## 2 Bearing Load Testing Method Based on Redundant Strain and Jacking Data Fusion
### Data Fusion Processing
#### 2.3.2 Jacking Data Fusion Based on Bayesian Estimation  
During the measurement phase after the bearing is disengaged, the support force measured by the RSGM is actually the force at the jacking point. A Bayesian estimation fusion method is used to fuse the jacking point force obtained by the RSGM and the jacking force obtained by the jacking force sensor, thereby improving the measurement accuracy of the JM. Overall Least Squares Fitting  
The measurement data of the jacking displacement \(S_{j}\) and the jacking force \(P_{j}\), as well as the support force \(F_{s,j}\) obtained from the RSGM, are plotted to create a jacking curve. Linear fitting is performed based on the total least squares method to determine the jack load \(P_{0}^{\prime}\) and \(P_{0}^{\prime\prime}\) at the point where the jacking displacement is zero. The jacking curve is shown in Figure 6.  
Incorporating the jacking curve, the calculation formula for the JM is as follows:  
\[F=K*P_{0} \tag{19}\]  
Figure 6: Jacking curve diagram. where \(P_{0}\) is the jack load at the point where the jacking displacement is zero; \(K\) is the correction factor. Taking into account the presence of measurement errors, the measurement equations for the jacking displacement \(S_{j}\) and the jacking force \(P_{j}\) can be expressed as:  
\[\left\{ \begin{array}{l}P_{j}=\overline{P}_{j}+k_{p}\overline{P}_{j}+\omega_{ pj}\approx\overline{P}_{j}+k_{p}P_{j}+\omega_{pj}\\ S_{j}=\overline{S}_{j}+k_{s}\overline{S}_{j}+\omega_{sj}\approx\overline{S}_{j} +k_{s}S_{j}+\omega_{sj}\end{array} \right. \tag{20}\]  
Performing a total least squares linear fitting on the data after the bearing is disengaged results in the following linear regression equation:  
\[y_{p}=a+bx_{s} \tag{21}\]  
According to reference [27], when \(x_{s}=0\), \(y_{p_{0}}{\sim}N(a+bx,\sigma_{r}^{2})\). Here, the least squares estimates of \(a\) and \(b\) and the unbiased estimate of \(\sigma_{r}\) are:  
\[\left\{ \begin{array}{l}b=\frac{s_{xy}}{s_{xx}}\\ a=\overline{y}-b\overline{x}\\ \sigma_{r}^{2}=\frac{1}{n-2}(s_{yy}-bs_{xy})\end{array} \right. \tag{22}\]  
These parameters can be calculated by:  
\[\left\{ \begin{array}{l}s_{xx}=\sum\limits_{i=1}^{n}x_{si}^{2}-\frac{1}{n}( \sum\limits_{i=1}^{n}{x_{si}})^{2}\\ s_{yy}=\sum\limits_{i=1}^{n}y_{pi}^{2}-\frac{1}{n}(\sum\limits_{i=1}^{n}{y_{pi }})^{2}\\ s_{xy}=\sum\limits_{i=1}^{n}x_{si}y_{pi}-\frac{1}{n}(\sum\limits_{i=1}^{n}{y_{pi }})(\sum\limits_{i=1}^{n}{x_{si}})\end{array} \right. \tag{23}\]  
Assuming that the systematic error coefficient \(k_{P}\) for the jacking force follows a normal distribution, \(k_{P}{\sim}N(0,\sigma_{k_{p}}^{2})\), according to the principle of maximum likelihood, by combining Equations (18), (20), and (22), the PDF of the \(P_{0}^{\prime}\) and \(P_{0}^{\prime\prime}\) can be derived as:  
\[\left\{ \begin{array}{l}f_{P_{0}^{\prime}}(x)=(2\pi\sigma_{P_{0}^{\prime}}^{ 2})^{-1/2}\exp(-\frac{(x-P_{0}^{\prime\prime})2}{2\sigma_{P_{0}^{\prime\prime} }^{2}})\\ f_{P_{0}^{\prime\prime}}(x)=(2\pi\sigma_{P_{0}^{\prime\prime}}^{2})^{-1/2}\exp (-\frac{(x-P_{0}^{\prime\prime})^{2}}{2\sigma_{P_{0}^{\prime\prime}}^{2}}) \end{array} \right. \tag{24}\]  
These parameters can be calculated by:  
\[\left\{ \begin{array}{l}\sigma_{P_{0}^{\prime}}^{2}=P_{0}^{\prime}\sigma_{k_{ p}}^{2}+\sigma_{r}^{\prime 2}\\ \sigma_{P_{0}^{\prime\prime}}^{2}=P_{0}^{\prime\prime}\sigma_{k_{s}}^{2}+\sigma_ {r}^{\prime 2}\end{array} \right. \tag{25}\]  
The Bayesian estimation fusion method is used to fuse the \(P_{0}^{\prime}\) and \(P_{0}^{\prime\prime}\). Bayesian estimation provides a probabilistic statistical approach to multi-sensor data fusion. It allows sensor information to be combined based on probability principles, with measurement uncertainties expressed as conditional probabilities. When sensor groups have consistent observation coordinates, their data can be directly fused; otherwise, Bayesian estimation is typically used for an indirect fusion of sensor measurement data [28].