## 4 Parameter Optimization
### Optimization Result  
#### 4.3.1 Coupling of Placement Parameters  
Fifteen optimizations were performed, and the nine checking parameters in each optimization are shown in Figure 9. The maximum angle error is approximately \(5.5^{\circ}\), which is impossible. These large angular errors imply coupling between the nine param eters. Coupling but not simple merging is the biggest difficulty in checking the error of airborne STILs.  
There are three directions in the rotation operation, corresponding to the three sets of coupling relationships of the checking parameters, shown as  
\[\left\{\begin{aligned} w&=0.9979r-0.0355\\ \Phi&=0.5632(s+v)+45.228\\ \Psi+\Omega&=-0.4965(t-u)+0.5391\\ \end{aligned}\right.. \tag{20}\]  
Their linear fitting diagram is shown as Figure 10.  
The parameter coupling phenomenon indicates a correlation before different placement angles and verifies that the particle swarm algorithm can solve the multi-parameter optimization problem.  
Based on the definitions in Table 1, the coupling parameters can be found to be related in orientation. The nine parameters come from three devices, the imaging system, the  
Figure 10: The coupling diagram of placement parameters.  
Figure 9: The distribution of each parameter after multiple PSO optimizations. In this boxplot, the boxes represent the range of data from the first to the third quartile.  
scanning system and the POS. Although the orientation of each parameter was not specified during modelling, based on the optimization results, it can be seen that \(\phi,s,\) and \(v\) represent the same orientation in all three coordinate systems. The same is true for other parameters. In addition, the scanning system was modelled with redundancy, resulting in \(\Psi\) and \(\Omega\) playing the same role.  
The coefficients between the parameters \(w\) and \(r\) in these three sets of couplings converge to 1, suggesting that the two may be condensed. The linear coupling coefficients for the other two sets of relationships are not 1, indicating that these parameters cannot be optimized by fitting the relationships. However, according to the model's definition, the parameters \(\Psi\) and \(\Omega\) are both used to describe the error of the motor rotation angle. The grating ruler has been installed co-axially to the rotating shaft so that its zero position is virtually the same physical quantity as the motor placement error in the tangential direction of the rotating shift.  
When one or more of these parameters are fixed, the parameters associated with them will also converge to fixed values during the optimization process. In traditional calibration methods, it is necessary to obtain the exact placement error. The phenomenon of parameter coupling, on the other hand, makes the number of parameters that have to be obtained by measurements much lower, which is very beneficial for the traditional checking methods.  
Considering that the purpose of checking is to improve the accuracy of the point cloud, the non-uniqueness of the optimum caused by parameter coupling is not a major issue. Even the exact physical meaning of the nine parameters seems less important, as long as they are substituted into the conformal equations to be able to invert a highly accurate point cloud.