It is instructive to begin analysing the general expression for the bending angle by considering

thin actuators made of relatively thin tube and patch

$(t\ll R$ and

$T\ll R)$ and characterized by parameters

$\rho \gg 1$ and

$\tau \ll \rho $. In this case

${f}_{n}\left(\tau \right)\approx {\rho}^{n-1}\tau $ and Equations (

12) and (

13) are simplified to the form

where

$\mu =\kappa \rho $ and

The bending performance of thin actuators is seen to be controlled by a single pressure-dependent structural parameter

$\mu $, which is the product of the pressure factor and the relative tube radius. From Equation (

11) the values of

$\mu $ are estimated to be limited by the ratio of yield stress to Young’s modulus,

$\mu \lesssim {\sigma}_{0}/E$. This ratio depends on the type of polymer (see

Table 1), ranging from about 0.28 for silicone elastomers (SE) to 17 for polyurethane elastomers (elPU) [

47]. From Equation (

16) it can be concluded that elongation of pneumatic actuators has little effect on their bending if they are made of polymers with

${\sigma}_{0}\lesssim 2E$ but may become crucial at high pressures for polymers with

${\sigma}_{0}\gg 2E$.

#### 3.1. Tubes of Equal Thicknesses

A thin pneumatic actuator can be fabricated by co-drawing a thin hollow tube of circular cross-section with a part of a similar tube serving as the patch. In this case

$\tau =1$ and the bending angle is a function of

$\alpha $ only. The optimal patch thickness, determined by the condition

${\vartheta}_{\alpha}^{\prime}(\alpha ,1)=0$, can be calculated from the transcendental equation

where

$s\left(\alpha \right)=sin(\alpha /2)/\chi (\alpha ,1)$. For

$\mu =0$ and

$\mu \gg 1$ the roots of this equation are given by

Figure 3 shows the optimal patch width plotted as function of

$\mu $. The monotonic decay of this function is accompanied by the growth of the maximal bending angle according to Equation (

16). The growth is linear at low pressures when

$\mu \lesssim 0.1$ and pure bending dominates,

${\vartheta}_{\mathrm{max}}\approx 0.133\phantom{\rule{0.166667em}{0ex}}(L/R)\mu $, and quadratic for larger

$\mu $,

${\vartheta}_{\mathrm{max}}\approx 0.066\phantom{\rule{0.166667em}{0ex}}(L/R){\mu}^{2}$. The insets in the figure show cross-sections of actuators corresponding to the optimal angles

${\alpha}_{0}$ and

${\alpha}_{\infty}$.

#### 3.3. Optimization in General Case

The preceding discussion was for the case of relatively thin actuators. In the instance in which either

t or

T (or both) is comparable to

R, it is not permissible to use Equation (

16) and we must recourse to the exact Equation (

12). This is evidenced by the fact that the bending angle of thin actuators does not have an absolute maximum but peaks over a hyperbola-shape ridge

$\tau \alpha =\mathrm{const}$ [c.f. Equations (

16) and (

21)]. On the other hand, the bending angle given by the exact expression has a single maximum, which determines the optimal values of

$\alpha $ and

$\tau $ as functions of

$\kappa $ and

$\rho $ for any parameters of the actuator.

The normalized bending angle of the pneumatic actuator [Equation (

12) with

$R=L$] is plotted in

Figure 5. In agreement with the limiting behavior expressed by Equations (

14) and (15), the bending of the actuator is the strongest for one set of optimal parameters

$({\alpha}_{\mathrm{opt}},{\tau}_{\mathrm{opt}})$ corresponding to the peak of function

$\vartheta (\alpha ,\tau )$ in

Figure 5a. The cross-section defined by the optimal actuator’s parameters is shown in

Figure 5b. Owing to the finite width of the peak and its ridge-like shape, it is possible to use a narrower patch at the expense of increasing its thickness without significantly reducing the bending angle. The approximate tradeoff between the two optimal parameters can be estimated from

Figure 4 and for

$\mu =1$ is given by

${\alpha}_{\mathrm{opt}}{\tau}_{\mathrm{opt}}\approx 2\pi \times 0.44$. This tradeoff is shown by the dashed hyperbola in

Figure 5a.

Figure 5c,d show the optimal width and thickness of the asymmetric patch as functions of relative tube radius

$R/t$ for different relative pressures

$P/E$. One can see that the optimal width monotonically decreases with

$R/t$ and that the higher the pressure applied to the actuator, the steeper the decrease. This trend is opposite to the monotonic growth of the optimal thickness, which becomes less and less steep with the buildup of pressure. The functional dependencies of the optimal dimensions of the patch on

$R/t$ suggest that bending of thinner tubes requires higher and higher asymmetry of the actuator’s cross-section as the pressure grows bigger. This conclusion is rather general and holds for actuators of other cross-sectional shapes.

The maximal bending angle

${\vartheta}_{\mathrm{max}}$ achievable with the optimal dimensions

${\alpha}_{\mathrm{opt}}$ and

${\tau}_{\mathrm{opt}}$ of the pneumatic actuator is shown in

Figure 5e. One can see that the optimized actuator can yield very high bending angles for even relatively low pressures. For example, in agreement with the colour scale of the contour plot, for

$\rho =10$ and

$\kappa =0.1$ we have

${\vartheta}_{\mathrm{max}}\approx 0.17$. This value corresponds to a 360°-bending of actuators with

$L\approx 37R$ and

$P=E/10$. It should be noted that since the pressure applied to the actuator is limited by the maximal circumferential stress it can withstand before breaking or significantly changing its cross-section, the maximal bending angle can be calculated from

Figure 5e only for

$\rho \lesssim ({\sigma}_{0}/E)/\kappa $ [see Equation (

11)]. Hence, the higher the yield stress of the polymer, the larger the bending that can be achieved with this polymer for a given actuation pressure.

Summarizing the above results, we can formulate the following general conclusions and design guidelines for soft pneumatic actuators made of hollow polymer tubes. First, the bending angle of soft pneumatic actuators scales linearly with the ratio of their length to the inner tube radius and can be maximized for a fixed pressure by tuning the cross-section of the asymmetric patch. Second, there is a tradeoff between the width and thickness of the optimal asymmetric patch allowing one to achieve almost the maximal bending angle using a wide range of patches. Third, the bending angle of thin pneumatic actuators (with $t\ll R$ and $T\ll R$) is determined by the pressure-dependent parameter $\mu =(P/E)(R/t)$, which should not exceed the ratio of yield stress to Young’s modulus E of the polymer. Fourth, if the thicknesses of the patch and the main hollow tube of the actuator are alike, the optimal angular width of the patch varies between 149° at low pressures, $P\lesssim (t/R)(E/10)$, and 112° at high pressures, $P\gg (t/R)(E/10)$. Fifth, the product $[{\alpha}_{\mathrm{opt}}/\left(2\pi \right)]({T}_{\mathrm{opt}}/t)$ of the optimal relative dimensions of the narrow patch of thin pneumatic actuators is a pressure-dependent constant, which varies from 1/2 at low pressures to about 0.323 at high pressures.