## 6 Simulation Results  
In this section, we provide two-joint robots to verify the effectiveness of the proposed NN attack reconstruction strategies and state-feedback secure control scheme against adversarial attacks. We expect to achieve the following control objective:  
1. The obtained NN attack reconstruction strategies in an insecure network with all information unavailable can effectively estimate unknown attacks. 2. The above secure control process reaches the expected trajectory, and all the closed-loop signals are bounded, i.e., \(\left\|z,\widetilde{\theta},\widetilde{\widetilde{w}},\widetilde{\widetilde{v }}\right\|\leq\sqrt{2C_{n}/D_{n}}\). 3. The output error of the system converges to a small neighborhood of zero, i.e., \(\lim_{t\rightarrow\infty}\|y-y_{0}\|=\epsilon_{0}\). \(\epsilon_{0}\) is an infinitesimal constant. **Example 1**. : _The dynamic equation of a two-joint manipulator is defined as follows:_  
\[ \begin{array}{l}M(q)\ddot{q}+C(q,\dot{q})\dot{q}+d=u\\ y=q\end{array}  \tag{54}\]  
_where \(q=\left[\theta_{1}\quad\theta_{2}\right]^{T}\), \(\dot{q}\) and \(\ddot{q}\in\mathbf{R}^{2}\) are the angle, angular velocity and angular acceleration of the manipulator, respectively. \(M(q)\in\mathbf{R}^{2\times 2}\) is an inertia; \(C(q,\dot{q})\in\mathbf{R}^{2\times 2}\) is a centripetal force and Coriolis moment; \(d\) refers to an external disturbance, which does not affect the system's convergence but allows for a more realistic representation of the system's behavior under practical conditions; \(u=\left[u_{1}\quad u_{2}\right]^{T}\) is a control input torque vector; \(y\) is an output vector. In order to apply the backstepping method, we define \(\mathbf{x}_{1}=\left[x_{1,1}\quad x_{1,2}\right]^{T}=q\) and \(\mathbf{x}_{2}=\left[x_{2,1}\quad x_{2,2}\right]^{T}=\dot{q}\), which is written (54) as_  
\[ \begin{array}{l}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=M^{-1}(x_{1})u-M^{-1}(x_{1})C(x_{1},x_{2})x_{2}-M^{-1}(x_{1})d\\ y=x_{1}\end{array}  \tag{55}\]_with_  
\[M^{-1}(\mathbf{x}_{1})= \begin{bmatrix}I_{1}+J_{2}+2m_{2}r_{2}l_{1}\cos\theta_{2}&J_{2 }m_{2}r_{2}l_{1}\cos\theta_{2}\\ J_{2}+m_{2}r_{2}l_{1}\cos\theta_{2}&J_{2}\end{bmatrix}  \tag{56}\]  
\[J_{1}=\frac{4}{3}m_{1}r_{1}^{2}+m_{2}l_{1}^{2},J_{2}=\frac{4}{3}m_{2}r_{2}^{2} \tag{57}\]  
\[C(\mathbf{x}_{1},\mathbf{x}_{2})= \begin{bmatrix}-2m_{2}r_{2}l_{1}\dot{\theta}_{2}\sin \theta_{2}&-m_{2}r_{2}l_{1}\dot{\theta}_{2}\sin\theta_{2}\\ m_{2}r_{2}l_{1}\dot{\theta}_{1}\sin\theta_{2}&0\end{bmatrix}  \tag{58}\]  
_where \(m_{1}\) is the mass of the first connecting rod and \(m_{2}\) is the mass of the second connecting rod ; \(l_{1}\) is the length of the first connecting rod and \(l_{2}\) is the length of the second connecting rod; \(r_{1}\) is the distance from the first joint to the center of gravity of the first connecting rod and \(r_{2}\) is the distance from the second joint to the center of gravity of the first connecting rod; \(\theta_{1}\) is the angle of the first connecting rod and \(\theta_{2}\) is the angle of the second connecting rod; \(l_{1}\) is the inertia matrix of the first connecting rod and \(l_{2}\) is the inertia matrix of the second connecting rod. We describe the two-joint robotic in simulation by means of a 3-dimensional model, which is shown in Figure 2._  
The FDI attacks model in (2) is defined as follows:  
\[ \begin{split}\widetilde{\mathbf{x}}_{1}(t)&=\mathbf{x}_{1}(t )+\left[\begin{array}{cc}a_{1}(t,0)&a_{1}(t,0)\end{array}\right]^{T}\\ \widetilde{\mathbf{x}}_{2}(t)&=\mathbf{x}_{2}(t)+\left[\begin{array}{cc}a_{2}(t,x_{2 })&a_{2}(t,x_{2})\end{array}\right]^{T}\end{split}  \tag{59}\]  
with  
\[a_{1}(t,0)= \begin{cases}0,&\text{if}\ 0\leq t<15\\ \sin(t)+0.5,&\text{if}\ t\geq 15\end{cases}  \tag{60}\]  
\[a_{2}(t,x_{2})= \begin{cases}0,&\text{if}\ 0\leq t<10\\ 0.5x_{2,1}^{2}-x_{2,1}\sin(x_{2,1}),&\text{if}\ t\geq 10\end{cases}