# Inertial-Navigation-Aided Single-Satellite Highly Dynamic Positioning Algorithm
## 1 Introduction  
Today's global navigation satellite systems (GNSS) have excellent performance and high positioning accuracy in most scenarios. With the support of GNSS constellation, they can satisfy the daily positioning requirements of military and civilian navigation. However, the inherent vulnerability of constellation GNSS significantly degrade their positioning ability or even make them completely unable to locate under extreme conditions, such as during wartime. To obtain locations precisely in emergency environments, a rapidly deployed navigation and positioning system is imperative. Single-satellite systems show apparent advantages with fast deployment and low cost compared to GNSSs. Thus, it is an essential supplement to GNSSs. Existing single-satellite positioning techniques adopt observation values at different times from the only satellite to locate. The literature [1] proposed a frequency-measurement-based single-satellite positioning method that utilized the frequency difference of arrival during satellite movement. However, this method has low positioning accuracy and is limited to stationary or small dynamic targets. The literature [2] found the geometric relationship between location and rate of phase difference, Doppler frequency and satellite ephemeris position to obtain positioning results but it is quite sensitive to the parameters.  
The literature [3] also proposed an angular-measurement-based single-satellite positioning method that utilized the angular-velocity relationship between satellite and target to achieve positioning. However, it requires high accuracy of satellite ephemeris. The literature [4; 5] proposed a Doppler-based single-satellite positioning method that utilized frequency difference to construct the positioning hyperboloid and integrated it with the inertial navigation system (INS) to achieve positioning. However, these methods are more suitable for static positioning, with low positioning accuracy and fast divergence for dynamic targets. The literature [6] proposed radial acceleration measurement based on a single-satellite positioning method. This method is simple since it only requires radial acceleration but, as the data source is limited, its reliability is poor. The literature [7] proposed a beam-scanning single-satellite positioning method with high positioning accuracy for static targets. However, it works in active location mode, which needs interaction with the data center. The literature [8] also proposed a BP neural-network-based single-satellite positioning method. The positioning results were stable but it takes a lot of time to perform the learning and updating process, so practical applications are limited. Researcher [9] also proposed an INS-based Doppler integration single-satellite positioning method that has a good positioning effect for both dynamic and static targets. However, since the final filtering result depends on the least-squares solution result, the stability of this method is limited. In the above algorithms, the main method is to transplant the GPS positioning method directly to the single-satellite positioning system. The problem of keeping the target immovable in single-satellite positioning systems is neglected. To answer this question, an inertial-navigation-aided single-satellite highly dynamic positioning algorithm is proposed. First, we computed and compensated the displacement of the moving target by INS [10]. Then, we utilized the integral Doppler pseudorange to construct an equal-frequency cone surface and use the height information of the moving target to establish the height surface. Next, we obtained the coarse positioning solution with these surfaces. Finally, the displacement value provided by INS was calculated and corrected in the UKF filter to improve the single-satellite positioning result.