## 2 Materials and Methods
### Robotic Arm Controller Design  
In this section, a controller is designed to enable the robotic arm to perform tracking tasks. In previous studies, proportional integral derivative (PID) controllers were the most widely used feedback controllers, which allow the robotic arm to achieve the tracking control effect by adjusting the parameter gain value. The self-adaptive PID controller consists of the least mean square algorithm to calculate the relationship between the error and the weight and then adjust the PID control parameters by itself. Due to the highly non-linear, hysteresis, and time-variant characteristics of pneumatic manual muscle cylinders, it is not appropriate and or efficient to achieve more accurate control performance under non-specific load and motion trajectories by simply adjusting the parameters through PIDcontrollers to obtain better control results. Iterative learning control provides an appropriate algorithm to ensure that the output of the controlled system tracks more accurately to the expected command when it is operated repeatedly.  
Figure 7 is the flow chart of the PID controller applied to the proportional directional control valve. Here, \(\theta_{d}\) is the default tracking task angle command, \(\theta\) is the actual angle response of the robotic arm rotation axis, and the control voltage command \(u\) is related to the calculation command \(u_{PID}\) of the PID controller, as in Equation (40), where \(u_{w}\) is the voltage at which the proportional directional control valve port stops operating:  
\[u=u_{w}+u_{PID} \tag{40}\]  
The self-adaptive PID controller design uses the LMS (least mean square) algorithm for the online identification of the PID controller parameters, which allows it to adjust the PID parameters by itself [31]. The LMS algorithm was proposed by [PERSON] and [PERSON] in 1960. It was initially applied to the noise processing of signals, and then was gradually applied to control, optimization, and neural-like networks [32]. The control flow is shown in Figure 8.  
Figure 8: Flow chart of LMS-PID controller applied to proportional directional control valve.  
Figure 7: Flow chart of PID controller applied to proportional directional control valve.  
The relationship between the control voltage command \(u\) and the calculation command \(u_{LMSPID}\) of the adaptive PID return controller is shown in (41):  
\[u=u_{w}+u_{LMS} \tag{41}\]  
where \(u_{w}\) is the voltage at which the proportional directional control valve stops feeding air, and the \(u_{LMSPID}\) is as in (42):  
\[u_{LMS}=W_{p}e(t)+W_{i}\int\limits_{0}^{t}e(t)dt+W_{d}\frac{de(t)}{dt} \tag{42}\]  
where \(W\) is the weight vector \(W=\begin{bmatrix}W_{p}&W_{i}&W_{d}\end{bmatrix}\) of the PID, \(X\) is the input signal vector \(X=\begin{bmatrix}X_{p}&X_{i}&X_{d}\end{bmatrix}\), the error \(e\) is defined as \(e=\theta_{d}-\theta=\theta_{d}-W^{T}XG\), and \(G\) is the dynamic system function. To minimize the error, the LMS defines the cost function \(J\) as in (43) by defining the root mean square value of the error as the minimum condition:  
\[J(t)=E\Big{[}(\theta_{d})^{2}\Big{]}-2P^{T}WG+W^{T}RWG^{2} \tag{43}\]  
In the above equation, \(E\) is the desired value symbol, \(P\) is the cross-correlation matrix of the proportional, integral, and differential inputs of the PID controller and the reference input, and \(R\) is the auto-correlation matrix of the proportional, integral, and differential inputs of the PID controller.