## 2 Theoretical Background and Problem Statement
### Preliminaries
Taking the time derivative of \(V_{1}\) yields \(\dot{V}_{1}\):  
\[\dot{V}_{1}=\frac{{e_{1}}^{T}\dot{e}_{1}-\frac{\dot{k}_{P}}{k_{P}}{e_{1}}^{T}e _{1}}{k_{P}^{2}-{e_{1}}^{T}e_{1}} =\frac{{e_{1}}^{T}\big{(}{\rm e}_{2}+\alpha-\dot{q}_{d}\big{)}- \frac{\dot{k}_{P}}{k_{P}}{e_{1}}^{T}e_{1}}{k_{P}^{2}-{e_{1}}^{T}e_{1}} \tag{32}\]  
\(\alpha\) is designed as follows:  
\[\alpha=-A_{11}\frac{{e_{1}}\Big{(}k_{p}^{2}-{e_{1}}^{T}e_{1}\Big{)}}{{e_{1}}^{ T}e_{1}}\Bigg{(}\frac{1}{2}ln\Bigg{(}\frac{k_{p}^{2}}{k_{p}^{2}-{e_{1}}^{T}e_{1}} \Bigg{)}\Bigg{)}^{\frac{1}{2}}-A_{12}\frac{{e_{1}}\Big{(}k_{p}^{2}-{e_{1}}^{T }e_{1}\Big{)}}{{e_{1}}^{T}e_{1}}\Bigg{(}\frac{1}{2}ln\Bigg{(}\frac{k_{p}^{2}} {k_{p}^{2}-{e_{1}}^{T}e_{1}}\Bigg{)}\Bigg{)}^{2}+\dot{q}_{d}-\bigg{(}\frac{1} {2}+\varepsilon\bigg{)}e_{1} \tag{33}\]  
In this context, \(A_{11}>0\) and \(A_{12}>0\) are design constants. Given that \({e_{1}}^{T}e_{1}\leq\frac{1}{2}{e_{1}}^{T}e_{1}+\frac{1}{2}{e_{2}}^{T}e_{2}\) and that we aim for the system output to remain within the predetermined range where \(\bigg{|}\frac{k_{P}}{k_{P}}\bigg{|}\leq\varepsilon\), by substituting (33) into (32), we obtain the following:  
\[\dot{V}_{1}\leq\frac{\frac{1}{2}{\rm e}_{2}^{T}{\rm e}_{2}}{k_{P}^{2}-{e_{1}} ^{T}e_{1}}-A_{11}\Bigg{(}\frac{1}{2}{\rm ln}\Bigg{(}\frac{k_{p}^{2}}{k_{p}^{2} -{e_{1}}^{T}e_{1}}\Bigg{)}\Bigg{)}^{\frac{1}{2}}-A_{12}\Bigg{(}\frac{1}{2}{\rm ln }\Bigg{(}\frac{k_{p}^{2}}{k_{p}^{2}-{e_{1}}^{T}e_{1}}\Bigg{)}\Bigg{)}^{2}\leq \frac{\frac{1}{2}{e_{2}}^{T}e_{2}}{k_{P}^{2}-{e_{1}}^{T}e_{1}}-A_{11}\sqrt{V_{ 1}}-A_{12}{V_{1}}^{2} \tag{34}\]  
The fixed-time control signal \(\beta\) is structured as follows:  
\[T_{d}\leq\frac{1}{\Gamma(p-1)}+\frac{1}{\Psi(1-q)} \tag{26}\]0  
Here, \(\beta^{a}\) and \(\beta^{b}\), based on errors \(e_{1}\) and \(e_{2}\), are defined as follows:  
\[T_{d}\leq\frac{1}{\Gamma(p-1)}+\frac{1}{\Psi(1-q)} \tag{26}\]1  
\[T_{d}\leq\frac{1}{\Gamma(p-1)}+\frac{1}{\Psi(1-q)} \tag{26}\]2  
Key parameters include design gains \(A_{21}\) and \(A_{22}\), both of which are positive values, and \(\lambda_{max}(\zeta)\) denotes the maximum eigenvalue of the matrix \(\zeta\). It is recommended that the latter not exceed \(0.712\). \(\hat{\Phi}\) denotes the estimation of \(\Phi\). To address the uncertainties in dynamics, we employ a radial basis function (RBF) neural network to help reduce system errors. \[T_{d}\leq\frac{1}{\Gamma(p-1)}+\frac{1}{\Psi(1-q)} \tag{26}\]3  
where \(\Phi\) symbolizes the network's optimal weight vector, and \(H\) encompasses the input parameters, including position \(q\) and velocity \(\dot{q}\). The function \(J(H)\) represents the basis function deployed within the network, while \(\sigma(H)\) indicates the bounded approximation error, with \(\sigma>0\) signifying the bound. To ensure the convergence of the speed error, we design \(V_{2}\) as follows:  
\[T_{d}\leq\frac{1}{\Gamma(p-1)}+\frac{1}{\Psi(1-q)} \tag{26}\]4The result of differentiating \(V_{2}\) obtained through Appendix A is as follows:  
\[T_{d}\leq\frac{1}{\Gamma(p-1)}+\frac{1}{\Psi(1-q)} \tag{26}\]5  
\(\widetilde{\Phi}\) represents the weight of the difference between the actual state and the desired value, and we design a rule for use updating the neural network as follows:  
\[T_{d}\leq\frac{1}{\Gamma(p-1)}+\frac{1}{\Psi(1-q)} \tag{26}\]6  
where \(\Lambda_{i}\) signifies the constant matrices, and \(\varepsilon_{i}\) denotes the constants employed to augment the robustness of the system. \(i=1,2,\ldots,\text{n}\). We designed \(V_{3}\) to handle system uncertainties and external disturbances, ensuring to improve the robustness of the system.