# Performance Analysis of Hybrid Kinematic Mechanism for Fusion Reactor Maintenance
## 5 Error Compensation  
### Variable Parameter Error Compensation  
The HKM has a large working space, large load capacity and high position accuracy requirements. To avoid excessive stress on the HKM, the center of gravity of the load is always located directly below the HKM end twist locks during operation. Therefore, the requirements for the HKM dexterity are not high and the focus is on positional accuracy. In order to compensate for the global errors (including piston rotation error, control error, rigid-flexible coupling error, etc.) and to meet the requirements for fast adjustment of end position accuracy of the HKM, an open-loop variable parameter error compensation method is proposed, as shown in Figure 11. From the analysis of the HKM end position error in Section 4, the HKM end position error \(\Delta\)P is obtained unevenly distributed in space. Therefore, it is not possible to compensate the error for the whole operating process by using a single parameter. Hence, the HKM workspace is meshed according to the error \(\Delta\)P distribution. Under certain position accuracy requirements, each grid can adopt a set of parameters at the center point of the grid for error compensation. Different grids are solved separately for the compensation parameters to achieve variable parameter error compensation for the HKM workspace. The flowchart of parameter error calculation within each grid is shown in Figure 12. First, we obtained the position error data of the HKM end-effector, and in order to simplify the calculation we take the end-effector position error caused by piston rotation as an example for variable parameter error compensation. Second, error modeling and preliminary meshing were performed. Since all kinds of errors such as piston rotation error, control error, and deformation error have certain influence on the end position accuracy of the HKM, and all of them are coupled with HKM kinematic parameters. Therefore, the errors of each parameter of the HKM can be expressed as a function of joint angle and piston  
Figure 10: HKM end position error during the left outboard (**a**) and inboard (**b**) BB lifting. stroke without considering the influence of random errors. The actual position \(\mathbf{Pr}\) of the HKM end-effector can be expressed as  
\[\mathbf{P}_{r}=\mathbf{F}(\alpha+\Delta\alpha,\beta+\Delta\beta,\cdots,\delta_{6}+\Delta \delta_{6}), \tag{32}\]  
\[\Delta\mathbf{P}=\mathbf{P}_{r}-\mathbf{P}_{t}\] \[=F(\alpha+\Delta\alpha,\beta+\Delta\beta,\cdots,\delta_{6}+\Delta \delta_{6})-\mathbf{F}(\alpha,\beta,\cdots,\delta_{6})^{\prime} \tag{33}\]  
where \(\mathbf{P}_{t}\) is the theory position of the HKM end-effector, \(\Delta\mathbf{P}_{r}\) is the position error, and the above equation can be linearized by discarding the higher order terms:  
\[\Delta\mathbf{P}=\frac{\partial\mathbf{P}}{\partial\alpha}\Delta\alpha+\frac{\partial \mathbf{P}}{\partial\beta}\Delta\beta+\cdots+\frac{\partial\mathbf{P}}{\partial\delta _{6}}\Delta\delta_{6}. \tag{34}\]  
Figure 11: HKM workspace variable density meshing (**a**); a set of parameters is used to compensate for one grid region (**b**). Figure 12: Flowchart of parameter error calculation within each grid. By bringing in multiple sampling points for parameter identification of the parametric errors \([\Delta\alpha,\Delta\beta,\cdots,\Delta\delta_{6}]\), the need for position error compensation within a single grid can be achieved. Third, we performed the Levenberg-Marquardt method of parameter error identification [37]. For a single grid, a single parameter set can be used to compensate under the premise of satisfying a certain position accuracy. Parameter error calculation is a nonlinear parameter identification process. \[\Delta\mathbf{x}=\mathbf{J}^{-1}\Delta\mathbf{P}, \tag{35}\]  
where \(\Delta\mathbf{x}\) is the parametric error matrix and \(\mathbf{J}\) is the error Jacobian matrix.