# GNSS and LiDAR Integrated Navigation Method in Orchards with Intermitt GNSS Dropout
## 2 Materials and Methods
### GNSS-LiDAR Fusion Positioning Principle  
#### 2.2.1 Coordinate System Integration of GNSS and LiDAR  
When the orchard robot operates on open and unobstructed roadways, it primarily relies on the GNSS system for navigation. The positioning information output by this system is based on the WGS84 geodetic coordinate system, presented in the form of LLA (latitude, longitude, altitude). This type of LLA position information cannot be directly used for robot vehicle localization and navigation. It is necessary to convert the latitude and longitude information obtained by the robot into coordinate values in a planar rectangular coordinate system. The Gauss-Kruger projection is essentially an equirectangular projection, which can convert the latitude and longitude coordinates of the Earth into two-dimensional right-angle coordinates of the plane [27]. Its geometric schematic is shown in Figure 4.  
Figure 2: GNSS-LiDAR fusion navigation framework. This paper adopts the \(6^{\circ}\) projection, whereby every \(6^{\circ}\) is divided into a projection belt from the meridian of \(1.5^{\circ}\)E, the world is divided into a total of 60 projection belts from west to east, and the belt number is sequentially coded as the 1 st, 2 nd, \(\ldots\), 60 th belt. For the longitude of the central meridian of China's \(6^{\circ}\) band, totaling 11 bands from \(73^{\circ}\) every \(6^{\circ}\) to \(135^{\circ}\). The band number is denoted by \(n\) and the central meridian of longitude by \(L\). If the geodetic coordinate of a certain point \((E,N)\) is known, the \(6^{\circ}\) belt projection of the point can be obtained. The formula for belt number is  
\[n=\frac{L-1.5}{6} \tag{1}\]  
Assuming that the earth is a rotating ellipsoid, since the format of latitude and longitude information output by the GNSS system used for navigation and positioning is WGS-84, the WGS-84 ellipsoid model is used in this paper, and its long half-axis \(a=637\),8137 m, short half-axis \(b=6\),356,752.3142 m, and inverse of the oblateness is 298.257233563 m. If  
Figure 4: Geometric representation of the Gauss–Kröiger projection. Figure 3: Orchard robot vehicle software system. \(e_{1}\) is the first eccentricity of the earth's ellipsoid and \(e_{2}\) is the second eccentricity of the earth, then  
\[e_{1}=\sqrt{\frac{a^{2}-b^{2}}{a^{2}}} \tag{2}\]  
\[e_{2}=\sqrt{\frac{a^{2}-b^{2}}{b^{2}}} \tag{3}\]  
Suppose that the radius of curvature of the meridian circle of the earth is \(M\), the radius of curvature of the dolomite circle is \(S\), and that the length of the meridian from the equator to the latitude \(W\) of the point where it is located is \(L_{s}\). Then,  
\[M=\frac{a\big{(}1-e_{1}^{2}\big{)}}{\sqrt{\big{(}1-e_{1}^{2}sin^{2}N\big{)}^{3}}} \tag{4}\]  
\[S=\frac{a}{\sqrt{\big{(}1-e_{1}^{2}sin^{2}N\big{)}}} \tag{5}\]  
\[L_{s}=a(1-e_{1}^{2})\bigg{[}C_{1}N-\frac{C_{2}}{2}\text{sin}\,2N+\frac{C_{3}}{ 4}\text{sin}\,4N-\frac{C_{4}}{6}\text{sin}\,6N+\frac{C_{5}}{8}\text{sin}\,8N \bigg{]} \tag{6}\]  
\(C_{1}\), \(C_{2}\), \(C_{3}\), \(C_{4}\), and \(C_{5}\) in Equation (10) are the basic constants, which are calculated as shown in Equation (11):  
\[\left\{ \begin{array}{l}C_{1}=h_{0}+\frac{h_{2}}{2}+\frac{3}{3}h_{4}+\frac{5} {16}h_{6}+\frac{35}{128}h_{8}\\ C_{2}=\frac{h_{2}}{2}+\frac{h_{4}}{2}+\frac{15}{32}h_{6}+\frac{7}{16}h_{8}\\ C_{3}=\frac{h_{4}}{8}+\frac{3}{3}h_{6}+\frac{7}{32}h_{8}\\ C_{4}=\frac{h_{6}}{32}+\frac{h_{8}}{16}\\ C_{5}=\frac{1}{128}\end{array} \right. \tag{7}\]  
where \(h_{0}=a\big{(}1-e_{1}^{2}\big{)}\), \(h_{2}=\frac{3}{2}e_{1}^{2}h_{0}\), \(h_{4}=5e_{1}^{2}h_{2}\), \(h_{6}=\frac{7}{6}e_{1}^{2}h_{4}\), \(h_{8}=\frac{9}{8}e_{1}^{2}h_{6}\). Then, the algorithm formula to convert geodesic coordinates \((E,N)\) to planar coordinates \((x,y)\) by Gaussian projection orthographic formula is as follows:  
\[\left\{