Movie S1. The traditional DIDO mechanism
In tackling this challenge, it is important to note that most state-of-the-art m-SPAs utilize a "direct actuation" mechanism in which compressed air enters the actuator directly through an inlet and leaves directly through an outlet to an ambient environment (termed DIDO, Figure 1aFigure S1b; and Movie S1\cite{Joshi.2021b,Xavier.2021,Zhang.2021}. As a result, the energy stored in the actuator for the DIDO mechanism cannot be recovered, and more input work is required. As a result, the internal energy stored in the exhaust air for the DIDO mechanism cannot be recovered. For higher energy efficiency, an optional method is to reduce the input work of the system by improving the inlet pressure of a pump \cite{Chou.1996}. For example, an air pump can recompress the exhaust air with residual pressure to a target pressure through an external exhaust air recirculation (EEAR, Figure 1a) mechanism \cite{Wehner.2014,Lee.2021}, where an extra pneumatic buffer (e.g., Re-Air valve) can be used to reduce severe pressure fluctuation \cite{Lee.2021}. However, a key limitation is that this mechanism depends on the incorporation of pumps that typically have low electrical-to-mechanical energy conversion efficiency and, therefore, consume excess energy when recompressing the exhaust air. In addition, this mechanism lacks consideration of the actuation speed of m-SPAs (Figure S1b). Based on the architecture of EEAR, some researchers also achieve impressive programmable output pressure by replacing the pneumatic buffer with a group of air regulators, while actuation speed or efficiency is not considered in applications \cite{Zhang.2021}.  
In this work, we introduce an internal exhaust air recirculation (termed IEAR, Figure 1a) mechanism for high-speed and low-energy actuation of m-SPAs that overcomes these existing challenges. Through the rhythmic actuation of multiple chambers following a shortened energy path, our IEAR mechanism can recirculate the exhaust compressed air from one chamber to another through a specialized valve island. In contrast to other existing methods based on air recirculation, this approach avoids the need for pump-controlled air recompression. Moreover, we introduce a dynamic model of m-SPAs to guide the analysis of our IEAR mechanism, with theoretical predictions that reasonably agree with the experimental measurements. Building on previous studies that examined the dynamics of pneumatic actuators with a single chamber \cite{Joshi.2021b,Lee.2021,Joshi.2021,Xavier.2020}, our dynamic model is valid for actuators with multiple pneumatic chambers.
Characterization focuses on two exemplary embodiments of m-SPAs: an actuator with two chambers (Double Bellows) and an actuator with three chambers (Triple Bellows). These studies show that with IEAR, the actuation frequency for these two classes of m-SPAs can be improved by 82.4%-91.2%, while the energy consumption per cycle is reduced by 47.7%-51.2% under typical conditions. We further demonstrate the broad applications of the IEAR mechanism in various soft robotic systems, such as a robotic fin, fabric-based finger, and quadruped robot, for improving the actuation speed and reducing energy consumption. This work demonstrates that our IEAR mechanism plays a promising technology in the high-speed and low-energy actuation of m-SPAs for soft machines and robots.

2.Results

2.1 Working principle of the IEAR mechanism

The principle of our IEAR mechanism is illustrated in Figure 1. Unlike the traditional DIDO mechanism, the IEAR mechanism (Figure 1a) can recirculate the exhaust compressed air between the chambers. Taking a pair of chambers X and Y as an example, we can describe the actuation rhythms of our IEAR mechanism with the following steps (Figure 1b): (i) We begin by using compressed air stored in the air tank to inflate X to a set working pressure \(p_{high}\) ; (ii) Next, we open the valve between X and Y to transmit the compressed air from X to Y; (iii) When the pressure difference between X and Y is lower than a threshold \(\Delta p\) , we then close the valve; (iv) The chamber X continues deflating to \(p_{low}\) , while the air tank takes over the inflating process to pressurize Y to the set working pressure \(p_{high}\) . By exchanging the roles of X and Y, we repeat the steps i-iv to complete a working cycle.Since working cycles in practical applications can generally be approximated as an isothermal process (open system), the integral \(\int pdV=W_{load}+W_{material}+W_{dissipation}\), can be used to determine the input work. In this sense, our IEAR mechanism essentially shortens the working cycle (i.e., the energy path), reduces the
large requirement of input work (most becomes energy loss when deflating) observed in the traditional DIDO mechanism (Figure 1c), and leads to enhanced air supply, thus improving the actuation speed.