## 3 Results
### Stability Analysis  
The main result of this paper is given by the following theorem.  
**Theorem 1**.: _Consider the ball-balancing robot model given in (2) and (3). Under Assumptions 1 and 2, if the control inputs \(\mathbf{\kappa}\) and \(\mathbf{u}\) given in (14) and (17) with the update law (20) are applied to the ball balancing model, then for any initial conditions satisfying \(V_{T}(0)\leq p\), where \(p\) is a positive constant, it is ensured that all error signals in the closed-loop system are uniformly ultimately bounded and can be made arbitrarily small._  
Proof.: From (14), one can show that \(\|\dot{\mathbf{\kappa}}\|\leq\eta(\mathbf{z}_{1},\mathbf{z}_{2},\mathbf{\omega},\mathbf{\theta}_{ f},\tilde{\mathbf{\theta}}_{f})\) for some continuous functions \(\eta\). Consider a set \(Q:=\{(\mathbf{\theta}_{f},\tilde{\mathbf{\theta}}_{f},\tilde{\mathbf{\theta}}_{f}):\| \tilde{\mathbf{\theta}}_{f}\|+\|\tilde{\mathbf{\theta}}_{f}\|+\|\tilde{\mathbf{\theta}}_{ f}\|\leq K_{0}\}\) and a set \(A:=\{(\mathbf{z}_{1},\mathbf{z}_{2},\mathbf{\omega},\tilde{\mathbf{\omega}}_{1},\tilde{\mathbf{ \omega}}_{2}):\mathbf{z}_{1}^{\top}\mathbf{z}_{1}+\mathbf{z}_{2}^{\top}\mathbf{z}_{2}+\mathbf{\omega }^{\top}\mathbf{\omega}+\tilde{\mathbf{\omega}}_{1}^{\top}\mathbf{T}_{1}^{-1}\tilde{\mathbf{ \omega}}_{1}+\tilde{\mathbf{\omega}}_{1}^{\top}\mathbf{T}_{2}^{-1}\tilde{\mathbf{\omega}}_{ 2}\leq 2p\}\) where \(K_{0}\) is a positive constant. Since \(Q\) and \(A\) are compact sets, there exists a positive constant \(\eta_{M}\) such that \(\eta(\cdot)\leq\eta_{M}\) on \(A\times Q\). Then, by Assumptions 1 and 2 and [PERSON]'s inequality, (21) can be rewritten as:  
\[\dot{V}_{T}\leq -\sum_{i=1}^{2}\Bigl{[}\mathbf{z}_{i}^{\top}\Bigl{(}\mathbf{K}_{i}-\frac{1 }{2}\mathbf{I}_{2}\Bigr{)}\mathbf{z}_{i}+\frac{1}{2}\sigma_{i}\tilde{\mathbf{\omega}}_{i}^ {\top}\tilde{\mathbf{\omega}}_{i}\Bigr{]}\] \[-\mathbf{\omega}^{\top}\Bigl{(}\tau_{2}^{-1}-\frac{1}{2}\Bigr{)}\mathbf{I} _{2}\mathbf{\omega}+\varepsilon_{M}^{2}+\frac{1}{2}\sigma_{1}\omega_{M}^{2}+\frac {1}{2}\sigma_{2}\omega_{M}^{2}+\frac{1}{2}\eta_{M}^{2}, \tag{22}\]where \(\mathbf{I}_{2}\in\mathbb{R}^{2\times 2}\) denotes the identity matrix.  
Choosing \(\mathbf{K}_{1}=\dfrac{1}{2}\mathbf{I}_{2}+\mathbf{K}_{1}^{*}\), \(\mathbf{K}_{2}=\dfrac{1}{2}\mathbf{I}_{2}+\mathbf{K}_{2}^{*}\), and \(\tau_{2}^{-1}=\dfrac{1}{2}+\tau_{2}^{*}\) with \(\mathbf{K}_{1}^{*}\), \(\mathbf{K}_{2}^{*}\), \(\tau_{2}^{*}>0\) results in:  
\[\hat{V}_{T}\leq-c_{0}V_{T}+c_{1}, \tag{23}\]  
where \(c_{0}=\min\{\lambda_{\mathbf{K}_{1}^{*}},\ \lambda_{\mathbf{K}_{2}^{*}},\ \tau_{2}^{*},\ \sigma_{1} \lambda_{\mathbf{\Gamma}_{1}^{-1}},\ \sigma_{2}\lambda_{\mathbf{\Gamma}_{2}^{-1}}\}\), \(c_{1}=\varepsilon_{M}^{2}+\tfrac{1}{2}\sigma_{1}\omega_{M}^{2}+\tfrac{1}{2} \sigma_{2}\omega_{M}^{2}+\tfrac{1}{2}\eta_{M}^{2}\), and \(\lambda_{(\cdot)}\) denotes the minimum eigenvalue of the matrix \((\cdot)\).  
Integrating both sides of (23) yields:  
\[V(t)_{T}\leq \bigg{(}V_{T}(0)-\dfrac{c_{1}}{c_{0}}\bigg{)}e^{-c_{0}t}+\dfrac{c_{1 }}{c_{0}}, \tag{24}\]  
From (24), the boundedness of \(\mathbf{z}_{1}\) is guaranteed, and it leads to the boundedness of the virtual angle \(\mathbf{\theta}_{v}\).  
Integrating both sides of (9) yields:  
\[\|\mathbf{e}_{s}(t)\|\leq\min\{\|\mathbf{e}_{s}(0)\|,\|\lambda^{-1}\mathbf{\lambda}_{\mathbf{ \theta}}^{-1}\mathbf{\theta}_{v}\|\}. \tag{25}\]  
Therefore, all errors in the closed-loop system are uniformly ultimately bounded and can be made arbitrarily small by properly choosing the design parameters. This completes the proof.