## 5 Conclusions  
The traditional AGMs for tracking PIG have the disadvantages of low positioning accuracy and single tracking dimension. To solve the problem, the paper proposed a dual-receiver tracking system and a corresponding magnetic field sign-integration (MFSI) algorithm. The designed dual receivers tracking system is mainly composed of two vertically distributed receiving coils, which can detect the radiated magnetic field signals of the transmitting coils at two vertical distances. Based on the magnetic dipoles method, the paper gives an expression of the radiation magnetic field of the transmitting coil considering the attenuation effect of the pipeline. Based on the characteristics of the radiated magnetic field and the structure of the dual tracking receivers, the 4-D tracking model of the PIG is proposed in three cases, including the normal case, the lateral offset case and the pitch angle case. Considering that the underground pipeline may be one of the above three cases or a combination of the three, this paper proposes the MFSI algorithm that can be used to achieve 4-D tracking of the PIG in any case. The MFSI algorithm in the above three cases is verified by the finite element simulation. The tracking errors of the degrees of freedom (DOF) \(x\), \(y\), and \(z\) have been reduced from 20 mm in the references to within 8.17 mm in location. The tracking errors of the DOF pitch angle have been reduced from 6\({}^{\circ}\) in the reference to 0.4\({}^{\circ}\) in orientation. The paper also designs a set of physical experiments to verify the MFSI algorithm in a case that contains both lateral offset and the pitch angle. The tracking accuracy of the DOF \(x\), \(y\), \(z\), and pitch angle in experiments is also doubled compared to the references. Finally, the characteristics of the MFSI algorithm, such as high accuracy and high robustness, have been discussed and demonstrated.  
Compared with traditional ELF-AGMs, the dual receivers tracking system and the corresponding 4-D tracking algorithm based on the MFSI method have three advantages. First, a single radial coil instead of the orthogonal dual search coils can be used to acquire the same measurement results. The system complexity is reduced while the reliability is enhanced. Secondly, the system improves the accuracy by at least 2 times in terms of positioning and orientation. Thirdly, the system has high robustness and can perform accurate tracking when the receiver is tilted or in the lateral offset case. In summary, the dual receivers tracking system and corresponding MFSI algorithm proposed in this paper can realize the high-precision and four-dimensional tracking and positioning of PIG in the pipeline.  
Conceptualization, Y.L. and S.H.; methodology, Y.L. and W.W.; software, Y.L. and S.W.; validation, Y.L., L.P. and W.W.; formal analysis, Y.L., W.Z. and S.H.; investigation, Y.L.; resources, S.H. and L.P.; data curation, Y.L. and S.W.; writing--original draft preparation, Y.L.; writing--review and editing, Y.L., S.H., L.P., W.W., S.W. and W.Z.; visualization, Y.L. and S.W.; supervision, S.H., W.Z. and L.P.; project administration, S.H.; funding acquisition, S.H. and L.P. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Key R&D Program of China (grant number 2018 YFF01012802), National Natural Science Foundation of China (NSFC) (grant number 52077110 and grant number 52007088).  
Not applicable.  
The authors declare no conflict of interest.