The dynamic characteristics of the pneumatic system must be analyzed to determine the function of the musculoskeletal system of the frog-inspired robot and lay the foundation for its control system design. The pneumatic muscles used in the robot have apparent nonlinearity and hysteresis characteristics. The previous modeling of the pressure dynamic process is based on the ideal conditions, which can lead to different response results. The dynamic models of the source chamber, muscle volume, exhaust chamber, and switch valves are established to simulate the pressure process of the pneumatic system.

### 3.1 Dynamic Model of the Source Chamber and Exhaust Chamber

The source chamber is regarded as a variable mass system. The equation of the thermodynamic process can be written according to the first law of thermodynamics [25]:

$${\text{d}}Q_{\text{s}} + i_{\text{s}} {\text{d}}M_{\text{s}} = \text{d} U_{\text{s}} + {\text{d}}W_{\text{s}} + i{\text{d}}M,$$

(1)

where d*Q*
_{s}—Heat gained in the source chamber caused by the heat exchange between the inner gas and the outside world through wall, *i*
_{s}—Specific enthalpy of the gas flow into the source chamber, *i*—Specific enthalpy of the gas flow to the pneumatic muscle from the source chamber, d*M*
_{s}—Gas mass flow into the source chamber, d*U*
_{s}—Internal energy change of the gas in the source chamber, d*W*
_{s} —Expansion work done by the gas in the source chamber, d*M*—Gas mass flow into the pneumatic muscle.

According to thermodynamics, d*U*
_{s} = *C*
_{v}
*M*
_{s}d*T*
_{s}, d*W*
_{s} = *P*
_{s}d*V*
_{s} and *i*d*M* = (*C*
_{v} + *R*)*T*
_{s}d*M*, where *C*
_{v} is the constant volume specific heat of air, *R* denotes the gas constant of air, and d*T*
_{s} is the temperature differential in the source chamber.

Given that the gas flow in the air pipe is faster than the heat exchange rate between gas and external environment, energy loss in the air pipe is much lesser than total gas energy. Hence, heat exchange can be ignored. Therefore, rapid pressurizing and depressurizing processes of pneumatic muscles can be regarded as adiabatic processes, such that d*Q*
_{s} = 0. The volume of the source chamber is constant, such that d*V*
_{s} = 0. If the air supply to the source chamber is disregard, then d*M*
_{s} = 0. Therefore, the gas thermal process in the source chamber can be simplified as follows:

$${\text{d}}p_{\text{s}} = - \frac{{kRT_{\text{s}} {\text{d}}M}}{{V_{\text{s}} }},$$

(2)

where *T*
_{s}—Temperature in the source chamber, *p*
_{s}—Pressure in the source chamber, *R*—Gas constant of air, 287 J·kg/K, *K*—Specific heat ratio of air, *K* = 1.4.

The relationship between temperature and pressure in the adiabatic process is represented as

$$\frac{{T_{\text{S}} }}{{T_{0} }} = \left( {\frac{{p_{\text{S}} }}{{p_{0} }}} \right)^{{\frac{k - 1}{k}}} ,$$

(3)

where *T*
_{0}—Initial temperature in the source chamber, *p*
_{0}—Initial pressure in the source chamber.

Similarly, the thermal process of the pressurizing process in the exhaust chamber is derived as

$$kRT{\text{d}}m = V_{\text{e}} {\text{d}}p_{\text{e}} ,$$

(4)

where *T*—Initial temperature in the pneumatic muscle, *p*
_{e}—Pressure in the exhaust chamber, *V*
_{e}—Volume of the exhaust chamber, d*m*—Exhausted air from the pneumatic muscle.

### 3.2 Dynamic Model of the Pneumatic Muscle

When the pneumatic muscle is assumed as a cylinder and the noncylindrical joint at both ends of the muscle is disregarded, the pneumatic muscle structure can be modeled as shown in Figure 7. In the figure, *θ* is the angle between the braided thread and central axis, and *D* is the pneumatic muscle diameter. Muscle volume can be expressed as follows [26]:

$$V = \frac{{b_{0}^{2} - L^{2} }}{{4\pi n^{2} }}L,$$

(5)

where *L*—Pneumatic muscle length, *n*—Number of loops of braided thread in the muscle, *b*
_{0}—Length of the non-retractable braided thread, *V*—Internal volume of the muscle.

According to Eq. (1), the pressurizing process of the muscle can be simplified as

$$kRT_{\text{S}} {\text{d}}M = kP{\text{d}}V + V{\text{d}}P.$$

(6)

The thermal process of muscle depressurizing is

$$- kRT{\text{d}}m = kP{\text{d}}V + V{\text{d}}P,$$

(7)

where *P*—Pressure in pneumatic muscle, *V*—Volume of pneumatic muscle.

### 3.3 Dynamic Model of the Pneumatic System

The pneumatic system in this paper uses the switch valve, which has a simple structure and digital control signal. The flow characteristics of the pneumatic system mainly depend on the valve [27, 28]. Thus, we use the high-speed switch valve MHJ9-LF manufactured by FESTO. The valve Sonic conductance (*C*) is 0.4 L/s·bar, and the critical pressure ratio (*b*) is 0.38. The volume flow through the valve can be solved by the following equation:

$$\begin{aligned} Q_{\text{m}} = C\left( {P_{1} + P_{0} } \right)\sqrt {\frac{{T_{0} }}{{T_{1} }}} \omega (\sigma ,b), \hfill \\ \omega (\sigma ,b) = \left\{ {\begin{array}{*{20}c} {\;\;\;\;\;\;\;\;1\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\sigma \le {\text{b,}}} \\ {\sqrt {1 - \left( {\frac{\sigma - b}{1 - b}} \right)^{2} } \;\;\;\;\;\;\;\sigma \le {\text{b,}}} \\ \end{array} } \right.\; \hfill \\ \sigma = \frac{{P_{2} + P_{0} }}{{P_{1} + P_{0} }}, \hfill \\ \end{aligned}$$

(8)

where *P*
_{0}—Atmospheric pressure in standard condition, *P*
_{1}—Upstream pressure, *P*
_{2}—Downstream pressure, *T*
_{0}—Gas temperature in the standard state.

As the gas cylinder connects six branch pipes for the muscles from one main pipe simultaneously, the gas flow at saturation, and the gas flow in the main pipe can be obtained through the continuity equation:

where *Q*—Main pipe flow rate, *r*—Pipe radius, *v*—Flow velocity in the pipe.

On the basis of Eqs. (2), (3), (5), (6), and (8), a simulation model of the pneumatic system of the robot is established to simulate the dynamic characteristics of the process of pneumatic muscle pressurizing, as shown in Figure 8.

The saturation term is added in the simulation process because of the calculated maximum flow rate in the main pipe. The muscle length is derived from the joint angle. With muscle length and initial pressure from the source chamber, the pressurizing processes in the chambers and muscles can be simulated, and the rationality of the established model can be verified by comparing the results to lay the foundation for future designs of robot control systems.