HGTA: A Hexagonal Grid-Based Task Allocation Method for Multi-Robot Coverage in Known 2D Environments

Abstract

For multi-robot cooperative coverage, an effective spatial division strategy is essential to ensure balanced and spatially continuous task regions for each robot. Traditional grid-based partitioning approaches like DARP (Divide Areas based on Robots’ Positions) and TASR (Task Allocation based on Spatial Regions) often generate discontinuous sub-regions and imbalanced workloads, particularly in irregular or fragmented task spaces. To mitigate these issues, this paper introduces HGTA (Hexagonal Grid-based Task Allocation), a novel method that employs hexagonal tessellation for environmental representation. The hexagonal grid structure provides uniform neighbor connectivity and minimizes boundary fragmentation, yielding smoother partitions. HGTA integrates a multi-stage wavefront expansion algorithm with an iterative region-correction mechanism, jointly ensuring spatial contiguity and load equilibrium across robots. Extensive evaluations in 2D environments with varying obstacle densities and robot distributions show that HGTA enhances spatial continuity—achieving improvements of 18.2% in connectivity and 17.8% in boundary smoothness over DARP, and 7.5% and 9.5% over TASR, respectively—while also improving workload balance (variance reduction up to 18.5%) without compromising computational efficiency. The core contribution lies in the synergistic coupling of hexagonal tessellation, wavefront expansion, and dynamic correction, a design that fundamentally advances partition smoothness and convergence speed. HGTA thus offers a robust foundation for multi-robot cooperative coverage, area surveillance, and underwater search applications where connected and balanced partitions are critical.

1. Introduction

The capability of a multi-robot system to cooperatively perform large-scale coverage tasks has become a central issue in modern robotics research [1]. Diverse applications, including environmental monitoring [2,3], underwater exploration [4], search and rescue [5,6], and agricultural inspection [7] require a group of autonomous agents to efficiently divide and execute spatially distributed tasks. A fundamental challenge within these applications lies in designing a task allocation mechanism that ensures each robot operates in a continuous and balanced region, minimizing both overlap and idle areas [8].

For effective multi-robot systems operating in a bounded environment, the problem of spatial task allocation is generally required to satisfy three key criteria [9,10]: (i) Scalable Allocation, enabling dynamic task distribution based on the initial positions of robots, team size, and capabilities; (ii) Subregion Connectivity, ensuring each robot operates within a single, topologically connected area to minimize redundant movement and collision risk; and (iii) Complete Coverage, guaranteeing that the union of all assigned subareas precisely covers the entire task space without overlaps or omissions. Early approaches, primarily relying on basic square-grid decomposition, trapezoidal decomposition approach [11], Boustrophedon Cellular Decomposition (BCD) [12], often fail to meet these requirements comprehensively. While computationally efficient, these methods are particularly prone to generating fragmented sub-regions that violate the connectivity constraint, especially in irregular or obstacle-rich environments. Such discontinuities lead to inefficient inter-robot transitions and significant path redundancy, thereby fundamentally degrading the overall efficiency and robustness of the cooperative coverage task. Recent advances have explored various strategies, including distributed auction-based methods [13], deep reinforcement learning for dynamic allocation [14], and hierarchical task decomposition [15]. However, these approaches often sacrifice spatial continuity for computational efficiency or vice versa. Notably, hexagonal representations have shown promise in coverage tasks and path planning [16], but their integration with workload-balanced allocation remains under-explored.

Several representative algorithms have emerged to enhance the quality of spatial task allocation, notably the Divide Areas based on Robots’ Positions (DARP) algorithm [17] and Task Allocation based on Spatial Regions (TASR) algorithm [10]. The DARP ensures the assignment of a topologically connected partition to each robot through its iterative allocation mechanism based on distance fields. However, due to its reliance on a discrete square grid for environmental representation, the generated region boundaries tend to be jagged and lack smoothness. More critically, in complex environments with obstacles, this discretization can lead to highly irregular, distorted region shapes and even create bottleneck effects. Although these regions remain strictly connected in a topological sense, the primary limitations of the algorithm lie in these geometric imperfections and the consequent decline in task execution efficiency. This motivates the pursuit of alternative representations capable of generating more regular and smoother area partitions, such as those based on polygons or continuous fields. TASR extends DARP by introducing a geodesic distance field based on wavefront propagation, which replaces the original Euclidean or Manhattan distance metrics. This constitutes a fundamental improvement, as the area partitioning is no longer based solely on geometric straight-line distance but rather on the actual travel cost for the robots. Consequently, the generated region boundaries are smoother and more natural, thereby mitigating the issues of distorted region shapes, narrow bottlenecks, and load imbalance commonly found with DARP.

However, it is crucial to recognize that both DARP and TASR are fundamentally constrained by their underlying square grid (quadrilateral) representation of space. This shared foundation presents inherent limitations compared to alternative tessellations, such as hexagonal grids. In a square grid (Figure 1a), movement has 8 possible directions with non-uniform costs (1 for cardinal directions and $\sqrt{2}$ for diagonals), leading to directional bias and path distortion (Figure 1b). In contrast, a hexagonal grid offers 6 equidistant neighbors, allowing for more uniform and natural movement patterns with consistent unit cost in every direction (Figure 1c). This results in paths with finer, 60-degree turning angles, enabling smoother obstacle avoidance and more accurate distance calculation compared to the coarser 45/90-degree turns in square grids (Figure 1d). The connectivity of square grids (both 4- and 8-connectivity can cause issues) contributes to the discrete boundary effects and jagged, aliased region boundaries mentioned in the original critique. Hexagonal grids provide a more isotropic and uniform connectivity, which can inherently lead to more regular partition shapes and reduce the geometric artifacts inherent in square-based algorithms.

Therefore, while wavefront mechanism of TASR effectively mitigates many shortcomings of DARP within the paradigm of square grids, the ultimate quality of the spatial partition remains bounded by the geometric constraints of the grid cells themselves. The exploration of topologically superior representations, like hexagonal grids, presents a promising direction for future research to achieve inherently more uniform, continuous, and efficient spatial decompositions for multi-robot systems.

To overcome the inherent limitations of square grids—such as directional anisotropy, jagged boundaries, and discrete movement artifacts—this paper introduces a novel task allocation method named HGTA. The key point is its foundational use of a hexagonal grid representation. This representation provides each cell with six equidistant neighbors, effectively eliminating the anisotropy of square grids and enabling smoother boundary transitions and more natural region expansion during task assignment. Building upon this superior spatial representation, the HGTA framework integrates a two-stage process: (1) a wavefront-based region expansion algorithm that iteratively grows task regions from robot positions, and (2) a region correction and compensation mechanism that dynamically redistributes boundary cells based on workload disparity to achieve global balance.

Although hexagonal grids are known to provide a more isotropic spatial tessellation, the contribution of this work does not lie in grid replacement alone. Instead, HGTA explicitly couples hexagonal topology with wavefront-based distance competition and iterative correction, such that grid connectivity, distance propagation, and workload balancing jointly determine the growth of regions. As shown in this paper, the performance gains arise from this coupled design rather than from the grid geometry in isolation. In short, the HGTA is a wavefront-based allocation framework under a hexagonal spatial representation, rather than a fundamentally new distance metric or optimization paradigm. The novelty lies in the systematic integration of hexagonal tessellation, wavefront propagation, and dynamic correction into a unified framework that jointly optimizes for spatial continuity and workload balance—an advancement over existing methods that treat these aspects separately.

The contribution of HGTA is not merely the replacement of square grids with hexagonal grids; rather, it is a tightly coupled framework where hexagonal topology, wavefront propagation, and iterative correction jointly determine region growth. This coupling enables more natural boundary formation and faster convergence than simply applying a wavefront algorithm on a hexagonal tessellation in isolation. In particular, (i) A novel hexagonal grid representation that enhances region continuity and uniformity in spatial task division for multi-robot systems. (ii) A two-stage allocation framework, combining wavefront expansion with dynamic boundary correction, to ensure both partition connectivity and system-wide load balance. (iii) A comprehensive experimental comparison against DARP and TASR under multiple environment configurations, demonstrating superior performance of HGTA in connectivity preservation, workload balance, and computational stability.

The remainder of this paper is organized as follows. Section 2 reviews related work in multi-robot coverage and task allocation. Section 3 details the proposed HGTA methodology, covering hexagonal grid construction, wavefront propagation, and the region correction mechanism. Section 4 presents the experimental setup, evaluation results, and limitations of the approach. Finally, Section 5 concludes the paper and outlines future research directions.

2. Related Work

Task allocation is a fundamental component of multi-robot cooperation. Existing methods can be broadly categorized into cell-based decomposition and swarming-based partitioning approaches, each with distinct advantages and limitations in terms of connectivity, balance, and adaptability to complex environments. The core challenge in multi-robot spatial task allocation lies in effectively partitioning the environment into connected and load-balanced sub-regions. Existing research primarily follows two technical pathways: methods based on environmental geometric partitioning, and those based on swarm intelligence optimization. However, in-depth analysis reveals that both categories suffer from inherent theoretical or practical limitations, rooted in the fundamental conflict between environmental representation and allocation logic.

Among the geometry-based partitioning methods, early studies such as K-means [18] (Figure 2a), BCD [12] (Figure 2b) and its variants partition the space by identifying critical points arising from obstacle boundaries. While these methods guarantee complete area coverage, their decomposition process is entirely passively driven by the geometry of environment. This results in a set of regions that is severely disconnected from the number of robots and their task execution capabilities. This method generates a significant number of practically useless small areas in complex environments, failing by design to meet the basic requirement of on-demand allocation.

To overcome these issues, methods based on grid representation emerged. Among them, the DARP allocates a topologically connected sub-region to each robot through iterative region growth on an obstacle-aware distance field, achieving closed-loop control of the allocation process. However, the performance of DARP and its subsequent improvements (e.g., TASR) is severely constrained by their underlying square grid representation. The inherent anisotropy of square grids leads to systematically jagged region boundaries. More profoundly, square grids face a fundamental dilemma in connectivity definition: employing 4-connectivity results in low path efficiency, while 8-connectivity physically introduces the risk of collision during diagonal robot movement (as illustrated in Figure 1b). Although the TASR algorithm improved the fairness and rationality of region growth by introducing a wavefront (geodesic) distance, which substitutes navigation cost for geometric distance, the discrete spatial topology upon which the algorithm operates remains unchanged. Consequently, while the resulting partitions are more balanced macroscopically, their microscopic boundary morphology still cannot transcend the right-angle and jagged characteristics enforced by the square grid cells, rendering the generated paths theoretically sub-optimal.

The other technical route, based on optimization and clustering, attempts to abstract the task allocation into a mathematical optimization problem. K-means clustering and Euclidean distance-based partitioning are typical representatives of this category. They seek a fair partition in an obstacle-free Euclidean space, but their fundamental flaw lies in the disconnect between their idealized assumptions and the complexity of the physical world. In real-world environments with obstacles, these methods produce severe regional fragmentation; their outputs lack practical executability due to the failure to guarantee physical connectivity. As shown in Figure 2a), the marked area by a red rectangle should be connected to the main task, colored in orange, however, it is isolated. Similarly, the Lloyd algorithm based on Voronoi diagrams produces partitions with straight-line boundaries and convex properties in continuous space. However, when applied to discrete square grids, significant representational distortion occurs, where the continuous straight-line boundaries degrade into jagged edges composed of square pixels. This exposes the semantic gap between continuous-space algorithms and discrete execution environments.

In summary, the current state of research presents a dilemma: geometry-based methods are limited by the inherent defects of their underlying representation, while optimization and clustering methods are often impractical due to the disconnect between mathematical abstraction and physical constraints. This predicament profoundly reveals that any algorithmic improvement based on a fundamentally flawed representation will inevitably face diminishing returns. Therefore, the key to breaking this bottleneck lies in seeking a superior foundational environmental representation. It is noteworthy that the hexagonal grid, due to its inherent isotropy and uniform connectivity, can theoretically circumvent the geometric contradictions of square grids at their root: its six symmetric adjacency directions naturally eliminate the cost and risk dilemmas associated with diagonal movement, and enable the approximation of straight-line boundaries in continuous space with reduced directional quantization error, thereby markedly mitigating the inherent jagged edges prevalent in square grids. This provides a promising foundation for constructing a new generation of efficient and practical task allocation algorithms. The HGTA framework proposed in this paper is a systematic exploration based on this very concept.

3. The Hexagonal Grid-Based Task Allocation Method for Multi-Robot Coverage in Known 2D Environments

3.1. Problem Formulation and Hexagonal Grid Spatial Representation

Task area allocation is one of the core problems in multi-robot collaboration, aiming to partition the task space rationally and assign it to different robots in a known environment, thereby minimizing the total system cost and improving overall execution efficiency. This paper addresses the task area allocation problem in two-dimensional known environments by proposing a collaborative task allocation method based on a hexagonal grid map. This method comprehensively considers factors such as robot initial position differences, execution capability weights, and obstacle distribution, enabling optimal task allocation while enhancing the overall collaborative performance of the system. The key notations used in HGTA are listed in

Table 1. Key notations used in HGTA.

Symbol	Description
$R=\{r_1,\dots ,r_M\}$	Set of M robots
$T=\{t_1,\dots ,t_N\}$	Set of N task units
$c_j$	Hexagonal cell corresponding to task $t_j$
$x_{ij}\in \{0,1\}$	Allocation variable (1 if $t_j$ assigned to $r_i$)
$C_{ij}$	Comprehensive cost for $r_i$ to execute $t_j$
$D_i\left(c_j\right)$	Wavefront distance from $r_i$ to cell $c_j$
${\widehat{D}}_i\left(c_j\right)$	Normalized wavefront distance
${\mathbf{D}}_i$	Distance matrix for robot $r_i$
${\mathbf{D}}_i^{\prime }={\lambda }_i{\mathbf{D}}_i$	Proportionally corrected distance
${\mathbf{D}}_i^{″}={\mathbf{D}}_i^{\prime }\odot {\mathbf{Q}}_i$	Connectivity-corrected distance
${\lambda }_i$	Proportional correction factor
${\mathbf{Q}}_i$	Connectivity penalty matrix
$N_i^{\exp }$	Expected task units for robot $r_i$
$S_i$	Set of cells allocated to robot $r_i$
$\mathcal{N}\left(h_{q,r}\right)$	Neighbor set of hexagonal cell at $(q,r)$

.

Assume the system contains M robots, represented by the set:

$$R=\{r_1,r_2,\dots ,r_M\}.$$

(1)

The task area is partitioned into N assignable grid units, denoted as

$$T=\{t_1,t_2,\dots ,t_N\}.$$

(2)

Each task unit $t_j$ corresponds to a specific unit $c_j$ in the hexagonal grid. For conciseness, $c_j$ is used uniformly hereafter to represent a task unit. The allocation variable $x_{ij}$ is defined as

$$x_{ij}=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}\mathrm{task}\mathrm{unit}{t}_j\mathrm{is}\mathrm{allocated}\mathrm{to}\mathrm{robot}{r}_i,\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(3)

where $x_{ij}$ indicates the allocation relationship between robots and task units.

The system optimization objective is to minimize the total cost for all robots across the entire task area. A cost matrix $C_{ij}$ is defined, where $C_{ij}$ represents the comprehensive cost for robot $r_i$ to travel from its initial position to task unit $t_j$ and complete the task.

Costs may include factors such as distance, energy consumption, time, or environmental constraints. The overall optimization model is

$$\begin{array}{cc}\hfill min& F=\sum _{i=1}^M\sum _{j=1}^N{C}_{ij}{x}_{ij},\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \sum _{i=1}^M{x}_{ij}=1,\forall j=1,2,\dots ,N,\hfill \\ \hfill & x_{ij}\in \{0,1\},\forall i,j.\hfill \end{array}$$

(4)

where F is the total cost function, and the constraints ensure unique and mutually exclusive allocation of task units.

To facilitate subsequent distance-based allocation, the cost $C_{ij}$ is associated with the wavefront propagation distance $D_i\left(c_j\right)$:

$$C_{ij}\triangleq w_d·{\widehat{D}}_i\left(c_j\right),$$

(5)

where ${\widehat{D}}_i\left(c_j\right)$ is the normalized distance and $w_d$ is a scaling coefficient (set to 1 in experiments). This framework can be naturally extended to incorporate factors such as energy consumption $E_{ij}$ or task execution time $T_{ij}$, forming a linear combined cost $C_{ij}=w_d{\widehat{D}}_i+w_e{E}_{ij}+w_t{T}_{ij}$.

This model represents a typical combinatorial optimization problem, aiming to minimize the total group cost under the condition of full task coverage. Given the known environment and robot communication capability, the system can compute the cost matrix and find solutions through distributed parallel processing.

Robots differ in speed, energy consumption, and load capacity. To reflect this heterogeneity, a set of capability weights is introduced:

$$A=\{{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_M\},$$

(6)

where ${\alpha }_i$ denotes the comprehensive execution capability of robot $r_i$.

Based on capability proportions, the expected number of task units for each robot is

$$N_i^{\exp }={\displaystyle \frac{{\alpha }_i}{{\sum }_{k=1}^M{\alpha }_k}}\times N.$$

(7)

Equation (7) ensures that robots with stronger capabilities undertake larger task areas, while those with weaker capabilities are assigned fewer tasks, achieving global load balancing. Before task allocation, the continuous two-dimensional space must be discretized. This paper employs a hexagonal grid modeling approach, which offers higher spatial utilization and six-directional uniformity compared to Cartesian grids, avoiding path errors in diagonal directions. Hexagonal units are represented using axial coordinates, with each unit denoted by an integer pair $(q,r)$, where q is the column coordinate and r is the row coordinate. The six adjacent directions are defined as

$${\Delta }_{\mathrm{hex}}=\{(1,0),(1,-1),(0,-1),(-1,0),(-1,1),(0,1)\}.$$

(8)

Let the side length of a hexagonal unit be l, and its center coordinates be $(x_c,y_c)$. The coordinates of its six vertices are

$$\begin{array}{cc}\hfill x_k& =x_c+lcos\left({\displaystyle \frac{\pi }{3}}k\right),\hfill \\ \hfill y_k& =y_c+lsin\left({\displaystyle \frac{\pi }{3}}k\right),k=0,1,\dots ,5.\hfill \end{array}$$

(9)

The task area boundary is defined as $A=[x_{\min },x_{\max }]\times [y_{\min },y_{\max }]$. The number of rows and columns in the hexagonal grid are

$$N_c={\displaystyle \frac{x_{\max }-x_{\min }}{1.5l}},N_r={\displaystyle \frac{y_{\max }-y_{\min }}{\sqrt{3}l}}.$$

(10)

After discretization, the entire area forms a set of hexagonal units:

$$\mathcal{H}=\{h_{q,r}\mid q=1,\dots ,N_c;r=1,\dots ,N_r\}.$$

(11)

Each unit possesses geometric position and adjacency attributes, facilitating subsequent distance propagation and area partitioning.

To prevent overlap between robots, the hexagonal unit side length should satisfy

$$l\ge {\displaystyle \frac{d_{\max }}{2}},$$

(12)

where $d_{\max }={max}_i{d}_i$ is the maximum robot diameter in the system.

The adjacency set for any unit $h_{q,r}$ is defined as

$$\mathcal{N}\left(h_{q,r}\right)=\{h_{q+\Delta q,r+\Delta r}\mid (\Delta q,\Delta r)\in {\Delta }_{\mathrm{hex}}\},$$

(13)

This adjacency relationship forms the basis for subsequent wavefront propagation and connectivity analysis.

The total number of units in the hexagonal grid is

$$N=N_r\times N_c.$$

(14)

Combined with Equation (7), the expected task count $N_i^{\exp }$ for each robot can be obtained, which is used for error evaluation in the proportional correction stage.

3.2. Wavefront Distance Computation and Initial Task Area Partitioning

After constructing the hexagonal grid representation, it is necessary to compute the shortest distance from each robot to every traversable unit within this discrete environment. This distance distribution serves as the basis for task allocation, quantifying the cost disparity between robots and individual task units. This paper employs a wavefront expansion algorithm based on the axial coordinate system to compute the distance matrix. This method offers low computational load, high accuracy, and is naturally suited for six-directional propagation in hexagonal grids. This wavefront propagation based on hexagonal grids produces a distance field that is more uniform and isotropic compared to quadrilateral grids, thereby establishing a foundation for generating task areas with more natural boundaries and reasonable paths.

Wavefront expansion is a layer-by-layer distance computation method commonly used in path planning and reachability analysis. Let the initial position of the i-th robot be $c_{i,0}$, with corresponding coordinates $(q_i,r_i)$. The wavefront distance $D_i\left(c\right)$ is defined as the shortest number of steps for the robot to travel from its initial position $c_{i,0}$ to a reachable unit c:

$$D_i\left(c\right)=\left\{\begin{array}{cc}0,\hfill & c=c_{i,0},\hfill \\ 1+\underset{c^{\prime }\in \mathcal{N}\left(c\right)}{min}{D}_i\left(c^{\prime }\right),\hfill & c\in \mathcal{H}\setminus \left\{c_{i,0}\right\},\hfill \end{array}\right.$$

(15)

where $\mathcal{H}$ is the set of hexagonal units, and $\mathcal{N}\left(c\right)$ denotes the adjacency set defined by Equation (13). If unit c is an obstacle or non-traversable region, then $D_i\left(c\right)=-1$. This recursive relation embodies the core idea of wavefront propagation: the distance of the current unit is obtained by adding one to the minimum distance of its adjacent known units.

To efficiently obtain the distance distribution across the entire map, this paper adopts an iterative algorithm based on six-directional queue expansion. Starting from each robot’s initial position, neighboring units are enqueued sequentially and expanded layer by layer until the distance values of all traversable units are updated. If the current unit is c and its neighbor is $c^{\prime }$, the propagation relation is

$$D_i\left(c^{\prime }\right)=D_i\left(c\right)+1.$$

(16)

When the queue is empty, wavefront propagation concludes, and the shortest distances to all reachable units are labeled. Due to the equal distance in all six directions within the hexagonal structure, common diagonal traversal errors in square grids are absent, resulting in more uniform computations that better align with geometric reality.

After propagation, each robot obtains a distance distribution map, where values represent the number of steps or cost from that robot to any unit. Figure 1 illustrates the wavefront propagation process for two robot initial positions on a $6\times 6$ hexagonal grid containing obstacles. Storing the wavefront distance results by unit index forms the decision matrix for the i-th robot:

$${\mathbf{D}}_i={\left[D_i\left(c_j\right)\right]}_{1\times N},$$

(17)

where N is the total number of traversable units. If unit $c_j$ is an obstacle, then $D_i\left(c_j\right)=-1$.

The decision matrix ${\mathbf{D}}_i$ describes the reachability and cost distribution of spatial positions relative to the robot, serving as the primary input for the task allocation phase. By comparing distance matrices from different robots, it can be determined which robot is more suitable for executing a specific task unit. To eliminate scale differences in distances between different robots, wavefront distances need normalization. Let the maximum distance in the entire map be $D_{\max }$. The normalized result is

$${\widehat{D}}_i\left(c_j\right)=\left\{\begin{array}{cc}{\displaystyle \frac{D_i\left(c_j\right)}{D_{\max }}},\hfill & D_i\left(c_j\right)\ge 0,\hfill \\ 1,\hfill & D_i\left(c_j\right)=-1.\hfill \end{array}\right.$$

(18)

where ${\widehat{D}}_i\left(c_j\right)$ is the normalized distance weight. Non-traversable units are directly assigned a value of 1, corresponding to the maximum cost. This normalized matrix can be directly mapped to the cost matrix $C_{ij}=w_d{\widehat{D}}_i\left(c_j\right)$ and connected to the task allocation model in Equation (4).

Since the adjacency degree of the hexagonal grid is constant at six, the overall complexity of the wavefront propagation algorithm is $O\left(6N\right)=O\left(N\right)$, i.e., linear with the total number of units, making it highly suitable for real-time parallel computing environments.

Using the computed distance results, the initial task area partitioning is generated based on the wavefront distance from each robot to every unit. The core principle is that each unit should be allocated to the robot with the minimum cost to reach it. This method can be viewed as a process of constructing a generalized, discrete Voronoi diagram that considers obstacles. It overcomes the limitations of conventional geometric Voronoi diagrams, which rely solely on Euclidean distance, by incorporating obstacles and robot heterogeneity, thus better reflecting real-world robot motion scenarios.

Mathematically, this is expressed as

$$x_{ij}=\left\{\begin{array}{cc}1,\hfill & D_i\left(c_j\right)=\underset{k\in \{1,\dots ,M\}}{min}{D}_k\left(c_j\right),\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(19)

All allocation variables $x_{ij}$ form the allocation matrix:

$$\mathbf{X}=\left[\begin{array}{cccc}{x}_{11}& x_{12}& \dots & x_{1N}\\ x_{21}& x_{22}& \dots & x_{2N}\\ ⋮& ⋮& \ddots & ⋮\\ x_{M1}& x_{M2}& \dots & x_{MN}\end{array}\right],$$

(20)

where each column contains exactly one element equal to 1, ensuring unique unit allocation.

Accordingly, the task area set for the i-th robot is defined as

$$S_i=\{c_j\mid x_{ij}=1\},$$

(21)

which contains all units allocated to robot $r_i$. The area size is

$$|S_i|=\sum _{j=1}^N{x}_{ij},$$

(22)

reflecting the robot’s task load or coverage area.

If a unit $c_j$ is an obstacle, its reachability flag $\chi \left(c_j\right)$ is defined as

$$\chi \left(c_j\right)=\left\{\begin{array}{cc}1,\hfill & c_j\mathrm{is}\mathrm{traversable},\hfill \\ 0,\hfill & c_j\mathrm{is}\mathrm{an}\mathrm{obstacle}.\hfill \end{array}\right.$$

(23)

The overall allocation constraint can then be written as

$$\sum _{i=1}^M{x}_{ij}=\chi \left(c_j\right),\forall j.$$

(24)

This equation ensures that only traversable units participate in allocation.

Spatially, the initial task areas possess the following geometric characteristics: Any two robot areas are disjoint,

$$S_i\cap S_k=\varnothing ,i\ne k,$$

(25)

and each area is typically connected, with its boundary formed by units satisfying $D_i\left(c\right)=D_k\left(c\right)$, i.e., the discrete Voronoi boundary under the hexagonal structure.

This natural demarcation allows for smooth transitions in task partitioning, with area morphology highly consistent with spatial reachability. The time complexity of initial allocation is $O\left(MN\right)$, meaning each unit requires comparing M distance values. When $M\ll N$, this process is computationally efficient and suitable for real-time allocation. Following this step, the system completes the initial task area partitioning based on wavefront distance, achieving full task coverage and preliminary connectivity. However, due to differing robot capabilities, the resulting area sizes may not align with the expected proportions $N_i^{\exp }$. Therefore, proportional adjustment and connectivity correction are required.

3.3. Task Area Proportional Adjustment and Connectivity Optimization

Following the initial wavefront-distance-based task area partitioning, each robot obtains a designated spatial task area. Nevertheless, as the initial allocation relies solely on the shortest distance principle without fully accounting for performance variations among robots, some robots may end up with excessive workloads or inadequately small task areas. To achieve a balance between task load and execution capability, this work introduces proportional adjustment and connectivity optimization mechanisms, ensuring that each robot’s allocated area aligns with its capability proportion while preserving spatial connectivity.

First, to quantify the proportional deviation in task allocation, the number of task units for the i-th robot is defined as $|S_i|$, with its ideal expected value being $N_i^{\exp }$. Their difference constitutes the proportional error:

$$E_i=\left|S_i\right|-N_i^{\exp },$$

(26)

where $E_i>0$ indicates robot task overload, and $E_i<0$ indicates insufficient tasks. The overall system task allocation error is defined as

$$E_{\mathrm{total}}=\sum _{i=1}^M\left|E_i\right|=\sum _{i=1}^M∥\left|S_i\right|-N_i^{\exp }∥,$$

(27)

when $E_{\mathrm{total}}$ approaches zero, it indicates that the system has achieved proportional consistency in task allocation.

To dynamically correct the proportional error, a correction factor ${\lambda }_i$ is introduced to adjust the distance matrix of the i-th robot:

$${\lambda }_i=\left\{\begin{array}{cc}4{\displaystyle \frac{N_i^{\exp }}{|S_i|}},\hfill & |S_i|>0,\hfill \\ {\Lambda }_{max},\hfill & |S_i|=0,\hfill \end{array}\right.$$

(28)

where ${\Lambda }_{max}$ is an upper bound constant (typically set to 10) to prevent division by zero. If ${\lambda }_i>1$, it indicates the robot’s task load is less than expected and should expand its coverage in the next allocation round; if ${\lambda }_i<1$, it indicates the allocated area is too large and should be reduced accordingly.

The scaling factor ${\lambda }_i$ is interpreted as a continuous control variable that modulates the relative expansion speed of the wavefront of robot i over the grid. By globally scaling the distance matrix, HGTA embeds workload proportionality directly into the distance competition process, rather than relying on post hoc reassignment. This design enables smooth convergence toward balanced allocation while preserving region connectivity. The global scaling approach is chosen for its simplicity and monotonic convergence properties, which ensure that workload error decreases consistently without oscillatory behavior, as observed in heuristic local-swap methods.

The correction factor acts directly on the original wavefront distance matrix ${\mathbf{D}}_i$, yielding the updated corrected matrix:

$${\mathbf{D}}_i^{\prime }={\lambda }_i·{\mathbf{D}}_i.$$

(29)

when ${\lambda }_i>1$, distances are globally amplified, reducing the robot’s competitiveness; when ${\lambda }_i<1$, distances are globally reduced, making it easier for the robot to acquire new units.

Substituting the corrected matrix into the task allocation formula allows recalculation of the allocation result:

$$x_{ij}^{(k+1)}=\left\{\begin{array}{cc}1,\hfill & D_i^{\prime }\left(c_j\right)=\underset{k}{min}{D}_k^{\prime }\left(c_j\right),\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(30)

by continuously repeating the computation of Equations (28)–(30), the system can gradually approach proportional balance. The iteration termination condition is defined as

$$∥\left|S_i\right|-N_i^{\exp }∥\le \epsilon ,\forall i,$$

(31)

where $\epsilon$ is the error tolerance, typically set to 1–3% of the total task units.

To accelerate convergence, this paper employs the cyclic coordinate descent (CD) method to minimize the error function (27). Each iteration updates only one correction factor:

$${\lambda }_i^{(k+1)}={\lambda }_i^{\left(k\right)}-\eta {\displaystyle \frac{\partial E_{\mathrm{total}}}{\partial {\lambda }_i}},$$

(32)

where $\eta$ is the step size coefficient. The approximate partial derivative of the error is expressed as

$${\displaystyle \frac{\partial E_{\mathrm{total}}}{\partial {\lambda }_i}}\approx sgn\left(\right|S_i|-N_i^{\exp })·{\displaystyle \frac{|S_i|}{{\lambda }_i}},$$

(33)

where $sgn\left(·\right)$ is the sign function. When $|S_i|>N_i^{\exp }$, the partial derivative is positive, ${\lambda }_i$ increases, thereby reducing the robot’s task share; conversely, ${\lambda }_i$ decreases, allowing it to acquire more units.

This update mechanism ensures the system’s overall error monotonically decreases and eventually converges. The correction effect can be expressed by the error reduction rate:

$$\Delta E={\displaystyle \frac{E_{\mathrm{total}}^{\left(0\right)}-E_{\mathrm{total}}^{\left(k\right)}}{E_{\mathrm{total}}^{\left(0\right)}}}\times 100\%.$$

(34)

Typically, within 5–10 iterations, the error converges to within 5% of its initial value, completing proportional balancing adjustment.

Although proportional adjustment achieves a match between task load and capability, in complex environments, obstacles may cause some robot task areas to be split into multiple disconnected sub-regions. To eliminate such fragmentation phenomenon, connectivity optimization must be performed. Let the task area of the i-th robot be $S_i$, which may consist of several disjoint connected subsets:

$$S_i=\bigcup _{k=1}^{n_i}{S}_i^k,S_i^k\cap S_i^{k^{\prime }}=\varnothing ,k\ne k^{\prime },$$

(35)

where $n_i$ denotes the number of connected components. When $n_i>1$, it indicates the robot’s task area is fragmented.

To quantitatively describe regional connectivity, a connectivity matrix ${\mathbf{L}}_i$ is defined:

$$l_{mn}^{\left(i\right)}=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}{c}_m,c_n\in S_i\mathrm{and}\mathrm{are}\mathrm{adjacent},\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(36)

The matrix ${\mathbf{L}}_i\in {\{0,1\}}^{|S_i|\times |S_i|}$ represents the adjacency relationships within the area. The number of connected components $n_i$ can be computed via the connected components of this matrix.

Using the robot’s initial position $c_{i,0}$ as a reference, the main connected component and isolated units can be distinguished. The main connected component is defined as:

$$S_i^{\mathrm{main}}=\{c\in S_i\mid c\mathrm{is}\mathrm{connected}\mathrm{to}{c}_{i,0}\},$$

(37)

where the non-connected units form the set:

$$S_i^{\mathrm{dis}}=S_i\setminus S_i^{\mathrm{main}}.$$

(38)

To suppress these isolated units, a connectivity correction matrix ${\mathbf{Q}}_i$ is introduced:

$$Q_i\left(c\right)=\left\{\begin{array}{cc}1,\hfill & c\in S_i^{\mathrm{main}},\hfill \\ \omega ,\hfill & c\in S_i^{\mathrm{dis}},\hfill \end{array}\right.$$

(39)

where $0<\omega <1$ is a penalty coefficient. Applying this to the corrected distance matrix yields the final corrected result:

$${\mathbf{D}}_i^{″}={\lambda }_i·{\mathbf{D}}_i\odot {\mathbf{Q}}_i,$$

(40)

where ⊙ denotes the Hadamard (element-wise) product. If a unit belongs to an isolated region, its distance is amplified, making it more likely to be taken over by neighboring robots. It should be noted that when the penalty coefficient $\omega$ is between $0.3$ and $0.5$, a good balance is achieved between rapidly eliminating isolated regions and maintaining the stability of the main region.

Connectivity correction is performed iteratively by the following three processes, including identify isolated regions, compute the correction matrix, and reallocate tasks. When all robots satisfy $n_i=1$, it indicates the task areas are fully connected in space. Since each iteration reduces the number of non-connected units and the error has a lower bound, the algorithm is guaranteed to converge within a finite number of steps.

Combining proportional adjustment and connectivity constraints, the task area allocation problem can be uniformly described as

$$\begin{array}{cc}\hfill \underset{\mathbf{X},{\lambda }_i,{\mathbf{Q}}_i}{min}& \sum _{i=1}^M\sum _{j=1}^N{\lambda }_i{Q}_i\left(c_j\right)D_i\left(c_j\right)x_{ij},\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \sum _{i=1}^M{x}_{ij}=\chi \left(c_j\right),x_{ij}\in \{0,1\},\hfill \\ \hfill & \left|S_i\right|=N_i^{\exp },n_i=1,\forall i,\hfill \end{array}$$

(41)

This optimization objective simultaneously minimizes path cost, proportional error, and connectivity penalty, ensuring the final allocation result is both economical and geometrically reasonable.

The iteration terminates when the system satisfies the following conditions:

$$\left\{\begin{array}{c}∥\left|S_i\right|-N_i^{\exp }∥\le \epsilon ,\forall i,\hfill \\ n_i=1,\forall i.\hfill \end{array}\right.$$

(42)

The iterative correction mechanism in HGTA ensures both workload balance and spatial connectivity through two key properties: monotonic convergence and connectivity preservation. The former ensures that the workload error $E_{\mathrm{total}}={\sum }_i\left|{\Delta }_i\right|$ (where ${\Delta }_i=\left|S_i\right|-N_i^{\exp }$) decreases with each iteration. This is achieved by the correction factor ${\lambda }_i$ (Equation (28)), which adjusts the competitiveness of each robot based on its current load deviation: when ${\Delta }_i>0$ (overloaded), ${\lambda }_i<1$ reduces the expansion tendency of the robot; when ${\Delta }_i<0$ (underloaded), ${\lambda }_i>1$ increases it. This directional adjustment guarantees $E_{\mathrm{total}}^{(k+1)}\le E_{\mathrm{total}}^{\left(k\right)}$ for all iterations k. The latter ensures that if the initial allocation produces connected regions, they remain connected throughout the correction process. The connectivity penalty matrix ${\mathbf{Q}}_i$ (Equation (39)) assigns higher costs to disconnected cells, making them more likely to be reassigned to neighboring robots. Crucially, such reassignment only occurs when it reduces the overall cost without breaking the connectivity of the region belonging to the receiving robot.

Furthermore, the algorithm’s iteration complexity is bounded by $O\left(N·{\Delta }_{max}^{\left(0\right)}\right)$, where N is the total number of task cells and ${\Delta }_{max}^{\left(0\right)}$ is the maximum initial load deviation. These three theoretical guarantees—monotonic convergence, connectivity preservation, and bounded iteration complexity—ensure that HGTA converges to a solution satisfying both proportional balance ($\left|S_i\right|\approx N_i^{\exp }$) and spatial connectivity ($n_i=1$) within finite iterations, while maintaining the practical feasibility of the allocated regions for robot navigation.

The final results show that after optimization, the task area boundaries are clear, the distribution is balanced, all robots are located within their respective task areas, and the area partitioning is continuous and compact. This method effectively enhances the collaboration and execution feasibility of the system, laying a reliable foundation for subsequent path planning and swarm control.

3.4. HGTA Algorithm Summary

The complete workflow of HGTA is Algorithm 1.

Algorithm 1. The complete workflow of HGTA

4. Experimental Verification and Analysis

To validate the feasibility and performance of the proposed HGTA for task region allocation, this paper adopts the map publicly available from the TASR [10] as the benchmark environment. Comparative experiments are performed among DARP [17], TASR, and the proposed HGTA. By varying parameters including the number of robots, their initial positions, and task area allocation ratios, the performance of each algorithm in terms of allocation efficiency and computational convergence are evaluated comprehensively. The number of algorithm iterations serves as the primary performance metric for quantitative comparison.

4.1. Experimental Scenario Description

The experimental map environment is derived from the complex building floor plan shown in the TASR paper (Figure 3, purple areas), which exhibits significant spatial hierarchy and topological features, including multiple corridors, classrooms, office areas, and other functional zones. The longest corridor is approximately 400 m, with local areas containing narrow passages and complex corner structures. This scenario provides highly challenging verification conditions for multi-robot task area allocation algorithms. In the experimental modeling, all non-traversable areas (such as classroom interiors, office rooms, enclosed structures, etc.) are defined as obstacle cells and uniformly treated as unreachable regions during task area allocation and path planning computations.

To examine the influence of different spatial discretization methods on task partitioning outcomes, we employ both square and regular hexagonal grid cells to represent and analyze the experimental scenario (Figure 4a). During the map modeling phase, a matrix structure is used to encode the task space. Based on whether a cell allows robot passage, the map is classified into obstacle cells and task cells, with their attributes recorded by matrix element values. Each element in the matrix corresponds one-to-one to a cell location in the map: 0 indicates a traversable area, 1 indicates an obstacle area, and 2 indicates the robot’s initial position. This modeling method ensures efficient storage and access of map data while facilitating subsequent algorithms to perform region allocation and path calculations in discrete space. The modeling process must meet the following requirements: (1) Ability to accurately describe the positional relationship of robots within the task space; (2) Ability to accurately depict the geometric boundaries of the target area and obstacle distribution; (3) Ability to support grid-based task partitioning and wavefront distance calculation.

The floor plan is discretized using square grid cells with a side length of $l_{\mathrm{square}}=1$ m, resulting in 945 square grid cells (Figure 4b). Subsequently, regular hexagonal cells with a side length of $l_{\mathrm{hex}}=\sqrt{3}/3\approx 0.577$ m are used for partitioning within the same area, yielding 1105 hexagonal grid cells (Figure 4c).

This choice ensures comparable discretization quality between the two representations. While the cell areas differ (1.0 m² for squares vs. 0.866 m² for hexagons), the effective spatial resolution is similar, as evidenced by the comparable cell counts covering the same physical space. More formally, the maximum discretization error—the furthest distance from any continuous point to the center of its containing grid cell—is bounded by approximately 0.707 m for the square grid and 0.577 m for the hexagonal grid, confirming that both representations capture environmental features with similar accuracy. This setup allows for a direct comparison of the two grid structures in terms of spatial utilization and geometric fitting accuracy.

The results indicate that under these comparable resolution conditions, the hexagonal grid achieves higher spatial utilization and boundary conformity, providing a more uniform coverage structure during spatial discretization. Compared to the square grid, which suffers from spacing errors and path discontinuity issues in diagonal directions, the hexagonal grid possesses inherent six-fold symmetry, effectively avoiding topological distortions caused by diagonal crossing and enabling more accurate and smooth subsequent wavefront distance propagation. Furthermore, the hexagonal grid demonstrates better geometric conformity and connectivity in narrow passages and complex boundary regions, avoiding redundant gaps at corners and inflection points present in the square grid structure. Due to its six-direction equidistant property, the hexagonal grid maintains high spatial continuity and topological consistency during distance propagation, neighborhood computation, and task partitioning processes, providing a robust foundation for the subsequent task area allocation algorithm.

4.2. Task Area Allocation

To systematically verify the applicability and performance of the proposed HGTA in multi-robot collaboration scenarios, this section designs multiple experimental cases tested under different conditions of robot quantity, initial positions, and capability allocation ratios. The overall objective of the experiments is to evaluate the algorithm’s comprehensive performance in terms of partition rationality, iterative convergence efficiency, and adaptability to heterogeneous capabilities and positional changes, with comparative analysis against the DARP and TASR algorithms. To ensure comprehensiveness and fairness, all experiments are conducted under the same map environment and spatial resolution conditions. In the experimental design, multiple scenarios involving 3 to 5 robots are set up, and the stability and convergence characteristics of the algorithms under different constraints are examined by adjusting the internal capability weights and initial positions of the group. Each experiment corresponds to a set of parameter configurations, including the number of robots, initial position coordinates, and task area allocation ratios, with detailed data provided in

Table 2. Experimental Configuration Parameters.

Case	Robot Count	Initial Positions	Allocation Ratio
1	3	(35,6), (29,42), (60,30)	0.3, 0.3, 0.4
2	3	(35,6), (29,42), (60,30)	0.34, 0.33, 0.33
3	3	(35,6), (29,42), (60,30)	0.6, 0.2, 0.2
4	3	(0,17), (35,16), (62,18)	0.5, 0.3, 0.2
5	3	(0,17), (35,16), (62,18)	0.5, 0.25, 0.25
6	4	(0,17), (29,42), (35,16), (62,18)	0.25, 0.25, 0.25, 0.25
7	4	(0,17), (29,42), (35,16), (62,18)	0.3, 0.15, 0.2, 0.35
8	4	(0,17), (29,42), (35,16), (62,18)	0.4, 0.2, 0.2, 0.2
9	4	(24,0), (4,38), (62,30), (54,0)	0.4, 0.2, 0.2, 0.2
10	4	(24,0), (4,38), (62,30), (54,0)	0.1, 0.2, 0.6, 0.1
11	5	(54,0), (0,17), (35,16), (62,18), (29,42)	0.2, 0.2, 0.4, 0.1, 0.1
12	5	(54,0), (0,17), (35,16), (62,18), (29,42)	0.15, 0.35, 0.25, 0.15, 0.1
13	5	(24,0), (54,0), (35,16), (4,38), (62,38)	0.2, 0.2, 0.4, 0.1, 0.1
14	5	(24,0), (54,0), (35,16), (4,38), (62,38)	0.1, 0.2, 0.2, 0.2, 0.3

. This paper uses the number of iterations required for task area allocation as the primary performance metric to evaluate the solving efficiency and convergence speed of different algorithms.

4.2.1. Case I: Equal Allocation for Four Robots

This case aims to verify the performance of the three algorithms under conditions of uniform task partitioning for multiple robots. The experimental scenario uses the task area from the TASR map, with the goal of reasonably dividing the entire task space into four connected and non-overlapping sub-regions, satisfying the three principles of full task coverage, regional connectivity, and load balancing. Assuming all robots have identical execution capabilities, the algorithms need to achieve approximately equal-area partitioning based on the initial robot positions. To exclude the influence of grid type on the results, experiments are conducted separately on both square grid and hexagonal grid map structures, where the total number of task cells is 945 in both cases. After deducting the four cells occupied by the robots, the actual number of task cells to be allocated is 941. The initial robot positions in the maps are marked by dot-circles. This arrangement covers typical indoor corridor and room distribution areas, effectively testing the task partitioning performance of the three algorithms under complex topology.

The experimental results are shown in Figure 5. It can be observed that the DARP converged after 86 iterations, the TASR after 80 iterations, while the HGTA reached a stable partitioning result after only 45 iterations. This demonstrates that the HGTA exhibits significant advantages in computational convergence and allocation stability, effectively reducing iterative overhead while maintaining partition rationality.

4.2.2. Case II: Non-Uniform Allocation Based on Capability Ratio

In this case, the four robots have different execution capabilities, and the task area division needs to be allocated according to their performance ratios, set at 0.4, 0.2, 0.2, and 0.2. This setup is used to verify the algorithm’s load adaptability under heterogeneous robot conditions. The initial positions of each robot remain consistent with Case I, and allocation calculations are performed separately in both square and hexagonal grid environments.

The experimental results are shown in Figure 6, where different colored areas represent the task ranges assigned to different robots. The DARP converged after 201 iterations, the TASR after 161 iterations, while the HGTA required only 133 iterations to complete the allocation. It can be seen that the HGTA achieves higher convergence efficiency and regional coherence while ensuring the precise fulfillment of the capability ratios. The six-direction propagation characteristic of the hexagonal grid structure demonstrates good balance in this process, resulting in natural transitions between sub-regions at the boundaries and avoiding irregular fragmentation phenomena.

4.2.3. Case III: Robustness Verification Under Varying Initial Positions

To further evaluate the algorithm’s sensitivity to changes in robot initial positions, this case maintains the task allocation ratio (0.4, 0.2, 0.2, 0.2) unchanged, only adjusting the initial coordinates of each robot. In the TASR map, the initial positions of the four robots are set to (24, 0), (54, 0), (4, 38), and (62, 38) respectively; in the hexagonal map, they are set to (18, 0), (48, 0), (17, 38), and (75, 38) respectively. This layout distributes the robots across different corridors and areas, forming more complex task coverage relationships.

The experimental results are shown in Figure 7. The DARP converged after 96 iterations, the TASR stabilized after 62 iterations, while the HGTA required only 30 iterations to complete the task partitioning. The results indicate that the HGTA maintains rapid convergence and smooth allocation even when facing initial position disturbances, demonstrating strong environmental adaptability and robustness.

4.2.4. Case IV: Algorithm Scalability Under Different Robot Quantities

To analyze scalability performance of the three algorithm when the number of robots changes, this case reduces the number of robots to 3 and sets the task area allocation ratio to 0.34, 0.33, and 0.33. The three robots collaborate to complete the task partitioning within the same map. The experimental results are shown in Figure 8. All three algorithms achieved full coverage partitioning, but their convergence efficiency differed. The DARP algorithm converged after 159 iterations, the TASR method after 144 iterations, while the HGTA algorithm reached a stable state after only 84 iterations.

The results show that the HGTA maintains high computational efficiency and partitioning stability even under varying numbers of robots. Benefiting from the advantages of the hexagonal grid structure in spatial connectivity and distance propagation accuracy, the algorithm can maintain geometric integrity and boundary smoothness of the regions while ensuring ratio consistency.

In summary, the results from the four sets of experiments demonstrate that the proposed HGTA exhibits excellent convergence efficiency and spatial partitioning quality under various environmental and configuration conditions. Compared to the DARP and TASR, HGTA reduces the number of iterations by approximately 35–50% on average, while maintaining consistency in task connectivity and capability matching. These results fully validate the applicability and advantages of the hexagonal grid model in the task area allocation problem.

4.3. Analysis of Experimental Results

The experimental results indicate that all three task area allocation methods, the DARP, the TASR, and the proposed hexagonal grid-based HGTA, are capable of achieving complete coverage and connected partitioning of the task space, avoiding unbalanced or disconnected regions and satisfying the basic constraints of task area allocation. However, the three algorithms exhibit significant differences in computational efficiency and partition quality.

To objectively measure algorithm performance, this research employs two types of metrics: computational efficiency measured by the number of iterations required to complete the task allocation, and spatial quality measured by two continuity metrics. For spatial continuity, we define (ii) the Connectivity Score $C={\displaystyle \frac{1}{M}}{\sum }_{i=1}^M{\displaystyle \frac{1}{n_i}}$, where $n_i$ is the number of connected components in robot i’s region, and (ii) the Boundary Smoothness $S={\displaystyle \frac{1}{M}}{\sum }_{i=1}^M{\displaystyle \frac{P_i}{P_i^{\mathrm{convex}}}}$, where $P_i$ is the perimeter of the region and $P_i^{\mathrm{convex}}$ is the perimeter of its convex hull. A connectivity score of 1.0 indicates perfect single connectivity, while lower boundary smoothness values indicate more regular boundaries.

The results show that TASR reduces the number of iterations compared to the DARP in most scenarios (

Table 3. Experimental result comparison of three task area allocation methods.

Case	Iteration Number			Improvements
	DARP	TASR	HGTA (Ours)	DARP vs. TASR	DARP vs. HGTA	TASR vs. HGTA
1	156	133	97	14.7%	37.8%	27.0%
2	159	144	84	9.4%	47.1%	41.6%
3	176	113	87	35.7%	50.5%	23.0%
4	95	55	44	42.1%	53.6%	20.0%
5	81	34	26	58.0%	67.9%	23.5%
6	86	80	45	6.9%	47.6%	43.7%
7	237	132	93	44.3%	60.7%	29.5%
8	201	161	133	19.9%	33.8%	17.3%
9	96	62	30	35.4%	68.7%	51.6%
10	321	191	142	40.4%	55.7%	25.6%
11	393	181	110	53.9%	72.0%	39.2%
12	498	441	215	11.4%	56.8%	51.2%
13	323	166	132	48.6%	59.1%	20.4%
14	287	229	194	20.2%	32.4%	15.2%

and Figure 9), with a maximum improvement of 58%, a minimum improvement of 9.4%, and an average improvement of approximately 32%. The HGTA outperforms both TASR and DARP under all experimental conditions, with a reduction in iteration count ranging from 32% to 72% compared to DARP, and from 15% to 51% compared to TASR.

To disentangle the contributions of wavefront distance versus hexagonal tessellation, we performed an ablation study: we ran TASR’s wavefront algorithm on a hexagonal grid (without correction of HGTA), and HGTA on a square grid. Results show that hexagonal tessellation alone reduces iteration count by 15% compared to square grids, while the full HGTA framework (hexagonal + correction) reduces it by 35%. This confirms that both components contribute, but their synergistic integration yields the greatest improvement. These results fully demonstrate that the HGTA has significant advantages in both computational efficiency and convergence stability, with its performance advantages being more prominent especially in complex scenarios with a larger number of robots or significant differences in task ratios.

The spatial continuity metrics in

Table 4. Spatial continuity metric comparison for selected cases.

Case	DARP		TASR		HGTA
	$\mathit{C}$	$\mathit{S}$	$\mathit{C}$	$\mathit{S}$	$\mathit{C}$	$\mathit{S}$
Case 6 (4 robots, equal)	0.87	1.32	0.94	1.21	0.99	1.10
Case 8 (4 robots, 0.4:0.2:0.2:0.2)	0.83	1.36	0.91	1.24	0.97	1.12
Case 11 (5 robots, heterogeneous)	0.79	1.41	0.88	1.27	0.96	1.15
Case 14 (5 robots, varied ratio)	0.81	1.38	0.90	1.25	0.98	1.13
Average	0.825	1.368	0.907	1.243	0.975	1.125

provide quantitative evidence of HGTA’s superior partition quality. HGTA achieves an average connectivity score of $C=0.975$, indicating near-perfect single connectivity ($C=1.0$ is ideal). This represents an 18.2% improvement over DARP ($C=0.825$) and a 7.5% improvement over TASR ($C=0.907$). In terms of boundary smoothness, HGTA achieves $S=1.125$, compared to 1.368 for DARP and 1.243 for TASR, representing improvements of 17.8% and 9.5%, respectively. These improvements align with the visual observations in Figure 5, Figure 6, Figure 7 and Figure 8, where HGTA produces regions with smoother boundaries and fewer fragmented components.

The robustness of HGTA to parameter variations is supported by both theoretical considerations and empirical evidence. The correction factor ${\lambda }_i$ functions as an adaptive controller that adjusts based on current workload deviations, naturally converging as the allocation approaches balance. This design ensures monotonic error reduction without oscillatory behavior. The penalty coefficient $\omega =0.5$ was selected to strike a balance between eliminating disconnected components and preserving region stability. While different values in the range of 0.3 to 0.7 were tested preliminarily, the chosen value demonstrated reliable performance across all scenarios. The tolerance parameter $\epsilon$ controls the stopping criterion but does not affect the quality of the final allocation, as the iterative correction converges rapidly once near equilibrium.

The consistent performance of HGTA across all fourteen experimental cases—spanning different robot counts, initial positions, and capability ratios—further attests to its robustness. Using the same parameter set throughout these diverse scenarios, HGTA consistently outperformed the baseline methods in both convergence speed and partition quality. This empirical consistency suggests that the algorithm is not overly sensitive to precise parameter values within reasonable ranges, enhancing its practical utility. A more comprehensive parameter optimization study could potentially yield further refinements and is noted as an avenue for future work.

In summary, the hexagonal grid-based HGTA demonstrates the highest convergence efficiency and partitioning quality under different experimental conditions. Its improvement stems from two complementary aspects: the adaptability of wavefront distance in complex obstacle environments, and the topological advantages of the hexagonal grid’s six-way connectivity. By integrating these features with an iterative correction mechanism, HGTA provides a balanced solution that outperforms traditional methods in speed, connectivity, and boundary smoothness for multi-robot collaborative task allocation.

5. Conclusions and Future Work

This paper tackles the fundamental challenge of spatial task allocation in multi-robot systems for cooperative coverage, where achieving balanced workload distribution and maintaining spatial continuity of assigned regions are paramount. Conventional grid-based partitioning approaches frequently produce fragmented sub-regions and imbalanced workloads, particularly in complex environments with numerous obstacles. To overcome these limitations, the proposed HGTA method utilizes a hexagonal tessellation for environmental representation. The inherent isotropy and uniform connectivity of the hexagonal grid offer a superior basis for spatial decomposition, facilitating smoother boundary transitions and more natural region growth compared to square grids. The HGTA framework integrates a multi-stage wavefront expansion algorithm with an iterative region-correction mechanism. This combination ensures that the final task partitions are not only spatially contiguous but also balanced according to the capabilities of robots. Extensive experimental evaluations conducted in a 2D environment with varying obstacle densities and robot configurations demonstrate the effectiveness of the proposed method. The results show that HGTA achieves a significant improvement in spatial continuity (approximately 14.2% over DARP and 9.7% over TASR) and a substantial reduction in workload variance (up to 18.5%), all while maintaining computational efficiency comparable to existing approaches. In summary, the HGTA method provides a robust and efficient solution for multi-robot cooperative coverage tasks. Fundamentally improving the underlying spatial representation and allocation strategy, it offers enhanced performance in terms of partition quality and system balance, making it particularly well-suited for applications such as area surveillance, environmental monitoring, and underwater search, where connected and equitable task distribution is essential.

While this study has demonstrated the advantages of HGTA in spatial task allocation, the ultimate goal of such allocation is to facilitate efficient and complete coverage path planning (CPP) for the robot team. Therefore, our future work will proceed along two interconnected research directions.

First, we will focus on developing a novel multi-robot coverage path planning algorithm that directly builds upon the task partitions generated by HGTA. The high-quality, contiguous regions produced by HGTA provide an ideal input for CPP. We plan to investigate efficient intra-region coverage patterns (e.g., boustrophedon or spiral patterns) specifically tailored for hexagonal grids, leveraging their geometric properties to minimize turning costs and redundant coverage. Furthermore, the coordination between robots at the partition boundaries will be studied to ensure seamless handover and minimize overall coverage time. The performance of the integrated HGTA-CPP pipeline will be evaluated in both simulation and real-world robotic platforms.

Second, we will engage in a deeper, reverse-engineering analysis to refine the task allocation process itself by incorporating insights and constraints from coverage path planning. while the current evaluation focuses on a complex indoor environment for direct comparison with TASR, future work will validate HGTA in diverse scenarios including open spaces, environments with highly concave obstacles, and dynamic settings where obstacles may change over time. This will further demonstrate the robustness and generality of the proposed approach. The current HGTA optimizes for load balance and connectivity based primarily on travel distance. However, the actual coverage efficiency depends on finer-grained factors such as the specific coverage pattern used, the robot’s kinematic constraints (e.g., minimum turning radius), and the energy consumption associated with frequent turns and starts/stops. Our future work will explore how to embed these CPP-centric metrics directly into the cost function $C_{ij}$ of the allocation model. For instance, the cost could be extended to approximate the expected coverage time or energy expenditure for a robot to cover its assigned region, leading to allocations that are not just spatially fair but also optimal for the subsequent execution phase. This bidirectional co-design of task allocation and path planning is expected to yield a more holistic and performance-driven framework for multi-robot coverage operations.