## 2 Theoretical Background and Problem Statement
### Preliminaries  
**Lemma 1 [38]**. : _This asserts that for any real numbers \(a_{1},a_{2},\ldots,a_{n}\), and for \(0\leq x\leq 1\), the inequality \(\left(\left|a_{1}\right|+\cdots\left|a_{n}\right|\right)^{x}\leq\left|a_{1} \right|^{x}+\cdots+\left|a_{n}\right|^{x}\) is valid. This lemma is essential for bounding the cumulative errors in nonlinear control environments._  
**Lemma 2 [20]. **_The Cauchy-Schwarz inequality is formalized as \(\left(\sum_{j=1}^{n}x_{j}\right)^{2}\leq n\cdot\sum_{j=1}^{n}x_{j}^{2}\). It plays a critical role in assessing the stability and convergence properties of the system._  
**Lemma 3 [39]. **_The inequality \(xy\leq\frac{\mu^{a}}{a}|x|^{a}+\frac{1}{b\mu^{k}}|y|^{b}\)for \(x,y\in R\), \(\mu>0\),\(a>0\),\(b>0\), and \((a-1)(b-1)=1\) is a form of [PERSON]'s inequality, which is useful for controlling product terms in analysis._  
**Lemma 4 [40]. **_In Lemma 4, it is posited that \(P\), an \(n\times n\)-dimensional real symmetric positive definite matrix, adheres to a crucial inequality for all \(a\in R^{n}\). Specifically, the quadratic form \(a^{T}Pa\) for any vector \(a\) is bounded by \(\lambda_{min}(P)\|a\|^{2}\) and \(\lambda_{max}(P)\|a\|^{2}\), where \(\lambda_{min}(P)\) and \(\lambda_{max}(P)\) represent the minimum and maximum eigenvalues of \(P\), respectively. This lemma highlights essential properties of linear algebra and matrix theory, facilitating the estimation of the impact matrix operations have on vectors._  
**Lemma 5 [14]. **_It is necessary to assume that for a system \(x(t)\), the origin is an equilibrium point, meaning the initial condition is \(x(0)=x_{0}\), and the dynamics of the system are described by \(\dot{x}(t)=f(t,x)\). If there exists a positive function \(V(x)\) and constants \(\Gamma,\Psi>0\), with \(0<q<1\) and \(q>1\), the \(f\) is determined as follows:_  
\[\dot{V}(x)\leq-\Gamma V^{p}(x)-\Psi V^{q}(x) \tag{25}\]  
The origin point is stable, the convergence speed does not change with time, and the upper limit of the system's stable time \(T_{d}\) follows the following relationship:  
\[T_{d}\leq\frac{1}{\Gamma(p-1)}+\frac{1}{\Psi(1-q)} \tag{26}\]  
**Remark 1 [41]. **_There are various definitions for the settling time function \(T_{d}\), and this article adopts an approach that is distinct from that used in previous research. By employing our definition, we demonstrate the practical efficacy and benefits of the fixed-time control strategy._  
**3. Fixed-Time Control Design and Lyapunov Function Proof Process**  
_3.1. Fixed-Time Control Design_  
The error signal between the actual state and the desired trajectory is determined as follows:  
\[e_{1}=q-q_{d} \tag{27}\]  
The derivative of \(e_{1}\) is given by the following:  
\[\dot{e}_{1}=e_{2}+\alpha-\dot{q}_{d} \tag{28}\]  
The error signal between the actual velocity and the virtual control variable is determined as follows:  
\[e_{2}=\dot{q}-\alpha \tag{29}\]  
where \(\alpha\) is the virtual control and \(q_{d}\) is the desired trajectory. The barrier Lyapunov function (BLF) is formulated as follows:  
\[V_{1}=\frac{1}{2}\text{ln}\frac{k_{P}^{2}}{k_{P}^{2}-{e_{1}}^{T}e_{1}} \tag{30}\]  
The selection of this function aims to ensure that the system state remains within the specified constraints by employing \(k_{P}\) control. It also aims to achieve system stabilitythrough the regulation of the error \(e_{1}\). The parameter \(k_{P}>0\) serves to enforce the constraints on the system state. \[k_{P}=ae^{-ct}+b \tag{31}\]  
In this formula, \(a\), \(b\), and \(\varepsilon\) are real numbers that satisfy the conditions \(a\geq 0\), \(b\geq 0\), and \(\varepsilon\geq 0\). These parameters are chosen and tuned based on the required error convergence trajectory.