## 2 System Description
### Prescribed Performance Constraint  
In this section, a PPC theory is applied for the control scheme to ensure the stability of the chassis in vertical displacement. First, the tracking error of the system state \(x_{1}\) is defined by  
\[e_{1}=x_{1}-x_{d} \tag{15}\]  
where \(x_{d}\) is the reference trajectory.  
**Definition 1**.: _A positive decreasing smooth function is used as the prescribed performance function [35]:_  
\[\rho(t)=(\rho_{0}-\rho_{\infty})e^{-\varphi t}+\rho_{\infty} \tag{16}\]  
_where \(\varphi>0\) represents the convergence rate, \(\rho_{0}\) denotes the initial state of PPF, and \(\rho_{\infty}\) is the allowable steady-state error, which are selected to meet the initial conditions \(\underset{t\to 0}{\lim}\rho(t)=\rho_{0}>0\), \(\underset{t\to\infty}{\lim}\rho(t)=\rho_{\infty}>0\), and \(\rho_{0}>\rho_{\infty}\)._  
According to (16), the tracking error of chassis movement will be ensured within the bounded performance by the transform inequality  
\[-\underline{\kappa}\rho(t)<e_{1}<\overline{\kappa}\rho(t),\ t>0 \tag{17}\]  
where \(\underline{\kappa}\,\overline{\kappa}>0\) are the constant parameters.  
**Remark 2**.: _From (16) and (17), it can be seen that \(-\underline{\kappa}\rho(0)\) plays a role as the lower bound of the undershoot while \(\overline{\kappa}\rho(0)\) representing the upper bound of the overshoot of \(x_{1}\). The steady-state performance of the system can be guaranteed by accordingly choosing the design parameters \(\underline{\kappa}\), \(\overline{\kappa}\), \(\rho_{0}\), \(\rho_{\infty}\), \(\varphi\)._  
Under the condition (17), the control strategy is designed by using an output transformation, which is used to construct the constraints into an equality form. By suggesting a smooth and strictly increasing function \(S(z_{1})\), we can obtain [35]  
\[S(z_{1})=\frac{\overline{\kappa}e^{z_{1}}-\underline{\kappa}e^{-z_{1}}}{e^{z _{1}}+e^{-z_{1}}} \tag{18}\]  
where the two following conditions are satisfied  
1. \(-\underline{\kappa}<S(z_{1})<\overline{\kappa}\)
2. \(\underset{z_{1}\to\infty}{\lim}S(z_{1})=\overline{\kappa}\), \(\underset{z_{1}\to-\infty}{\lim}S(z_{1})=-\underline{\kappa}\)  
Therefore, we can write the performance (17) as follows  
\[e_{1}=\rho(t)S(z_{1}) \tag{19}\]  
From the above definition, prescribed performance can be designed to establish the transform errors of system states. Since \(S(z_{1})\) is strictly monotonically increasing and prescribed performance parameters are selected to satisfy \(\rho(t)>\rho_{\infty}>0\), we can conclude an inverse transfer function \(z_{1}\) as follows  
\[z_{1}=S^{-1}\bigg{(}\frac{e_{1}}{\rho(t)}\bigg{)} \tag{20}\]  
So, we have the transform function of \(z_{1}\)  
\[z_{1}=\frac{1}{2}\ln\bigg{(}\frac{\mu+\underline{\kappa}}{\overline{\kappa}- \mu}\bigg{)}, \tag{21}\]  
where \(\mu=\frac{e_{1}}{\rho(t)}\).  
**Lemma 1**.: [35]_. From the above procedure, the system state (14) is transformed by the smooth function \(S(z_{1})\), so the tracking error signal \(e_{1}\) is guaranteed under the prescribed performance boundary (17)._  
**Remark 3**.: _Based on the above analysis, as the parameters \(\rho_{0},\underline{\xi},\overline{\kappa}\) are designed such that the initial condition satisfies \(-\underline{\kappa}\rho(0)<x_{1}(0)<\overline{\kappa}\rho(0)\) and \(z_{1}\) will be restrained within the boundaries, the condition \(-\underline{\kappa}<S(z_{1})<\overline{\kappa}\) is guaranteed. Therefore, the control problem (14) with the condition \(-\underline{\kappa}\rho(t)<x_{1}(t)<\overline{\kappa}\rho(t)\) is ensured by the transformation (20)._