## 3 Results
### The Physical Experiments
#### 3.2.1 The Design of the Physical Experiment  
The designed experiment is a comprehensive case combining the lateral case and the pitch angle case, as shown in Figure 9. Take the forward direction of the transmitter as the \(z\)-axis direction and establish a Cartesian coordinate system. The lateral offset of the dual receivers is 0.91 m. \(d_{1}\), the distance in the \(y\)-axis between the PIG transmitter and the tracking receiver 1 is 3 m. The distance \(r\) between the dual receivers is 0.90 m. The pitch angle \(\alpha\) of the dual receivers is 22.885\({}^{\circ}\). The structure of the PIG transmitter is consistent with the simulation model, and it also includes a coil and a magnetic core, which is marked with a red solid line in Figure 9. The frequency of the excitation current is 22 Hz. To accurately measure the position of the transmitter, the transmitter is not placed in the pipeline during the lab experiment. Figure 8: The radiated magnetic field of the transmitting coil in the pitch angle case. \(B_{\rm d1}\) and \(B_{\rm d2}\) are the measured magnetic field on the line ((0, 3, \(-\)4.5), (0, 3, 4.5)) and line ((0, 3.5, \(-\)4.5), (0, 3.5, 4.5)) in meters, respectively. \(CompensationB_{\rm d0}\) is the compensated magnetic field by MFSI method, while \(ActualB_{\rm d0}\) is \(y\)-axis component of the original magnetic field on the line ((0, 3, \(-\)4.5), (0, 3, 4.5)) in meter. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**DOF** & **Pitch Angle** & \(z\) & \(y\) \\ \hline MFSI method & 29.6038\({}^{\circ}\) & \(-\)8.17 (cm) & 291.83 (cm) \\ True value & 30.0000\({}^{\circ}\) & 0.00 (cm) & 300.00 (cm) \\ Error & 0.3962\({}^{\circ}\) & \(-\)8.17 (cm) & \(-\)7.15 (cm) \\ Error (\%) & 1.32 & - & \(-\)2.38 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The tracking error of the algorithm in the pitch angle case. Coils in dual tracking receivers wound around a cylindrical core made of Mn-Zn Ferrrite. The geometric dimensions of the coils and cores are shown in Table 4. Coils are made of copper wire with a diameter of 0.2 mm and a number of turns of 3600. Due to the attenuation effect of the pipeline and the large burial depth of the pipeline, it can be seen from the theoretical calculations and simulation results that the magnetic field intensity at the receiver is much lower than the geomagnetic field. Therefore, the signals received from the tracking receivers need to be preprocessed by a signal processing circuit, which is implemented as the printed circuit board (PCB) in Figure 9. As shown in Figure 10, the signal processing circuit is mainly used for amplification and filtering the measured magnetic field signals and contains five modules. The preamplifier and amplifier module both amplify the weak signal detected by the receivers, and the total amplification of the two modules is 10,000 times. Both amplifiers use high-precision instrumentation amplifiers from Analog Devices. The two bandpass filter modules mainly filter out environmental interference and circuit noise. Their center frequency \(f_{0}\) is 22 Hz, and the bandwidth \(f_{\text{bw}}\) is 3 Hz, so the quality factor is 7.33. The bandstop filter is mainly used to deal with the power frequency electromagnetic interference caused by various instruments in the laboratory environment. To improve the quality factor and reduce the bandwidth, all filters use the state variable filter. In Figure 10, the output signals of the Receiver PCB are screenshots of the oscilloscope during the actual measurement process. In contrast, the effective signal amplitude of the signal received by the tracking receiver is too small to be directly observed without signal processing. Therefore, the received signal shown in the figure is actually the signal received when the transmitter and receiver are only 0.1 m apart, mainly to show the characteristics of the received original signal and not equal to the actual detected signal by the dual receivers in experiments.