Although empirical formulas [43,51,56–59] have approximated the flow function of an air passage conveniently, these models work well only when the pressure on one side remains constant. In our actuator, the pressures in both chambers change rapidly. It is more appropriate to obtain a data-driven flow function for greater accuracy.
We use two steel air tanks (10 L) to characterize the flow function of the air passage between a pair of chambers, as is shown in Figure S10b. The characterization has three steps: i) Pressurize air tank-1 to over 130 kPa. ii) Open the valve in the middle of the air passage and make the compressed air in tank-1 flow to tank-2. iii) When \(p_{high}\)  decreases to have a difference of less than 2 kPa from \(p_{low}\) , close the valve. We try to keep downstream pressure \(p_{low}\)  constant using a customized pneumatic testing system. To completely characterize the mass flow under various working conditions, we repeat those steps under different \(p_{low}\)  from 0 kPa to 120 kPa with a step length of 10 kPa. For those air passages deflating the compressed air into the ambient, tank-2 is removed during the test. The large 10 L air tanks reduce air pressure fluctuation and make the pressures change smoothly. All the data are recorded when the solenoid is fully open. We can obtain the experimental results as shown in Figure S10c.