Agricultural Machinery Path Tracking with Varying Curvatures Based on an Improved Pure-Pursuit Method

Abstract

The current research on path tracking primarily focuses on improving control algorithms, such as adaptive and predictive models, to enhance tracking accuracy and stability. To address the issue of low tracking accuracy caused by variable-curvature paths in automatic navigation within agricultural environments, this study proposes a fuzzy control-based path-tracking method. Firstly, a pure-pursuit model and a kinematic model were established based on a Four-Wheel Independent Steering and Four-Wheel Independent Driving (4WIS-4WID) structure. Secondly, a fuzzy controller with three inputs and one output was designed, using the lateral deviation, d_e; heading deviation, θ_e; and bending degree, c, of the look-ahead path as the input variables. Through multiple simulations and adjustments, 75 control rules were developed. The look-ahead distance, Ld, was obtained through fuzzification, fuzzy inference, and defuzzification processes. Next, a speed-control function was constructed based on the agricultural machinery’s pose deviations and the bending degree of the look-ahead path to achieve variable speed control. Finally, field tests were conducted to verify the effectiveness of the proposed path-tracking method. The tracking experiment results for the two types of paths indicate that under the variable-speed dynamic look-ahead distance strategy, the average lateral deviations for the variable-curvature paths were 1.8 cm and 3.3 cm while the maximum lateral deviations were 10.1 cm and 10.5 cm, respectively. Compared to the constant-speed fixed look-ahead pure-pursuit model, the average lateral deviation was reduced by 56.1% and the maximum lateral deviation by 50.4% on the U-shaped path. On the S-shaped path, the average lateral deviation was reduced by 56.0% and the maximum lateral deviation by 58.9%. The proposed method effectively improves the path-tracking accuracy of agricultural machinery on variable-curvature paths, meeting the production requirements for curved operations in agricultural environments.

1. Introduction

Path-tracking control is a key component in implementing automatic navigation technology for agricultural machinery [1,2,3]. Traditional field-operation paths are mostly straight lines, leading to extensive research on straight-line path tracking [4,5,6]. However, when operating on curved paths, particularly those with variable curvatures, changes in the curvature often result in poor tracking performance. This is reflected in significant lateral deviations, reduced tracking stability, severe oscillations, and other related issues [7,8], causing significant insufficient tilling and incomplete harvesting. In practical agricultural environments, paths with varying curvatures are widely distributed [9,10,11] and it is essential to improve the tracking accuracy of agricultural machinery on variable-curvature paths to better meet the demands of mechanized and autonomous production.

The pure-pursuit control model is a geometric path-tracking control method that simulates manual driving and is widely used in agricultural machinery automatic navigation systems [12,13,14,15,16]. The selection of the look-ahead distance, Ld, directly affects the path-tracking performance of the pure-pursuit model. Consequently, numerous scholars have conducted extensive research on dynamically adjusting look-ahead distance. Zhang et al. [17] employed a particle swarm optimization algorithm to dynamically determine look-ahead distance based on real-time lateral tracking errors, thereby improving the straight-line tracking accuracy of tractor automatic navigation; Xu et al. [18] used the pose deviation of agricultural machinery as input for a fuzzy controller to adaptively adjust the look-ahead distance and validated the feasibility of their method using a Four-Wheel Independent Steering and Four-Wheel Independent Driving (4WIS-4WID) prototype. However, both studies overlooked factors such as path bending degree and speed, which significantly influence path-tracking accuracy, resulting in poor performance when tracking variable-curvature paths.

Some scholars have conducted research on curved-path tracking. Wu et al. [19] proposed a pure-pursuit control strategy in which look-ahead distance adaptively changes based on the current speed and the bending degree of the reference path, and optimization algorithms were employed to fine-tune the look-ahead distance under different operating conditions. However, this approach did not account for the influence of vehicle body deviation on variable-curvature path tracking. Nagasaka et al. [20], under the assumption of a fixed turning radius, utilized heading errors to achieve headland-turning path tracking for agricultural machinery. However, due to limited decision-making parameters, this method exhibited significant overshoot at the transition points between straight and curved paths. Ahn et al. [21] selected preview points based on a prototype’s position, orientation, and path and placed the preview points outside the path to reduce the “corner-cutting” phenomenon during curved-path tracking; Yang et al. [22] proposed a curved-path-tracking control method based on the optimal preview point, achieving an average curve tracking error of 6.7 cm when a tractor was traveling at 1 m/s. However, both methods failed to incorporate speed as a factor. These studies on dynamic look-ahead distance focused on certain factors, such as lateral deviation, heading deviation, vehicle speed, and path bending degree, but none considered all four factors comprehensively.

This paper investigates unmanned mobile platforms used in agricultural machinery operating in hilly and mountainous environments such as orchards and forests. These areas often contain paths with varying curvatures. To enhance the tracking accuracy of agricultural machinery on such paths, this paper proposes a fuzzy control-based path-tracking method tailored for variable-curvature paths. The lateral deviation, d_e; heading deviation, θ_e; and path bending degree, c, were input into the fuzzy controller to dynamically adjust the look-ahead distance, Ld. A speed-control model was developed to enable real-time speed adjustment during path tracking. Field path-tracking experiments verified that the proposed method effectively enhances the tracking accuracy of agricultural machinery on variable-curvature paths. This study aims to provide new insights and technical references for agricultural-machinery path-tracking technology.

2. Materials and Methods

2.1. Experimental Platform

2.1.1. Structure

The experimental prototype adopted a fully electric 4WIS-4WID mechanical structure, with body dimensions (length × width × height) of 130 cm × 158 cm × 66 cm. The prototype was primarily assembled using 40 mm × 40 mm square steel tubing, with a track width of 130 cm, a wheelbase of 100 cm, and a ground clearance of 55 cm. As shown in Figure 1, four drivers (AQMD6015BLS; Chengdu Aikong Electronics Technology Co., Ltd., Chengdu, China) independently drive the corresponding hub motors (9-inch, 24 V) to enable the prototype’s movement. Four steering gears (DH-03X, 120°/s, 38 N·m) control the steering of each drive wheel within a range of −90° to 90°. The pose-state information of the prototype was provided by a positioning Global Navigation Satellite System (GNSS) (PA-3 receiver; Shanghai Huace Navigation Technology Co., Ltd., Shanghai, China) and a 9-axis electronic gyroscope (BWT901CL; Shenzhen WitMotion Intelligent Technology Co., Ltd., Shenzhen, China) with pose-information error ranges of ±5 cm and ±0.1°, respectively. The main controller (STM32F103ZET6; Guangzhou Xinying Electronic Technology Co., Ltd., Guangzhou, China) is responsible for receiving and processing data as well as sending control commands. The entire system is powered by a 24 V, 20 Ah lithium battery with a full charge capacity of 480 Wh. The prototype is capable of carrying equipment for agricultural tasks such as spraying and harvesting. It can operate both autonomously and via remote control, with a working speed of 0.8 m/s and a maximum speed of 2 m/s. The average autonomy is 2 h (currently sufficient to complete the experiments, and future upgrades to the battery specifications can extend the autonomy).

2.1.2. Control System

When tracking the reference path, a PC receives coordinate information (x,y) from the positioning GNSS and the heading angle, θ, from the electronic gyroscope. It then converts the position and pose information into the lateral deviation, d_e, and heading deviation, θ_e, relative to the reference path. A fuzzy controller with three inputs and one output, along with a speed-control model, was designed with the PC. For the fuzzy controller, the three input signals were the lateral deviation, d_e; the heading deviation, θ_e; and the path bending degree, c, while the output signal was the look-ahead distance, Ld. The output look-ahead distance, Ld, from the fuzzy controller was fed into the pure-pursuit model to calculate the steering radius, R, of the prototype under the current deviation state. Subsequently, the Ackermann four-wheel low-speed steering model calculated the steering angles, δ, for each wheel [23]. The speed-control model calculated the current vehicle speed, v, based on the degrees of the lateral deviation degree, a; heading deviation degree, b; and path bending degree, c. The main controller received the steering angles, δ, of each wheel and the vehicle speed, v, provided by the PC. Two encoders respectively fed back the current actual speed, v_f, and steering angle, δ_f, to the controller, forming a closed-loop system with the motors and other components to ensure control accuracy. The corrected steering angle and speed signals, adjusted through the feedback of the closed-loop system, were converted into Pulse Width Modulation (PWM) signals and input into the steering gears and hub motors, respectively, enabling the prototype to drive and steer and completing the path-tracking control. The control-system block diagram is shown in Figure 2.

2.2. Improved Pure-Pursuit Model

2.2.1. Definition of Vehicle Deviation

Pure-pursuit control is a path-tracking algorithm based on geometric principles. It calculates the turning radius between the current vehicle position and the look-ahead point on a reference path in real time, enabling the vehicle to follow the desired trajectory. The improved pure-pursuit model is shown in Figure 3.

In Figure 3, A_i represents the geometric center of the prototype; P₁ and P₂ represent the reference path, which is composed of several equally spaced discrete points; P_i is the closest discrete point on the path to the geometric center of the prototype; P_i+1 is the next-closest discrete point on the path to the geometric center; Ld_max is the preset maximum look-ahead distance; P_i+n is the discrete point on the path corresponding to the maximum look-ahead distance, Ld_max; the lateral deviation, d_e, is the distance between the geometric center, A_i, and the nearest path point, P_i; the heading deviation, θ_e, is the angle between the forward direction of the vehicle body and the line segment P_i P_i+1; Ld_i is the look-ahead distance; S_i is the preview point; and O_i is the steering center of the prototype.

2.2.2. Definition of Path Bending Degree

The path bending degree has a significant impact on the look-ahead distance. The greater the path bending degree, the smaller the look-ahead distance and the higher the tracking accuracy. However, the global average curvature of a path cannot accurately represent the local path bending degree. To better adapt to changes in the upcoming path, it is necessary to obtain the bending degree of the look-ahead path. In this study, the look-ahead path is defined as the segment between P_i and P_i+n, where P_i is the starting point and P_i+n is the endpoint. We calculated the straight-line distance from the starting point to the endpoint of the look-ahead path and the total sum of the distances between the adjacent discrete points within the look-ahead path [24]. This can be expressed as

$$c=1-\exp \left(k_{\mathrm{c}}\times \left(‖P_{\mathrm{i}+\mathrm{n}}-P_{\mathrm{i}}‖-{\displaystyle \sum _{j=0}^{n-1}‖P_{\mathrm{i}+\mathrm{j}+1}-P_{\mathrm{i}+\mathrm{j}}‖}\right)\right)$$

(1)

In Equation (1), the closer c is to 1, the more curved the look-ahead path is; conversely, the closer c is to 0, the straighter the path is. k_c is the preset adjustment coefficient. In this study, kc is set to 3.

2.2.3. Kinematic Model of the Vehicle

As shown in Figure 3, in ∆A_i O_i S_i, the sine rule gives

$${\displaystyle \frac{L{d}_{\mathrm{i}}}{\sin (2\alpha )}}={\displaystyle \frac{R_{\mathrm{i}}}{\sin (\frac{\pi }{2}-\alpha )}}$$

(2)

$$R_{\mathrm{i}}={\displaystyle \frac{L{d}_{\mathrm{i}}}{2\sin \alpha }}$$

(3)

In Equation (2), α represents the angle between the forward direction of the prototype and the look-ahead distance, rad, and R_i represents the turning radius of the prototype, m.

Combined with the four-wheel steering model, the steering angles of each wheel were obtained as follows:

$$\left\{\begin{array}{l}{\delta }_{\mathrm{fr}}=-{\delta }_{\mathrm{rr}}=\arctan ({\displaystyle \frac{L}{2R_{\mathrm{i}}-W}})\\ {\delta }_{\mathrm{fl}}=-{\delta }_{\mathrm{rl}}=\arctan ({\displaystyle \frac{L}{2R_{\mathrm{i}}+W}})\end{array}\right.$$

(4)

In Equation (3), L represents the wheelbase of the prototype, m; W represents the track width of the prototype, m; and δ_fr, δ_fl, δ_rr, and δ_rl represent the steering angles of the four wheels of the prototype, rad.

The look-ahead distance, Ld, is the only adjustable parameter in the pure-pursuit model. A smaller look-ahead distance will enable the prototype to respond more quickly to path changes such as sharp turns or paths with high curvature variation rates. However, it may cause frequent changes in the wheel steering angles, resulting in less smooth driving. Conversely, a larger look-ahead distance will require more time and distance for the prototype to adjust its path, leading to slower responses to curved paths [25,26]. The look-ahead distance is influenced by the lateral deviation, d_e; heading deviation, θ_e; and path bending degree, c, of the prototype. To dynamically adjust the look-ahead distance based on these factors, a fuzzy controller was designed and integrated into the system.

2.3. Design of the Fuzzy Controller

The fuzzy controller is an intelligent control method based on fuzzy logic, which converts uncertainties and empirical rules in complex systems into operable mathematical models to achieve precise control. It utilizes fuzzy set theory to transform linguistic variables into numerical operations, thereby automatically generating control outputs to optimize system performance [27,28,29,30].

Using the fuzzy control method, the look-ahead distance is dynamically adjusted according to different conditions, thereby improving the accuracy and stability of the prototype’s path tracking and enhancing its adaptability to complex road conditions. The fuzzy control system is shown in Figure 4. The Mamdani fuzzy inference method is used to generate fuzzy outputs, which are then converted into precise values through defuzzification. The defuzzification process employs the centroid method, which provides relatively smooth output calculations.

2.3.1. Input and Output Variables

Based on the above analysis, fuzzification was applied to the input and output variables. The domain of the lateral deviation was [−0.3 m, 0.3 m], with five fuzzy levels: Negative Big (NB), Negative Small (NS), Zero (ZO), Positive Small (PS), and Positive Big (PB). The domain of the heading deviation was [−30°, 30°], also with five fuzzy levels. The definitions for the signs of the lateral and heading deviations are as follows: when the prototype is on the left side of the reference path’s forward direction, the lateral deviation is positive; when it is on the right side, the deviation is negative. The heading deviation was positive in a counterclockwise direction and negative in a clockwise direction. The domain of the path bending degree was [0, 1], with three fuzzy levels: Small (S), Medium (M), and Big (B). If the input values exceeded the domain boundaries, they were set to the boundary values. The domain of the look-ahead distance was [0.5 m, 2.5 m], with five fuzzy levels. The membership functions for the input and output variables are shown in Figure 5.

This study adopted non-uniform quantization membership functions instead of uniform quantization. The reason is that when the prototype’s pose deviation or the path bending degree was large, a larger-scale quantization level was used to ensure good control stability. Conversely, when the pose deviation or path bending degree was small, a smaller-scale quantization level was employed to achieve more precise adjustments to the prototype’s pose.

2.3.2. Control Rules

To ensure the fuzzy controller would reliably and quickly guide the prototype to converge to the reference path, reasonable control rules needed to be designed. The following principles were observed when formulating the control rules:

(1) When d_e and θ_e have the same sign: This indicates that the prototype will move further away from the reference path if it continues along its current direction. In such cases, if the lateral deviation, d_e, or the heading deviation, θ_e, is large, a smaller look-ahead distance should be selected to enable the prototype to converge more quickly to the reference path, thereby improving path-tracking efficiency.

(2) When d_e and θ_e have opposite signs: This indicates that the prototype is approaching the reference path if it continues along its current direction. In this case, when the deviations (d_e, θ_e) are small, the look-ahead distance can be appropriately increased to enhance the stability of the path tracking.

(3) Meanwhile, to enhance the prototype’s responsiveness to path-curvature variations and reduce the “corner-cutting” phenomenon caused by a larger local path bending degree, c, a smaller look-ahead distance should be selected to enable the prototype to adapt to path-curvature changes.

Based on the above driving principles and after multiple simulations and adjustments, the designed fuzzy controller included a total of 75 control rules. The fuzzy control rules are shown in

Table 1. Fuzzy control rule table (path bending degree, c = S).

Ld		θe
		NB	NS	ZO	PS	PB
de	NB	NB	NB	NS	ZO	PS
	NS	NS	NS	ZO	PS	PS
	ZO	ZO	PS	PB	PS	ZO
	PS	PS	PS	ZO	NS	NS
	PB	PS	ZO	NS	NB	NB

,

Table 2. Fuzzy control rule table (path bending degree, c = M).

Ld		θe
		NB	NS	ZO	PS	PB
de	NB	NB	NB	NS	NS	ZO
	NS	NB	NS	NS	ZO	ZO
	ZO	NS	ZO	PS	ZO	NS
	PS	ZO	ZO	NS	NS	NB
	PB	ZO	NS	NS	NB	NB

and

Table 3. Fuzzy control rule table (path bending degree, c = B).

Ld		θe
		NB	NS	ZO	PS	PB
de	NB	NB	NB	NB	NB	NS
	NS	NB	NB	NB	NS	NS
	ZO	NB	NS	ZO	NS	NB
	PS	NS	NS	NB	NB	NB
	PB	NS	NB	NB	NB	NB

, and the fuzzy rule surfaces are illustrated in Figure 6.

2.3.3. Defuzzification

Defuzzification is a key step in fuzzy control, used to convert fuzzy sets into precise and actionable output values. Among various defuzzification methods, the centroid method is the most commonly used. Essentially, the centroid method is a weighted average approach where the weights are the membership degrees of each element in the fuzzy inference conclusion. In this study, the centroid method was employed for defuzzification, and the formula is shown as follows:

$$u={\displaystyle \frac{{\displaystyle \int u\times {\mu }_{\mathrm{N}}\left(u\right)}du}{{\displaystyle \int {\mu }_{\mathrm{N}}\left(u\right)du}}}$$

(5)

The formula obtained after discretization is as follows:

$$u={\displaystyle \frac{{\displaystyle \sum u\times {\mu }_{\mathrm{N}}\left(u_i\right)}}{{\displaystyle \sum {\mu }_{\mathrm{N}}\left(u_i\right)}}}$$

(6)

where u represents the target value obtained after defuzzification, μ_N represents the membership function, and u_i represents the possible output values.

2.4. Speed-Control Model

Vehicle speed, v, significantly impacts the accuracy of path tracking and navigation efficiency. This study proposes a nonlinear speed-control model based on the degree of vehicle deviation and path curvature. The model integrates the effects of the lateral deviation degree, a; the heading deviation degree, b; and the path bending degree, c, to dynamically adjust the prototype’s speed. If an input value exceeded the domain range, it was set to the boundary value of the domain:

$$\left\{\begin{array}{l}a=\left|{\displaystyle \frac{d\mathrm{e}}{0.3}}\right|\\ b=\left|{\displaystyle \frac{\theta \mathrm{e}}{30}}\right|\end{array}\right.$$

(7)

$$v\left(a,b,c\right)=v_{\min }+\left(k_1\times {\left(1-a\right)}^2\right.+k_2\times {\left(1-b\right)}^2+\left.k_3\times {\left(1-c\right)}^2\right)\times \left(v_{\max }-v_{\min }\right)$$

(8)

where v(a,b,c) represents the dynamically adjusted speed, m·s⁻¹; v_min is the minimum travel speed, m·s⁻¹; v_max is the maximum travel speed, m·s⁻¹; and k₁, k₂, and k₃ are the weighting coefficients for the lateral deviation degree, heading deviation degree, and path curvature, respectively. The exponents of (1 − a), (1 − b), and (1 − c) were set to 2 in order to enhance the nonlinear influence of these terms on speed. With this nonlinear speed-control model, the system can appropriately increase v to maintain high driving efficiency when the lateral deviation, heading deviation, and path curvature are small. However, when any of these deviations increases, the system can respond quickly and effectively reduce the risk of the vehicle deviating from the reference path. Based on the above analysis and multiple tuning experiments, the values of k₁, k₂, and k₃ were set as 0.4, 0.2, and 0.4, respectively.

2.5. Experiment Design

To validate the effectiveness of the proposed algorithm, path-tracking experiments were conducted in October 2024 at the Agricultural Experiment Base of Zhejiang A&F University. During the experiment, the test site was open, with stable and accurate GNSS signals. The prototype’s forward speed was set to both constant and variable modes. The variable-speed range was configured as [0.4 m/s, 1.2 m/s] based on practical requirements, while the constant speed was set to 0.8 m/s. Path-tracking experiments were performed using variable-curvature paths, with a navigation system data-sampling frequency of 5 Hz. Two classic variable-curvature paths commonly found in agricultural environments were selected for the experiment: a “U”-shaped path and an “S”-shaped path.

Experiment 1: Under the condition of constant vehicle speed, a comparative experiment was conducted on the traditional fixed-look-ahead pure-pursuit model and the improved dynamic-look-ahead pure-pursuit model proposed in this paper. The purpose was to verify the effectiveness of the dynamic look-ahead distance based on the fuzzy controller in improving the path-tracking accuracy for variable-curvature paths.

Experiment 2: Under the premise of using the dynamic look-ahead distance strategy in both cases, a comparative experiment was conducted between constant speed and variable speed to verify the effectiveness of the proposed speed-control model in improving path-tracking accuracy for variable-curvature paths.

To better evaluate the quality of the variable-curvature path tracking, metrics such as average lateral deviation, maximum lateral deviation, standard deviation, and path-tracking time were used. The average lateral deviation refers to the mean absolute value of all lateral deviations from the start to the end of the experiment; the maximum lateral deviation refers to the maximum absolute lateral deviation observed throughout the experiment; the standard deviation reflects the degree of dispersion between the tracked path and the desired path; and the path-tracking time refers to the total time required to track the desired path from the start to the end of the experiment. The average and maximum lateral deviations are the key indicators of the path-tracking accuracy, the standard deviation reflects the path-tracking stability, and the path-tracking time indicates the tracking efficiency.

3. Results and Discussion

For the convenience of subsequent discussions, the fixed look-ahead distance path-tracking method under a constant speed is defined as Method 1, the dynamic look-ahead distance path-tracking method under a constant speed is defined as Method 2, and the dynamic look-ahead distance path-tracking method under variable speed is defined as Method 3.

3.1. Results of Experiment 1

After Experiment 1, relevant data can be obtained for better discussion of the constructed fuzzy controller. The data from Experiment 1 are shown in

Table 4. Result statistics (Experiment 1).

Tracking Path	Look-Ahead Distance (m)	Average Lateral Deviation (cm)	Maximum Lateral Deviation (cm)	Standard Deviation (cm)	Path-Tracking Time (s)
U-shaped	1.5	4.1	20.4	4.8	32.9
	Dynamic Ld	2.3	12.5	2.5	33.8
S-shaped	1.5	7.5	25.6	8.1	31.9
	Dynamic Ld	4.5	15.9	4.9	32.7

.

3.2. Discussion of Experiment 1

3.2.1. Tracking Accuracy (Experiment 1)

In Experiment 1, the U-shaped path tracking using Method 2 resulted in an average lateral deviation of 2.3 cm and a maximum lateral deviation of 12.5 cm. For the S-shaped path tracking with Method 2, the average lateral deviation was 4.5 cm and the maximum lateral deviation was 15.9 cm. Compared to Method 1, Method 2 reduced the average lateral deviation and maximum lateral deviation in the U-shaped path by 43.9% and 38.7%, respectively. In the S-shaped path, Method 2 reduced the average lateral deviation and maximum lateral deviation by 40% and 37.9%, respectively.

Method 2 achieved better path-tracking accuracy for the agricultural machinery on variable-curvature paths. This is because Method 2 dynamically adjusts look-ahead distance in real-time using the fuzzy controller based on the vehicle’s pose and the path curvature. The above data demonstrate that on variable-curvature paths, the dynamic look-ahead pure-pursuit model outperformed the fixed look-ahead distance pure-pursuit model.

3.2.2. Tracking Stability (Experiment 1)

In Experiment 1, on the U-shaped path, Method 2 improved the path-tracking stability by 47.9% compared to Method 1. On the S-shaped path, Method 2 improved the path-tracking stability by 39.5% compared to Method 1.

Figure 7 and Figure 8 provide a clear visual indication that Method 2 significantly reduces the “corner-cutting” phenomenon. The fuzzy controller effectively addresses “corner-cutting” and overshooting by outputting smaller look-ahead distances, enabling more efficient trajectory convergence for the prototype. The difference between the tracked trajectory and the reference path, as shown in Figure 7 and Figure 8, and the lateral deviation change curve shown in Figure 9, indicates that on variable curvature paths, the dynamic look-ahead pure-pursuit model achieves better path tracking stability compared to the fixed look-ahead distance pure-pursuit model.

3.2.3. Tracking Time (Experiment 1)

In Experiment 1, the tracking times of Method 1 and Method 2 for both the U-shaped and S-shaped paths were approximately the same.

As shown in Figure 7 and Figure 8, Method 1, due to the fixed look-ahead distance, exhibits a more pronounced “corner-cutting” phenomenon. While this resulted in reduced tracking time for some parts of the path, additional time was required to converge back to the desired path after turning. In contrast, Method 2, with its dynamic look-ahead distance, significantly reduces the “corner-cutting” phenomenon. As a result, it requires covering more path distance, which increases time compared to Method 1. However, Method 2 converges to the desired path more quickly after a turn, thereby reducing time compared to Method 1.

3.3. Results of Experiment 2

After Experiment 2, relevant data can be obtained for better discussion of the speed-control model. The data from Experiment 2 are shown in

Table 5. Result statistics (Experiment 2).

Tracking Path	Vehicle Speed (m·s⁻¹)	Average Lateral Deviation (cm)	Maximum Lateral Deviation (cm)	Standard Deviation (cm)	Path-Tracking Time (s)
U-shaped	0.8	2.3	12.5	2.5	33.8
	Variable-Speed	1.8	10.1	2.1	28.8
S-shaped	0.8	4.5	15.9	4.9	32.7
	Variable-Speed	3.3	10.5	3.6	32.1

.

3.4. Discussion of Experiment 2

3.4.1. Tracking Accuracy (Experiment 2)

In Experiment 2, the U-shaped path tracking using Method 3 resulted in an average lateral deviation of 1.8 cm and a maximum lateral deviation of 10.1 cm. For the S-shaped path tracking with Method 3, the average lateral deviation was 3.3 cm and the maximum lateral deviation was 10.5 cm. Compared to Method 2, Method 3 reduced the average lateral deviation and maximum lateral deviation on the U-shaped path by 21.7% and 19.2%, respectively. On the S-shaped path, Method 3 reduced the average lateral deviation and maximum lateral deviation by 26.7% and 33.9%, respectively.

Method 3, which takes speed variations into account, effectively reduced the maximum lateral deviation during path tracking while maintaining high path-tracking accuracy. Method 3 built upon Method 2 by additionally adjusting the speed dynamically, which further improved the tracking of the desired path. The above data demonstrate that in variable-curvature paths, speed control enhances path-tracking accuracy.

3.4.2. Tracking Stability (Experiment 2)

In Experiment 2, on the U-shaped path, Method 3 improved the path-tracking stability by 16.0% compared to Method 2. On the S-shaped path, Method 3 improved the path-tracking stability by 26.5% compared to Method 2.

From Figure 10 and Figure 11, it can be clearly observed that Method 3 further improved the “corner-cutting” phenomenon compared to Method 2. The speed-control model effectively addresses both “corner-cutting” and overshooting by adjusting the vehicle speed, which enables more efficient trajectory convergence of the prototype. As shown in Figure 10 and Figure 11, the difference between the tracked trajectory and the reference path, along with the lateral deviation change curve shown in Figure 12, indicates that speed control can improve path tracking stability on variable curvature paths.

3.4.3. Tracking Time (Experiment 2)

In Experiment 2, the tracking time for Method 3 for the U-shaped path was 28.8 s, which was 14.8% shorter compared to Method 2. The tracking times for Method 2 and Method 3 for the S-shaped path were approximately the same.

In the U-shaped path tracking, Method 3 required the least time; in the S-shaped path tracking, the tracking time of Method 3 was similar to the other two methods while path tracking accuracy and stability were improved. When the prototype encountered a path with changing curvature, the speed was reduced to enhance tracking accuracy and further mitigate the “corner-cutting” phenomenon. When the path ahead was relatively straight, the speed was increased to improve the tracking efficiency. The data presented above demonstrate that speed control can enhance path-tracking efficiency while ensuring tracking accuracy and stability.

3.5. Discussion of the Comparison of the Proposed Method to Other Methods

The method proposed in this paper, compared to other improved pure-pursuit methods [14,15,16,17,18,19], offers the following key advantages: It defines the curvature of the look-ahead path in the pure-pursuit model, thus enhancing the model with foresight. It comprehensively considers the effects of lateral deviation, d_e; heading deviation, θ_e; path bending degree, c; and velocity, v, on path tracking and constructs a fuzzy controller and velocity control model to improve the accuracy and efficiency of path tracking. In contrast, the methods mentioned in the introduction typically only consider a subset of these parameters. Therefore, the method proposed in this paper demonstrated better robustness and is more adaptable to paths with varying curvature.

4. Conclusions

(1) This paper addresses the issue of low path-tracking accuracy in agricultural machinery caused by the varying curvatures of paths by proposing a dynamic look-ahead distance pure-pursuit model based on a fuzzy controller. The model uses lateral deviation, heading deviation, and path curvature as fuzzy inputs and generates the look-ahead distance as a fuzzy output, which is then defuzzified to determine the optimal look-ahead distance. Compared to the fixed look-ahead distance path-tracking method, the fuzzy control-based approach reduced the average lateral deviation and maximum lateral deviation by 43.9% and 38.7%, respectively, while improving the path-tracking stability by 47.9% in U-shaped path tracking. For S-shaped paths, the average lateral deviation and maximum lateral deviation were reduced by 40% and 37.9%, respectively, and the path-tracking stability increased by 39.5%. These results demonstrate the effectiveness of the fuzzy controller proposed in this study.

(2) This paper presents a variable speed-control function that adjusts the prototype’s speed in real-time based on the vehicle’s pose deviation and the path curvature in the look-ahead area. Comparative experiments were conducted to evaluate the path-tracking performance under constant-speed and variable-speed conditions for paths with varying curvatures. With the same dynamic look-ahead distance, the variable speed-control method reduced the average lateral deviation and maximum lateral deviation by 21.7% and 19.2%, respectively, while improving the path-tracking stability by 16% and reducing the tracking time by 14.8% for U-shaped paths compared to constant speed control. For S-shaped paths, the average lateral deviation and maximum lateral deviation were reduced by 26.7% and 33.9%, respectively, and the path-tracking stability increased by 26.5%. These findings validate the effectiveness of the speed controller proposed in this study.

(3) The quantitative scale design of the fuzzy control rules can be further optimized. In future research, more refined fuzzy control rules can be used by quantizing the input and output variables into additional fuzzy subsets, allowing for the identification of more deviation states and providing more accurate dynamic look-ahead distance. Additionally, the design of the speed-control model can be further refined to improve the path-tracking accuracy of the agricultural machinery’s autonomous navigation system.

(4) Improving the tracking accuracy of agricultural machinery on variable-curvature paths can enhance the technological competitiveness of manufacturers. For farmers, it can reduce labor demands and improve operational precision. Future research in this area could focus on further optimization of path-tracking algorithms; integration of multi-sensor technologies; and optimization of energy efficiency to enhance battery life.