## 2 Autonomous Global Positioning Method
### Global Positioning Method Equation System  
The system of equations of the global positioning method is defined as follows:  
\[\cos A_{\rm s}=\frac{\sin\delta-\sin h_{\rm s}\sin\psi}{\cos h_{\rm s}\cos\psi}, \tag{1}\]  
\[\sin h_{\rm s}=\sin\psi\sin\delta+\cos\psi\cos\delta\cos(\eta+(UTC+E_{\omega}) \times 15^{\circ}-180^{\circ}), \tag{2}\]  
where \(A_{\rm s}\) is the solar azimuth angle; \(h_{\rm s}\) is the solar altitude angle; \(\delta\) is the solar declination; \(\psi\) is the geographic latitude; \(\eta\) is the geographic longitude; \(E_{\omega}\) is the difference between  
Figure 1: The global position calculation flowchart.  
the true and mean solar hours; _UTC_ is the UTC value; and \(\omega\) is the solar hour angle, which is calculated by \(\eta+(UTC+E_{\omega})\times 15^{\circ}-180^{\circ}\).  
The heading angle (\(Y\)), pitch angle (\(P\)), and roll angle (\(R\)) are determined using a true north measurement device. A three-dimensional rotation matrix created by the three angles is defined by Equation (3). This matrix can be used to convert a vector from the body coordinate system to the ENU coordinate system. The body coordinate system \(O_{b}X_{b}Y_{b}Z_{b}\) is defined as follows. The coordinate origin is set at the center of gravity of a vehicle; the \(x\)-axis points to the right side of the vehicle; the \(y\)-axis points to the front of the vehicle; and the \(z\)-axis is perpendicular to the vehicle pointing upwards, forming a right-handed coordinate system. The ENU coordinate system \(O_{n}X_{n}Y_{n}Z_{n}\) is defined as follows. The origin of this coordinate system is set at the observation point, the \(x\)-axis points to true east, the \(y\)-axis points to true north, and the \(z\)-axis is perpendicular to the horizontal plane, pointing upwards, also forming a right-handed coordinate system.  
\[\mathbf{C}_{n}=\begin{bmatrix}\cos Y&\sin Y&0\\ -\sin Y&\cos Y&0\\ 0&0&1\end{bmatrix}\begin{bmatrix}1&0&0\\ 0&\cos P&-\sin P\\ 0&\sin P&\cos P\end{bmatrix}\begin{bmatrix}\cos R&0&\sin R\\ 0&1&0\\ -\sin R&0&\cos R\end{bmatrix}. \tag{3}\]  
A solar vector obtained from the polarization information of polarized sky light is denoted by \(\mathbf{S}_{b}\) and defined in the body coordinate system. The solar vector in the ENU coordinate system is expressed as follows:  
\[\mathbf{S}_{n}=\mathbf{C}_{n}\mathbf{S}_{b}=\begin{pmatrix}S_{x}\\ S_{y}\\ S_{z}\end{pmatrix}=\begin{pmatrix}\cos h_{\rm s}\sin A_{n}\\ \cos h_{\rm s}\cos A_{n}\\ \sin h_{\rm s}\end{pmatrix}. \tag{4}\]  
Using Equation (4) and the solar altitude angle value, a solar altitude angle \(h_{\rm s}\) and an azimuth angle \(A_{n}\) in the ENU coordinate system can be determined. The solar azimuth angle is defined in the plane formed by \(X_{n}\) and \(Y_{n}\), with the geographic true north direction (\(Y_{n}\)) as a reference direction and clockwise as positive; therefore, it holds that  
\[A_{\rm s}=90^{\circ}-A_{n}. \tag{5}\]  
The autonomous global positioning method determines the global position of the carrier by solving the system of equations formed by Equations (1) and (2). This study aims to develop a global error model that can provide the global distribution of deviations in the geographic longitude and latitude, which are caused by the measurement angle errors of the two independent variables at different times. In addition, this study aims to translate the angle errors of the geographic longitude and latitude (expressed in degrees) into the corresponding distance errors of the Earth's surface (expressed in kilometers).