Path Planning for Full Coverage of Farmland Operations in Hilly and Mountainous Areas Based on the Dung Beetle Optimization Algorithm

Abstract

This study aims to address the issues of full-coverage path planning in single fields and optimal traversal order in multi-fields in hilly, mountainous areas. To this end, it proposes a full-coverage path planning method based on an improved DBO algorithm. Using the digital elevation model to construct the farmland model, the energy consumption model is introduced into single-field planning to determine the optimal operating direction angle for full-coverage path planning with optimal energy consumption. To address the issues of the traditional DBO algorithm easily falling into a local optimum and the lack of information interaction among populations, a multi-strategy improved DBO algorithm is proposed to determine the optimal traversal sequence for multiple fields. Tent chaotic mapping is used to initialize the population and the Osprey optimization algorithm and adaptive T-perturbation distribution strategy are integrated to enhance the foraging behavior of small dung beetles. This gives the algorithm good global exploration capabilities in the initial stage and strong local exploitation capabilities in the later stage. The simulation results show that the total energy consumption of energy-optimal path planning is 5.62 × 10⁴ J, which is 19.93% less than the optimal path length. The traversal order solved by the improved DBO algorithm saves 9.2% more energy than the original algorithm, demonstrating a significant energy-saving effect.

1. Introduction

Hilly mountainous areas for China’s grain and characteristics of agricultural products, covering 19 provinces, autonomous regions, and cities, with more than 1400 counties and municipalities, arable land area, and crop sowing area, accounted for 1/3 of the country, involving nearly 300 million people in agriculture. However, the comprehensive mechanization rate of crop ploughing, planting, and harvesting in hilly and mountainous areas is less than 50%, and only 29% in hilly and mountainous regions of the southwest of China, which is far lower than the national level of 69% [1]. Terrain features such as fine fragmentation of land, slopes and ridges, irregular shape, and other topographical features make it difficult for large and medium-sized agricultural machinery to operate; machinery that operates on the ground is threatening to become a significant bottleneck restricting the development of mechanization [2]. The whole-area path planning for farmland operation in hilly areas, the optimization of operation routes for farm machinery, and the reduction of work area repetition rate are the keys to realizing the automation and intelligent operation of farm machinery [3,4,5].

With the advancement of appropriate mechanization transformation and the development of agricultural mechanization in hilly mountainous areas, intelligent agricultural machinery operation technology has become a research hotspot. Autonomous navigation technology is the core technology of intelligent farm machinery; its innovation research is the primary task of developing intelligent farm machinery in hilly and mountainous areas [6].

Path planning is the core component of autonomous navigation technology for agricultural machinery and is crucial for improving the operational efficiency of intelligent agricultural machinery [7]. Path planning is primarily divided into point-to-point path planning and full-coverage path planning. Full-coverage path planning is an algorithm that provides an executor with a path to ensure it traverses all points within a work area while avoiding obstacles on the map [8]. Most agricultural machinery operations require farmland coverage to complete harvesting, ploughing, sowing, and fertilizing tasks. Therefore, full-coverage path planning technology has been widely applied in agricultural machinery automation. Compared to traditional methods where operators make subjective judgments, full-coverage operations help reduce path repetition rates during operations, improve agricultural machinery operational efficiency and quality, and reduce energy consumption [9]. Traditional full-coverage path planning methods are primarily applied in indoor and flat terrain environments. However, conventional methods cannot address the issue of increased energy consumption caused by terrain undulations for farmland environments in hilly regions.

Regarding algorithm optimization, researchers have studied the coverage range of the working area based on the Boustrophedon cell decomposition method or the internal spiral method [10,11]. Subsequently, they optimized the algorithms on the single-field block planning path using traditional algorithms such as the gravity search algorithm [12], ant colony algorithm [13], genetic algorithm [14], and simulated annealing algorithm [15]. For example, I.A. Hameed [16] designed a three-dimensional terrain farmland full-coverage path planning algorithm based on the genetic algorithm. This algorithm uses the genetic algorithm to find the optimal field travel direction in three-dimensional farmland, enabling agricultural machinery to complete the traversal task with minimal energy consumption. Shen Mengwei [17] designed a path planning algorithm for full coverage in hilly and mountainous outdoor environments. He constructed an energy consumption model and used an exhaustive algorithm to search for the optimal travel direction for a single field block. Finally, he used a genetic algorithm to plan the coverage sequence for multiple fields. However, this method only applies to terraced farmland, sloped but flat internally. Some scholars have further optimized the sequence of operations across multiple fields based on single-field full-coverage path planning, employing intelligent algorithms such as the whale algorithm [18], beetle algorithm [19], genetic algorithm [18], particle swarm algorithm [20], ant colony algorithm [21], and reinforcement learning path planning algorithm [22]. These approaches aim to identify the shortest paths between field transitions and reduce energy consumption.

However, the aforementioned studies have primarily focused on two-dimensional path planning for large-scale sites in flat terrain, with limited research on farmland in hilly or mountainous regions that are extremely uneven and have slopes [23]. Additionally, these methods are ineffective in reducing energy consumption for crop coverage. For large-scale hilly terrain farmland, where terrain changes occur more frequently, factors such as elevation and slope must be taken into account. Adopting an appropriate full-coverage path planning method can significantly reduce the costs of sowing and harvesting processes, thereby improving operational efficiency and production levels.

Based on the above analysis, this paper adopts an improved beetle optimization algorithm to address the issues of reducing energy consumption in path planning for comprehensive coverage of farmland operations in hilly and mountainous areas and optimizing the optimal traversal order between multiple fields. Firstly, the remote sensing image is constructed by a UAV, and the farmland environment is constructed by a geometrical method and the energy consumption model is introduced into the single-field full-coverage path planning to search for the optimal operating direction angle, so as to achieve the full-coverage path planning with optimal energy consumption for single-field operation. Then, the dung beetle optimization algorithm is improved to improve the convergence speed, global path optimization ability, and search path smoothing degree for the multi-field traversal sequential optimization solution in view of the weak global search ability of the traditional dung beetle algorithm. Finally, the effectiveness and feasibility of the method are verified by simulation experiments.

2. Farmland Environment Construction

Full-coverage path planning aims to achieve complete traversal of the operational area and eliminate path duplication. Optimization of path planning algorithms based on accurate map construction and obstacle identification in farmland operations can ensure complete coverage of farmland areas and reduce redundant paths. Farmland environment modelling methods mainly include 3D modelling methods, topological methods [24], geometric methods [25], and raster methods [26], as shown in Figure 1.

2.1. Modelling of the Farmland Environment

Based on the geospatial cloud platform to obtain the digital elevation model data of the farmland operation area (106°47′ E, 23°09′ N), ArcGIS10.8 software was used to process the elevation information and extract the geographic coordinates of the target fields. The precise contour data set was constructed by marking the field boundary feature points point by point. The MATLAB2023a platform was used to read the geographic information of the field and execute the coordinate system conversion algorithm to convert the latitude and longitude coordinates to the Cartesian coordinate system to generate the field contour vector image. Figure 2 shows the satellite image of the test field with the extracted contour results.

2.2. Energy Consumption Model Construction

When operating in hilly and mountainous areas, the energy consumption of agricultural machinery is mainly due to terrain and soil resistance, as well as changes in gravitational potential energy. The key factor influencing energy consumption is the slope of the terrain: uphill operation requires overcoming gravitational force, leading to a significant increase in energy consumption. In the downhill process, although some of the gravitational potential energy can be recovered, braking still generates additional energy loss. The slope factor increases the complexity of the path planning algorithm in a 3D terrain environment compared to flat farmland. Figure 3 shows the angle relationship during the traveling process of the unmanned farm machine.

$$\tan \theta ={\displaystyle \frac{z}{x}}$$

(1)

$$\mathit{sin}\beta ={\displaystyle \frac{x}{\sqrt{x^2+y^2}}}$$

(2)

$$\mathit{tan}\phi ={\displaystyle \frac{\sqrt{x^2+y^2}}{y}}$$

(3)

$$\mathit{tan}\alpha ={\displaystyle \frac{z}{\sqrt{x^2+y^2}}}=\mathit{tan}\theta ·\mathit{sin}\beta$$

(4)

$$\alpha ={\mathit{tan}}^{-1}\left(\mathit{tan}\theta ·\mathit{sin}\beta \right)$$

(5)

In the above equation, θ is the travelling slope angle, φ is the heading angle, δ is the angle between the horizontal direction and the boundary of the farmland, η is the slope angle of the farmland, which has been determined during the collection of the farmland data, α is the angle between the projection of the travelling direction in the horizontal plane and the boundary of the farmland, and β is the inclination angle of the operating direction.

2.2.1. Energy Consumption of Agricultural Machinery in the Area of Operation of Linear Segments

The energy consumption of agricultural machinery while driving is determined by both the distance travelled and the driving force. Based on the energy consumption calculation demand, this paper makes the following assumptions:

(1)

The driving force of agricultural machinery remains constant during a specific driving phase; that is to say, the driving force does not change over time under a given working condition.

(2)

When travelling at a constant speed, the driving force is equal to the resistance, meaning the energy consumption is equal to the work done to overcome rolling, slope, and rotary tiller resistance. Based on the above assumptions, the energy loss of agricultural machinery is quantitatively analyzed. The path differentiation method is used to divide the overall path into unit linear segments, and the total energy consumption is solved by integration. The unit path force analysis is shown in Figure 4.

On a unit path, the distance travelled by agricultural machinery is calculated from the Euclidean distance [27]:

$$S_i=\sqrt{{\left(x_i-x_{i+1}\right)}^2+{\left(y_i-y_{i+1}\right)}^2+{\left(z_i-z_{i+1}\right)}^2}$$

(6)

According to the vehicle longitudinal dynamics modelling equation, as well as Equation (6), it can be obtained that the energy consumption per unit of vehicle travelling is

$$W_i=\sqrt{{\left(x_i-x_{i+1}\right)}^2+{\left(y_i-y_{i+1}\right)}^2+{\left(z_i-z_{i+1}\right)}^2}\left[b\left(c{z}_0{K}_{pc}+0.5z_0^2{\gamma }_s{K}_{pr}\right)+{\displaystyle \frac{1}{{\left(3-n\right)}^{\frac{2n+2}{2n+1}}\left(n+1\right){\left(k_c+b{k}_{\phi }\right)}^{\frac{1}{2n+1}}}}{\left[{\displaystyle \frac{3G\mathit{cos}{\theta }_i}{\sqrt{D}}}\right]}^{{\displaystyle \frac{2n+2}{2n+1}}}+G{f}_i\mathit{cos}{\theta }_i+G\mathit{sin}{\theta }_i+R\right]$$

(7)

Under the conditions of fixed vehicle parameters and soil parameters, the travelling slope angle θ_i of the agricultural machinery is determined by the slope of the farmland and the heading angle, and the pushing resistance and compaction resistance depend only on the travelling slope and the travelling length. Accordingly, the energy consumption can be expressed as a function W (φ, L) of the initial heading angle φ and the path length L, which is applicable to the calculation of the energy consumption in the straight section and the turning section.

$$W\left(\phi ,L\right)=\sum W_i$$

(8)

2.2.2. Energy Consumption for Ground Header Turns

The headland area provides space for the turning of agricultural machines to realize the connection of adjacent straight-line operation paths. The turning mode is determined by the headland width W_h and the minimum turning radius R of the farm machine, and different turning modes correspond to different constraints and path cost calculation methods. Figure 5 shows several typical turning methods.

Let the operating row spacing ω, the angle β between the operating direction and the farmland boundary, the minimum turning radius R, and the headland width W_h be known, and the four turning methods and their cost calculations are as follows [28]:

When R < ω/2, a ‘Bow turning’ is used, the path cost of the curved section is πR, and the total path cost C_BT is shown in Equation (9).

When R = ω/2, a ‘Semicircular turning’ is used, the path cost of the curved section is ωπ/2, and the total path cost C_ST is shown in Equation (10).

When R > ω/2, a ‘Pear-shaped turning’ type of turn can be used, the straight-line segment path cost is ω tanβ, and the total path cost C_PT is shown in Equation (11).

When R > ω/2 and the head space is limited, a Fishtail turning’ is used, and the total path cost C_FT is shown in Equation (12).

$$C_{BT}=\omega \mathit{tan}\beta +\omega -2R+R\pi$$

(9)

$$C_{ST}=\omega \mathit{tan}\beta +\omega \pi /2$$

(10)

$$C_{PT}=R+4R{\mathit{cos}}^{-1}\left({\displaystyle \frac{\omega +2R}{4r}}\right)+\omega \mathit{tan}\beta$$

(11)

$$C_{FT}=\left(\pi -2\right)R-\omega \mathit{tan}\beta$$

(12)

where

ω:

operating row spacing, m;

β:

angle between operation direction and farmland boundary, °;

R:

minimum turning radius, m;

W_h:

reserved width at the headland.

Combining the above analyses leads to a decision model for ground head turning, as shown in Figure 6.

Assuming that the farm machine is travelling at an even speed while turning in the field, without considering wheel slip, and that the driving force is balanced against friction, there is

$$F=F_f=\mu \mathit{mgcos}\alpha$$

(13)

The energy consumption WT1 consumed by an agricultural machine for a single turn is

$$W_{T1}=F·C=\mu m\mathrm{g}\mathit{cos}\alpha ·C$$

(14)

The total number of turns in a single field is

$$N_T={\displaystyle \frac{{\sum }_{i=1}^{n_e}\left|\mathit{sin}{\delta }_i\right|L_i}{2\omega }}$$

(15)

where

n_e: number of field edges;

L_i: the length of the i-th edge, m;

δ_i: angle between the i-th edge and the direction of operation of the agricultural machine;

ω: width of the agricultural machine.

Then the total energy consumption WT consumed by the farm machine when turning in a single field is

$$W=W_{T1}·N_T$$

(16)

3. Single Field Full Coverage Path Planning

3.1. Path Planning Performance Metrics

Currently, path length is still the main optimization objective for full-coverage path planning. However, the energy consumption of agricultural machines in real farmland operations not only depends on the travelling distance, but also is significantly affected by the slope of the terrain and the angle of the operating direction. In order to evaluate the effectiveness of the energy consumption model constructed in this paper in reducing the energy consumption of planning, this study compares and analyses the total energy consumption, the energy consumption reduction rate (P_E), and the path reduction rate (P_L) of unmanned agricultural machines under two scenarios of optimal energy consumption and optimal path length.

$$P_E={\displaystyle \frac{\left(E_L-E_P\right)}{E_P}}$$

(17)

$$P_L={\displaystyle \frac{\left(L_P-L_L\right)}{L_L}}$$

(18)

where E_P is the operational energy consumption of the unmanned agricultural machine in the energy consumption optimal case, J; L_P is the operational path length of the unmanned agricultural machine in the energy consumption optimal case, m; E_L is the operational energy consumption of the unmanned agricultural machine in the path length optimal case, J; and L_L is the operational path length of the unmanned agricultural machine in the path length optimal case, m.

3.2. Travelling Path Selection

There are three modes in which unmanned agricultural aircraft can operate in the field: random, spiral, or reciprocating. This study selects the reciprocating operation mode, which considers the contours of the farmland and the minimum turning radius of the agricultural machinery. The aim is to generate an operation path with strong continuity and minimum repetitive coverage. Additionally, the shortest path criterion is employed to plan the farm machine’s routes in and out of the field. A heuristic search algorithm is then used to determine the optimal entry and exit points, effectively reducing non-operational travel distance.

3.2.1. Access Routes

The travelling distance of the agricultural machine from the point of the lower field to the initial operating position is defined as the entry path, and its planning process is as follows:

(1)

Read the coordinate data of the vertices of the polygon at the boundary of the farmland.

(2)

Set the starting point of the farmland for the farm machine to go down to the field.

(3)

Calculate the shortest distance from this field launching point to each boundary line segment and determine the nearest boundary.

(4)

If the initial operation point is located at the boundary, the farm machine can travel directly to the target operation point in a straight line; otherwise, it is necessary to travel along the boundary of the farmland to the initial operation position using a heuristic search strategy. The entry path planning process of the reciprocating operation mode is detailed in Figure 7.

3.2.2. Exit Path

The distance travelled by a farm machine to drive back to the lower field point from the terminated operation position after completing a full coverage operation is defined as the exit path. The planning strategy is implemented according to the following principles:

Calculate the clockwise encircling distance (d_s) and counterclockwise encircling distance (d_n) between the termination operation location and the lower field point. When the condition d_s ≤ d_n is satisfied, the farm machine returns in the clockwise direction; conversely, it chooses to travel in the anti-clockwise direction. This decision-making mechanism is not affected by the relative orientation of the termination point and the downfield point. The specific flow of exit path planning is detailed in Figure 8.

3.3. Realization of a Single-Field Full-Coverage Path Planning Scheme

The flowchart of the three-dimensional terrain single-field block full coverage path planning scheme is shown in Figure 9. The main contents include: input of farmland and agricultural machinery data, visualization of farmland data, construction of energy consumption optimization model and search for driving angles, construction of turning decision model at the field edge, and visualization of search paths, etc.

4. Multi-Strategy Improvement of the Dung Beetle Optimization Algorithm

In practice, multi-field operations need to determine the optimal traversal sequence based on the task requirements. In this study, the multi-field traversal sequence problem is modelled as a traveler’s problem (TSP) and solved using an improved dung beetle optimization algorithm.

The dung beetle optimization algorithm (DBO) was first proposed by Xue and Shen in 2022 [21], and its mathematical model simulates five types of natural behaviors of dung beetles: dung ball rolling, navigational dancing, food searching, resource stealing, and reproduction. The algorithm can efficiently handle complex engineering optimization problems by balancing global exploration and local exploitation capabilities with the advantages of fast convergence and high solution accuracy.

4.1. Improving the Dung Beetle Optimization Algorithm

Although the dung beetle optimization algorithm has the advantages of strong optimization searching ability and fast convergence speed, an in-depth study reveals that it has the problems of imbalance between global exploration and local exploitation ability, easy to falling into local optimum, and insufficient global exploration ability. Aiming at the above defects, this study improves the initialization stage and convergence factor.

4.1.1. Improved Tent Chaos Mapping Initialization Population

Similar to most population intelligence algorithms, the original DBO algorithm adopts a random approach to generate the initial population distribution, which easily leads to insufficient population diversity and a low convergence rate. In order to enhance the algorithm’s ability to explore the global world in the early stages, this study introduces the chaotic mapping operator for population initialization.

Chaos, as a typical nonlinear dynamical phenomenon, is widely used in the field of optimization search because its sequences have traversal characteristics and random features. Commonly used chaotic perturbation models include Logistic mapping and Tent mapping; Logistic mapping presents a high probability density distribution in the boundary region of the solution space, which reduces the search efficiency when the global optimal solution is located in the central region; in contrast, Tent chaotic mapping has more uniform traversal distribution characteristics and a faster iteration convergence rate. Therefore, this paper adopts Tent mapping to generate uniformly distributed chaotic sequences, which effectively reduces the sensitivity of the algorithm to the initial solution.

The expression of the Tent chaotic mapping is as follows:

$$x_i+1=\left\{\begin{array}{c}2x_i, 0\le x\le {\displaystyle \frac{1}{2}}\\ 2\left(1-x_i\right), {\displaystyle \frac{1}{2}}<x\le 1\end{array}\right.$$

(19)

Analyzing the chaotic iterative sequence of the Tent, it can be found that there are small cycles and unstable cycle points in the sequence. In order to avoid the Tent chaotic sequence falling into the small cycle points and unstable cycle points during the iteration, a random variable rand (0, 1), N is introduced into the original Tent chaotic mapping expression, then the improved Tent chaotic mapping expression is as follows:

$$x_i+1=\left\{\begin{array}{c}2x_i+rand\left(\mathrm{0,1}\right)\times {\displaystyle \frac{1}{N}}, 0\le x\le {\displaystyle \frac{1}{2}}\\ 2\left(1-x_i\right)+rand\left(\mathrm{0,1}\right)\times {\displaystyle \frac{1}{N}}, {\displaystyle \frac{1}{2}}<x\le 1\end{array}\right.$$

(20)

where N is the number of particles within the sequence. Introducing the random variable rand (0, 1), N not only maintains the randomness, traversal, and regularity of Tent chaotic mapping, but also effectively avoids the iteration falling into small and unstable cycle points. In this paper, the improved Tent chaotic mapping is used to replace the random initialization, improve the distribution quality of the initial population in the search space, enhance the global search capability, and thus improve the algorithm solution accuracy.

4.1.2. Fusion Osprey Optimization Algorithm

The global exploration strategy of the Osprey optimization algorithm in the first stage is used to replace the position update formula of the original dung beetle algorithm for the rolling ball stage. The primitive dung beetle rolling ball stage formula is shown in (21).

$$\begin{array}{c}{x}_i\left(t+1\right)=x_i\left(t\right)+\alpha \times k\times x_i\left(t-1\right)+b\times ∆x\\ ∆x=\left|x_i\left(t\right)-X^w\right|\end{array}$$

(21)

where

t: represents the current iteration number;

x_i(t): denotes the position information of the ith dung beetle at the tth iteration;

k ∈ (0, 0.2] denotes a constant value which indicates the deflection coefficient;

b: indicates a constant value belonging to (0, 1);

α: is a natural coefficient which is assigned −1 or 1;

X_W: indicates the global worst position;

Δx: is used to simulate changes of light intensity;

The global exploration strategy of the Osprey optimization algorithm can make up for the shortcomings of the dung beetle algorithm’s rolling behavior, which relies only on the worst value, lacks inter-individual information interactions, and has more parameters. The global exploration strategy of the Osprey algorithm is used to randomly detect the position of the dung ball and roll it, and its first-stage global exploration strategy is formulated as follows:

$$x_{i,j}^{P1}=x_{i,j}+r_{i,j}·\left(S{F}_{i,j}-I_{i,j}·x_{i,j}\right)$$

(22)

where

$x_{i,j}^{P1}$ is the jth dimension of the position of the ith dung beetle after rolling the ball;

$x_{i,j}$ is the jth dimension of the current position of the ith dung beetle rolling the ball;

$r_{i,j}$ is the degree of position offset, whose value is a random number in the interval (0,1);

$F_{i,j}$: The jth dimension of the position determined by the i-th dung beetle rolling the ball;

$I_{i,j}$: The direction of the dung beetle’s rolling behavior, randomly taking values of 1 or 2.

4.1.3. Adaptive T-Perturbation Distribution Strategy

Perturbation of the t-distribution of the foraging behavior of small dung beetles during the dung beetle foraging phase. The primitive small dung beetle foraging behavior equation is shown in (23):

$$x_i\left(t+1\right)=x_i\left(t\right)+C_1\times \left(x_i\left(t\right)-L{b}^b\right)+C_2\times \left(x_i\left(t\right)-U{b}^b\right)$$

(23)

where

x_i (t) indicates the position information of the ith small dung beetle at the tth iteration;

C₁ represents a random number that follows a normal distribution;

C₂ denotes a random vector belonging to (0, 1);

Lb^b and Ub^b mean the lower and upper bounds.

Bounds of the optimal foraging area, respectively, and other parameters are defined in (3).

The t-distribution variational perturbation with the number of iterations as the parameter of degrees of freedom is used to improve the foraging behavior of small dung beetles, so that the algorithm has a strong global exploration ability in the early iteration period and a good local exploitation ability in the later period, and improves the convergence speed. The specific position update method is as follows:

$$X_{new}^j=X_{best}^j+t\left(C\_iter\right)·X_{best}^j$$

(24)

where

X_new: The latest location of foraging dung beetles;

X_best: Best location for foraging dung beetles;

C_iter: T-distribution variation operator for freedom degree parameters.

4.1.4. Multi-Strategy Improved Dung Beetle Optimization Algorithm

To address the issue of imbalance between global exploration and local exploitation capabilities in the iterative process of the DBO optimization algorithm, which can lead to getting stuck in local optima and weak global exploration capabilities, the Tent chaos mapping is introduced during population initialization to ensure uniform distribution of initial population members; additionally, the algorithm integrates the Fish Eagle Optimization Algorithm and an adaptive T-disturbance distribution strategy to perturb the foraging behavior of small dung beetles, enabling the dung beetle algorithm to exhibit strong global development capabilities in the early stages of iteration and excellent local exploration capabilities in the later stages, while also improving the algorithm’s convergence speed. The pseudocode for the improved DBO optimization algorithm combining the Osprey optimization algorithm and adaptive T-distribution strategy (Hereinafter referred to as OTDBO) is as follows, and Figure 10 shows the flowchart of the OTDBO optimization algorithm Algorithm 1.

Algorithm 1 OTDBO optimization algorithm

Input: Maximum iteration count Tmax, population size N Output: Optimal position $X_{best}$, fitness value fb 1: Randomly initialize the population and define algorithm-related parameters 2: while (t ≤ Tmax) do 3: for i = N:1 do 4: if i ∈rolling ball beetle then 5: if random number < $\delta$ then 6:    Update the rolling ball beetle position using Formula (21) 7: else 8:    Update the rolling beetle’s position using Formula (22) 9: end if 10:    end if 11: if i ∈ foraging beetle then 12: Update the foraging beetle’s position using Formula (24) 13:      end if 14: if i ∈ stealing beetle then 15:    Update the position of the stealing beetle using Formula (7) [21] 16:        end if 17: if i ∈brooding beetle then 18:    Update the position of the brood beetle using Formula (7) from reference [21] 19: end if 20: end for 21. If there is a value better than the current global optimal value, update the optimal solution and optimal value 22.    t = t + 1 23. end while 24. Output the fitness value fb

Figure 10. Flowchart of the improved dung beetle optimization algorithm [21].

Figure 10. Flowchart of the improved dung beetle optimization algorithm [21].

4.2. Performance Validation of OTDBO Algorithm

In order to verify the effectiveness of the improved strategy, the OTDBO algorithm was compared with six intelligent algorithms, including the original dung beetle algorithm (DBO), the subtractive optimizer algorithm (SABO), the Northern Goshawk algorithm (NGO), the Whale algorithm (WOA), the Harris Hawk Optimization algorithm (HHO), and the Grey Wolf Optimization algorithm (GWO).

The CEC2005 test set is a widely used benchmark test set in the field of evolutionary computation, primarily used to evaluate the performance of optimization algorithms. It was introduced by the IEEE International Conference on Evolutionary Computation (CEC) in 2005 [24]. These test functions cover a variety of characteristics, including single-peaked, multi-peaked, high-dimensional, and low-dimensional, enabling effective testing of algorithm performance across different levels of problem complexity, including search capability, convergence speed, and solution accuracy. Its design objective is to provide a unified evaluation standard for optimization algorithms, facilitating researchers in comparing the performance of different algorithms.

Therefore, this paper compares the optimal values, average values, and standard deviations of each algorithm on the 23 test functions in the CEC2005 test set. Experimental setup: 30 independent runs, maximum iteration counter of 500, and a population size of 30. The comparison algorithm parameters are set according to the original literature. Among these, F1~F7 are single-peak functions, and F8~F23 are multipeak functions. The 23 test functions are listed in

Table 1. CEC2005 Test Function Information.

Function Category	Test Function	Dimension	Value Range	Theoretical Value
Single peak	$F_1\left(x\right)={\sum }_{i=1}^n{x}_i^2$	30	[−100, 100]	0
	$F_2\left(x\right)={\displaystyle {\sum }_{i=1}^n}\left|x_i\right|+{\displaystyle {\prod }_{i=1}^n}\left|x_i\right|$	30	[−10, 10]	0
	$F_3\left(x\right)={\sum }_{i=1}^n{\left({\sum }_{j=1}^i{x}_j\right)}^2$	30	[−100, 100]	0
	$F_4\left(x\right)={max}_i\left\{\left|x_i\right|\right.,\left.1\le i\le n\right\}$	30	[−100, 100]	0
	$F_5\left(x\right)={\displaystyle {\sum }_{i=1}^n}\left|x_i\right|+{\displaystyle {\prod }_{i=1}^n}\left|x_i\right|$	30	[−30, 30]	0
	$F_6\left(x\right)={\sum }_{i=1}^n{\left[x_i+0.5\right]}^2$	30	[−100, 100]	0
	$F_7\left(x\right)={\sum }_{i=1}^{n}i{x}_i^4+random\left(\mathrm{0,1}\right)$	30	[−128, 128]	0
Multipeak	$F_8\left(x\right)={\sum }_{i=1}^n-x_i\mathit{sin}\sqrt{\left|x_i\right|}$	30	[−500, 500]	418.982 9 × D
	$F_9\left(x\right)={\sum }_{i=1}^n\left[x_1^2-10\mathit{cos}\left(2\pi x_i\right)\right]+20+e$	30	[−512, 512]	0
	$F_{10}\left(x\right)=-20exp\left(-0.2\sqrt{{\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{x}_i^2}\right)-exp\left({\displaystyle \frac{1}{n}}{\displaystyle {\sum }_{i=1}^n}\mathit{cos}\left(2\pi x_i\right)\right)+20+e$	30	[−32, 32]	0
	$F_{11}\left(x\right)={\displaystyle \frac{1}{4000}}{\sum }_{i=1}^n{x}_i^2-{\displaystyle {\prod }_{i=1}^n}\mathit{cos}\left({\displaystyle \frac{x_i}{\sqrt{i}}}\right)+1$	30	[−600, 600]	0
	$F_{12}\left(x\right)={\displaystyle \frac{\pi }{n}}\left\{10\mathit{sin}\left(\pi y_1\right)+\right.{\sum }_{i=1}^{n-1}{\left(y_i-1\right)}^2\left[1+10\mathit{sin}2\left(\pi y_i+1\right)\right]+{\left(y_n\_1\right)}^2+\left.{\sum }_{i=1}^{n}u\left(x_i,\mathrm{10,100,4}\right)\right\}u\left(x_i,a,k,m\right)=\left\{\begin{array}{c}k{\left(x_i-a\right)}^m\\ 0,-a<x_i<a\\ k{\left({-x}_i-a\right)}^m,x_i<-a\end{array}\right.$	30	[−50, 50]	0
	$F_{13}\left(x\right)=0.1\left\{{\mathit{sin}}^2\left(3\pi x_1\right)+{\sum }_{i=1}^n{\left(x_i-1\right)}^2\left[1+{\mathit{sin}}^2\left(3\pi x_i+1\right)\right]\right.\left.+{\left(x_n-1\right)}^2\left[{1+\mathit{sin}}^2\left(2\pi x_n\right)\right]\right\}+{\displaystyle {\sum }_{i=1}^n}u\left(x_i,\mathrm{5,100,4}\right)$	30	[−50, 50]	0
	$F_{14}\left(x\right)={\left({\displaystyle \frac{1}{500}}+{\sum }_j^{25}{\displaystyle \frac{1}{j+{\sum }_{i=1}^2{\left(x_i-a_{ij}\right)}^6}}\right)}^{-1}$	2	[−65, 65]	1
	$F_{15}\left(x\right)={\sum }_{i=1}^{11}{\left[a_i-{\displaystyle \frac{x_1\left(b_i^2+b_i^2{x}^2\right)}{b_i^2+b_i{x}_3+x_4}}\right]}^2$	4	[−5, 5]	0.000 3
	$F_{16}\left(x\right)=4x_1^2-2.1x_1^4+{\displaystyle \frac{1}{4}}{x}_1^6+x_1{x}_2-4x_2^2+4x_2^4$	2	[−5, 5]	−1.301 6
	$F_{17}\left(x\right)={\left(x_2-{\displaystyle \frac{5.1}{4{\pi }^2}}{x}_1^2+{\displaystyle \frac{5}{\pi }}-6\right)}^2+10\left(1-{\displaystyle \frac{1}{8\pi }}\right)\mathit{cos}x+10$	2	[−5, 5]	0.398
	$F_{18}\left(x\right)=\left[1+{\left(x_1+x_2+1\right)}^2\left(19-14x_1+6x_1{x}_2+3x_2^2\right)\right] \times \left[30+{\left(2x_1-3x_2\right)}^2 \times \left(18-3{2x}_1+12x_2^2+48x_2-36x_1{x}_2+27x_2^2\right)\right]$	2	[−5, 5]	3
	$F_{19}\left(x\right)=-{\sum }_{i=1}^{4}c exp\left(-{\sum }_{j=1}^{3}a{i}_j{\left(x_j-p_{ij}\right)}^2\right)$	4	[1, 3]	−3.86
	$F_{20}\left(x\right)=-{\sum }_{i=1}^{4}c exp\left(-{\sum }_{j=1}^{6}a{i}_j{\left(x_j-p_{ij}\right)}^2\right)$	6	[0, 1]	−3.32
	$F_{21}\left(x\right)=-{\sum }_{i=1}^5{\left[\left(X-a_i\right){\left(X-a_i\right)}^T+c_i\right]}^{-1}$	4	[0, 10]	−10.153 2
	$F_{22}\left(x\right)=-{\sum }_{i=1}^7{\left[\left(X-a_i\right){\left(X-a_i\right)}^T+c_i\right]}^{-1}$	4	[0, 10]	−10.402 8
	$F_{23}\left(x\right)=-{\sum }_{i=1}^{10}{\left[\left(X-a_i\right){\left(X-a_i\right)}^T+c_i\right]}^{-1}$	4	[0, 10]	−10.536 3

, the experimental results are shown in

Table 2. Test results of OTDBO in comparison with other group intelligence algorithms.

		OTDBO	DBO	SABO	GWO	NGO	WOA	HHO
F1	min	0.00 × 1000	9.39 × 10−199	7.07 × 10−204	4.52 × 10−35	2.83 × 10−90	4.42 × 10−97	9.23 × 10−116
	std	0.00 × 1000	4.15 × 10−95	0.00 × 1000	5.05 × 10−33	2.78 × 10−88	4.12 × 10−83	8.76 × 10−102
	avg	0.00 × 1000	7.57 × 10−96	8.27 × 10−201	3.02 × 10−33	1.79 × 10−88	1.28 × 10−83	1.90 × 10−102
F2	min	1.99 × 10−248	6.95 × 10−89	3.07 × 10−115	2.09 × 10−20	8.89 × 10−47	1.95 × 10−59	1.24 × 10−61
	std	0.00 × 1000	6.19 × 10−55	7.34 × 10−114	4.42 × 10−20	7.25 × 10−46	2.61 × 10−52	6.64 × 10−52
	avg	2.62 × 10−300	1.13 × 10−55	7.57 × 10−114	7.27 × 10−20	7.69 × 10−46	4.81 × 10−53	1.46 × 10−52
F3	min	0.00 × 1000	2.09 × 10−159	5.90 × 10−88	2.79 × 10−10	1.74 × 10−28	7.83 × 1003	1.06 × 10−107
	std	0.00 × 1000	9.49 × 10−67	1.47 × 10−28	2.90 × 10−07	1.14 × 10−21	1.27 × 1004	1.48 × 10−80
	avg	0.00 × 1000	1.75 × 10−67	2.68 × 10−29	9.48 × 10−08	2.94 × 10−22	3.05 × 1004	2.71 × 10−81
F4	min	3.57 × 10−237	7.91 × 10−82	1.71 × 10−79	3.62 × 10−09	1.29 × 10−38	1.61 × 10−05	1.60 × 10−56
	std	0.00 × 1000	1.79 × 10−57	5.51 × 10−78	2.22 × 10−08	1.09 × 10−37	2.88 × 1001	1.71 × 10−49
	avg	8.10 × 10−291	3.93 × 10−58	4.27 × 10−78	2.53 × 10−08	1.20 × 10−37	4.16 × 1001	3.27 × 10−50
F5	min	2.28 × 1001	2.47 × 1001	2.72 × 1001	2.54 × 1001	2.48 × 1001	2.69 × 1001	4.78 × 10−05
	std	5.68 × 1000	1.80 × 10−01	4.23 × 10−01	8.98 × 10−01	3.53 × 10−01	4.58 × 10−01	5.01 × 10−03
	avg	9.29 × 10−06	2.51 × 1001	2.84 × 1001	2.68 × 1001	2.56 × 1001	2.74 × 1001	3.91 × 10−03
F6	min	5.61 × 10−12	6.50 × 10−10	1.34 × 1000	3.70 × 10−05	5.15 × 10−07	1.78 × 10−02	8.36 × 10−08
	std	9.21 × 10−12	2.60 × 10−08	5.11 × 10−01	2.71 × 10−01	2.84 × 10−06	6.45 × 10−02	6.51 × 10−05
	avg	2.42 × 10−13	1.78 × 10−08	2.29 × 1000	4.56 × 10−01	1.39 × 10−06	7.53 × 10−02	6.46 × 10−05
F7	min	3.06 × 10−04	1.82 × 10−04	2.43 × 10−06	2.97 × 10−04	1.09 × 10−04	9.64 × 10−05	1.02 × 10−06
	std	2.57 × 10−04	1.76 × 10−03	1.15 × 10−04	3.68 × 10−04	2.25 × 10−04	3.17 × 10−03	1.22 × 10−04
	avg	2.08 × 10−06	2.12 × 10−03	8.87 × 10−05	1.04 × 10−03	4.38 × 10−04	2.38 × 10−03	1.02 × 10−04
F8	min	−1.19 × 1004	−1.19 × 1004	−4.32 × 1003	−7.38 × 1003	−8.77 × 1003	−1.26 × 1004	−1.26 × 1004
	std	9.41 × 1002	1.29 × 1003	4.29 × 1002	7.66 × 1002	5.42 × 1002	1.64 × 1003	3.09 × 1002
	avg	−1.26 × 1004	−9.02 × 1003	−3.22 × 1003	−6.18 × 1003	−7.75 × 1003	−1.10 × 1004	−1.25 × 1004
F9	min	0.00 × 1000	0.00 × 1000	0.00 × 1000	5.68 × 10−14	0.00 × 1000	0.00 × 1000	0.00 × 1000
	std	0.00 × 1000	5.25 × 1001	0.00 × 1000	3.64 × 1000	0.00 × 1000	2.47 × 10−14	0.00 × 1000
	avg	0.00 × 1000	2.48 × 1001	0.00 × 1000	2.55 × 1000	0.00 × 1000	7.58 × 10−15	0.00 × 1000
F10	min	4.44 × 10−16	4.44 × 10−16	4.00 × 10−15	2.89 × 10−14	4.00 × 10−15	4.44 × 10−16	4.44 × 10−16
	std	0.00 × 1000	0.00 × 1000	0.00 × 1000	4.73 × 10−15	1.66 × 10−15	2.91 × 10−15	0.00 × 1000
	avg	4.44 × 10−16	4.44 × 10−16	4.00 × 10−15	4.36 × 10−14	5.06 × 10−15	3.52 × 10−15	4.44 × 10−16
F11	min	0.00 × 1000	0.00 × 1000	0.00 × 1000	0.00 × 1000	0.00 × 1000	0.00 × 1000	0.00 × 1000
	std	0.00 × 1000	1.79 × 10−02	0.00 × 1000	6.82 × 10−03	0.00 × 1000	1.57 × 10−02	0.00 × 1000
	avg	0.00 × 1000	3.26 × 10−03	0.00 × 1000	3.23 × 10−03	0.00 × 1000	2.87 × 10−03	0.00 × 1000
F12	min	6.91 × 10−03	1.23 × 10−11	7.14 × 10−02	1.30 × 10−02	3.85 × 10−08	4.89 × 10−04	1.26 × 10−09
	avg	1.14 × 10−14	3.52 × 10−03	1.97 × 10−01	3.03 × 10−02	1.78 × 10−07	6.09 × 10−03	2.57 × 10−06
	min	6.59 × 10−03	4.05 × 10−08	7.96 × 10−01	5.72 × 10−05	1.29 × 10−05	4.47 × 10−02	2.13 × 10−07
F13	std	1.28 × 10−02	6.51 × 10−02	6.73 × 10−01	1.92 × 10−01	6.06 × 10−02	1.24 × 10−01	1.02 × 10−04
	avg	2.38 × 10−14	5.92 × 10−02	2.63 × 1000	3.55 × 10−01	5.34 × 10−02	1.89 × 10−01	5.33 × 10−05
	min	9.98 × 10−01	9.98 × 10−01	9.98 × 10−01	9.98 × 10−01	9.98 × 10−01	9.98 × 10−01	9.98 × 10−01
F14	std	8.25 × 10−17	3.62 × 10−01	1.38 × 1000	4.71 × 1000	0.00 × 1000	1.88 × 1000	1.26 × 1000
	avg	9.98 × 10−01	1.06 × 1000	2.58 × 1000	4.68 × 1000	9.98 × 10−01	1.79 × 1000	1.36 × 1000
	min	3.37 × 10−04	3.08 × 10−04	3.28 × 10−04	3.07 × 10−04	3.07 × 10−04	3.08 × 10−04	3.09 × 10−04
F15	std	4.78 × 10−05	3.01 × 10−04	8.01 × 10−04	8.12 × 10−03	2.51 × 10−08	3.70 × 10−04	1.78 × 10−04
	avg	3.07 × 10−04	7.39 × 10−04	6.87 × 10−04	4.40 × 10−03	3.07 × 10−04	6.89 × 10−04	3.75 × 10−04
	min	−1.03 × 1000	−1.03 × 1000	−1.03 × 1000	−1.03 × 1000	−1.03 × 1000	−1.03 × 1000	−1.03 × 1000
F16	std	6.58 × 10−16	6.65 × 10−16	1.07 × 10−02	9.70 × 10−09	6.71 × 10−16	9.12 × 10−10	1.41 × 10−10
	avg	−1.03 × 1000	−1.03 × 1000	−1.02 × 1000	−1.03 × 1000	−1.03 × 1000	−1.03 × 1000	−1.03 × 1000
	min	3.98 × 10−01	3.98 × 10−01	3.98 × 10−01	3.98 × 10−01	3.98 × 10−01	3.98 × 10−01	3.98 × 10−01
F17	std	0.00 × 1000	0.00 × 1000	5.65 × 10−02	1.53 × 10−06	0.00 × 1000	1.43 × 10−06	6.14 × 10−07
	avg	3.98 × 10−01	3.98 × 10−01	4.32 × 10−01	3.98 × 10−01	3.98 × 10−01	3.98 × 10−01	3.98 × 10−01
	min	3.90 × 1000	3.00 × 1000	3.00 × 1000	3.00 × 1000	3.00 × 1000	3.00 × 1000	3.00 × 1000
F18	std	4.93 × 1000	1.67 × 10−15	1.27 × 1000	1.97 × 10−05	5.34 × 10−16	7.39 × 10−05	2.39 × 10−08
	avg	3.00 × 1000	3.00 × 1000	3.80 × 1000	3.00 × 1000	3.00 × 1000	3.00 × 1000	3.00 × 1000
	min	−3.86 × 1000	−3.86 × 1000	−3.86 × 1000	−3.86 × 1000	−3.86 × 1000	−3.86 × 1000	−3.86 × 1000
F19	std	2.63 × 10−15	1.98 × 10−03	1.57 × 10−01	1.44 × 10−03	2.71 × 10−15	2.90 × 10−03	7.97 × 10−04
	avg	−3.86 × 1000	−3.86 × 1000	−3.67 × 1000	−3.86 × 1000	−3.86 × 1000	−3.86 × 1000	−3.86 × 1000
	min	−3.32 × 1000	−3.32 × 1000	−3.32 × 1000	−3.32 × 1000	−3.32 × 1000	−3.32 × 1000	−3.30 × 1000
F20	std	1.36 × 10−15	7.39 × 10−02	7.97 × 10−02	8.03 × 10−02	1.41 × 10−15	9.37 × 10−02	6.64 × 10−02
	avg	−3.32 × 1000	−3.25 × 1000	−3.24 × 1000	−3.26 × 1000	−3.32 × 1000	−3.24 × 1000	−3.20 × 1000
	avg	1.14 × 10−14	3.52 × 10−03	1.97 × 10−01	3.03 × 10−02	1.78 × 10−07	6.09 × 10−03	2.57 × 10−06
F21	min	−1.02 × 1001	−1.02 × 1001	−5.36 × 1000	−1.02 × 1001	−1.02 × 1001	−1.02 × 1001	−1.02 × 1001
	std	6.02 × 10−15	2.63 × 1000	2.44 × 10−01	2.06 × 1000	2.37 × 10−09	1.93 × 1000	1.55 × 1000
	avg	−1.02 × 1001	−6.36 × 1000	−4.98 × 1000	−9.14 × 1000	−1.02 × 1001	−9.30 × 1000	−5.56 × 1000
F22	min	−1.04 × 1001	−1.04 × 1001	−8.65 × 1000	−1.04 × 1001	−1.04 × 1001	−1.04 × 1001	−1.02 × 1001
	std	9.33 × 10−16	2.82 × 1000	8.10 × 10−01	1.61 × 1000	1.62 × 10−12	3.03 × 1000	9.41 × 10−01
	avg	−1.04 × 1001	−8.48 × 1000	−5.09 × 1000	−9.87 × 1000	−1.04 × 1001	−7.83 × 1000	−5.26 × 1000
F23	min	−1.05 × 1001	−1.05 × 1001	−5.13 × 1000	−1.05 × 1001	−1.05 × 1001	−1.05 × 1001	−5.13 × 1000
	std	1.32 × 10−15	2.96 × 1000	4.37 × 10−01	7.46 × 10−04	2.80 × 10−15	3.04 × 1000	1.08 × 10−03
	avg	−1.05 × 1001	−7.67 × 1000	−4.74 × 1000	−1.05 × 1001	−1.05 × 1001	−8.29 × 1000	−5.13 × 1000

(bold text indicates the best results), and Figure 11 displays the convergence iteration curves for each algorithm.

4.2.1. Experimental Analysis of the OTDBO Algorithm on Single-Peak Benchmark Test Functions

In Table 2, F1~F7 are the experimental results of the OTDB algorithm and the control algorithm for the single-peak test function, and the optimal results are shown in bold. It can be seen from the test results that the OTDBO algorithm can find the theoretical optimal value on F1~F4, which is far ahead of the other comparison algorithms. Although the proposed algorithm fails to find the theoretical optimal value on functions F6 and F7, the average value of the optimization results is the best among all the compared algorithms. The test results on F5 are also ranked among the top of several algorithms. From the variance results in Table 1, it can be seen that the OTDBO algorithm has very good robustness. The convergence plots of the OTDBO algorithm and the comparison algorithms on the single-peak test function are shown in Figure 11. From the figure, it is easy to see that the OTDBO algorithm shows a very fast convergence speed and high convergence accuracy. Therefore, the OTDBO algorithm shows a strong local development ability on the single-peak test function.

4.2.2. Experimental Analysis of the OTDBO Algorithm on Multiple-Peak Benchmark Test Functions

Table 2 shows the experimental results of the OTDBO algorithm and other comparative algorithms on the multipeak test functions F8~F23, and the optimal results are shown in bold. The OTDBO algorithm finds the theoretical optimal values on F9~F11, F14, F16, and F19~F23, and the mean results on F8 are the best relative to the other algorithms, and the convergence accuracy is higher on F12 than the other comparative algorithms, and only inferior to the HHO algorithm on F13. The convergence accuracy is higher on F12 than other comparative algorithms, and only inferior to the HHO algorithm on F13. From the variance of several test functions in Table 2, DBO has a smaller variance and shows stronger robustness. The iterative convergence of the OTDBO algorithm and the comparison algorithms on the multipeak test functions is shown in Figure 11. There are many algorithms that can find the optimal value on F9~F11, but the proposed algorithm, OTDBO, has a much faster convergence speed and is ahead of the other comparison algorithms. The above analysis proves that the OTDBO algorithm has a better ability to balance global exploration performance with local exploitation.

Analysis of the above experimental results shows that the OTDBO algorithm outperforms the others in terms of optimization accuracy, convergence speed, and robustness for most functions, indicating that the improved DBO algorithm has stronger optimization performance.

5. Based on the Actual Farmland Simulation Test Analysis

In order to verify the feasibility and effectiveness of the algorithm in solving practical problems, this study selects a field (Figure 1) located in the hilly area of southwestern Gui at 23°09′ N latitude and 106°47′ E longitude as the object of study, and acquires the digital elevation model (DEM) data and carries out simulation tests. According to the distribution of field roads and farmland characteristics, the whole operation area was divided into 12 sub-areas.

Farmland parameter information is shown in

Table 3. Field parameters.

Field Number	Area/m2	Perimeter/m	Highest Altitude/m	Lowest Elevation/m	Altitude Difference/m	Maximum Slope/°	Minimum Slope/°	Slope Difference/°
1	738.84	3369.05	721	714	7	13.19	5.58	7.61
2	569.71	2871.3	718	714	4	13.19	6.43	6.76
3	586.84	2547.8	718	714	4	14.98	6.96	8.02
4	585.13	2584.33	720	717	3	18.45	6.96	11.49
5	412.52	8982.09	724	720	4	19.40	6.96	12.44
6	425.00	3493.14	724	719	5	11.43	6.96	4.47
7	503.87	5903.96	724	715	9	19.40	6.96	12.44
8	253.44	4030.14	718	713	5	11.43	6.42	5.01
9	364.31	4363.6	723	716	7	9.09	7.12	1.97
10	263.43	4457.44	723	716	7	14.58	9.08	5.5
11	256.54	4410.28	724	717	7	21.62	9.09	12.53
12	528.69	6015.20	724	715	9	10.86	3.54	7.32

, with all data sourced from the geospatial data cloud [29]. The 12 work areas vary in size and shape. The smallest work area has an area of 253.44 m² (Field 8), while the largest work area has an area of 738.84 m² (Field 1). Among the 12 fields, Field 7 has a relatively small area but the greatest elevation difference and slope difference, indicating that the terrain of this hilly mountainous region is relatively complex.

5.1. Optimum Operating Direction Angle for a Single Field

In single-field full-coverage path planning, the operating direction angle β directly affects the quality of agricultural machine operation and energy consumption. MATLAB2023a was used to simulate the full-coverage path planning for 12 sub-fields, and the operation direction angle was searched from 0° to 360° in 5° steps to analyze the energy cost and path length at each angle. The optimal working direction angle was determined using energy consumption cost and path length as evaluation criteria. The simulation results are shown in Figure 12. Taking Field 1 as an example, Figure 12a shows that the energy consumption cost is lowest when the working direction angle is 320°, and highest when it is 225°. Figure 12b shows that the path length is shortest when the working direction angle is 360°, and longest when it is 315°. Figure 13 shows the full coverage path planning results for operating direction angles of 320°, 225°, 360°, and 315°. The corresponding number of turns in Figure 13a, Figure 13b, Figure 13c, and Figure 13d are 52, 44, 67, and 44, respectively, with energy consumption of 5.25 × 10³, 1.26 × 10⁴, 8.62 × 10³, and 5.25 × 10³ J, and path lengths of 1.4 × 10³, 1.05 × 10³, 0.98 × 10³, and 1.53 × 10³ m, respectively.

The above simulation results show that different operating direction angles lead to different path planning results, which directly affect the length of the operating path and the energy consumption of agricultural machines.

Table 4. Full coverage path planning simulation results.

Field Number	Optimal Energy Consumption			Path-Optimal			Energy Consumption Reduction Rate/%	Path Reduction Rate/%
	Optimum Operating Direction Angle/(°)	Power Consumption/(× 103 J)	Path Length/(m)	Optimum Operating Direction Angle/(°)	Power Consumption/(× 103 J)	Path Length/(m)
1	320	5.2	1404	360	8.6	983	65.38	42.82
2	290	4.3	800	5	5.2	775	20.93	3.22
3	295	4.5	778	295	4.5	772	0	0.77
4	25	4.1	806	355	5.8	745	41.46	8.19
5	40	4.1	613	30	4.1	607	0	0.99
6	25	4.5	613	25	4.5	727	0	−15.68
7	280	5.5	1046	25	6.0	1022	9.09	2.34
8	340	3.2	466	340	3.2	466	0	0
9	25	4.8	1255	5	6.1	1250	31.70	0.40
10	25	3.2	523	25	3.2	523	0	0
11	290	3.1	389	320	3.7	379	19.35	2.63
12	135	4.1	831	360	6.2	737	51.22	12.75

shows the energy consumption and optimal operating direction angle for each field under energy-optimal and path-optimal conditions.

Field 1 showed the greatest difference in energy consumption between the energy-optimal and path-optimal objectives. Operating in the energy-optimal condition reduced energy consumption by 65.38%. Fields 1, 2, 4, 9, and 12 had energy consumption reduction rates of over 20% when operating under the energy-optimal objective. However, when the goal is to minimize the path length, only Fields 1 and 12 achieve a reduction rate of more than 10%, with an average reduction rate across the 12 farmlands of just 4.87%.

The results show that planned paths with energy consumption optimization as the target significantly outperform those with path length as the target in hilly, mountainous farmland operations. Using the optimal energy consumption index for path planning can effectively reduce energy consumption, making it more valuable for autonomous agricultural machine path planning.

5.2. Coordinates for the Import and Export of Single-Field Blocks

The optimal operating direction angle for each field is determined by analyzing the operating energy consumption of the agricultural machine in each field. Figure 14 shows the simulation results for the optimal coverage path for each field based on the optimal operating direction angle for single-field full-coverage path planning simulation.

Table 5. Optimal Import and Export Coordinates of the Field.

Field Number	Entrance Coordinates	Exit Coordinates
1	[106′48717, 23′09756]	[106′48746, 23′09752]
2	[106′48760, 23′09728]	[106′48780, 23′09729]
3	[106′48721, 23′09730]	[106′48800, 23′09736]
4	[106′48794, 23′09753]	[106′48882, 23′09746]
5	[106′48805, 23′09734]	[106′48819, 23′09743]
6	[106′48798, 23′09733]	[106′48789, 23′09707]
7	[106′48764, 23′09696]	[106′48780, 23′09726]
8	[106′48759, 23′09721]	[106′48735, 23′09724]
9	[106′48754, 23′09710]	[106′48761, 23′09694]
10	[106′48719, 23′09718]	[106′48741, 23′09712]
11	[106′48730, 23′09727]	[106′48718, 23′09722]
12	[106′48733, 23′09726]	[106′48740, 23′09745]

shows the position coordinates of the agricultural machine entering and leaving each field.

5.3. Full Coverage Path Planning

Assuming that the optimal entry and exit coordinates for agricultural machinery, as well as the entrance and exit positions for each field, have been determined based on the energy consumption model, the OTDBO optimization algorithm can be used to calculate the optimal traversal sequence for multiple fields. The results of the traversal are shown in Figure 15a. To verify the performance of the OTDBO optimization algorithm when solving practical problems, both the original OTDBO optimization algorithm and the OTDBO optimization algorithm were used to solve the optimal traversal sequence for multiple fields simultaneously. The traversal results obtained using the original DBO algorithm are shown in Figure 15b.

The traversal optimal sequence of the OTDBO optimization algorithm is 3→4→5→6→7→8→9→10→11→12→1→2, while the traversal optimal sequence of the original DBO optimization algorithm is 2→3→4→5→6→7→8→9→10→11→12→1. From the distribution positions of each field, it can be seen that both the OTDBO optimization algorithm and the DBO optimization algorithm form a snake-shaped operation path. From the simulation results, the total energy consumption of OTDBO and DBO optimization algorithms is 5.62 × 10⁴ J and 6.13 × 10⁴ J, respectively. The energy consumption of the OTDBO optimization algorithm is 9.2% lower than that of the original DBO optimization algorithm, and OTDBO has the fastest convergence speed during the optimization process.

6. Conclusions and Limitations

6.1. Conclusions

(1)

This study analyses the effects of different operating directions on tractor operating paths and energy consumption in hilly, mountainous areas. Through simulation tests, it determines the optimal operating direction angle for each field, achieving an energy-optimal, full-coverage operating path for a single field.

(2)

The dung beetle optimization algorithm has an imbalance between global exploration and local exploitation ability during the iteration process. It is prone to falling into local optima and has weak global exploration ability. The population is initialized through Tent chaotic mapping. The Osprey optimization algorithm and the adaptive T-perturbation distribution strategy are integrated to perturb the foraging behaviors of small dung beetles. This improves the global exploitation ability of the dung beetle algorithm in the early iteration period and its local exploration ability in the late iteration period, while also improving the convergence speed of the algorithm.

(3)

This study demonstrates that determining the optimal operational path based on minimizing energy consumption can significantly reduce energy usage in a simulated hilly, mountainous environment comprising 12 specific field plots. Field 1 achieved a 65.38% reduction in energy consumption under both energy minimization and path optimization conditions. Reduction rates exceeding 25% were achieved in Fields 1, 2, 4, 9, and 12, with an average reduction across the 12 plots of 19.93%. These results suggest that the proposed method has significant energy-saving potential in these scenarios and with these samples.

(4)

Under the condition that the coordinates of the optimal entry and exit positions of agricultural machinery and the position information of the entrances and exits of each field have been determined based on the energy consumption model, the DBO optimization algorithm and the OTDBO optimization algorithm are simultaneously used to solve the optimal traversal sequence for multiple fields. The total energy consumption of the OTDBO and DBO optimization algorithms is 5.62 × 10⁴ J and 6.13 × 10⁴ J, respectively. The energy consumption of the OTDBO optimization algorithm is 9.2% lower than that of the original DBO optimization algorithm, and the OTDBO optimization algorithm has the fastest convergence speed during the optimization process.

This paper proposes a terrain energy consumption model combined with an improved beetle optimization algorithm (OTDBO) to overcome the limitations of current technologies in agricultural machinery path planning in hilly and mountainous areas. The model considers the impact of slope and elevation on energy consumption and reveals the significant influence of the working direction angle on energy consumption. It also provides a universal, single-field optimization scheme for farmland with arbitrary slopes. This research opens up new avenues for path planning in hilly and mountainous farmland.

6.2. Limitations and Future Work

This paper provides a preliminary summary of research into path planning algorithms for agricultural machinery. Due to limitations in terms of time, resources, capabilities, and experimental conditions, there are certain shortcomings and further research is required. Future studies could explore the following areas:

(1)

Conduct in-depth research into how path planning influences the overall efficiency of agricultural operations. This could involve analyzing factors such as operation duration or overall efficiency in order to address time-sensitive agricultural scenarios, such as urgent harvesting or planting.

(2)

Conduct multi-scenario, real-vehicle testing (e.g., varying crop row spacing, slope terrain, and obstacle distribution) by combining GNSS/IMU positioning data with agricultural machinery parameters (e.g., steering angle and fuel consumption) to validate the effectiveness of the algorithm.

(3)

Extend current single-machine path planning to cluster-based collaborative optimization by studying collision-avoidance strategies and task allocation mechanisms among agricultural machinery to enhance the efficiency of large-scale farmland operations.