## 5 Simulation  
In the simulations, we consider a 2-DoF robot manipulator moving in the vertical plane with gravity. The robot links are assumed to be a rod of length 1 m and 0.8 m with concentrated mass at the rod end of 1 kg and 1.5 kg, respectively. The dynamics of the 2-DoF robot manipulator can be described as  
\[M(q)=\left[\begin{array}{cc}(m_{1}+m_{2})l_{1}^{2}+m_{2}l_{2}^{2}+2m_{2}l_{ 1}l_{2}\cos(q_{2})&m_{2}l_{2}^{2}+m_{2}l_{1}l_{2}\cos(q_{2})\\ m_{2}l_{2}^{2}+m_{2}l_{1}l_{2}\cos(q_{2})&m_{2}l_{2}^{2}\end{array}\right], \tag{21}\]  
\[C(q,\hat{q})=\left[\begin{array}{cc}-m_{2}l_{1}l_{2}\hat{q}_{2}\sin(q_{2})&- m_{2}l_{1}l_{2}(\hat{q}_{1}+\hat{q}_{2})\sin(q_{2})\\ m_{2}l_{1}l_{2}\hat{q}_{1}\sin(q_{2})&0\end{array}\right], \tag{22}\]  
\[G(q)=\left[\begin{array}{cc}(m_{1}+m_{2})gl_{1}\cos(q_{2})+m_{2}gl_{2}\cos( q_{1}+q_{2})\\ m_{2}gl_{2}\cos(q_{1}+q_{2})\end{array}\right], \tag{23}\]  
where \(m_{1}=1\), \(m_{2}=1.5\), \(l_{1}=1\), \(l_{2}=0.8\), \(w(t),v(t)\) are white Gaussian noise signals with mean 0 and variance 0.01.  
We set \(u(t)=[2,2]^{T}\), \(k=10\), \(\hat{\theta}(0)=[1,1]^{T}\), \(P(0)=I_{2}\), \(\mathrm{Y}(0)=[\begin{array}{cc}0&I_{2}\end{array}]^{T}\), \(x(0)=[1,1,1,1]^{T}\) and \(\hat{x}(0)=[0,0,0,0]^{T}\). The simulation results are presented in Figure 1.  
As discussed in Section 3, the actual dynamic robotic system can be regarded as a drive system, and our proposed ASO is treated as the response system. The errors between estimated state values and true values can be seen in Figure 1. The four states eventually converge to the real values, and the errors approach zero. This demonstrates the convergence of the proposed algorithm.  
In Figure 2, we testify the performance of our algorithm for the observation of unknown parameters. For comparison, we introduce the extended state observer (ESO) proposed in [23]. Before 10 th second, the real system runs at the expected state, i.e., \(\theta=[1,1]\). At 10 th second, the fault occurs due to several unpredictable external or internal disturbances, and \(\theta=[0.6,0.8]\). From Figure 2, it can be seen that due to the random fluctuations in the system, the estimated parameters based on both ASO and ESO will eventually converge to a value near the true value. However, the proposed ASO shows a faster convergence rate than the ESO. This also verifies the effectiveness of our fault detection algorithm.  
Choose \(T=1\), we can obtain \(\bar{\theta}_{1}\), \(\bar{\theta}_{2}\) based on the health assessment approach proposed in Section 4. Compared with Figure 2, the mean value of the estimated parameters over a period of time is closer to the true value as shown in Figure 3. This is consistent with our theorem. In addition, this also shows that a health assessment \(\bar{\theta}\) based on the mean estimation over a period of time is more effective than that based on only one estimation.  
Figure 1: The state estimation errors of four states.  
Finally, we select \(\alpha=0.05\), \(h=100\) and \(\omega=[0.5,0.5]^{T}\) to quantify the health degree. Given a nonlinear function \(g(\cdot)\) as a quadratic function \(g(x)=500x^{2}\), the evolution curve of the health degree of the 2-DoF robot manipulator over time is shown in Figure 4. As the fault occurs, the health degree drops sharply to around 60, which verifies the rapidity of our fault detection algorithm and the applicability of our health degree assessment index.  
It is noted that the 2-DoF robot manipulator in the simulations is used only as an illustrative example for testifying the performance of the proposed fault detection algorithm and health degree  
Figure 4: The health degree of the 2-DoF robot manipulator.  
assessment index. Since our theoretical analysis in Sections 3 and 4 is based on the dynamic model of an n-DoF robot manipulator (1), the results of Figures 1-4 can be generalized to robot manipulators with arbitrary DoF.