# Research on the Dynamic Control Method of CFETR Multi-Purpose Overload Robot
## 3 CMOR Dynamic Control Method  
### CMOR Closed-Loop Control Strategy  
From Equation (17) the dynamics equation of the 8-joint robotic arm of CMOR can be expressed as  
\[T=M(q)\ddot{q}+V\big{(}q,\dot{q}\big{)}\dot{q}+G(q)+\Delta\big{(}q,\dot{q} \big{)}+dt \tag{20}\]  
where \(\Delta\big{(}q,\dot{q}\big{)}\) is the modeling uncertainty part and \(dt\) is the bounded unknown perturbation. The ideal position command is \(q_{d}\), the tracking error is \(e=q-q_{d}\), and the design control law is  
\[T=u+M(q)\ddot{q}+V\big{(}q,\dot{q}\big{)}\dot{q}+G(q) \tag{21}\]  
where \(u\) is the feedback control law. Taking Equation (20) into the CMOR dynamics model Equation (20) gives  
\[u=M(q)\ddot{e}+V\big{(}q,\dot{q}\big{)}\dot{e}+d \tag{22}\]  
where \(\ddot{e}=\ddot{q}-\ddot{q}_{d},\dot{e}=\dot{q}-\dot{q}_{d},d=\Delta\big{(}q, \dot{q}\big{)}+dt\). Define the sliding mode function as  
\[s=\dot{e}+ce \tag{23}\]where \(c>0\). Bringing Equation (23) into Equation (22) gives  
\[\left\{ \begin{array}{l}\dot{e}=s-ce\\ M\dot{s}=-Vs+\omega-d+u\end{array} \right. \tag{24}\]  
The sliding mode allows the feedback control law \(u\) to be designed according to demand, and the sliding mode motion of the system is independent of the parameter changes of the control object and the external disturbances of the system. Therefore, the robustness of the sliding mode variable structure control system is stronger than that of the general conventional continuous system. However, the sliding-mode variable-structure control is by nature discontinuous switching characteristics, which makes the variable-structure control generate high-frequency vibration in the sliding mode, affecting the accuracy of control, increasing energy consumption, generating oscillation or instability, and damaging controller components. Adaptive control is a control method that can modify its characteristics to adapt to changes in the dynamic characteristics of objects and disturbances. The combination of adaptive control and sliding mode variable structure control can solve the parameter uncertainty and time-varying parameter system control problems, increase the system robustness and achieve a very good control system performance. To improve the robustness of the CMOR control system and enhance the stability of the control system under uncertainty random error parameter uptake, this paper designs an adaptive robust sliding mode controller (ARSMC) for CMOR based on the Hamilton Jacobi equation to enhance the immunity to disturbances, as shown in Figure 4. Consider the following system model  
\[\left\{ \begin{array}{l}\dot{x}=f(x)+g(x)d\\ z=h(x)\end{array} \right. \tag{25}\]  
where \(d\) is the disturbance and \(z\) is the evaluation signal of the system. To achieve the adaptive robust control requirement of the CMOR system, the evaluation signal of the sliding mode function is defined as  
\[z=s=\dot{e}+c\dot{e} \tag{26}\]  
where \(c>0\). Define the 2-Norm for modelling uncertainty and position perturbation \(d\) as  
\[\left\|d\right\|_{2}=\left\{\int_{0}^{\infty}d^{T}ddt\right\}^{\frac{1}{2}} \tag{27}\]  
To express the robustness of the system against interference, the following performance indicators are defined  
\[J=\sup_{\left\|d\right\|_{2}\
eq 0}\frac{\left\|z\right\|_{2}}{\left\|d \right\|_{2}} \tag{28}\]  
Figure 4: CMOR adaptive robust sliding mode control principle. where \(J\) is the gain of the system \(\left\|d\right\|_{2}\), and the smaller the value of \(J\), the better the robust performance of the system. The Hamilton Jacobi equation [36; 37; 38] can be described as follows: for any given positive real number \(\gamma\), if there exists a positive definite and differentiable function \(L(x)\geq 0\) and for any interfering signal \(d\) satisfying  
\[\dot{L}\leq\frac{1}{2}\Big{\{}\gamma^{2}\|d\|^{2}-\|z\|^{2}\Big{\}} \tag{29}\]  
Then we can obtain \(J\leq\gamma\). To take advantage of the Hamilton Jacobi equation principle, Equation (24) is rewritten in the following form  
\[\left\{ \begin{array}{l}\dot{x}=f(x)+g(x)d\\ z=\dot{e}+ce\end{array}