# Research on 2D Laser Automatic Navigation Control for Standardized Orchard
## 2 Materials and Methods
### Navigation Controller Design
#### Adaptive Pure Tracking Model Controller Design  
After research, we found that the magnitude of the forward-looking distance was affected by both the lateral deviation \(d\) and the heading deviation \(\theta\). Therefore, \(d\) and \(\theta\) were used as the input of the fuzzy controller and \(L_{d}\) was used as the output. The \(L_{d}\) was adjusted in real time by the fuzzy controller. When \(d\) and \(\theta\) were large, the smaller \(L_{d}\) was used to make the vehicle approach the tracking path quickly, reduce the system adjustment time and improve the system response speed. When \(d\) and \(\theta\) were small, a larger \(L_{d}\) was used to prevent system overshoot and improve system stability.  
We set the basic domain of the lateral deviation \(d\) to [\(-0.5\) m, \(0.5\) m] and divide it into 13 levels with quantization levels: {\(-6\), \(-5\), \(-4\), \(-3\), \(-2\), \(-1\), \(0\), \(1\), \(2\), \(3\), \(4\), \(5\), \(6\)}, the corresponding quantization factor \(K_{d}\) = 0.12. The basic domain of the heading deviation \(\theta\) was [\(-\)\(\pi\)/6, \(\pi\)/6], which is divided into 13 levels with quantization levels: {\(-6\), \(-5\), \(-4\), \(-3\), \(-2\), \(-1\), \(0\), \(1\), \(2\), \(3\), \(4\), \(5\), \(6\)}, quantization factor \(K_{\theta}\) = 0.087. The basic domain of the forward-looking distance \(L_{d}\) is [\(1\) m, \(6\) m], which is divided into 6 levels, the quantization levels: {1, 2, 3, 4, 5, 6}, and the quantization factor \(K_{Ld}\) = 1.  
According to the positional relationship between the orchard intelligent vehicle and the orchard navigation center line, the lateral deviation \(d\) and the heading deviation \(\theta\) are divided into seven levels: left big (LB), left medium (LM), left small (LS), zero (ZO), right small (RS), right medium (RM), right big (RB). The forward-looking distance \(L_{d}\) is divided into seven levels: very small (VS), small (S), little small (LS), medium (M), little big (LB), big (B), very big (VB). Then, the distribution of this fuzzy variable membership function can be seen in the following Figure 12.  
Figure 12: The distribution of the membership function. (**a**) lateral deviation membership function expression. (**b**) heading deviation membership function expression. (**c**) forward-viewing distance membership function expression.  
After several anti-simulation experiments, the optimal 49 fuzzy control rules were determined as shown in Table 2; the fuzzy control surface was plotted as shown in Figure 13.  
In this study, [PERSON]'s fuzzy reasoning method and extremely small fuzzy logic inference synthesis rules were used; the defuzzification work is completed by the center of gravity method [42]. The fuzzy control query table is shown in Table 3.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(\mathbf{\theta}\)} & \multicolumn{10}{c}{_d_} \\ \cline{2-13}  & **LB** & **LM** & **LS** & **Z** & **RS** & **RM** & **RB** \\ \hline LB & VS & S & S & VS & S & S & VS \\ LM & VS & LS & M & LS & M & M & VS \\ LS & VS & M & LB & LB & LB & M & VS \\ Z & VS & LB & VB & VB & VB & LB & VS \\ RS & VS & M & LB & LB & LB & M & VS \\ RM & VS & LS & M & LS & M & LS & VS \\ RB & VS & S & S & VS & LS & S & VS \\ \hline \hline \end{tabular}
\end{table}
Table 2: Fuzzy Control Rule Table.  
Figure 13: Fuzzy control surface.