A common six-module dual-origami soft fluidic bending actuator that is initially folded and deploys through dual-origami unfolding is shown in Figure 1A and Movie S1. Two integral components of the dual-origami design are the origami fluidic network and the origami strain-limiting layer based on conventional deployable origami architectures. We selected Miura-ori polyhedron and zigzag folded Miura-ori as the parent origami frame of two components, respectively.[44] These origami architectures have large potential deployment ratios springing from their asymmetric geometries where the facets are serially connected via crease lines at each end, as well as additional advantages of flat-foldability and rigid-foldability. A single- Miura-ori polyhedron module is comprised of four parallelogram facets connected in a closed form, and when folded, each layer contains two parallelogram facets and a diamond shaped crease line loop (Figure 1B). A single-module zigzag folded Miura-ori is identical to half of the Miura-ori polyhedron, which is comprised of two parallelograms not in a closed form (Figure 1B). For soft robotic application, we modified/determined their geometry as follows: (1) For both origami architectures, we merged two parallelograms on the same layer by removing the shared crease line because the folding of the removed crease line can be replaced by the bending of the merged flexible facet. (2) The Miura-ori polyhedron’s diamond shaped crease-line-loops at the top and bottom were enclosed by plates of the same shape for fluidic actuation. (3) We optionally made a V-cut at the crease lines of the zigzag folded Miura-ori to tune the folding stiffness. As shown in Figure 1A and B, the proposed dual-origami design is in the initial folded form where the zigzag folded Miura-ori strain-limiting layers are interposed between the facets of the Miura-ori polyhedron fluidic network.
Both origami components are built to be C-channel-shaped geometry in which their facets are serially connected in the direction of stacking via C-channel shaped crease lines. When bending moments are applied at the C-channel shaped crease lines, bending intensively occurs as if origami unfolding occurs. Because the applied fluidic pressure at the Miura-ori polyhedron fluidic network produces bending moments at entire crease lines directly (fluidic network) or indirectly (strain-limiting layer), dual-origami components unfold simultaneously. We defined a single module dual-origami unit as the structure consisting of a flat fluidic network with two parallel facets and a strain-limiting layer of a single facet connected to it. A multi-module dual-origami can be readily designed by the stacking module units and connecting them via C-channel-shaped crease lines.
The quasi-sequential deployment and bending behavior of a representative six-module dual-origami soft fluidic bending actuator was experimentally measured and plotted in Figure 1C (red line). This peculiar motion can be distinguished into two modes that quasi-sequentially appear in accordance with the applied fluid pressure level (Figure 1D). The first mode, a deployment-dominant mode, appears at relatively low fluid pressure (\(P\) < 40 kPa), and both the origami fluidic network and the origami strain-limiting layer unfold simultaneously. Therefore, the deployment ratio (\(\lambda=\frac{\left(L-L_0\right)}{L_0}\), where \(L\) is the effective layer length and \(L_0\) is its initial value), in response to applied fluidic pressure, rapidly increases compared to the bending angle (\(\phi\)) (Figure 1E, red area). As the applied fluid pressure increases, the second mode, a bending-dominant mode appears, and the unfolding speed of origami fluidic network overwhelms the unfolding speed of the origami strain-limiting layer. This is because the strain-limiting layer is nearly completely unfolded yet the fluidic network is not, and thus \(\phi\) surges while \(\lambda\) slowly increases (Figure 1E, gray area). When the applied pressure is decreased, the deployed soft body retracts due to its own elasticity. We also built a conventional soft bending actuator (widely known as PneuNet design),[19,20,22] and plotted its behavior for comparison (Figure 1B, black line). Although the initial length \(L_0\) of the dual-origami design (29.5 mm) was much lower than the \(L_0\) of the conventional design (107 mm), the dual-origami actuator was unfolded similarly in scale to the conventional actuator, and the bending trajectories overlapped near 16\(\degree\)<\(\phi\)<80 °  (the conventional design is still advantageous for large bending because of the trade-off relationship between deployment and bending, which is further discussed in section 2.2). The aspect ratio of the dual-origami bending actuators was initially 0.657 and increased up to 3 when unfolded, which is similar to the conventional soft bending actuator’s aspect ratio of 2.89 (dimensions of both actuators are shown in Figure S1).