## 4 Nonsingular Predefined-Time Attitude Controller Design  
In the traditional predefined-time terminal SMC design process, it can be seen that the negative fractional power term always appears in the control law, which may result in a singularity if \(e_{1}=0\), \(e_{2}\
eq 0\)[20]. Thus, in this section, a novel nonsingular terminal sliding controller without using piecewise continuous functions is proposed. Firstly, a novel predefined-time terminal SMS is expressed as follows:  
\[S=\text{sig}^{2+2a/3}e_{1}+\text{sig}^{\frac{12+4a}{3-2a}}\left(\frac{(3+2a)T_{ c}}{\sqrt{3}\pi+2\sqrt{3}\text{arctan}\Big{(}1/\sqrt{3}\Big{)}}\mathbf{e_{2}}+ \text{sig}^{2+2a/3}\mathbf{e}_{1}\right), \tag{24}\]  
where \(T_{c}>0\) and \(-1.5<a<-0.75\). **Theorem 3**. : _The phenomenon of the singularity is avoided. Once the tracking error \(\mathbf{e}_{1}\) and its derivative \(\mathbf{e}_{2}\) make \(\mathbf{S}=[S_{1},\ S_{2},\ S_{3}]^{\mathrm{T}}=\mathbf{0},\mathbf{e}_{1}\) and \(\mathbf{e}_{2}\) will converge to zero within the predefined time \(T_{c}\)._  
Proof of Theorem 3.: When \(\mathbf{e}_{1}=[e_{11},\ e_{12},\ e_{13}]^{\mathrm{T}}\) and \(\mathbf{e}_{2}\) evolve on the expected SMS, we have  
\[ \begin{array}{l}\text{sig}^{1/2-a/3}e_{1i}+\frac{(3+2a)T_{c}}{\sqrt{3}\pi+2 \sqrt{3}\text{arctan}\frac{1}{\sqrt{3}}}\mathbf{e}_{2i}+\text{sig}^{2+2a/3}e_{1i}= 0\\ \Downarrow\\ -\frac{(3+2a)T_{c}}{\sqrt{3}\pi+2\sqrt{3}\text{arctan}\frac{1}{\sqrt{3}}}\bm {e}_{2i}=\text{sig}^{2+2a/3}e_{1i}+\text{sig}^{1/2-a/3}e_{1i}\\ \Downarrow\\ e_{2i}\text{sign}(e_{1i})=-\frac{\sqrt{3}\pi+2\sqrt{3}\text{arctan}\frac{1}{ \sqrt{3}}}{(3+2a)T_{c}}\Big{(}|e_{1i}|^{2+2a/3}+|e_{1i}|^{1/2-a/3}\Big{)}\end{array} , \tag{25}\]  
where \(i=1,2,3\). Choosing the Lyapunov function, \(V_{i}=|e_{1i}|\) and the derivative of \(V_{i}\) can be expressed as  
\[ \begin{array}{l}\dot{V}_{i}=e_{2i}\text{sign}(e_{1i})\\ =-\frac{\sqrt{3}\pi+2\sqrt{3}\text{arctan}\frac{1}{\sqrt{3}}}{(3+2a)T_{c}} \Big{(}|e_{1i}|^{2+2a/3}+|e_{1i}|^{1/2-a/3}\Big{)}\\ \leq-\frac{\sqrt{3}\pi+2\sqrt{3}\text{arctan}\frac{1}{\sqrt{3}}}{(3+2a)T_{c}} \Big{(}V_{i}^{1/2-a/3}+V_{i}^{2+2a/3}\Big{)}\end{array} . \tag{26}\]  
By applying Theorem 1, the proposed nonsingular SMS is predefined-time stable within \(T_{c}\). Meanwhile, \(\mathbf{e}_{1}\) and \(\mathbf{e}_{2}\) will reach the SMS \(\mathbf{S}=\mathbf{0}\), whose proof is given in Theorem 4. The proof of Theorem 3 is completed. \(\square\)  
The simulation below affirms the effectiveness of the proposed predefined-time SMS. The different values of \(a\) and \(T_{c}\) are taken when the initial values satisfy \(e_{1i}=1\), \(e_{2i}=0\), and \(S_{i}=0\), \(i=1,\ 2,\ 3\). In Figure 4, we can see that \(e_{1i}\) converges to zero within \(T_{c}\). Then, a predefined-time extended state observer (PTESO) is introduced, which can effectively estimate the disturbances \(\mathbf{D}\). The form of PTESO is expressed as follows:  
\[ \begin{array}{l}\mathbf{\varphi}_{1}=\mathbf{e}_{1}-\hat{\mathbf{e}}_{1}\\ \hat{\mathbf{e}}_{1}=\mathbf{e}_{2}+\Psi\zeta\overline{\zeta}\overline{\Xi}_{1}\mathbf{ \varphi}_{1}+(1-\Psi)k_{1}\gamma^{1/3}\text{sig}^{2/3}(\mathbf{\varphi}_{1})\\ \hat{\mathbf{\hat{e}}}_{2}=\hat{\mathbf{\Lambda}}+\Psi\zeta^{2}\overline{\Xi}_{2}^{2} \mathbf{\varphi}_{1}+(1-\Psi)k_{2}\gamma^{2/3}\text{sig}^{1/3}(\mathbf{\varphi}_{1})+ \mathbf{H}+\mathbf{U}\\ \hat{\mathbf{\hat{\Lambda}}}=\Psi\zeta^{3}\overline{\Xi}_{1}^{2}\mathbf{\varphi}_{1}+ (1-\Psi)k_{3}\gamma\text{sig}^{0}(\mathbf{\varphi}_{1})\end{array} , \tag{27}\]  
\[\begin{array}{l}\text{where }\mathbf{H}=\mathbf{P}\mathbf{e}_{1}+\mathbf{A}\left( \begin{array}{c }(\hat{\mathbf{R}}\mathbf{R}^{-1}+\mathbf{R}\mathbf{J}^{-1}\mathbf{\Omega}\mathbf{J}^{-1})\mathbf{z}_{2}- \tilde{\mathbf{\Lambda}}_{\mathrm{C}}-\mathbf{\Phi}\\ +(\hat{\mathbf{R}}\mathbf{R}^{-1}+\mathbf{R}\mathbf{J}^{-1}\mathbf{\Omega}\mathbf{J}^{-1})\hat{\mathbf{ \Lambda}}_{\mathrm{C}}\end{array}