## 4 Simulation and Application
### Application in Pushing Distance Estimation  
In addition to the simulation, the proposed method is also applied to the application of pushing distance estimation, based on the real parameters of the hydraulic pushing cylinder, theoretically, the velocity and the pushing distance of the pushing link. In this application, we use a high-precision displacement sensor to measure the pushing distance. The built-in electro-hydraulic control system of a hydraulic powered roof support is used to obtain the observable pushing distance. An inertial sensor is used to measure the inertial information of the pushing link. The detail of the equipment configuration is shown in Table 4.  
The inertial sensor, as shown in Figure 9A, is mounted on the pushing link, and the displacement sensor is installed along the direction of pushing operation. The theoretical pushing distance, velocity, and acceleration, as shown in Figure 9B, reveals the basic movement features of the pushing distance. The measured pushing distance from the high-precision displacement sensor, computed velocity via differential equation, and the measured acceleration by the inertial sensors are also presented.  
\begin{table}
\begin{tabular}{c c c} \hline
**Index** & **Equipment** & **Information** \\ \hline
1 & Hydraulic powered roof Support & ZY9000/15/28D \\
2 & Displacement sensor & PR-500 mm \\
3 & Data acquisition card & Smacq USB-4432 \\
4 & Inertial Sensor & ADIS 16448 \\ \hline \end{tabular}
\end{table}
Table 4: Equipment configuration.  
Figure 8: The estimation error of different optimization algorithms.  
Through the proposed IFOA-KF, the measured acceleration can be used to estimate pushing distance, as shown in Figure 10. The IFOA optimized the processing and observation covariance, and the final optimized parameters are listed in Table 5.  
For a pushing operation of the hydraulic powered roof support, the maximum pushing distance, theoretically, should be 500 mm, as shown in Figure 10A. If the proposed IFOA-KF is not utilized, the directly computational result (the blue curve) will be terribly affected by cumulative errors, and the estimated pushing distance increases rapidly to an unreasonable magnitude. By using the proposed  
Figure 9: Experiment configuration and data collection. (**A**) The sensor installation on the hydraulic powered roof support; and (**B**) the theoretical data and measured data.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Covariances** & **Values** \\ \hline \(Q\) & 0.1758 \\ \(R\) & 0.4935 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Optimized parameters.  
IFOA-KF, the estimated pushing distance, the green curve, is rather close to the real measured pushing distance. Figure 10B also presents the details of the estimation performance at the final stage of the pushing operation. It can be seen that the cumulative error is very limited.  
With careful design of the application conditions, the final estimation error can be perfectly reduced; as shown in Figure 11, the error of measurement is 14.7467, while the estimation error of the proposed IFOA-KF is only 1.0425. Thus, the application proves the special performance of the proposed method.