Combing Equation (S20) -(S24) and Equation (S34), the numerical solutions of \(R\)  and \(\varphi\)  can be obtained.

Supplementary Note 4. Experimental tests of  Double Bellows and Triple Bellows

·Experiments on Double Bellows
The experimental parameters are set to be  \(p_{low}\)=10 kPa, \(p_{tank-low}\) = 120 kPa, and \(p_{tank-high}\) =125 kPa. We use the experimental setup in Figure S2a to measure the actuator performance obeying the following procedures.
    i) Turn on the pump to pressurize the air tank (see Results-2.1. Working principle of the IEAR       
mechanism)
    ii) For a specified \(p_{high}\)  and \(\Delta p\)  , actuate Double Bellows following the working flow chart Figure 2d for DIDO or Figure 2e for IEAR, whose timings of solenoid valves are presented in Figure S2d.
    iii) Wait for  Double bellows to enter the steady-state pressure response.
    iv) Record the actuation frequency \(f\) , air tank pressure \(p_{tank}\) , system power \(P\) , and the air temperature in the air tank by calculating their average on a window of 20 s.
    v) Stop the actuation and exhaust the air in the actuator. For a new specified  \(p_{high}\)  and \(\Delta p\)   , repeat step ii-v.
    vi) Each specified working condition is measured seven times to reduce random error.
In this work, the working pressure \(p_{high}\)  with 50,60,70,80,90,100, and 75 kPa are covered, and the IEAR control parameter \(\Delta p\)=5, 10, and 15 kPa are measured (Figure 2f-h, Figure S5a and b, Figure S6). Based on the experiments, performance indicators are further calculated by the method in Table 1 and Supplementary Note 13.
Notably, there is a deformation difference between DIDO and IEAR, as shown in Figure 2d and e (Movie S3). We may mention that these differences have limited influence on the comparison between DIDO and IEAR since the volume difference of PVC tube does not exceed 4.6% even if the bending angle difference is up to 30° (the difference in this work is markedly less than this value), which should not have a remarkable influence on the dynamic pressure governed by our model (\(\boldsymbol{\dot{p}}=\xi \left[ diag\left( \boldsymbol{V} \right) +diag\left( \boldsymbol{p} \right) \frac{\partial \boldsymbol{V}}{\partial \boldsymbol{p}} \right] ^{-1}\boldsymbol{Q} \) ).
 
·Experiments on Triple Bellows
The experimental setup and parameters of Triple Bellows are the same as Doube Bellows with the following procedures.
    i) Turn on the pump to pressurize the air tank (see Results-2.1. Working principle of the IEAR mechanism)
    ii) For a specified \(p_{high}\)  and \(\Delta p\)  , actuate Triple Bellows following the working flow chart Figure 3d for DIDO or Figure 3e for IEAR, whose timings of solenoid valves are presented in Figure S2d.
    iii) -vi) These steps are the same as Double Bellows.
In this work, the working pressure \(p_{high}\)  with 50,60,70,80,90,100, and 75 kPa are covered, and the IEAR control parameter \(\Delta p\)= 5, 10, and 15 kPa are measured (Figure 3f-h, Figure S5c and d, Figure S7). Based on the experiments, performance indicators are further calculated by the method in Table 1 and Supplementary Note 13.