# Research on the Dynamic Control Method of CFETR Multi-Purpose Overload Robot
## 2 CMOR Modelling
### CMOR Dynamics Model  
CMOR can withstand high loads for the maintenance of vacuum chamber components. The CMOR structure with 8 degrees of freedom as shown in Figure 1b can flexibly adjust its attitude in a limited space to transport the end-effector to any maintenance location within the vacuum chamber. The end-effector has multiple degrees of freedom and is capable of performing a variety of maintenance operations. The CMOR dynamics equations are established by the Lagrangian method. The first step is to calculate the CMOR kinetic energy including the link dynamic energy \(K_{l}\), the drive motor rotor kinetic energy \(K_{r}\), and the load kinetic energy \(K_{p}\). Therefore, the total CMOR kinetic energy is  
\[K=K_{r}+K_{l}+K_{p}+\Delta T \tag{5}\]  
where \(\Delta T\) is the calculated error under unknown disturbance. \[K_{r}=\frac{1}{\dot{\theta}}^{T}I_{a}\dot{\theta}=\frac{1}{2}\big{(}N\dot{q} \big{)}I_{a}\big{(}N\dot{q}\big{)}=\frac{1}{2}\sum_{i=1}^{8}N_{i}^{2}\dot{q}_{i }^{2}I_{ai} \tag{6}\]  
Figure 3: CMOR remote handling maintenance tasks (**a**) Dual arm end-effector (**b**) First wall target plate removal tool (**c**) Multi-joint endoscopic robotic arm (**d**) Vacuum chamber pipe cutting (**e**) First wall target plate removal (**f**) Endoscopic inspection of the bottom of the divertor. Figure 2: The working process of the CMOR system (**a**) CMOR installed in CASK transport vehicle (**b**) CMOR transport to the middle port of the vacuum chamber (**c**) CMOR is locked with the middle port (**d**) Open the screen doors (**e**) CMOR transport to vacuum chamber via the mobile platform (**f**) CMOR completes maintenance tasks on top and bottom of the vacuum chamber. where \(I_{ai}\) is the equivalent rotational inertia of the \(i\)-th joint actuator. \(\dot{q}_{i}\) is the velocity of joint \(i\). Assume that the mass of any mass point \(P\) on joint \(i\) is \(dm\), then its kinetic energy is  
\[dk_{i}=\frac{1}{2}v_{p}^{2}dm=\frac{1}{2}Trace[\sum_{j=1}^{i}\sum_{k=1}^{i} \frac{\partial T_{i}}{\partial q_{j}}r_{p}^{i}(r_{p}^{i})^{T}(\frac{\partial T_ {i}}{\partial q_{k}})\dot{q}_{j}\dot{q}_{k}]dm \tag{7}\]  
where \(T_{i}\) is the \(i\)-th joint transformation matrix, including rotation and translation transformations, and \(r_{p}^{i}\) is the position vector of the local coordinate system (relative to joint \(i\)). The kinetic energy of joint \(i\) can be obtained by integrating the kinetic energy of the mass points on joint  
\[K_{i}=\int_{link_{i}}dk_{i}=\frac{1}{2}\overline{T}race[\sum_{j=1}^{i}\sum_{k =1}^{i}\frac{\partial T_{i}}{\partial q_{j}}(\int_{link_{i}}r_{p}^{i}(r_{p}^{ i})^{T}dm)(\frac{\partial T_{i}}{\partial q_{k}})\dot{q}_{j}\dot{q}_{k}] \tag{8}\]  
Thus, the total kinetic energy of each link in the CMOR can be written as follows  
\[K_{l}=\sum_{i=1}^{8}K_{i}=\frac{1}{2}\sum_{i=1}^{8}Trace[\sum_{j=1}^{i}\sum_{k =1}^{i}\frac{\partial T_{i}}{\partial q_{j}}I_{i}(\frac{\partial T_{i}}{ \partial q_{k}})\dot{q}_{j}\dot{q}_{k}] \tag{9}\]  
where \(I\) denotes the pseudo-inertia matrix expressed as follows  
\[I_{i}=\int_{link_{i}}r^{i}(r^{i})^{T}dm=\left[ \begin{array}{cccc}\int_{i}x_ {4}^{2}dm&\int_{i}x_{i}y_{4}dm&\int_{i}x_{i}zdm&\int_{i}x_{i}dm\\ \int_{i}x_{i}y_{i}dm&\int_{i}y_{i}^{2}dm&\int_{i}y_{i}z_{i}dm&\int_{i}y_{i}dm\\ \int_{i}x_{i}z_{i}dm&\int_{i}y_{i}z_{i}dm&\int_{i}z_{i}^{2}dm&\int_{i}z_{i}dm\\ \int_{i}x_{i}dm&\int_{i}y_{i}dm&\int_{i}z_{i}dm&\int_{i}dm\\ \end{array} \right] \tag{10}\]  
The CMOR load is the self-weight of the end-effector and load weight. It is assumed that all external loads are concentrated at one point on the end joint of the CMOR, so that the end load has only translational energy and no rotational energy  
\[K_{p}=\frac{1}{2}m_{p}Trace[\sum_{j=1}^{8}\sum_{k=1}^{8}\frac{\partial T_{8}} {\partial q_{j}}r_{8}^{T}(\frac{\partial T_{8}}{\partial q_{k}})\dot{q}_{j} \dot{q}_{k}] \tag{11}\]  
where \(m_{p}\) is the mass of the end-effector and the load. The second step calculates the CMOR potential energy.