## 2 Materials and Methods
### WIPR Model
,\]  
By defining \(A(q_{m})=[\sin\theta,-\cos\theta,0]\), the incompleteness constraint of Equation (1) yields the following result:  
\[A(q_{m})q_{m}=0, \tag{3}\]  
The WIPR system's incomplete constraint force is \(A^{T}(q_{m})\lambda\), where \(\lambda\) is the Lagrange Multiplier. To eliminate the constraint forces in the system, we seek to find a matrix \(S(q_{m})\in\mathbb{R}^{3\times 2}\) that satisfies \(S^{T}(q_{m})A^{T}(q_{m})=0\). By defining \(\
u=[v,w]^{T}\), therefore, Equation (4) can be deduced. \[\dot{q}_{m}=S(q)\
u, \tag{4}\]  
To eliminate the incompetent constraint forces, a new vector \(\dot{\zeta}= \begin{bmatrix}\dot{\zeta}_{1},\dot{\zeta}_{2},\dot{\zeta}_{3} \end{bmatrix} ^{T}= \begin{bmatrix}v,w,\dot{\alpha}\end{bmatrix} ^{T}\) is defined and used to transform the equation. The transformation involves multiplying both sides of the equation by a scalar \(S^{T}(q_{m})\), resulting in Equation (5):  
\[M(q)\ddot{\zeta}+C\big{(}q,\dot{q}\big{)}\dot{\zeta}+G(q)=\tau+\tau_{d}, \tag{5}\]  
The dynamics of the system can be described using the following equation, in which \(M(q)\in\mathbb{R}^{3\times 3}\) represents the inertia matrix, \(C\big{(}q,\dot{q}\big{)}\dot{q}\in\mathbb{R}^{3\times 3}\) is the Coriolis force matrix, \(G(q)\in\mathbb{R}^{3\times 1}\) is the gravity matrix, \(\tau\) is the control input matrix, and \(\tau_{d}\) is the total unknown disturbance. The detailed expressions of each vector or matrix are presented below. \[M(q)= \begin{bmatrix}S^{T}M_{m}S&S^{T}M_{ma}\\ M_{am}S&M_{\alpha}\end{bmatrix} = \begin{bmatrix}m_{11}&0&m_{13}\\ 0&m_{22}&0\\ m_{31}&0&m_{33}\end{bmatrix} ,\;C\big{(}q,\dot{q}\big{)}= \begin{bmatrix}S^{T}C _{m}S+S^{T}M_{m}\dot{S}&C_{m\alpha}\\ C_{am}S+M_{am}\dot{S}&C_{\alpha}\end{bmatrix} \]  
\[ \begin{bmatrix}0&0&c_{13}\\ 0&c_{22}&c_{23}\\ 0&c_{32}&0\end{bmatrix} ,\;G(q)= \begin{bmatrix}S^{T}G_{m}\\ G_{\alpha}\end{bmatrix} = \begin{bmatrix}0\\ 0\\ g_{3}\end{bmatrix} = \begin{bmatrix}0\\ 0\\ -MgL\sin\alpha\end{bmatrix} ,\tau= \begin{bmatrix}S^{T}B_{m}(q_{m})\tau_{m}\\ 0\end{bmatrix} = \begin{bmatrix}\tau_{1}\\ \tau_{2}\\ 0\end{bmatrix} \]  
\[\tau_{d}= \begin{bmatrix}S^{T}d_{m}\\ d_{\alpha}\end{bmatrix} = \begin{bmatrix}d_{1}\\ d_{2}\\ d_{3}\end{bmatrix} .\]  
The value of each variable in the expression is indicated as \(m_{11}=2m+2I_{M}/r^{2}+M\), \(m_{13}=m_{31}=ML\cos\alpha\), \(m_{22}=d^{2}m/2+I_{M}d^{2}/2r^{2}+I_{\omega}+ML^{2}\sin^{2}\alpha\), \(m_{33}=ML^{2}+I_{M}\), \(c_{22}=(1/2)ML^{2}\dot{\alpha}\sin^{2}(2\alpha)\), \(c_{23}=-(1/2)\omega ML^{2}\sin(2\alpha)\), \(c_{13}=-ML\dot{\alpha}\sin\alpha\), \(c_{32}=-(1/2)\omega ML^{2}\sin(2\alpha)\). Due to the coupling of the state variables in the system, Equation (5) is decoupled into Equation (6). \[\left\{ \begin{array}{l}(m_{11}m_{33}-m_{13}m_{31})\ddot{\zeta}_{1}-m_{13}c_ {32}\dot{\zeta}_{2}+m_{33}c_{13}\dot{\zeta}_{3}-m_{13}g_{3}\\ m_{33}(\tau_{1}+d_{1})-m_{13}d_{3}\\ m_{22}\ddot{\zeta}_{2}+c_{22}\dot{\zeta}_{2}+c_{23}\dot{\zeta}_{3}=\tau_{2}+d_{2} \\ (m_{11}m_{33}-m_{13}m_{31})\ddot{\zeta}_{3}+m_{11}c_{32}\dot{\zeta}_{2}-m_{31}c_ {13}\dot{\zeta}_{3}+m_{11}g_{3}\\ m_{11}d_{3}-m_{31}(\tau_{1}+d_{1})\end{array} \right., \tag{6}\]By defining \(\zeta_{1}=[\zeta_{1},\,\zeta_{2},\,\zeta_{3}]^{T}\), \(\zeta_{2}=\left[\dot{\zeta}_{1},\dot{\zeta}_{2},\dot{\zeta}_{3}\right]^{T}\), Equation (6) is converted to Equation (7). \[\left\{ \begin{array}{l}\dot{\xi}_{1}=\xi_{2}\\ \dot{\xi}_{2}=f(\xi)+g(\xi)\tau+D\end{array} \right., \tag{7}\]  
where  
\[