## 2 Autonomous Global Positioning Method
### Number of Solutions to Global Positioning Method Equation System  
Before establishing the error model, it is necessary to identify the number of solutions to the autonomous global positioning method because two or more solutions can significantly increase the difficulty of the error model design. First, based on the symmetry of the trigonometric functions, there are four sets of solutions to the system of equations. Therefore, supplementary criteria are required to eliminate redundant or unreasonable solutions. The specific steps for solving the system of equations are as follows:1. Solve the latitude \(\psi\) using Equation (1). Let \(\tan(\psi/2)=t_{\rm M}\), so \(\sin\psi=2t_{\rm M}/(1+t_{\rm M}^{2})\) and \(\cos\psi=(1-t_{\rm M}^{2})/(1+t_{\rm M}^{2})\) according to the universal equations of trigonometric functions, and a quadratic equation in terms of \(t_{\rm M}\) is expressed by \[(\sin\delta+\cos A_{\rm s}\cos h_{\rm s})t_{\rm M}^{2}-2\sin h_{\rm s}t_{\rm M }+\sin\delta-\cos A_{\rm s}\cos h_{\rm s}=0.\] (6)
2. A quadratic equation generally has two solutions, which means that there are also two \(\psi\). 3. Substitute \(\psi\) into Equation (2) to calculate \(\eta\); there is only one cosine function that includes \(\eta\) in Equation (2), so \(\eta\) can be solved directly. However, due to the cosine function's symmetry, each \(\psi\) value corresponds to two \(\eta\) values. To eliminate ambiguity, this study introduces the sine value of the solar azimuth angle as follows: \[\sin A_{\rm s}=-\frac{\sin(\eta+15^{\circ}\cdot(UTC+E_{\omega})-180^{\circ}) \cdot\cos\delta}{\cos h_{\rm s}}.\] (7)  
Then, a longitude \(\eta\) can be solved using Equations (2) and (7); there are two equations and only one unknown. Based on Equations (2) and (7), the sine and cosine values of a solar hour angle, \((\eta+15^{\circ}\cdot(UTC+E_{\omega})-180^{\circ})\), can be directly calculated, which allows for the determination of a unique value of the solar hour angle; subsequently, a unique longitude \(\eta\) can be obtained. In summary, each latitude \(\psi\) corresponds to only one longitude \(\eta\). 4. The analysis of Steps 1 and 2 indicates that the system of equations has two sets of solutions, so the global positioning method will provide two locations. 5. Exclude solutions that have no practical significance based on the latitude and longitude value ranges. The initial range of \(t_{\rm M}\) obtained by Equation (6) is from negative infinity to positive infinity; therefore, the range of \(\psi/2\) is \([-90^{\circ}\), \(90^{\circ}]\); this further indicates that the range of geographic latitude obtained by the global positioning method is \([-180^{\circ}\), \(180^{\circ}]\). However, the actual value range of the Earth's latitude \(\psi\) is \([-90^{\circ}\), \(90^{\circ}]\), so solutions that exceed this range have no practical significance and can be excluded. When this situation occurs, the system of equations has only one set of solutions. In summary, the system of equations typically has two sets of solutions. However, in certain cases, one set of solutions can be excluded due to the actual value range of the Earth's latitude, resulting in a unique solution. Subsequently, this study uses a combination of numerical and graphical methods to analyze the mathematical characteristics of the system of equations for autonomous global positioning. It also defines the conditions under which the two sets of solutions exist, as well as the conditions under which there is only a unique solution. This study uses the time moment of 27 May 2024, \(UTC=21\) h, as a research sample. It has been found from the astronomical calendar that this time moment corresponds to the solar declination of \(\delta=21.4849^{\circ}\) and \(E_{\omega}=0.0456\) h. The contour color maps of the solar azimuth angle and solar altitude angle are shown in Figures 2 and 3, respectively.