Experimental Section/Methods
Simplified kinematic model
The six-module dual-origami soft fluidic robot can be represented as connected linkage model, and its lengths, angles, and positions were defined as shown in Figure 20. With the assumption that the stretching of the material is negligible (E~10 MPa, strain is ~ 1% at applied pressure of 100 kPa), it can also be assumed that lengths (\(L,\ h_1,\ h_2\)) are constant.
For positions \(\vec{p_n}\), following relationships can be established:
\[\begin{equation} \vec{p_0}= \left[ \begin{array}{cc} 0\\ 0\\ \end{array} \right] \end{equation}\]
\[\begin{equation} \vec{p_{4n-3}}= \vec{p_{4n-4}}+ \left[ \begin{array}{cc} h_1 \cos{\beta_{2n-1}}\\ h_1 \sin{\beta_{2n-1}} \end{array} \right] \end{equation}\]
\[\begin{equation} \vec{p_{4n-2}}= \vec{p_{4n-3}}+ \left[ \begin{array}{cc} L \cos{\alpha_{2n-1}}\\ L \sin{\alpha_{2n-1}} \end{array} \right] \end{equation}\]
\[\begin{equation} \vec{p_{4n-1}}= \vec{p_{4n-2}}+ \left[ \begin{array}{cc} h_2 \cos{\beta_{2n}}\\ h_2 \sin{\beta_{2n}} \end{array} \right] \end{equation}\]
\[\begin{equation} \vec{p_{4n}}= \vec{p_{4n-1}}+ \left[ \begin{array}{cc} L \cos{\alpha_{2n}}\\ L \sin{\alpha_{2n}} \end{array} \right] \end{equation}\]
For angles \(\alpha_n\) and \(\beta_n\), following relationships can be established:
For n is odd,
\(\begin{equation} \alpha_{n+1}-\alpha_n = \{ \begin{array}{cc} 2 \theta_n + \pi,\; for \; \alpha_n<0 \;and\; \alpha_{n+1}>0\\ 2 \theta_n - \pi,\; else\\ \end{array} \end{equation}\)
\(\begin{equation} \beta_{n+1}= \{ \begin{array}{cc} -\frac{\pi}{2} + \theta_n + \alpha_n + 2 \pi,\; for \; \alpha_n<0 \;and\; \beta_{n+1}>0\\ -\frac{\pi}{2} + \theta_n + \alpha_n,\; else\\ \end{array} \end{equation}\)
For n is even,
\(\begin{equation} \alpha_{n+1}-\alpha_n = \{ \begin{array}{cc} -2 \theta_n - \pi,\; for \; \alpha_n>0 \;and\; \alpha_{n+1}<0\\ -2 \theta_n + \pi,\; else\\ \end{array} \end{equation}\)
\(\begin{equation} \beta_{n+1}= \{ \begin{array}{cc} \frac{\pi}{2} - \theta_n - \alpha_n - 2 \pi,\; for \; \alpha_n>0 \;and\; \beta_{n+1}<0\\ \frac{\pi}{2} - \theta_n - \alpha_n,\; else\\ \end{array} \end{equation}\)
Initial conditions \(\alpha_n\) and \(\beta_n\):
\[\begin{equation} \begin{array}{cc} \alpha_1 = \pi - \theta_0 \end{array} \end{equation}\]
\(\begin{equation} \begin{array}{cc} \beta_1 = \frac{\pi}{2} \end{array} \end{equation}\)
By solving for (6-11), angles \(\alpha_n\) and \(\beta_n\)\(\) can be expressed as follows:
For \(n\ge2\), \(n\in N\), \(\)\(m_{1,}m_2\in Z\),
\(\begin{equation} \begin{array}{cc} a_n=2\sum_{k=1}^{n-1}\left\{\left(-1\right)^{k+1}\theta_k\right\}+\frac{1+\left(-1\right)^n}{2}\pi+\pi-\theta_0+2\pi m_1 \end{array} \end{equation}\)
\(\begin{equation} \begin{array}{cc} \beta_n=-\frac{\pi}{2}+\left(-1\right)^n\left[2\sum_{k=1}^{n-2}\left\{\left(-1\right)^{k+1}\cdot\theta_k\right\}+\theta_{n-1}+\pi-\theta_0\right]+2\pi m_2 \end{array} \end{equation}\)
When all modules are same, following relationship for \(\theta_n\) is established:
\(\begin{equation} \theta_n = \{ \begin{array}{cc} \theta_{ori},\; n \; is \; odd\\ \theta_{free},\; n \; is \; even\\ \end{array} \end{equation}\)
Finally, positions can be obtained as a function of \(\theta_n\) by substituting (1-5) and (10-14).
For 2N-module structure, bending angle \(\phi\) can be calculated from end-tip position \(\vec{p_{4N}}\):
\[\begin{equation} \vec{p_{4N}}= \left[ \begin{array}{cc} \sum_{k=1}^N\left\{L\left(\cos\alpha_{2k-1}+\cos\alpha_{2k}\right)+h_1\cos\beta_{2k-1}+h_2\cos\beta_{2k}\right\}\\ \sum_{k=1}^N\left\{L\left(\sin\alpha_{2k-1}+\sin\alpha_{2k}\right)+h_1\sin\beta_{2k-1}+h_2\sin\beta_{2k}\right\}\\ \end{array} \right] \end{equation}\]
In case of all modules are same and \(L>>h_1,h_2 \), \(\phi\) can be calculated from (15),
\[\begin{equation} \begin{array}{cc} \phi\approx\tan^{-1}\left\{\frac{\sum_{k=1}^N\cos\alpha_{2k-1}+\cos_{2k}}{\sum_{k=1}^N\sin\alpha_{2k-1}+\sin\alpha_{2k}}\right\}\\ \\ =\tan^{-1}\left\{\frac{\sum_{k=1}^N\sin\left\{\left(2k-1\right)\left(\theta_{free}-\theta_{ori}\right)\right\}}{\sum_{k=1}^N\cos\left\{\left(2k-1\right)\left(\theta_{free}-\theta_{ori}\right)\right\}}\right\}\\ \\ =N\left(\theta_{free}-\theta_{ori}\right)\\ \end{array} \end{equation} \]
In Figure 20C, the model was compared with the experimental result, and it was well suited for low applied pressure.
Change in effective layer length \(\Delta L\) can be approximated as a length of an arc:
\[\begin{equation} \begin{array}{cc} \Delta L \approx 2N\theta_{ori}l_{network} \end{array} \end{equation}\]
A ratio of the bending angle to the deployment ratio can be derived from (16) and (17):
\[\begin{equation} \begin{array}{cc} \frac{\phi}{\lambda}=\frac{\phi}{\Delta L / L_0}\\ \\ =\frac{L_0}{2l_{network}}\left(\frac{\theta_{free}}{\theta_{ori}}-1\right)\\ \\ =CA_0\\ \end{array} \end{equation}\]
Where we defined bending-to deployment ratio factor \(C\) as:
\[C=\frac{1}{2}\left(\frac{\theta_{free}}{\theta_{ori}}-1\right)\]
and \(A_0=\frac{L_0}{l_{network}}\)is an initial aspect ratio.