connection of links. (**a**) Five-sections link with joints. (**b**) Joint between links. (**c**) Joined and reconfigurable links.

The joint between links is very wide, and ideally should rotate along the three axes *XYZ* and allow mobility in the angles yaw *ψ*, pitch *ρ* and roll *φ* (Figure 3b). The ideal joints will enable as many links to be joined as required. The links should always be joined in an orderly sequence, so that there is an *IN joint* at the beginning of the chain and an *OUT* *joint* at the end. Using this pattern, each link has corresponding position within the chain, with the first link acting as the main or master link.

The ideal case with three rotation axes allows the kinematic chain to move either in a plane or in the three dimensional space. If the yaw angle *ψ* is rotated in a plane, a snake-like movement will be obtained, while the rotation of roll angle *φ* through space can be assimilated to the behavior of a finger or an arm, according to the number and length of the connected links, as shown in Figure 3c.

#### *Link Prototype*

Our aim was to build a prototype to validate the idea of modular and scalable links, designing a paperless origami model. With this purpose, all the parts were modeled using CAD design applications and made using 3D printing technologies. Figure 4 illustrates the basic parts to generate the triangular polyhedron for a cylinder with six faces (*n* = 6). The base with constant length *a* is shown in Figure 4a. The *lAC* (Figure 4b) was designed as a variable length piston with spherical bearings at its ends for mobility. Finally, the length *lBC* is a soft material spring that can be warped with spherical bearings as well (Figure 4c). Values *lBC* and *a* have been deemed constant, following Jianguo et al.'s suggestion [33], and *lAC* is the only variable. In this case *lBC* = *LBC*. This design allows for operation while keeping constant values, but at the same time, its flexibility enables the free movement of the link. Design dimensions are *a* = 35 mm, *α* = 38◦, *β* = 30◦, *r* = 30◦ and planar state lengths *LAC* = 64.90 mm, *LBC* = 43.09 mm. The piston length *lAC* decreases when the structure deploys and its size increases during the collapse.

**Figure 4.** Components of the triangulated polyhedron prototype. (**a**) Constant base *a*. (**b**) Length *lAC*, greater displacement. (**c**) Length *lBC*, minor displacement.

Figure 5 illustrates the single link prototype, which represents an assembly of the components shown in Figure 4. In addition, connecting couplings and pins were required to keep the link assembled but still able to move. The triangles were assembled according to the design in Figure 1, where *lAC* is folded inside the link, and *lBC* is outside. Therefore, the bistable movement from the deployed to the collapsed state generates a clockwise rotation of the angle *θ*.

**Figure 5.** Single link CAD prototype. (**a**) Collapsed state. (**b**) Folding state. (**c**) Deployed state.

Several single links can be nested between them to obtain a cylinder with multiple sections. The union of two (*h*2) is shown in Figure 6a, whereas a cylinder with three sections (*h*3) is shown in Figure 6b. Each of these groupings constitutes an independent link, which is capable of modifying its length by varying the *h* value in each section, in an adjustable way. As discussed before, another option providing a relative 3 *DoF* link movement is the two-part ball socket. This configuration features two ball joints connected through a fixed union, attached to both links. The spherical bearing allows free rotation between the axes. However, for the construction and operation of this coupling, a more complex design is required.

**Figure 6.** Nested links CAD prototype. (**a**) two-sections link. (**b**) three-sections link. (**c**) Two single links with a joint. (**d**) Two single links with a joint rotated.

Figure 6c illustrates the union of two links through the spherical joint, where both bodies are aligned with each other in a starting position, with each body consisting of a single link and fully deployed. On the other hand, in Figure 6d the chain of cylinders is horizontal and shows a slight rotation in the roll angle. The double spherical bearing represents the *IN joint* and *OUT joint*, respectively, as described in Figure 3b.

Finally, the prototype components were built in a 3D printer and assembled. The hexagonal bases (length *a*), the couplings, and the pistons (*lAC*) were made of Polylactic Acid (PLA) plastic material. The bar *lBC* was manufactured with a flexible material (NinjaFlex) from the manufacturer NinjaTek, to allow short displacements and keep the structure stable. Metric (M2) screws and 2 mm nuts were used for the final assemble of the prototype.

Figure 7 shows the bistate of the single link prototype assembly. The completely unfolded polyhedron is shown in Figure 7a. In this state the *lAC* pistons are compressed and the *lBC* soft links are extended. The final position of the deployment depends on *lAC*, as mentioned before, and *lBC* is adapted to that length.

In addition, the nested links were assembled to validate the design, as shown in Figure 8. The changing link size feature has been checked; each single link is able to fold and deploy in a two-sections link according to Figure 8a,b. The spherical joint shown in Figure 8c allows the union and the rotation of two single links while keeping the bistable operation.

**Figure 7.** Single link first prototype. (**a**) Deployed single link prototype. (**b**) The single link is in collapsed state, and the height has changed. The *lAC* pistons are extended and *lBC* is slightly compressed. (**c**) Top view of the deployed prototype. (**d**) Top view of the collapsed prototype; the condition of the pistons and *θ* rotation are clearly shown.

**Figure 8.** Assembled nested links prototype. (**a**) Two-sections link collapsed. (**b**) Two-sections link deployed. (**c**) Two single links with a joint. (**d**) Two single links with a joint, vertically rotated and one of them extended. (**e**) Two single links with a joint, horizontally rotated and collapsed.
