The bending angle of the two-module dual-origami soft fluidic bending actuator is geometrically derived as \(\phi \approx (\theta_{free}-\theta_{ori})\) and the change in effective layer length can be written as \(\Delta L \approx 2\theta_{ori}l_{network}\) (See Experimental Section/Methods for details on Simplified kinematic model. Variables are defined in Figure 5A;  \(\theta_{ori}\) and \(\theta_{free}\) are the angles between the two-module dual-origami soft fluidic bending actuators with an attached and unattached origami strain-limiting layer, respectively. \(l_{layer}\) and \(l_{network}\) are side lengths of the origami layer and the fluidic network, respectively). The ratio of the bending angle to the deployment ratio can be derived as  \(\frac{\phi}{\lambda}=\frac{\phi}{\Delta L / L_0} =\frac{L_0}{2l_{network}}\left(\frac{\theta_{free}}{\theta_{ori}}-1\right)=CA_0\), where \(A_0=\frac{L_0}{l_{network}}\)  is an initial aspect ratio and \(C=\frac{1}{2}\left(\frac{\theta_{free}}{\theta_{ori}}-1\right)\). We named \(C\) as the bending-to-deployment ratio factor because it represents the dominance of bending to deployment during the quasi-sequential deployment and bending motion. A high value of \(C\) means that the dominance of bending is large; when the robot only bends without deployment, \(C\to\infty\) (e.g., when the inextensible strain-limiting layer is attached, like conventional soft bending robots), and when the robot only deploys without bending, \(C=0\) (e.g., origami fluidic network without a strain-limiting layer). For common dual-origami soft fluidic bending actuators, it was observed that \(C\) rapidly increase at the transition pressure at which the dominance shift occurs (Figure 5B).
In order to better understand the quasi-sequential deployment and bending motion, we investigated the relationship between the geometric parameters of the origami strain-limiting layer and   \(\theta_{ori}\) or \(\theta_{free}\)  by performing experiments and FEA simulations (Figure 5B-E). A crease line width of one side (\(w\)) and a ratio between   \(l_{layer}\) and \(l_{network}\) (\(\Lambda \)) were considered as the important geometric parameters because it was expected that the rotational stiffness of the crease line would increase with increasing \(w\) and the maximum value of   \(\theta_{ori}\)  would be geometrically determined by   \(\Lambda \) (Figure 5A). In both the experiment and simulation, the response of   \(\theta_{free}\) to applied fluid pressure was constant even though geometric parameters of the origami layer at the nearby modules were changed (Figure 5C). On the other hand, the response of   \(\theta_{ori}\) to applied fluid pressure was affected by \(w\) and \(\Lambda\). It was noteworthy that the change in \(w\) gradually shifts the response while \(\Lambda\) changes it significantly (Figure 5D and E). For example, the increase in \(w\) from 3 to 12 mm decreased \(\theta_{ori}\) from 58.16\(\degree\)to 49.56\(\degree\) at 100 kPa (Figure 5D, for \(\Lambda\)= 0.476), and the increase in \(\Lambda\) from 0.238 to 0.476 increased \(\theta_{ori}\) from 25.47\(\degree\) to 54.43\(\degree\)at 100 kPa (Figure 5E, for \(w\)= 6 mm). The result also indicates that the amount of deployment and bending should be selectively pre-programmed with consideration of their tradeoff relationship; \(\phi\) decreases as \(\theta_{ori}\) increases but \(\lambda\) is proportional to \(\theta_{ori}\). Accordingly, \(C\) was 0.158 at 20 kPa for the most deployable case, while \(C\) was 0.825 at 20 kPa for the most bendable case, which is about 5.22 times difference (Figure 5B). 
To confirm the pre-programmability of the quasi-sequential deployment and bending motion, six-module dual-origami soft fluidic bending actuators with different geometric parameters were built (Figure 5F-M). As shown in Figure 5F, J, Figure 6 and 7, we could design clear differences in motion determined by \(w\) and \(\Lambda\) ; their end-tip trajectories were plotted in Figure 5G and K, respectively. \(\lambda\) decreased from 4.78 to 4.32 and \(\phi\) increased from 90.9\(\degree\) to 106.9\(\degree\) as \(w\) increased from 3 mm to 12 mm, whereas increase of \(\Lambda\) from 0.238 to 0.476 greatly increased \(\lambda\) from 2.35 to 4.75 and decreased \(\phi\) from 144.75\(\degree\) to 99.66\(\degree\) (at P= 200 kPa, Figure 5H, I, L and M). The result confirms that the quasi-sequential deployment and bending motion is more sensitive to \(\Lambda\) than \(w\). Therefore, it would be recommended to determine \(\Lambda\) for coarse adjusting in precedence to \(w\) for fine adjusting. Additionally, thickness of the strain-limiting layer (\(t\)) can be considered for pre-programming of the motion. However, it should be noted that a small change in \(t\) would change motion dramatically because the bending stiffness of the layer is proportional to \(t^3\), and therefore it is not recommended to tune \(t\) for motion pre-programming. (Figure 8shows simulation result that small change of \(t\)= 0.6 mm to 0.8 mm increases \(C\) significantly from 0.115 to 0.296 at 18 kPa).