## 2 Materials and Methods
### Design of the Recursive Least Squares Observer
The fitting process involves conducting a series of simulations under different conditions of \(T_{\mathrm{L}}\), \(J\), and \(\theta_{\mathrm{ini}}\) to obtain the optimal value of \(c\), where the position response does not exhibit overshoot (i.e., the system's position response can quickly follow the reference value without overshooting). Once the optimal value is determined, curve fitting is performed using existing data to derive the relationship between \(T_{\mathrm{L}}\), \(J\), \(\theta_{\mathrm{ini}}\), and \(c\). To simplify the fitting process, \(\theta_{\mathrm{ini}}\) is initially kept constant to obtain the relationship between \(c\) and \(J\) under different inertia conditions. Then, keeping \(J\) constant, the relationship between \(c\) and \(\theta_{\mathrm{ini}}\) under different inertia conditions is determined. Finally, the two relationships are combined to derive the overall fitting equation. When \(\theta_{\mathrm{ini}}\) rad and \(J\) are \(1.75\times 10^{-5}\) kg\(\cdot\)m\({}^{2}\) and \(2.85\times 10^{-5}\) kg\(\cdot\)m\({}^{2}\), respectively, the relationship between \(c\) and \(\theta_{\mathrm{ini}}\) determined by the fitting method is shown in Table 1. The relationship between \(c\) and \(T_{\mathrm{L}}\), \(J\) can be expressed as Equation (21). \[\left\{ \begin{array}{l}c=100000(T_{\mathrm{L}})^{3}-9400(T_{\mathrm{L}})^{2} +1240T_{\mathrm{L}}+242\\ (J=1.75\times 10^{-5}\mathrm{kg}\cdot\mathrm{m}^{2})\\ c=-100000(T_{\mathrm{L}})^{3}+(100T_{\mathrm{L}})^{2}+550(T_{\mathrm{L}})+200\\ (J=2.85\times 10^{-5}\mathrm{kg}\cdot\mathrm{m}^{2})\end{array} \right. \tag{21}\]  
At different moments of inertia, when \(T_{\mathrm{L}}\) = 0 N\(\cdot\)m and \(J\) is constant, the relationship between \(c\) and \(\theta_{\mathrm{ini}}\) is expressed as Equation (22). \[c=\sqrt{\left[c_{0}\right]^{2}/\theta_{\mathrm{ini}}} \tag{22}\] \begin{table}
\begin{tabular}{c c c} \hline \multirow{2}{*}{\(T_{\mathrm{L}}\)/Nm} & \multicolumn{2}{c}{\(c\)} \\ \cline{2-3}  & \(J\) = \(1.75\times 10^{-5}\) kg\(\cdot\)m\({}^{2}\) & \(J\) = \(2.85\times 10^{-5}\) kg\(\cdot\)m\({}^{2}\) \\ \hline
0 & 242 & 198 \\ \hline
0.01 & 255 & 206 \\ \hline
0.02 & 268 & 212 \\ \hline
0.03 & 282 & 222 \\ \hline
0.04 & 286 & 232 \\ \hline
0.05 & 305 & 240 \\ \hline
0.06 & 318 & 249 \\ \hline \end{tabular}
\end{table} 
Table 1: Relationship between parameter c and load T.  
When \(J\) = 1.75 \(\times\) 10\({}^{-5}\) kg\(\cdot\)m\({}^{2}\), \(c_{0}\) = 245, and when \(J\) = 2.85 \(\times\) 10\({}^{-5}\) kg\(\cdot\)m\({}^{2}\), \(c_{0}\) = 200. Combining Equations (21) and (22), the overall relationship between parameter \(c\) and \(T_{\rm L}\), \(J\), \(\theta_{\rm ini}\) can be obtained. \[\left\{ \begin{array}{l}J=1.75\times 10^{-5}\rm{kg}\cdot\rm{m}^{2}\\ \sqrt{\left[c_{0}\right]^{2}/\theta_{\rm ini}}=100000\left(T_{\rm L}\right)^{3 }-9400\left(T_{\rm L}\right)^{2}\\ \qquad\qquad+1240T_{\rm L}+242\end{array} \right. \tag{23}\]  
\[\left\{ \begin{array}{l}J=2.85\times 10^{-5}\rm{kg}\cdot\rm{m}^{2}\\ \sqrt{\left[c_{0}\right]^{2}/\theta_{\rm ini}}=-100000\left(T_{\rm L}\right)^{ 3}\\ \qquad\qquad+\left(100T_{\rm L}\right)^{2}+550\left(T_{\rm L}\right)+200\end{array} \right. \tag{24}\]  
Under different moments of inertia conditions, the relationship surface between \(c\), \(T_{\rm L}\), and \(\theta_{\rm ini}\) determined by the fitting method or analytical analysis is shown in Figure 6.