Movie S3. Experimental verification of IEAR on the Double Bellows
Here, the Double Bellows has a symmetric structure to produce bidirectional bending by alternatively inflating these two chambers (Figure 2a and Figure S4). According to Equation  (1), we establish the dynamic model of the Double Bellows to obtain the theoretical pressures by analyzing its deformation geometry and the force equilibrium (Figure 2bcSupplementary Note 2). 
The rhythmic actuation of the Double Bellows with our IEAR mechanism has different ON/OFF timings of the solenoid valves compared to the working cycle with a conventional DIDO mechanism (Figure 2d, e; Figure S2d). Following the initial actuation cycles, the Double Bellows exhibits a rapid transition and enters a steady-state pressure response (Figure S5a, b). Unlike DIDO, our IEAR mechanism prevents a sharp decrease in the supplied air pressure \(p_{tank}\)  during the transition. The comparative results demonstrate that our dynamic model of the Double Bellows agrees well with the experimental results under both the DIDO and IEAR mechanisms (Figure 2f and Figure S6). Such agreement suggests that our model can facilitate analysis of the actuation performance, including actuation speed and energy consumption. (see Table 1 in Methods)
We obtain the theoretical actuation frequency and energy consumption per cycle of the Double Bellows with \(p_{high}\)  ranging from 50 kPa to 100 kPa, and experimentally validate these results (Figure 2g, h). With the increase of \(p_{high}\) , both actuation frequencies with the DIDO and IEAR mechanisms decrease. However, the actuation frequency with our IEAR mechanism is markedly higher than that with DIDO for the full range of pressures (Figure 2g). At the same time, the energy consumption per cycle with IEAR is far less than that with DIDO (Figure 2h). For example, when  \(p_{high}\)=75 kPa, the actuation frequency and energy consumption with DIDO are 0.49 ± 0.01 Hz and 8.08 ± 0.15 mWh·cycle-1, respectively. Our IEAR mechanism improves the actuation frequency to 0.93 ± 0.02 Hz (91.2%↑) while reducing the energy consumption to 3.94 ± 0.09 mWh·cycle-1 (51.2%↓). In addition, we change \(\Delta p\)  (5, 10, 15 kPa) to investigate its influence on the actuation performance (Figure S6). It demonstrates that the IEAR mechanism is widely feasible in various working conditions to achieve high-speed and low-energy actuation. Also, the dynamic model of m-SPAs shows a satisfying capability of actuation performance analysis on the Double Bellows.