# GNSS and LiDAR Integrated Navigation Method in Orchards with Intermitt GNSS Dropout
## 2 Materials and Methods
### Path-Tracking Control System  
#### 2.3.1 Kinematic Model of Robotic Vehicle  
To adapt to the complex, uneven, and soft soil ground in the orchard, the robotic vehicle adopts a track structure for its chassis. This structure has a large contact area with the ground, providing greater friction and stronger off-road performance compared to wheel drive, making it more suitable for the orchard environment. The chassis used in this study is based on the principle of differential motion, whereby steering is achieved by controlling the speed difference between the left and right tracks. We assume that the  
Figure 7: The laser odometer usage effect diagram. (**a**) The original laser odometer usage effect; (**b**) the laser odometer usage effect after GNSS revision.  
surface of the orchard roads is approximately a two-dimensional plane, and the mass of the mobile platform is uniformly distributed. Its center of mass is located on the geometric longitudinal symmetry line, and it will not deflect or slip during driving. Based on this, a differential drive kinematic model is established.  
The robot kinematics model is shown in Figure 9. The point \(O_{L}(x_{0},y_{0})\) is the center of mass of the moving platform, XOY is the world coordinate system, and \(X_{L}O_{L}Y_{L}\) is the robot coordinate system. The forward motion direction is the positive direction of the \(y\)-axis, the right motion direction is the positive direction of the \(x\)-axis, and the \(z\)-axis is perpendicular to the paper surface outward. At this time, the position of the robot is represented by the vector \(\left[x\quad y\quad\theta\right]^{T}\), which serves as the reference coordinate point of the moving platform in the orchard. \(O_{c}\) is the center of rotation center of mass; \(v\) is the linear velocity of the center of mass; \(r_{c}\) is the turning radius of the center of mass; \(v_{l}\), \(v_{r}\) are the linear velocities of the left and right driving wheels; \(\omega\) is the angular velocity of the center of mass with respect to the rotational center \(O_{c}\); and \(L_{r}\) is the distance between the centers of the two tracks.  
Figure 8: Schematic diagram of orchard robotic vehicle operation.  
Figure 9: Kinematic modeling of tracked differential movement.  
The linear velocity of the orchard moving platform can be expressed as  
\[V=\frac{v_{l}+v_{r}}{2} \tag{20}\]  
The angular velocity of the orchard moving platform can be expressed as  
\[\omega=\frac{V}{L_{c}}=\frac{v_{r}}{L_{c}+L_{r}/2}=\frac{v_{l}}{L_{c}-L_{r}/2}= \frac{v_{r}-v_{l}}{L_{r}} \tag{21}\]  
The derivative of the positional state of the orchard moving platform is  
\[\begin{bmatrix}\dot{x}\\ \dot{y}\\ \dot{\theta}\end{bmatrix}=\begin{bmatrix}\cos\theta&0\\ \sin\theta&0\\ 0&1\end{bmatrix}\begin{bmatrix}v\\ \omega\end{bmatrix} \tag{22}\]  
According to Equations (20) and (21), \(\dot{x},\ \dot{y},\ \dot{\theta}\) are, respectively,  
\[\left\{\begin{array}{l}\dot{x}=\frac{v_{l}+v_{r}}{2}\cos\theta\\ \dot{y}=\frac{v_{l}+v_{r}}{2}\sin\theta\\ \dot{\theta}=\frac{v_{r}-v_{l}}{L_{r}}\end{array}\right. \tag{23}\]  
The above equation shows that controlling the rotational speed of the tracks on both sides of the robotic vehicle can realize the control of its steering and driving speed, which can be obtained by discretization:  
\[\left\{\begin{array}{l}x(t+1)=x(t)+V\cos\theta\times T\\ y(t+1)=y(t)+V\sin\theta\times T\\ \theta(t+1)=\theta(t)+\omega\times T\end{array}\right. \tag{24}\]  
In summary, it can be seen that by controlling the rotational speed of the tracks on both sides of the robot, it is possible to control the robot's steering and straight-line travel.