Optimization of Farmland Cultivated Land Path Based on Hybrid Adaptive Neighborhood Search Algorithm

Abstract

Path planning for large-scale agricultural fields faces challenges such as irregular field shapes, uncertain boundaries, and the need to balance path efficiency, energy consumption, and coverage quality. To address these problems, this research introduces a strategy-aware hierarchical hybrid optimization framework (HANS) for autonomous agricultural operations. This framework introduces a global principal axis extraction method based on Principal Component Analysis (PCA), utilizing the statistical distribution of field boundaries to guide path direction, thereby improving robustness against boundary noise and irregular geometries. The framework integrates Adaptive Large Neighborhood Search (ALNS) for global exploration and Tabu Search (TS) for local optimization, forming a tightly coordinated hybrid structure. The framework further employs a Pareto-set-based decision support selection strategy to solve a multi-objective optimization model encompassing machine kinematics, turning patterns, and energy-aware cost evaluation. This strategy provides three methods: weighted preference-based compromise solution selection, crowding distance-based diversified solution selection, and single-objective extreme value-based dedicated optimization solution selection. To balance the impact of path length, energy consumption, and coverage rate, we assigned equal or nearly equal weights to them (i.e., (0.33, 0.33, 0.34)). Furthermore, the framework incorporates operators and feedback learning mechanisms specific to agricultural coverage path problems to enable adaptive operator selection and reduce reliance on manual parameter tuning. Simulation results under three representative field scenarios show that compared to fixed-direction planning, HANS improves the average coverage rate by 0.51 percentage points and reduces fuel consumption by 4.34%. Compared to Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Tabu Search (TS), and Simulated Annealing (SA), the proposed method shortens the working path length by 0.37–0.83%, improves coverage rate by 0.34–1.11%, and reduces energy consumption by 0.61–1.03%, while maintaining competitive computational costs. These results demonstrate the effectiveness and practicality of HANS in large-scale autonomous farming operations.

1. Introduction

The cultivation of new-quality productive forces and the advancement of smart agriculture have emerged as the predominant trajectories for the future of agricultural development. The deep integration of agricultural technology and information technology effectively changes traditional farming methods, achieving a high-quality, high-efficiency, and low-energy consumption farming model. With the development of satellite and navigation technologies, unmanned autonomous driving has also been widely applied in the agricultural field. Currently, agricultural machinery autonomous driving technology mainly focuses on autonomous planning of operation paths. Agricultural path planning (ARP) in farm management aims to design or arrange the movement routes of agricultural machinery within the field. Good path design can minimize the travel distance of agricultural machinery, improve operation quality, save a lot of human resources, and effectively solve the problem of high labor costs. Therefore, optimizing path planning for machinery used in agricultural operations within the field is crucial [1]. High-efficiency path planning algorithms can significantly improve the efficiency of agricultural machinery operations, especially in large farms where this technology is highly needed. However, existing path planning algorithms still have limitations when dealing with irregularly shaped fields. On the one hand, traditional method performs operations based on the longest side of the field [2], which leads to a large number of extremely short operation rows at the edges when the field has non-right-angled slanted edges, curved boundaries, or concave polygonal areas. It is particularly important to note that, in this study, the challenges posed by irregular field shapes are primarily focused on in the context of full-area coverage tasks, as the boundary geometry directly affects path decomposition, turning operations, and coverage efficiency. In contrast, for simple point-to-point path planning from one location to another, irregular fields can usually be treated as regular areas with boundary obstacles, and therefore do not present the same level of difficulty. Agricultural machinery needs to turn frequently at the headlands, resulting in wasted operation time and increased unnecessary fuel consumption. On the other hand, most current algorithms only focus on finding the shortest working path, neglecting fuel consumption and coverage rate. Especially in irregular fields, unreasonable path planning leads to frequent turning of agricultural machinery at the headlands. These phenomena are even more pronounced in large, irregularly shaped fields. Therefore, studying methods for determining the optimal travel direction for agricultural machinery planting in irregularly shaped and large-area fields is of great significance for enhancing the quality and efficiency of working.

Currently, many researchers have developed various complete coverage path planning (CPP) algorithms. Based on whether the boundaries of the farmland work area are pre-divided, CPP algorithms can be classified into offline and online operations. The latter primarily relies on sensors for operation, as it requires real-time acquisition and processing of surrounding environmental data. Furthermore, based on the dimensionality of the workspace, CPP algorithms are divided into two-dimensional and three-dimensional types. When agricultural machinery operates in areas with slopes, the influence of the slope needs to be considered, such as the coverage path planning in three-dimensional space proposed by Hameed [3]. In addition, multi-robot solutions have received widespread attention in recent years, as they are very suitable for large-area farmland operations and can significantly reduce operation time [4]. This study focuses on the CPP problem for a single field with flat terrain and no slopes; therefore, a two-dimensional, offline algorithm is used.

Regarding CPP algorithms for individual fields, Hameed proposed a real-time method for generating field geometric representations for operation planning using a method that generates a geometric representation of the field. This geometric representation of the field provides a map on which advanced optimization methods for agricultural machinery path planning problems can be applied for field operation planning. Bochtis and Ruggioukas [5] proposed minimizing headland distance and Han et al. [6] considered the operating travel angle as a decision variable, allowing for the adaptive determination of path direction based on field boundary characteristics, thereby augmenting the overall straight-line path length. Jin and Tang [7] proposed a partitioning planning strategy for large fields, effectively reducing the complexity of the problem. However, most of these methods only consider the single objective of path length, neglecting other important factors such as energy consumption and coverage rate.

Regarding the generation of turning paths, Bochtis and Vougioukas systematically analyzed the impact of different turning patterns on non-working distance in their 2008 study, laying the foundation for subsequent research. Spekken and de Bruin [8] improved agricultural machinery path planning by optimizing turning and service times, proposing a comparative analysis method for various turning strategies. However, although these traditional geometric turning methods are simple to implement, the turning paths are not smooth enough, leading to frequent acceleration and deceleration of agricultural machinery. This not only increases energy consumption and operating time but may also cause additional soil compaction.

To reduce energy consumption and excessive soil compaction during turns, Jensen et al. [9] designed a smooth turning path generation method Built upon B-spline curves. Conesa-Muñoz et al. [10] designed a new path operator to optimize turning paths, reducing the time and energy losses caused by sharp turns. Chamen et al. [11] further quantified the impact of different turning patterns on soil compaction, finding that smooth turning paths can reduce soil compaction depth by more than 30%.

The problem of agricultural machinery path planning is essentially a complex combinatorial optimization problem, characterized by multiple constraints, multiple objectives, and large scale, belonging to the NP-hard class of problems [12]. While exact algorithms can guarantee optimal solutions, their computational complexity increases exponentially with the problem size, making it difficult to solve real-world problems within a reasonable time. Therefore, heuristic and metaheuristic algorithms have become the main methods for solving such problems [13]. These include evolutionary algorithms such as Genetic Algorithms (GA) [14], NSGA-II [15], and MOEA/D [16]; swarm intelligence algorithms such as Ant Colony Optimization (ACO) [17], Particle Swarm Optimization (PSO) [18], and Grey Wolf Optimization [19]; and trajectory-based local search algorithms such as Simulated Annealing (SA) [20], Whale Optimization Algorithm (WOA) [21], and Tabu Search (TS) [22]. Simultaneously, hybrid metaheuristic algorithms have also attracted researchers’ attention. Utamima et al. [23] proposed an evolutionary hybrid neighborhood search method, combining the global search capabilities of evolutionary algorithms with the local optimization capabilities of neighborhood search, achieving better results than single algorithms in field logistics path planning. Seyyedhasani and Dvorak [24] used a hybrid heuristic method to handle path adjustments in dynamic environments, improving the robustness and adaptability of the system. In recent years, algorithms based on Adaptive Large Neighborhood Search (ALNS) have demonstrated powerful solving capabilities and broad applicability in the field of logistics and manufacturing system optimization. Nasri et al. [25] studied the Vehicle Routing Problem with Time Windows (VRPTW) and proposed an improved Adaptive Large Neighborhood Search (MALNS) method, which replaces the traditional roulette wheel selection mechanism commonly used in the ALNS framework with an improved selection function (MCF). MCF comprehensively considers the historical performance of each operator, the dependencies between operator pairs, and the idle time of the operators, and uses dynamic parameter adjustment to balance search intensity and solution diversity, thereby improving the solution quality on standard benchmark instances. In urban delivery, Himstedt and Meisel [26] developed an ALNS-based heuristic algorithm for a two-level vehicle routing problem involving collaborative multi-delivery systems, combining satellite facility management with dynamic operator weight adjustment. Their method aims to coordinate autonomous delivery agents (such as robots and drones) with traditional trucks and validate its effectiveness through practical case studies in complex urban delivery environments.

However, the above ALNS-based methods and their improved variants are mainly developed for urban logistics or manufacturing optimization problems and have not been applied to full-coverage path planning in agricultural environments, explicitly considering the combined effects of irregular field geometry, vehicle kinematic constraints, and energy consumption during headland turns, which are characteristic features of agricultural full-coverage path planning. Existing metaheuristic algorithms each have their advantages in agricultural full-coverage path planning problems, but they generally suffer from insufficient characterization of path continuity, turning costs, and agricultural machinery operation constraints. Evolutionary algorithms and swarm intelligence algorithms have strong global exploration capabilities but limited solution quality, while single local search methods are susceptible to local optima entrapment. Hybrid algorithms that improve individual modules lack a reasonable coordination mechanism. Therefore, it is necessary to design a hybrid metaheuristic algorithm that integrates global search and problem-specific local optimization mechanisms to further improve the quality and stability of the solution set for agricultural full-coverage path planning.

In view of the current state of coverage path planning (CPP) algorithms under agricultural machinery constraints, research attention has gradually shifted from traditional geometric coverage toward integrated optimization frameworks that deeply couple vehicle kinematic characteristics with operational environment constraints. Existing studies primarily focus on practical challenges such as non-smooth turning maneuvers, low operational efficiency, and energy limitations. To address the non-holonomic constraints inherent to agricultural machinery, researchers commonly adopt improved algorithms based on polynomial curves (e.g., Bézier curves or B-splines) combined with Dubins or Reeds–Shepp curves, ensuring curvature continuity and compliance with minimum turning radius requirements. For instance, studies published between late 2023 and 2024 proposed a four-stage decomposition approach that integrates the generalized traveling salesman problem (GTSP) with curvature constraints, enabling smooth and feasible coverage paths in complex farmland environments [27]. Meanwhile, deep reinforcement learning (DRL) has demonstrated significant advantages in handling complex constraints. Recent studies in 2025 and 2026 have combined bidirectional long short-term memory (Bi-LSTM) networks with deep dueling double Q-networks (D3QN) to explicitly address energy consumption constraints arising from payload variations during agricultural operations. By constructing effective payload–energy models, these methods achieve coordinated optimization between coverage efficiency and energy safety [28,29]. In addition, considering the prevalence of irregularly shaped fields with dense obstacles in real-world farmland, Markov-chain-enhanced genetic algorithms (MCGA) have been shown to effectively reduce path overlap rates and improve search efficiency under constrained environments, achieving up to a 21.8% reduction in path length compared to traditional methods [30]. In summary, contemporary CPP algorithms are evolving from single-objective path-length minimization toward multi-objective optimization frameworks that integrate vehicle kinematic constraints, dynamic payload-dependent energy efficiency constraints, and cooperative obstacle avoidance constraints in multi-vehicle scenarios [31].

In response to the inherent challenges of agricultural coverage path planning, including strongly coupled multiple objectives, stringent path–structure constraints, and highly non-convex solution spaces, multi-objective heuristic algorithms have attracted increasing attention due to their ability to simultaneously ensure solution quality and search stability. Therefore, this paper proposes a hybrid optimization framework combining TS and ALNS. TS effectively suppresses cycles and premature convergence during the search process by constructing a tabu mechanism based on (θ, s, τ) solution features, and achieves fine-grained optimization of working angle, swath spacing, and turning strategy within the local neighborhood. ALNS, through the destruction and reconstruction of the current solution, introduces large-scale search perturbations across structural levels, significantly enhancing the algorithm’s global exploration capabilities. Furthermore, by maintaining a Pareto optimal solution archive and combining dominance relationships with crowding distance criteria, the proposed method maintains diversity and convergence of the solution set in the multi-objective space. Experimental results show that this hybrid strategy exhibits superior or comparable overall performance to comparison algorithms in terms of path length, energy consumption, and coverage completeness.

The novelties and key contributions of this paper are outlined below:

• A PCA-based global principal axis extraction method is proposed to guide coverage path planning in cultivated fields. By exploiting the global statistical distribution of boundary points rather than local geometric features, the method provides improved robustness against boundary noise, mapping errors, and degenerate or irregular field shapes.
• A strategy-aware hierarchical hybrid optimization framework (HANS) is developed, in which Adaptive Large Neighborhood Search (ALNS) is responsible for global exploration, while Tabu Search performs local refinement. This explicit separation of optimization roles enables effective coordination between exploration and exploitation.
• We designed unique operators specifically for the full-coverage path planning problem in agricultural fields and a feedback learning mechanism. The feedback learning mechanism is embedded into the ALNS component to dynamically adapt the selection probabilities of destroy and repair operators according to their historical performance. This learning-driven design reduces sensitivity to manual parameter tuning and improves robustness across diverse field geometries.
• A multi-objective optimization model incorporating realistic agricultural constraints—such as machine kinematics, turning patterns, and energy-aware cost evaluation—is formulated and solved using a Pareto-based archive. Extensive simulation and ablation studies demonstrate that the proposed framework achieves consistent performance improvements over conventional heuristic and metaheuristic methods with manageable computational cost.

It should be emphasized that the area coverage path planning problem addressed in this study is formulated as a global, offline optimization problem under static field conditions. The primary objective is to obtain high-quality, near-optimal solutions in regard to coverage rate, total working length, and energy consumption, rather than to provide real-time adaptability to dynamic or uncertain environments. Model predictive control (MPC)-based approaches and other local planning strategies are effective for online trajectory generation and dynamic obstacle avoidance, and they can incorporate metrics such as time or fuel consumption. However, due to their inherently local and receding-horizon nature, such methods are generally far from globally optimal when applied to large-scale area coverage tasks. Therefore, these approaches are not considered as direct competitors in this work.

2. Materials and Methods

2.1. Operational Modeling

This subsection introduces the acquisition and modeling of farmland field information, formulates the objective functions considered in this study, and discusses the turning maneuvers, thereby providing the foundation for the subsequent research.

2.1.1. Agricultural Information Processing

This study selected farmland fields from Huaxing Farm in Changji, Xinjiang, China. A DJI Mavic 3 (SZ DJI Technology Co., Ltd., Shenzhen, China) drone was used as the data collection device, with an operational altitude of 110 m and a constant velocity of 15 m/s. The collected data was processed and stored in a database. The collected map data was also divided into regions, with the boundaries between different fields precisely marked. Since the geographical coordinates of the map data collected by the data acquisition equipment were in WGS84 format, coordinate transformation was used to convert the collected map data to the UTM (Universal Transverse Mercator) coordinate system for subsequent modeling purposes, and the farmland areas were divided accordingly. The farm’s satellite map and its regional division are shown in Figure 1.

2.1.2. Farm Field Modeling

In practical applications, to ensure that agricultural machinery can operate smoothly in the field, It is essential to partition the field into sensible sub-regions. In this study, a field is divided into two sub-regions: the headland area and the working area. The headland area is used for the agricultural machinery to perform turning operations when reaching the edge of the field, while the working area is the main area for tillage. The formula for calculating the width of the headland area is shown in Equation (1).

$$W_h=R_{\mathrm{m}\mathrm{i}\mathrm{n}}+{\displaystyle \frac{w}{2}}+R_{\mathrm{m}\mathrm{i}\mathrm{n}}(1+\cos \theta ),$$

(1)

Here, $\theta$ corresponds to the angle of incidence between the direction of the working trajectory and the inward normal of the field boundary. When the field boundary is locally horizontal, this angle coincides with the angle between the trajectory direction and the vertical axis. $W_h$ is the working width, and $R_{\mathrm{m}\mathrm{i}\mathrm{n}}$ is the minimum turning radius of the agricultural machinery. Equation (1) is used in the field modeling process to calculate the width of the headland, and the boundary is offset inward by a distance equal to the headland width. The local boundary is then smoothed to generate the dividing line between the headland area and the working area, ensuring sufficient turning space for the agricultural machinery at the headland. The field area division modeling is shown in Figure 2.

2.1.3. Objective Function

This study aims to develop a complete coverage path planning strategy for tractor-based farmland tillage operations. Due to the specific characteristics of cultivated land, cross-row operations may damage field flatness and adversely affect soil compaction. Therefore, during tillage operations, the tractor follows a boustrophedon motion pattern [32], ensuring land flatness and a low rate of missed plowing. This study uses a Hybrid Adaptive Neighborhood Search (HANS) algorithm to search for the optimal driving angle of the tractor within the range of 0° to 180°, and incorporates two decision variables: spacing ratio and turning strategy. The spacing ratio balances coverage and efficiency by adjusting the distance between adjacent paths, while the turning strategy changes fuel consumption and turning distance through different turning methods. Parallel scanning technology is then used to generate working tracks parallel to the tractor’s operating direction, and these tracks are traversed sequentially using a boustrophedon motion to cover the working area. In the formulation stage of the model, multiple objectives are considered for optimization: high coverage rate, low fuel consumption, and minimized travel path. The HANS algorithm is used to find a path that provides the optimal solution for these three objectives. The objective function is defined as follows:

(1) Maximum Tillage Coverage Rate: When performing tillage operations in farmland, the coverage rate serves as a pivotal metric for evaluating path planning efficacy. To maintain a high coverage rate and avoid missed areas during tillage operations, this study proposes a method to improve coverage by adjusting the spacing ratio, allowing the tractor to exhaustive cover the target working field during autonomous tillage operations. The formula for calculating the coverage rate is shown in Equation (2).

$$F_{cov}={\displaystyle \frac{F_{\overline{hl}}\cap \left\{{\bigcup }_i{P}_i\right\}}{F_{\overline{hl}}}}$$

(2)

where $F_{cov}$ represents the coverage rate, $F_{\overline{hl}}$ is the area of the working area without headland, $P_i$ is the i-th working swath, $\cap$ is the intersection operator, and $\cup$ is the union of all working swaths.

(2) Minimizing Fuel Consumption: In tillage operations, fuel consumption is a key factor affecting operating costs and energy efficiency. Since tillage processes typically involve low-speed, high-load conditions, fuel consumption is not only related to the distance traveled but is also significantly affected by the number of turns and driving strategy. Unreasonable path planning can lead to extra non-working travel and frequent turns, thus increasing fuel consumption per unit area. Therefore, minimizing fuel consumption as an optimization objective in tillage path planning helps reduce operating costs, improve energy efficiency, and is of great significance for achieving energy conservation and emission reduction in agriculture. The calculation formula is given by Equations (3)–(5).

$$E_l^{\left(i\right)}={\displaystyle \frac{(P_e\times B_e)}{3.6\times {10}^6\times v\times {\rho }_f}}$$

(3)

$$E_{turn}^j=k·f_{idle}·{\displaystyle \frac{T_{turn}^j}{3600}}$$

(4)

$$Q_{total}={\sum }_i{E}_l^{\left(i\right)}·P_i+{\sum }_j{E}_{turn}^j$$

(5)

Here $Q_{total}$ represents fuel consumption, $E_l^{\left(i\right)}$ refers to the fuel consumption during straight-line travel in the i-th working swath, $P_e$ represents the tractor’s engine power (kW), $B_e$ is the fuel consumption rate [g/(kW·h)], $v$ is the straight-line travel speed (m/s), ${\rho }_f$ is the fuel density (kg/L), $E_{turn}^j$ is the fuel consumption during the i-th turn, $k$ is the turning correction coefficient, $f_{idle}$ is the idling fuel consumption, and $T_{turn}^j$ is the turning time for the j-th turn.

(3) Minimum working swath length: In tillage operations, the distance traveled in straight working swaths directly determines the main energy consumption during the operation phase. Redundant or unreasonable straight-line travel significantly increases fuel consumption under high-load conditions. By minimizing the travel distance of straight working swaths, the cumulative energy consumption during the operation phase can be effectively reduced, thereby improving overall operational efficiency and economic performance. The calculation formula is given by Equations (6) and (7).

$$P_i=\sqrt{{(x_i^{end}-x_i^{start})}^2+{(y_i^{end}-y_i^{start})}^2}$$

(6)

$$L_p={\sum }_i{P}_i$$

(7)

where $L_p$ is the sum of the lengths of all work swaths, $P_i$ is the path length of the i-th work swath, ($x_i^{end}$, $y_i^{end}$) are the coordinates of the last point of the i-th work swath, and ($x_i^{start}$, $y_i^{start}$) are the coordinates of the first point of the i-th work swath.

2.1.4. Turning Path Planning

During the plowing process with a tractor, the turning method significantly affects the fuel consumption and operational efficiency of the agricultural machinery. Due to factors such as the minimum turning radius of the machinery, the working width, and the terrain of the field, choosing an appropriate turning method is crucial for improving work efficiency and reducing fuel consumption. The common turning methods for agricultural machinery mainly fall into three categories: flat turn, bulb turn, and fishtail turn, as shown in Figure 3.

When the minimum turning radius of the agricultural machinery r < w/2, a flat turn is used (Figure 3a). When r > w/2, the vehicle does not have enough space for a flat turn and requires a bulb-shaped turn (Figure 3b). When W_h < r(1 + cos θ) + w/2, there is not enough space at the headland for the agricultural machinery to make a normal turn, so a fish-tail turn with reversing capability must be chosen (Figure 3c). The distance calculation methods for the above three turning methods are shown in Equations (8)–(10).

2.1.5. Constraints

Equations (8)–(10) provide the distance calculation methods for flat turns, bulb turns, and fishtail turns. Equations (11)–(16) ensure that each work strip is visited only once. Equation (17) ensures that once the agricultural machine enters a point, it must leave that point, preventing path interruptions Constraint (18) enforces a feasible coverage sequence among all swaths. The auxiliary variable $u_i$ represents the position of swath [i] in the coverage order. For any pair of swaths [i] and [j], the constraint ensures that the relative ordering is consistent with the assigned positions, preventing subtours and guaranteeing a single continuous traversal. This mechanism is analogous to subtour elimination in TSP formulations, adapted for agricultural fields with headland and turning constraints. As a result, $u_i$ provides a systematic way to convert the positional indices of swaths into a valid, operational path for coverage planning. The parameters involved in the constraint formulas are listed in

Table 1. Parameters in the constraint formulas and their meanings.

Parameters	Meaning
$V$	The set of all work swath sequence numbers $V=V_{task}\bigcup \{S,E\}$
$V_{task}$	The set of work swath numbers excluding the starting and ending work swath numbers.
$N$	The total number of work swaths.
$P_{ij}$	Binary variable, Pij = 1, indicates a transition from work swath i to swath j; otherwise, Pij = 0.
$S$	Indicates the starting work swath number.
$E$	Indicates the sequence number of the last work swath.
$u_i$	An auxiliary variable artificially introduced to solve logical problems, representing the order of access.

.

$$flat\left(i,j,\theta \right)=\left|i-j\right|w+r(\pi -2)$$

(8)

$$bulb\left(i,j,\theta \right)=\pi r+{\displaystyle \frac{4r^2-\left|i-j\right|w\left(4r+w{cot}^2\theta +w\right)}{4r-2w}}\times {\sin }^{-1}{\displaystyle \frac{\left|i-j\right|w\left(cot\theta -2wcot\theta \right)}{4r^2-\left|i-j\right|w\left(4r+w{cot}^2\theta +w\right)}}$$

(9)

$$fishtail\left(i,j,\theta \right)=r(\alpha +\beta +\gamma )$$

(10)

$${\sum }_{j\in V,j\ne i}{P}_{ij}=1,\forall i\in V_{task}$$

(11)

$${\sum }_{i\in V,i\ne j}{P}_{ij}=1,\forall j\in V_{task}$$

(12)

$${\sum }_{j\in V}{P}_{Sj}=1$$

(13)

$${\sum }_{i\in V}{P}_{iS}=0$$

(14)

$${\sum }_{j\in V}{P}_{Ej}=0$$

(15)

$${\sum }_{i\in V}{P}_{iE}=1$$

(16)

$${\sum }_{i\in V}{P}_{ik}={\sum }_{j\in V}{P}_{kj}, \forall k\in V_{task}$$

(17)

$$u_i-u_j+N·P_{ij}\le N-1,\forall i,j\in V_{task},i\ne j$$

(18)

2.2. Algorithm Description

First, the input KML geographic data is converted into mathematical polygon objects, and immediately preprocessed geometrically to separate the field boundaries from the internal working areas. To provide a physically meaningful initial search direction for the subsequent optimization process, the algorithm performs Principal Component Analysis (PCA) on the boundary point set of the internal working area. By calculating the covariance matrix of the coordinate points and solving for the eigenvectors, the direction of maximum variance in the field is identified, and based on this, an initial solution set containing working angles, row spacing ratios, and turning strategies is generated with random perturbations. Entering the core iterative loop, the algorithm employs a two-layered nested evolutionary strategy. In each iteration, the system first selects a reference solution from the Pareto archive, and then activates the Adaptive Large Neighborhood Search (ALNS) mechanism. This mechanism dynamically selects a set of “destruction” and “repair” operators using a roulette wheel algorithm to perform topological mutations on the current solution. The destruction operator randomly removes specific dimensions from the solution vector; subsequently, the repair operator reconstructs these dimensions using heuristic rules. The generated neighborhood solution is then filtered by tabu search. This module maintains a tabu list recording recently visited features, aiming to prevent the algorithm from oscillating repeatedly near local optimal solutions. However, if the new solution satisfies the “amnesty criterion” (i.e., its performance is better than existing solutions in the archive), it will be forcibly accepted even if it is within the tabu period. Once a new solution vector containing angles, spacing, and strategies is generated, the system calls the path generator, which uses the ray casting method to construct parallel working paths inside the field polygon, and dynamically adjusts the swath width according to the current spacing coefficient. Subsequently, the turning planner intervenes, using the Reeds-Shepp curve algorithm or geometric turning models (flat/bulb/fishtail) based on the strategy parameters in the solution vector to generate connecting trajectories between path endpoints that comply with the minimum turning radius constraints of the agricultural machinery. Based on this generated complete topological structure, the system simultaneously calculates the total path length, energy consumption, and the area of the uncovered region obtained through polygon difference operations, forming a three-dimensional objective function vector. After obtaining the three-dimensional objective vector of the candidate solution, the algorithm performs strict Pareto dominance detection. The new solution is compared against each existing solution in the archive: if the new solution is not inferior to an existing solution on all objectives and is superior on at least one objective, the existing solution is eliminated; conversely, if the new solution is dominated by an existing solution, it is discarded. To maintain the diversity of the solution set and prevent the archive from growing indefinitely, when the archive capacity reaches its limit, the algorithm calculates the crowding distance of the solutions in the objective space, prioritizing the removal of solutions that are densely distributed and highly similar to each other, thus preserving individuals distributed at both ends of the Pareto front and in sparse regions. This process continues iteratively until the preset number of iterations is reached, finally outputting a set of optimal solutions that cannot be replaced by each other, which are then used by the decision module for weighted selection based on the user’s preference weights for length, energy consumption, or coverage. Algorithm 1 details the step-by-step implementation of the proposed method, and the core flowchart is shown in Figure 4.

Algorithm 1. The pseudocode of HANS

1:    procedure MULTI_OBJECTIVE_HANS 2:          Initialize the Pareto archive and the tabu list 3:          Initialize the DestroyWeights and RepairWeights 4:          InitialSolutions ← GENERATE_INITIAL_SOLUTIONS 5:          for each solution in InitialSolutions do 6:                objectives ← EVALUATE_SOLUTION 7:                UPDATE_PARETO_ARCHIVE 8:        end for 9:          for iteration ← 1 to max_iterations do 10:                  (current_solution, current_objectives) ← SELECT_FROM_ARCHIVE 11:                  destroy_idx ← SELECT_OPERATOR(DestroyWeights) 12:                  repair_idx ← SELECT_OPERATOR(RepairWeights) 13:                  destroyed ← DESTROY_OPERATORS[destroy_idx] 14:                  neighbors ← REPAIR_OPERATORS[repair_idx] 15:                  best_neighbor ← NULL 16:                  best_objectives ← NULL 17:                  best_score ← 0 18:                  for each neighbor in neighbors do 19:                         if IS_TABU(neighbor, TabuList) then 20:                                    neighbor_obj ← EVALUATE_SOLUTION 21:                                    if not ASPIRATION_CRITERION then 22:                                          continue 23:                                    end if 24:                          else 25:                                  neighbor_obj ← EVALUATE_SOLUTION 26:                          end if 27:                          is_new_pareto ← UPDATE_PARETO_ARCHIVE 28:                          if is_new_pareto then 29:                                 score ← 50 30:                          else if DOMINATES(neighbor_obj, current_objectives) then 31:                                  score ← 20 32:                          else if not IS_DOMINATED_BY_ARCHIVE then 33:                                 score ← 5 34:                          else 35:                                 score ← 0 36:                          end if 37:                          if score > best_score then 38:                                  best_neighbor ← neighbor 39:                                  best_objectives ← neighbor_obj 40:                                  best_score ← score 41:                          end if 42:                  end for 43:                  if best_neighbor ≠ NULL then 44:                          UPDATE_SCORE(DestroyWeights, RepairWeights) 45:                          ADD_TABU(best_neighbor, TabuList, tenure = 10) 46:                  end if 47:                  UPDATE_TABU_LIST(TabuList) 48:                  if iteration mod 100 = 0 then 49:                          UPDATE_WEIGHTS(DestroyWeights) 50:                          UPDATE_WEIGHTS(RepairWeights) 51:                          PRINT_STATISTICS(iteration, ParetoArchive) 52:                  end if 53:          end for 54:          return ParetoArchive 55:    end procedure

2.3. Algorithm Improvements

2.3.1. Initialization Strategy Improvement

In the path optimization for complete coverage of cultivated land, the initial solution is crucial. Traditional genetic algorithms generate the initial population completely randomly, with each individual gene sampled independently and randomly. This directly leads to extremely low-quality initial solutions, wasting a large amount of computational resources on evaluating low-quality solutions. The simulated annealing algorithm uses a single random initial solution and is extremely sensitive to the initial solution’s position. If the initial solution is of poor quality, it takes a very long time to find a good solution, and it is not suitable for multi-objective optimization. Furthermore, traditional algorithms often use the direction of the longest side as the driving direction.

This study proposes a multi-level hybrid initialization method. Principal Component Analysis (PCA) is used to identify the main direction of the field. By performing covariance matrix decomposition on the field boundary points, the eigenvector corresponding to the largest eigenvalue is extracted as the principal axis direction. This is based on the characteristic that agricultural fields are usually elongated along the principal axis direction. Driving along the principal axis or perpendicular direction can minimize the number of paths, maximize the length of a single path, and reduce boundary effects. Angle calculation is shown in Equations (19)–(22).

$$\overline{b}=\left[{\displaystyle \frac{1}{n}}\sum x_i,{\displaystyle \frac{1}{n}}\sum y_i\right]$$

(19)

$$b_i={(x_i,y_i)}^{\mathrm{T}}$$

(20)

$$M={\displaystyle \frac{1}{n-1}}\sum (b_i-\overline{b}){(b_i-\overline{b})}^{\mathrm{T}}$$

(21)

$${ \theta }_{pca}=\left\{\begin{array}{l}\arctan \left({\displaystyle \frac{v_y}{v_x}}\right), \mathrm{if}{ v}_x\ne 0\\ {\displaystyle \frac{\pi }{2}} , \mathrm{if}{ v}_x=0\end{array}\right.$$

(22)

where ${\theta }_{pca}$ is the angle of the principal direction of the field being solved, ($v_x$, $v_y$) is the eigenvector corresponding to the largest eigenvalue of the covariance matrix, When ${ v}_x=0$, it means the direction of the main axis is vertical, and $b_i$ represents the boundary points of the land parcel.

Robustness to noise is an important consideration when determining the working direction from field boundary data. Unlike approaches that rely solely on the longest polygon edge, the PCA-based method utilizes all boundary points to estimate the principal direction. Since PCA is based on the global variance of the data, local boundary noise—such as small positional deviations caused by GPS signal drift affecting only a few points—has a negligible influence on the resulting covariance matrix and principal axis. With respect to mapping errors, the PCA computation is performed on centralized boundary data, making the principal axis invariant to global translations of the field. Consequently, uniform shifts in the mapped field position do not affect the estimated working direction. The principal axis is only significantly altered when mapping errors change the overall aspect ratio or symmetry of the field geometry. For example, if an originally elongated rectangular field is incorrectly mapped as a near-square polygon, the dominant eigenvalues of the covariance matrix may become similar, leading to an unstable principal direction. To mitigate this issue, the HANS framework does not rely on a single PCA-derived direction. Instead, multiple offset angles around the principal axis are automatically generated to form the initial population. As a result, even if the PCA-based direction is biased by several degrees, the subsequent adaptive large neighborhood search (ALNS) can iteratively correct this deviation across the entire global optimization phase.

Based on the principal axis direction, the system generates multiple high-quality candidate angles, including the principal axis, perpendicular direction, diagonal directions, and perturbed angles near these directions. This is also effective for irregular fields, with a computational complexity that is only linear. To prevent the initial solutions from being overly concentrated in the directions identified by PCA, several completely random solutions are additionally generated. This balances heuristic and randomness, preventing PCA from being misleading in special cases (such as square fields), increasing the robustness of solution space exploration, and providing diverse starting points for subsequent searches. Faced with multiple candidate solutions, a two-stage evaluation strategy is employed. The first stage uses a simplified model to quickly estimate the quality of the solutions, with constant time complexity. The second stage sorts the solutions based on the approximate evaluation, uses Pareto dominance relationships for filtering, and selects the top 30 non-dominated solutions for full evaluation, including generating actual paths, calculating precise objective values, and checking constraint satisfaction. This strategy ensures the quality of the initial solutions while controlling computational overhead.

2.3.2. Improved Neighborhood Generation Mechanism

Traditional genetic algorithms typically employ random perturbations in their mutation operations, such as adding Gaussian noise or uniform random increments. This perturbation method does not utilize the structural information of the problem, does not consider the characteristics of the current solution, and does not learn from the search history. For path planning problems with complex constraints and objective relationships, undirected random perturbations are inefficient. Furthermore, using a fixed neighborhood generation strategy, the crossover rate, mutation rate, or perturbation magnitude remains constant throughout the search process or changes in a predefined manner. This static strategy cannot adapt to the different needs of the search process. Early in the search, large-scale exploration is needed to discover promising regions, while later in the search, small-scale refinement is needed to improve the quality of the solution. A fixed strategy struggles to meet both of these needs simultaneously.

To address these two problems, the algorithm presented in this paper designs repair operations that integrate multi-layered knowledge. The angle repair operation utilizes the dominant direction of the fields identified by principal component analysis, references the angle parameters of high-quality solutions in the Pareto archive, and adaptively adjusts the perturbation magnitude based on the search stage. The spacing repair operation dynamically adjusts the generation strategy based on the coverage performance of the nearest solutions, achieving feedback-based adaptive search. The strategy repair operation systematically explores all turning strategy options. These repair operations organically combine problem structure knowledge, search history knowledge, and feedback information. An adaptive destruction mechanism is also introduced, where the degree of destruction is adaptively adjusted based on the search progress. During this process, the algorithm uses a roulette wheel selection operator, whose weights are updated after the k-th iteration according to an exponential smoothing formula (Equation (23)).

$$w_i^{(k+1)}=\left(1-p\right)w_i^{\left(k\right)}+p{\displaystyle \frac{S_i}{N_i}}$$

(23)

Here, $p$ is the reaction factor, representing the sensitivity of the control algorithm to recent performance; $S_i$ represents the cumulative score obtained by operator i in this round; and $N_i$ represents the number of times operator i was selected in this round.By monitoring the ratio of the current number of iterations to the total number of iterations, the algorithm dynamically determines the number of decision variables to be disrupted. A larger degree of disruption is applied during the initial search phase to promote exploration, a moderate degree of disruption is used in the middle stages to balance exploration and exploitation, and a smaller degree of disruption is used in the later stages for local refinement. This mechanism enables a fluid transition between global search and local exploitation.

2.3.3. Improvements to Local Optimization Problems

Under the selective pressure of genetic algorithms, the genes of superior individuals rapidly spread throughout the population, leading to a gradual loss of population diversity. Although mutation operations provide some diversity, it is difficult to counteract the effects of selective pressure with a low mutation rate.

To address this problem, the algorithm presented in this paper employs a tabu search mechanism to maintain a memory of the search history. Each visited solution is added to a tabu list and is not revisited within the tabu period. This mechanism forces the algorithm to continuously explore new areas, preventing repetitive searches in known regions. The tabu mechanism provides an explicit means of diversification that does not rely on randomness or population size, but systematically guides the search away from previously visited areas through a memory structure.

2.3.4. Improvements to Address Missed Planting Areas

Traditional boustrophedon methods leave many unplowed areas at the boundaries of irregularly shaped fields. The algorithm presented in this paper employs a dynamic gap-filling technique. After initial path generation, the algorithm calculates the uncovered areas and recursively applies path planning (attempting different angles) to these remaining areas to complete the coverage.

3. Results

3.1. Simulation Experiment

This study is based on plot modeling at Huaxing Farm in Changji, Xinjiang. Six representative plots were selected from Huaxing Farm as experimental research subjects. Three representative plots were chosen for visual demonstration, as shown in Figure 5, Detailed information provided in

Table 2. Detailed information of the three farmlands.

Field No.	Perimeter /m	Area /m2	Altitude /m	Headland /m	WorkingWidth /m	Turning Radius/m
18	2473.19	364,720.12	495.85	6.0	2.5	5.1
15	3225.27	622,509.63	490.84	6.0	5.1	5.1
14	3692.59	797,432.04	487.67	6.0	2.5	5.1
13	2120.47	261,985.85	490.55	6.0	2.5	5.1
12	2914.54	528,742.94	490.71	6.0	2.5	5.1
5	2930.69	487,157.85	491.67	6.0	2.5	5.1

.

Among the three fields, Field 18 has a typical irregular convex polygonal geometric structure, making path planning moderately difficult. Field 14 has a geometric structure similar to a rectangle, which is relatively regular, making path planning relatively simple. Field 5 has multiple irregular concave and convex sections, which are prone to missed areas during cultivation and present difficulties during turns, making path planning more challenging. Therefore, these three fields were selected to verify the reliability of this algorithm.

3.2. Evaluation Results

This algorithm demonstrates varying degrees of improvement over traditional tillage methods in terms of path planning length, operational energy consumption, and coverage efficiency.

Under the conventional approach, operators typically select the longest straight edge of a field as a reference line, performing parallel back-and-forth operations along this long edge (plow-style tillage). While this method yields acceptable results for regular rectangular fields, it proves inefficient for irregular polygonal fields (as shown in Figure 5c). this simple alignment leads to significant “corner residue” at field ends or frequent short-distance reversals, drastically reducing efficiency while increasing path length and fuel consumption. In traditional tillage, row spacing is typically fixed to the machine’s working width. When field widths are not integer multiples of this width, the final row often suffers from substantial overlap (wasting fuel) or leaves uncovered narrow strips. In turn planning, traditional methods disregard complex kinematic curves, typically executing in-place reversals or wide-radius U-turns at headlands. This not only creates excessively large headland widths, reducing effective tillage area, but also causes soil compaction at headlands. Furthermore, inefficient back-and-forth reversals consume additional energy.

Improved Algorithm: Compared to the traditional method, this algorithm employs scientifically rigorous kinematic constraints using Reeds-Shepp curves to generate optimized turning paths. This reduces fuel consumption at headlands and shortens turning distances while increasing coverage through adjusted row spacing ratios and dynamic blind-spot compensation techniques.

To ensure a fair comparison, all experiments were performed within a unified simulation environment, with both methods using the uniform working width and turning radius constraints for the agricultural machinery. Performance evaluation was based on several key indicators, including total path length of tillage operation, coverage rate, and fuel consumption. These indicators scientifically reflect the efficacy of different path planning methods in realistic operations, covering aspects such as operational efficiency, economic benefits, and adaptability. The experimental results are summarized in

Table 3. Comparison between HANS and traditional method in terms of path length, coverage, energy consumption, and working time. Each row corresponds to one farmland instance, and values represent deterministic optimization results. (↑ indicates that a higher value is better, ↓ indicates that a lower value is better).

	$\mathbf{Path} \mathbf{Length}/\mathbf{m}$↓		$\mathbf{Coverage} \mathbf{Rate}/\%$↑		$\mathbf{Energy}/\mathbf{L}$↓		Work $\mathbf{Time}/\mathbf{min}$↓
Field No.	HANS	Tra-method	HANS	Tra-method	HANS	Tra-method	HANS	Tra-method
18	137,969.71	139,891.72	99.81	97.91	868.73	890.51	1136.72	1168.46
15	235,369.97	238,900.52	99.90	97.81	1445.63	1484.21	1915.15	1946.81
14	307,574.14	311,856.16	99.92	98.33	1900.3	1981.62	2436.48	2529.73
13	100,616.79	102,118.45	99.77	97.42	648.78	667.96	838.99	852.59
12	201,661.31	204,932.46	99.84	97.26	1260.69	1300.55	1655.21	1687.19
5	185,352.93	191,529.12	99.87	97.64	1188.6	1246.83	1533.23	1594.50

The differences were compared using the Wilcoxon signed-rank test, n = 6, p = 0.031.

.

The experimental setup for this study was configured as follows: Windows 11 (64-bit) operating system, Intel(R) Core(TM) i7-11800H processor, and 16GB of RAM. In the experiment, a John Deere 7M-2204 wheeled tractor was used, pulling a Lemken 1LFTT-550 five-furrow plow. The tractor’s engine power was 161.8 KW, its turning radius was 5.1 m, and its working width was 2.5 m.

To assess whether the observed improvements are consistent across instances rather than driven by individual cases, We used the Wilcoxon signed-rank test to compare the differences, with a sample size of n = 6, yielding p = 0.031, indicating a statistically significant difference at the 5% significance level. As shown in the table, this method demonstrates superior performance across all key indicators compared to the traditional method. In the traditional method, agricultural machinery operators often rely on personal experience to conduct field operations. The working direction is typically aligned with the longest boundary of the field, while turning maneuvers are executed empirically without explicit optimization for total travel distance, coverage rate, or fuel consumption. Comparative analysis revealed that HANS reduced the average path length by 3447.26 m, about increased 0.3%, achieving a coverage rate of 99.87% on the irregularly shaped plot 3. In terms of energy savings, HANS saved an average of 43.16 L of fuel. This reduction in fuel consumption not only improved the economic efficiency of agricultural production but also mitigated the environmental footprint. It also effectively shortened the duration of field operations. These results empirically substantiate that the proposed method significantly curtails operational path length and the number of turning maneuvers. Its superior performance are particularly evident in complex and irregularly shaped fields, highlighting the algorithm’s adaptability and robustness. This method not only improves the efficiency of agricultural machinery operations but also significantly reduces labor costs. Meanwhile, we conducted a Wilcoxon signed-rank test to assess the paired differences between the HANS method and the traditional method across three agricultural field scenarios.

The Figure 6 shows a visualization of the complete coverage operation paths generated by the HANS algorithm for three exemplary fields (Field 18, Field 14, and Field 5). Figure 7 presents the visualized complete coverage operation paths produced by the traditional method for three representative fields (Fields 18, 14, and 5). This method is guided by the driver’s experience.

Of particular importance is that this experiment was conducted in a simulated environment, and the data used were field boundary data extracted from drone imagery. No field testing has been conducted yet. Therefore, In complex real-world scenarios, factors such as navigation drifts and terrain-induced discrepancies, and uncertainties in agricultural machinery operation may affect operational efficiency. Further field trials are recommended to comprehensively evaluate the adaptability and stability of the proposed method under actual agricultural operating conditions.

3.3. Comparative Experiments of the HANS Algorithm in Path Optimization

To verify the practical performance of the proposed algorithm (HANS) in agricultural machinery path optimization, this paper conducted comparative experiments with other algorithms. The evaluation metrics included average working path length, mean coverage rate, and mean fuel consumption. The parameters of the agricultural machinery were: working width of 2.5 m, minimum turning radius of 5.1 m, and engine power of 161.8 KW. The comparison algorithms included the genetic algorithm, particle swarm optimization algorithm, tabu search algorithm, and simulated annealing algorithm.

Table 4. Extensive simulation data of different algorithms for complete coverage path planning in 20 fields. (↓) indicates that a lower value of the metric is better, an (↑) indicates that a higher value of the metric is better.

Algorithm	$\mathbf{Path} \mathbf{Length}/\mathbf{m}$↓	$\mathbf{Coverage} \mathbf{Rate}/\%$↑	$\mathbf{Energy}/\mathbf{L}$↓	Time Complexity
Tra	226,269.28	99.31	1408.71	-
GA	219,718.22	98.71	1361.53	$O(I_{GA}\cdot P\cdot N)$
PSO	219,445.41	98.89	1358.91	$O(I_{PSO}\cdot S\cdot N)$
TS	219,173.26	99.26	1357.33	$O(I_{TS}\cdot N^2)$
SA	218,901.79	99.48	1355.70	$O(I_{SA}\cdot N)$
HANS	218,093.44	99.82	1347.46	$\mathit{O}({\mathit{I}}_{\mathit{A}}·\mathit{M}·{\mathit{I}}_{\mathit{T}}·\mathit{N})$

shows the comprehensive experimental results of these six methods on 20 fields. Because fields 10 and 11 are adjacent and lack a clear boundary, they were combined into a single field for the planning experiment. The same applies to fields 19 and 20.

Table 5. Parameter Settings for All Algorithms: Values, justifications, and literature references for hyperparameters used in the comparative study.

Algorithm	Hyperparameter	Value	Reason
GA	population size	100	3 variables × 30 ≈ 100, balancing diversity and efficiency
	generations	50	Standard setting
	crossover_rate	0.8	Emphasis on information exchange, classic value.
	mutation_rate	0.1	Maintain diversity and avoid disrupting the delicate balance.
PSO	swarm_size	50	PSO requires fewer individuals than GA.
	max_iterations	100	Standard setting
	Inertia weight	0.7	Balancing exploration and development
	Individual learning factors	1.5	Approaching the theoretical optimum of 1.49618
	Social learning factor	1.5	Balancing individual and social learning
SA	initial_temp	1000	Initial acceptance rate: 80–90%
	final_temp = 1	1	Provides 135 cooling cycles
	cooling_rate	0.9	Moderate cooling rate
TS	max_iterations	100	Standard setting
	tabu_tenure	10	Short-term memory, prevents looping
	neighborhood_size	20	Balancing search breadth and depth
HANS	max_iterations	100	Standard setting
	tabu_tenure	10	Standard setting
	Pareto archive capacity	100	Store the complete Pareto front
	New Pareto solution reward	50	Strong incentives for new discoveries
	Improved solution reward	20	Moderate incentive improvements
	Accepting rewards for solving puzzles	5	Mild incentives for exploration
	Weight update factor	0.1	Balancing stability and responsiveness

shows the core hyperparameters of the proposed algorithm and the comparison algorithms. We also analyzed the time complexity of these intelligent algorithms, where $N$ is the number of decision variables.

(1)

Genetic Algorithm (GA):

Population size: $P$

Number of generations: $I_{GA}$

Fitness evaluation cost per individual: $O\left(N\right)$

Genetic operators (selection, crossover, mutation): at most $O\left(N\right)$ per individual

Per generation cost:

Fitness evaluation: $P\cdot O\left(N\right)$

Genetic operators: $P\cdot O\left(N\right)$

Total complexity: $T_{GA}=O(I_{GA}\cdot P\cdot N)$

(2)

Particle Swarm Optimization (PSO)

Swarm size: $S$

Number of iterations: $I_{PSO}$

Position/velocity update: $O\left(N\right)$ per particle

Fitness evaluation: $O\left(N\right)$ per particle

Per iteration cost:

Update + evaluation: $S\cdot O\left(N\right)$

Total complexity: $T_{PSO}=O(I_{PSO}\cdot S\cdot N)$

(3)

Tabu Search (TS)

Number of iterations: $I_{TS}$

Neighborhood size per iteration: $O\left(N\right)$ (e.g., swap, insertion, or reordering of swaths)

Cost to evaluate one neighbor: $O\left(N\right)$

Per iteration cost:

Neighborhood evaluation: $O\left(N\right)\cdot O\left(N\right)=O\left(N^2\right)$

Total complexity: $T_{TS}=O(I_{TS}\cdot N^2)$

This quadratic dependence is intrinsic to TS when full neighborhood evaluation is required.

(4)

Simulated Annealing (SA)

Number of iterations: $I_{SA}$

One neighbor generated per iteration

Cost to generate neighbor: $O\left(1\right)$–$O\left(N\right)$

Cost to evaluate neighbor: $O\left(N\right)$

Per iteration cost:

Neighbor evaluation: $O\left(N\right)$

Total complexity: $T_{SA}=O(I_{SA}\cdot N)$

(5)

Hybrid Adaptive Neighborhood (HANS)

Number of ALNS iterations: $I_A$

Number of candidate solutions generated per ALNS iteration: $M$

Number of Tabu Search iterations applied to each candidate: $I_T$

Per iteration cost:

Neighbor evaluation: $O\left(N\right)$

Total complexity: $T_{HANS}=O(I_A·M·I_T·N)$

Although HANS has a higher theoretical complexity, under certain conditions, it can achieve comparable or even shorter computation times than simpler metaheuristic algorithms. This phenomenon is primarily attributed to structural differences in how search effort is allocated.

Table 6. The computation time of the heuristic algorithm on three land parcels (Parcels 18, 14, and 5).

Algorithm	Field 18	Field 14	Field 5
GA	16.35 min	19.14 min	17.58 min
PSO	15.27 min	18.35 min	15.83 min
TS	16.81 min	18.89 min	18.46 min
SA	13.63 min	16.96 min	15.41 min
HANS	15.85 min	18.62 min	16.12 min

shows the computation time of the heuristic algorithm on three land parcels (Parcels 18, 14, and 5).

First, HANS employs PCA-based global axis initialization, significantly reducing the effective search space from the outset. By providing a strong, geometrically informed initial solution, HANS avoids spending considerable time in early iterations exploring suboptimal or infeasible regions, unlike population-based methods such as Genetic Algorithms (GA) or Particle Swarm Optimization (PSO).

Second, the feedback learning mechanism in HANS gradually suppresses ineffective destruction-repair operators and focuses computational resources on those operators better suited to the current geometric structure of the problem domain. Therefore, in subsequent iterations, most evaluations are highly informative, whereas GA and PSO evaluate the entire population or particle swarm in every iteration regardless of progress.

Third, the tabu search component in HANS is selectively applied to a small number of promising candidate solutions generated by ALNS. In contrast, independent tabu search or simulated annealing algorithms might repeatedly explore larger neighborhoods or accept low-quality moves, leading to slower convergence and wasted computational resources.

Finally, for irregular or large-scale domains, simpler heuristic algorithms often require more iterations or larger population sizes to escape local optima. In such cases, HANS’s structured and strategically aware search approach can converge to high-quality solutions in fewer effective iterations, resulting in comparable or even lower overall computation time. The primary focus of this study is not to substantially reduce computation time, but rather to improve the effectiveness and practical applicability of coverage path planning for agricultural machinery operating in irregular fields.

As shown in Table 4, the path length planned by the HANS algorithm is 218,093.44 m, which is 8175.84 m shorter than the 226,269.28 m planned by traditional path planning algorithms, representing a reduction of 3.61% on average per field. Compared to other heuristic algorithms, HANS shows the following average improvements: a reduction of 1624.78 m compared to the Genetic Algorithm (GA, 219,718.22 m), and a reduction of 808.35 m compared to the Simulated Annealing algorithm (SA, 218,901.79 m).

In terms of energy consumption, the HANS algorithm planned a route with an energy consumption of 1347.46 L, significantly better than the 1408.71 L planned by traditional path planning algorithms, representing an average reduction of 4.34%. It also outperformed other heuristic algorithms, reducing energy consumption by 1.03% compared to the Genetic Algorithm (GA, 1361.53 L) and 0.84% compared to the Particle Swarm Optimization algorithm (PSO, 1358.91 L).

In terms of coverage rate, the HANS algorithm achieved a coverage rate of 99.82%, which is 0.51% higher than the 99.31% achieved by traditional path planning algorithms. It also outperformed the Genetic Algorithm (GA, 98.71%) and Particle Swarm Optimization algorithm (PSO, 98.89%), with improvements of 1.11% and 0.93%, respectively. Although the improvement is relatively small, the cumulative increase in covered area across 20 plots is considerable, which is of significant importance for improving agricultural efficiency. It should be noted that in some cases, HANS does not offer a significant performance advantage over simple heuristic methods. Particularly for approximately rectangular areas with simple geometries and few obstacles, a heuristic strategy based on the longest region edge can generate near-optimal coverage paths. In these low-complexity scenarios, the improvements offered by the additional global search performed by HANS are limited.

Similarly, when the area is small and the number of work strips is small, the solution space itself is limited, and simple heuristic methods can obtain competitive results at a lower computational cost. In these cases, the advantages of the metaheuristic framework are less pronounced.

3.4. Ablation Study and Convergence Analysis

3.4.1. Ablation Study

To rigorously evaluate the contribution of each core component within the HANS framework, we conducted an ablation study and a convergence analysis over 20 representative test fields. As illustrated in the ablation results (Figure 8), the Full HANS achieves the highest Hypervolume (HV) of 0.93, outperforming all ablated variants. The ablation study, quantified via the Hypervolume (HV) index and Relative Efficiency (RE), highlights the necessity of the proposed hybrid architecture:

(1)

The Dominance of the Full HANS: The full HANS configuration achieves a peak HV of $0.93$. This serves as the performance baseline (100%RE), demonstrating that the synergistic interaction between PCA-based orientation, adaptive neighborhood search, and tabu-based refinement is essential for navigating complex, non-convex Pareto fronts.

(2)

The Navigational Role of PCA: Removing the PCA-based initialization (w/o PCA) causes the HV to drop to 0.82, a significant 13.7% loss in efficiency. This sharp decline indicates that without PCA’s ability to align the search direction with the field’s principal axes, the algorithm wastes substantial computational budget exploring unproductive angular regions, leading to suboptimal coverage patterns even after 100 iterations. ALNS vs. Fixed Strategies: The variant without the adaptive mechanism (w/o Adaptive) yields an HV of 0.86. This 7.5% performance gap underscores the importance of the credit-based reward system (${\sigma }_1,{\sigma }_2{,\sigma }_3$), which allows the algorithm to learn which “destroy” and “repair” operators are most effective for specific field shapes in real-time.

(3)

The Precision of Tabu Search: Excluding the Tabu Search (w/o Tabu Search) results in an HV of 0.90. Although ALNS captures the broad structure of the optimal solution, TS provides the “last-mile” optimization required to fine-tune the heading angles within a constrained neighborhood, ensuring the paths are mathematically optimized for both energy efficiency and coverage.

Overall, the ablation and convergence results confirm that the superior performance of HANS does not arise from any single component, but from the coordinated interaction of global orientation, adaptive exploration, and local refinement, which together enable robust and efficient navigation of complex farmland optimization landscapes.

3.4.2. Convergence Analysis

Figure 9 provides a comparative convergence analysis between the full HANS algorithm and its ablated variants (TS-only and ALNS-only) in terms of hypervolume (HV). Since TS-only and ALNS-only share the same solution representation and operate within the same search space as HANS, their convergence curves provide a meaningful basis for comparison. In contrast, general-purpose heuristics such as GA, PSO, or SA follow fundamentally different search mechanisms and are not directly comparable in terms of iteration-based convergence. The HV value is the average of all representative farmland areas, reflecting the overall performance of the algorithm under different field geometries.

Several key observations can be drawn from the figure. First, in the early stages of iteration, the HV value of the algorithm using only ALNS (ALNS-only) increases rapidly. The curves of HANS (blue) and ALNS-only (orange) show a high degree of overlap, and the slopes of their HV indicators are significantly higher than that of TS-only (green). This reflects the powerful exploration capability provided by the adaptive large neighborhood search operator, strongly demonstrating the dominant role of the adaptive large neighborhood search (ALNS) mechanism in the early stages of the search. Through the global perturbation of the “destroy-repair” operator, the algorithm can overcome local optima and quickly locate high-quality Pareto regions in the solution space. However, its improvement rate slows down in the middle and later stages, indicating that without the guidance of tabu search, the algorithm may temporarily get stuck in local high-quality regions.

As the search progresses, the ALNS-only curve begins to show more significant “oscillations” and the growth trend tends to flatten. At the same time, the HANS curve shows stronger sustained upward resilience and forms a stable difference with ALNS-only around the 45th iteration. This stage reflects the fine-grained deep exploration capability of tabu search (TS) in the local neighborhood. Although ALNS can maintain global exploration capabilities, its randomness makes it difficult to converge stably during the fine-tuning stage. HANS cleverly utilizes the memory mechanism of TS to lock in the preferred angular direction, achieving a smooth transition from “extensive exploration” to “precise development”.

Finally, HANS stabilizes at the highest HV level of approximately 0.95, while ALNS-only is locked at around 0.90. Most significantly, TS-only can only reach approximately 0.74, and its curve basically enters a plateau in the later stages. The experimental data truthfully shows that pure local search operators are very prone to “premature convergence” in complex continuous angular spaces. HANS ultimately achieved a Pareto quality upper limit approximately 28.4% higher than the baseline version. This demonstrates that HANS is not simply a superposition of algorithms, but rather a multi-level optimization system where PCA provides high-quality principal component directions, ALNS handles global topological optimization, and TS is responsible for local parameter alignment. This allows it to achieve a level of global convergence unattainable by any single algorithm.

In summary, the convergence curves show that HANS’s hybrid design not only accelerates early exploration compared to algorithms using only TS, but also ensures continuous improvement and stability in later iterations compared to algorithms using only ALNS. This behavior confirms that the algorithm can maintain diverse, high-quality solutions while steadily approaching the Pareto optimal frontier.

4. Discussion

Xinjiang is one of the most important agricultural production regions in China and exhibits a strong demand for agricultural mechanization. However, farmland parcels in most farms are characterized by large areas formed through long-term cultivation and consolidation, which commonly results in irregular boundary shapes, complex polygonal structures, and non-uniform headland widths. These characteristics pose significant challenges to autonomous path planning for agricultural machinery and simultaneously create an urgent need for efficient path planning algorithms to improve operational quality and efficiency. To investigate the core technologies of path planning within intricate field environments, this study selects Fields No. 18, No. 15, No. 14, No. 13, No. 12,and No. 5 of Huaxing Farm in Xinjiang as exemplary fields and constructs a diverse dataset for evaluation.

Simulation results demonstrate that the proposed path planning algorithm achieves excellent performance across all three selected fields, yielding desirable outcomes in terms of coverage rate, fuel consumption, and working path length. The generated paths are more reasonable, well-organized, and compact, with smoother and more natural turning maneuvers. As a result, the algorithm effectively improves coverage while reducing fuel consumption and total working path length. In contrast, traditional farmland path planning methods, which are largely constrained by experience-based operations and typically align the working direction along the longest field edge, tend to produce rigid and inflexible paths. This leads to reduced coverage, increased redundant fuel consumption, and a higher rate of repeated tillage, thereby severely degrading operational efficiency.

Although the results obtained by the proposed algorithm are encouraging, certain limitations still remain. Specifically, when dealing with more complex concave fields, the algorithm exhibits deficiencies in field partitioning. How to achieve more effective field decomposition while keeping variations in coverage rate, fuel consumption, and working path length within reasonable bounds constitutes a key direction for future research. In addition, further field experiments are required to conduct on-site validation, taking into account stochastic navigation uncertainties and the actual motion behavior of agricultural machinery, in order to more comprehensively and realistically evaluate their impact on path planning performance in real-world tillage operations.

5. Conclusions

This paper addresses the challenges of path planning for agricultural machinery in irregularly shaped fields and proposes the HANS algorithm. Unlike conventional heuristic and hybrid metaheuristic approaches that loosely combine global and local search in a sequential manner, HANS introduces a tightly coupled and hierarchical optimization framework in which Adaptive Large Neighborhood Search and Tabu Search continuously interact throughout the search process. HANS further distinguishes itself by incorporating a feedback-driven operator learning mechanism, enabling dynamic adaptation of the search strategy to varying field geometries and problem instances without manual parameter tuning. In addition, the hybrid optimization is explicitly problem–structure-aware, with operators designed to target different decision layers of agricultural coverage path planning, rather than applying generic perturbations to monolithic solutions. By directly embedding operational constraints such as multi-modal turning behaviors and energy-aware cost evaluation into the optimization loop, HANS ensures consistent and practically meaningful optimization outcomes. It achieves excellent optimization results in terms of working path distance, coverage rate, and energy consumption.

(1) Simulation results show that the HANS algorithm achieved excellent performance in three test fields in terms of working path distance, coverage rate, and energy consumption. Compared with traditional path planning strategies, this method has competitive edge: the coverage rate increased by 2.12%, the average total path length decreased by 3447.26 m, and the average energy consumption decreased by 3.42%. The differences were evaluated using the Wilcoxon signed-rank test (n = 6), which indicates a statistically significant improvement of HANS over the traditional method (p = 0.031). This result suggests that the observed performance gains are consistent across the tested farmland instances rather than being caused by instance-specific outliers.

(2) Large-scale simulation results further attest to the practical significance of our approach. In a comprehensive test involving 20 test fields, HANS demonstrated superior optimization capabilities. Compared with traditional method, the average total path length optimization rate reached 3.61%, and the mean energy expenditure decreased by 4.34%. Comparative experiments with other heuristic optimization algorithms showed that HANS’s path optimization effect ranged from 0.37% to 0.83%, and energy consumption decreased by 0.34–1.11%. Although the HANS algorithm shows improvement percentages of less than 1% compared to other heuristic algorithms in some metrics, these seemingly small percentages translate into significant absolute savings in actual agricultural operations. Across multiple fields and repeated daily operations, these savings accumulate, significantly reducing time and fuel costs, thereby directly improving operational efficiency and cost-effectiveness. An ablation study and convergence analysis conducted on 20 representative farmland instances demonstrate that the full HANS framework consistently outperforms all ablated variants. The complete configuration achieves the highest Hypervolume (HV = 0.93), confirming that the synergistic integration of PCA-based orientation, adaptive large neighborhood search, and tabu-based refinement is essential for effectively navigating complex, non-convex solution spaces. Removing any core component leads to a noticeable performance degradation, highlighting that each module plays a distinct and complementary role in achieving robust and efficient coverage path planning.

Comprehensive analysis shows that the HANS algorithm has powerful optimization effects and good adaptability in handling large-scale irregular convex fields and simple concave fields, enabling full coverage of the cultivated land. Future inquiries could profitably explore further exploring the identification of complex concave structures and segmentation strategies for complex concave fields, and investigating field decomposition strategies to partition complex geometric domains into manageable sub-regions without incurring redundant traversal overhead. This facilitates a more exhaustive coverage path planning (CPP) and establishes a robust framework for future multi-machine coordination.