## 2 Materials and Methods
### Design of Adaptive Preview Control Algorithm
#### 2.4.2 Design of the Sub-Regulator  
1. Definition of sub-regulator  
The sub-regulator, as a supplement to the main regulator, and when the system oscillates or sideslips under perturbation conditions, the preview distance is adjusted adaptively according to the results of the deviation pattern identification, and the deviation is suppressed quickly.  
The output of the sub-regulator is set to be the preview distance adjustment value _f\({}_{sub}\)_, whose expression equation is shown in Equation (22).  
\[f_{sub}=f_{sub\_osc}+f_{sub\_slip}, \tag{22}\]  
where _f\({}_{sub\_osc}\)_, _f\({}_{sub\_slip}\)_ are the preview distance adjustment values for oscillation and sideslip (as a function of _deg\({}_{osc}\)_ and _deg\({}_{slip}\)_).  
Define the oscillation adjustment coefficient \(A\) and the sideslip adjustment coefficient \(B\), and rewrite Equation (23) as  
\[f_{sub}=A\cdot deg_{osc}+B\cdot deg_{slip}, \tag{23}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Index** & \(v\) **(m/s)** & _P\({}_{e}\)_ **(m)** & _\(\tau\)_ **(s)** & \(d\) **(m)** \\ \hline
1 & 0.90 & 0.04 & 0.1  
According to the rules and labels in Tables 4 and 5, \(deg_{osc}\in(1,2,3)\) and \(deg_{slip}\in(1,2,3)\) in Equation (23).
2. Setting constraints The length of the preview distance is related to the speed. The traveling speed \(v\) of a rice transplanter is generally in the range of 0.75\(\sim\)1.5 m/s. In order to ensure the system has a good control response, the preview distance should not be too large or too small. Therefore, on the basis of the main regulator, limit \(f_{sub}\), constituting the constraints as shown in Equation (24). From the above analysis, \(A\) and \(B\) should be taken in the range of 0 to 1.
3. Optimal adjustment factor test Further, in order to select the optimal adjustment coefficients A and B, a control test was carried out on the rice transplanter navigation control simulation platform: (1) At a \(v=0.8\) m/s, the control curve is set up as shown in Figure 10. Specifically, on the basis of the main regulator (the sub-regulator does not intervene), three degrees of oscillations are induced by narrowing the preview distance, and three degrees of sideslip are induced by applying different degrees of lateral velocity. The overall variance of the control curves \(VAR_{refer}=0.0022\) m. (2) Set \(A\) and \(B\) as multiple groups of level factors (value range 0-1), intervene the sub-regulator on the basis of the main regulator, i.e., add the preview distance adjustment value as shown in Equation (23), carry out the simulation test with different adjustment coefficients, and obtain the overall variance \(VAR_{test}\) of each tracking curve. (3) The declining variance \(Devr_{test}\) of each tracking curve is used as an evaluation index, as shown in Equation (25). \[Decr_{test}=VAR_{refer}-VAR_{test},\] (25) The test results are shown in Table 9.  
With \(A\) and \(B\) as the independent variables and \(Decr_{test}\) as the dependent variable, the expression in Equation (26) was fitted using the least squares method.  
\[Decr_{test}=\Big{(}2.37-1.35\cdot A+0.519\cdot B+0.046\cdot A\cdot B-0.228 \cdot B^{2}\Big{)}\cdot 10^{-4}, \tag{26}\]  
Figure 10: Control curve for actively induced path-tracking deviations.  
From the response surface analysis, it can be seen that the \(Devr_{test}\) decreases with the increase in the \(A\) and increases first and then decreases with the increase in the \(B\). In the range of the constraint interval, the values of \(A\) and \(B\) at the highest point of the selected surface are (0.288, 0.992), i.e., the optimal regulation coefficients, when the theoretical variance decrease value \(Devr_{test}\) of the tracking control curve is the largest.