## 1 Introduction  
In long-distance navigation tasks, the use of global positioning technology is necessary. Current global positioning technologies mainly include satellite navigation technology and inertial navigation technology. Satellite navigation technology has been extensively implemented into various transportation vehicles, including aircraft, maritime vessels, and automobiles. However, satellite signals are man-made and are susceptible to interference, particularly under extreme conditions, such as in Global Positioning System (GPS)-denied environments. In addition, the positioning error in inertial navigation accumulates over time. Bionic polarized light-based global positioning technology can effectively overcome the two limitations of satellite navigation and inertial navigation technologies. It employs a completely autonomous global positioning approach without error accumulation. The exploration of this technology can be traced back to 1947, when [PERSON] et al. reported that bees could perceive polarized light in the atmosphere [1], prompting scientific inquiry into bionic polarized light navigation. [PERSON] et al. focused on the compound eye structure of insects [2]. Over several decades, researchers have developed a comprehensive understanding of the way various organisms use polarized light from the sky to obtain information on orientation. In 1997, [PERSON] and [PERSON] developed a sky polarized light compass and performed trajectory control experiments using a four-wheeled robot [3,4]. [PERSON] et al. (2005) investigated the polarized light sensitivity model of insect compound eyes [5] and analyzed its angle measurement mechanism model in detail, subsequently developing a heading measurement sensor [6]. [PERSON] and [PERSON] (2012) developed and calibrated a polarized light compass and conducted flight tests on a fixed-wing drone; the results demonstrated that the heading accuracy was comparable to that of a traditional magnetic compass [7]. Further, in 2012, [PERSON] et al. reported that ancient Vikings might have employed calcite to detect polarized light in the sky for navigational purposes [8]. [PERSON] et al. analyzed the errors in the three navigation steps in detail for using polarized light like the Vikings [9, 10, 11]. The simulation results derived from extensive observational data suggested that it was indeed plausible for the Vikings to navigate from Norway in Northern Europe to Greenland using polarized light information from the sky [12]. In summary, previous studies have enabled researchers to derive heading information by detecting polarization data from scattered atmospheric light [13]. However, there are significant distinctions between the long- and short-distance movements along the Earth's surface. For short distances, such as 3 km, the destination can be seen. In contrast, for long distances, like 1000 km, the destination is out of sight from the starting point. When navigating over bodies of water or through the atmosphere, one might experience yaw because of the influence of water currents and airflows. Due to this phenomenon, it is necessary to perform continuous course adjustments according to one's global position. Therefore, it is insufficient to attain orientation only, and advancing global positioning technology is imperative. The technology of polarized light navigation has progressively transitioned in recent years from obtaining the heading to obtaining the global position. [PERSON] and [PERSON] (2015) developed an autonomous real-time global positioning navigation method based on the Earth's sky polarized light and magnetic field [14]. [PERSON] et al. (2015) designed a non-real-time autonomous global positioning method [15], which relies exclusively on the Earth's sky polarized light and requires at least two independent observations of the sky. However, both studies simulated positioning errors for specific times and locations without addressing the variability of errors across different times and geographical locations globally. [PERSON] et al.