# Multi-Lidar System Localization and Mapping with Online Calibration
## 2 Front-End Laser Odometry and Online Calibration
### Realization of Multi-Lidar Online Calibration
#### 2.2.2 Convergence Assessment of External Parameters  
The LM algorithm used in this paper transforms the nonlinear problem into a local linear problem by computing the Jacobian matrix J of the error function. Consequently, the linear problem of each row can be regarded as a hyperplane of the solution space. The resultant solution x is a point in this space, fulfilling the requirement of minimal Euclidean distance to every hyperplane. The degeneracy factor is established by investigating the geometric distribution of the plane, incorporating additional hyperplane constraints and perturbations, as shown in Figure 4.  
In Figure 4, the black line represents the plane constraint, the orange line denotes the additional hyperplane constraint, the blue point \(x_{0}\) represents the optimization problem's solution, \(c\) is the normal vector of the additional hyperplane constraint with a modulus length of 1, \(\delta d\) denotes the offset distance of the hyperplane after adding the disturbance, and \(\delta x_{c}\) is the offset distance of the constraint, i.e., the offset distance of \(x_{0}\) after adding the disturbance. The derivation of the degradation factor can be obtained. Since the constraint intersects with \(x_{0}\), the latter is not altered by the additional constraint, resulting in the following equation:  
\[c^{T}(x-x_{0})=0,\parallel c\parallel=1 \tag{13}\]  
If the constraint is moved along its normal direction \(c\) by a certain distance, then \(x_{0}\) will produce a displacement of \(\delta x_{c}\) in the same direction. For a given disturbance, let \(\delta x_{c}\) represent the maximum displacement:  
\[\delta x_{c}^{*}=\underset{c}{\max}\delta x_{c} \tag{14}\]  
The definition of the degradation factor is then expressed as follows:  
\[D=\frac{\delta d}{\delta x_{c}{}^{*}} \tag{15}\]  
Namely, the degradation factor is defined as the ratio of the hyperplane's movement distance to the solution's maximum displacement. Evidently, a smaller value implies greater system stability. Simultaneously considering (14) and (15), we obtain the following:  
\[ \begin{bmatrix}A\\ c^{T}\end{bmatrix} x= \begin{bmatrix}b\\ c^{T}x_{0}\end{bmatrix}  \tag{16}\]  
After introducing perturbations, we have the following:  
\[ \begin{bmatrix}A\\ c^{T}\end{bmatrix} x= \begin{bmatrix}b\\ c^{T}x_{0}+\delta d\end{bmatrix}  \tag{17}\]  
Substituting the left pseudo-inverse \( \begin{bmatrix}A\\ c^{T}\end{bmatrix} _{left}^{-1}=\left( \begin{bmatrix}A^{T}&c\end{bmatrix} \begin{bmatrix}A\\ c^{T}\end{bmatrix} \right)^{-1} \begin{bmatrix}A^{T}&c\end{bmatrix} \) into (16), we obtain the following:  
\[\delta x=\left(A^{T}A+cc^{T}\right)^{-1}c\delta d \tag{18}\]  
The displacement \(\delta x_{c}\) in the \(c\) direction can be calculated as the dot product of \(c\) and \(\delta x\):  
\[\delta x_{c}=c^{T}\delta x=c^{T}\Big{(}A^{T}A+cc^{T}\Big{)}^{-1}c\delta d \tag{19}\]  
Figure 4: Degradation factor definition model. This illustrates that the degradation factor only relates to the original constraint direction \(A\), independent of the initial constraint position b; parallel original constraint directions induce degradation. Converting \(c\) and \(c^{T}\) according to the pseudo-inverse \(c_{left}^{-1}=\left(c^{T}c\right)^{-1}c^{T}=c^{T}\), \(c_{right}^{-1}=c\left(c^{T}c\right)^{-1}=c\) yields the following:  
\[\delta x_{c}=\left(c^{T}A^{T}Ac+1\right)^{-1}\delta d \tag{20}\]  
For maximizing, (21) must be satisfied:  
\[ \begin{array}{l}c^{*}=\underset{c}{\mathrm{argmin}_{c}}^{T}A^{T}Ac\\ \parallel c\parallel=1.\end{array}  \tag{21}\]  
Since the modulus of \(c\) is 1, (21) can also be expressed as follows:  
\[c^{*}=\underset{c}{\mathrm{argmin}_{c}}\frac{c^{T}A^{T}Ac}{c^{T}c} \tag{22}\]  
which takes the form of a Rayleigh quotient, with the minimum value occurring when \(c\) is the minimum eigenvalue of \(A^{T}A\).