## 1 Introduction  
The exploration of the extremely distant, extremely dark, and extremely cold regions at the solar system boundary is a hot point in deep space exploration. After the human exploration of the Moon and Mars, which has progressively expanded to the solar system boundary, further space exploration could continuously open new windows for humans' understanding of the universe [1]. There are only a limited number of deep space missions that continue to explore the solar system boundary after completing the scheduled exploration mission, such as Pioneer 10-11 [2; 3; 4], Voyager 1-2 [5; 6], and New Horizons [7; 8]. With the continuous extension of the field of deep space exploration, China has put forward the SSBE program, which could build the capability of the probe to reach the whole area of the solar system and realize super long-distance detection from internal celestial bodies to interstellar space, providing us with the capability to independently explore interplanetary far-reaching space [1; 9]. Due to the characteristics of unknown and varied exploration environments, super long space exploratory distances, super long orbital flight times, and significant communication data delays between the SSBE and the ground, the implementation of the exploration mission is very difficult [10]. Because the probe is so far from the ground, it cannot rely on traditional navigation methods, such as the radio station aeronautical ground, very long baseline interferometry (VLBI), and tracking telemetry and command (TT&C) communication networks to provide real-time and high-accuracy navigation information, which presents new challenges to ultra-long-distance autonomous high-accuracy navigation [11; 12]. Regarding the solar system boundary probe, reliance on local ground-based tracking navigation would no longer be applicable and could not be used for autonomous real-time and high-accuracy navigation of the probe at great distances from the ground [12]. In order to improve the autonomous operation capability of the probe in orbit, autonomous navigation technology that is independent of ground navigation measurements has been actively studied in depth. X-ray pulsar navigation (XNAV) is a new kind of autonomous celestial navigation technology that can be applied to both near-Earth and deep space [13; 14; 15]. Based on the three X-ray pulsars, the autonomous position determination of the deep space probe could be realized. By collecting the X-ray radiation signals from the X-ray pulsar with the X-ray detector, the pulse time-of-arrival (TOA) can be obtained. The arrival time of the pulse reaching the detector and the time to reach the solar system barycenter (SSB) are predicted, which can determine the position of the probe by characterizing its position direction along the X-ray pulsar [16; 17]. [PERSON] et al. [18] and [PERSON] [19] stated that the position of the spacecraft is determined through the measurement of the phase of a periodic signal, and they speculated for the first time that the X-ray pulsar could be used to determine the time and position of the spacecraft in orbit [15]. [PERSON] et al. [20] proposed a method to estimate the comprehensive six DOF position navigation and time solutions for the deep space exploration spacecraft, which relied on the measurement of the arrival time and angles of the X-ray photons from the X-ray pulsars and other bright stars. [PERSON] et al. [21] proposed differential XNAV based on the time difference of the pulse arrival, which analyzed the impact of the ephemeris errors on correcting the photon TOA and observation equations, and three Mars orbits with different altitudes were simulated to validate the efficacy of the navigation scheme and its suitability for the different heights from the Martian surface. When the detector moves relative to stellar objects in far-reaching space exploration, the measured stellar objects spectrum also undergoes Doppler frequency offset, which reflects the velocity of the probe relative to the stellar objects [22].