## 3 A Model-Based TSMC Approach Using PPCA  
### Ppca  
The error variable for the position and velocity are defined as follows: \(e_{1i}=\alpha_{1 di}-\alpha_{1i}\) and \(e_{2i}=\alpha_{2 di}-\alpha_{2i}\), where \(i=1,\ldots,n\). Additionally, \(\alpha_{1 di}\) is the \(i^{th}\) desired position trajectory for the \(i^{th}\) joint of the robot. It is desired that the position error \(e_{1i}\) at robot joint \(i\) remains within a predetermined range:  
\[-\ell_{b}(t)<e_{1i}\text{sign}(e_{1i_{0}})<\ell_{t}(t)\;. \tag{10}\]  
Here, \(\mathbf{e}\) represents [PERSON]'s number, \(e_{1i_{0}}\) is the initial value of \(e_{1i}\) at \(t=0\), \(\ell_{t}(t)=(\ell_{0}-\ell_{\infty})\mathbf{e}^{-\phi t}+\ell_{\infty}\), and \(\ell_{b}(t)=(\ell_{1}-\ell_{\infty})\mathbf{e}^{-\phi t}+\ell_{\infty}\). These functions are smooth, decreasing, positive functions that map \(\mathbb{R}^{+}\) to \(\mathbb{R}^{+}\) and satisfy the conditions \(\lim_{t\to\infty}\ell_{t}(t)=\ell_{\infty}>0\) and \(\lim_{t\to\infty}\ell_{b}(t)=\ell_{\infty}>0\). The constants \(\ell_{0}\), \(\ell_{1}\), and \(\ell_{\infty}\) are chosen, such that \(\ell_{0}>|e_{1i_{0}}|>0\) and \(\ell_{0}\geq\ell_{1}\geq\ell_{\infty}\). The positive constant \(\phi\) is utilized to adjust the performance bounds. The position error can be transformed using the proposed error transformation function (ETF) as follows:  
\[\left\{ \begin{aligned} e_{1i}=\mathcal{P}_{i}(t) \mathcal{T}_{i}(E_{1i})\\ \mathcal{T}_{i}(E_{1i})=\frac{2}{\pi}\arctan(E_{1i})\end{aligned}  \right.\quad i=1,\ldots,n\;. \tag{11}\]  
Here, \(E_{1i}\) represents the \(i\)th transformed error. \[\mathcal{P}_{i}(t)=\left\{ \begin{aligned} &\ell_{t}(t)\quad\text{if sign}(e_{1i}.e_{1i_{0}})>0\\ &\ell_{b}(t)\quad\text{if sign}(e_{1i}.e_{1i_{0}})<0\end{aligned} \right.\,.\]  
The function \(T_{i}(E_{1i})\) exhibits the following characteristics: it is a smooth and strictly increasing function, satisfying \(-1<\mathcal{T}_{i}(E_{1i})<1\). Additionally, \(\mathcal{T}_{i}(E_{1i})=0\) when \(E_{1i}=0\), and as \(E_{1i}\) approaches negative infinity, \(\lim\limits_{E_{1i}\to-\infty}\mathcal{T}_{i}(E_{1i})=-1\), while as \(E_{1i}\) approaches positive infinity, \(\lim\limits_{E_{1i}\to+\infty}\mathcal{T}_{i}(E_{1i})=1\). If \(e_{1i_{0}}>0\) and \(e_{1i}>0\), then we have \(0\leq\mathcal{T}_{i}(E_{1i})<1\) and \(\ell_{t}(t)>0\). This implies that \(0\leq\ell_{t}(t)\mathcal{T}_{i}(E_{1i})<\ell_{t}(t)\). On the other hand, if \(e_{1i_{0}}>0\) and \(e_{1i}<0\), then \(\mathcal{T}_{i}(E_{1i})\) satisfies \(-1<\mathcal{T}_{i}(E_{1i})\leq 0\), and \(\ell_{b}(t)>0\). Consequently, we have \(-\ell_{b}(t)<\ell_{b}(t)\mathcal{T}_{i}(E_{1i})\leq 0\). Thus, when \(e_{1i_{0}}>0\), the position error \(e_{1i}\) falls within the range \(-\ell_{b}(t)<e_{1i}<\ell_{t}(t)\). Similarly, if \(e_{1i_{0}}<0\) and \(e_{1i}<0\), then \(\mathcal{T}_{i}(E_{1i})\) satisfies \(-\ell_{t}(t)<\mathcal{T}_{i}(E_{1i})<0\). Furthermore, if \(e_{1i_{0}}<0\) and \(e_{1i}>0\), then \(\mathcal{T}_{i}(E_{1i})\) falls within \(0<\mathcal{T}_{i}(E_{1i})<\ell_{b}(t)\). In both cases, we can conclude that when \(e_{1i_{0}}<0\), the position error \(e_{1i}\) lies between \(-\ell_{t}(t)\) and \(\ell_{b}(t)\). Consequently, by utilizing Equation (10), we can define the transformation of the position error in advance, encompassing both the transient and steady-state stages. From Equation (11), we can obtain \(E_{1i}\) and its derivative as follows:  
\[E_{1i}=\tan\biggl{(}\frac{\pi e_{1i}}{2\mathcal{P}_{i}(t)}\biggr{)}\, \tag{12}\]  
\[\dot{E}_{1i}=\frac{\pi\bigl{(}1+E_{1i}^{2}\bigr{)}}{2\mathcal{P}_{i}(t)} \biggl{(}e_{2i}-\frac{2\dot{\mathcal{P}}_{i}(t)}{\pi}\arctan(E_{1i})\biggr{)} \ . \tag{13}\]  
Here, we define \(A=\text{diag}(A_{i})\) with \(A_{i}=\frac{\pi\bigl{(}1+E_{1i}^{2}\bigr{)}}{2\mathcal{P}_{i}(t)}>0\) and \(\wp=\bigl{[} \begin{array}{cc}\wp_{1}&\ldots&\wp_{n}\end{array} \bigr{]}^{T}\) with \(\wp_{i}=\frac{2\dot{\mathcal{P}}_{i}(t)}{\pi}\arctan(E_{1i})\), for \(i=1,\ldots,n\). Let \(E_{2i}=e_{2i}-\wp_{i}\).