Thermodynamic diagram guide type multi-agent path planning method for high-density storage environment

Abstract

The invention relates to a thermodynamic diagram guiding type multi-agent path planning method for a high-density storage environment, and the method comprises the steps: employing a thermodynamic diagram guiding-based explicit estimation conflict search algorithm HEECBS, constructing a thermodynamic diagram to record the aggregation degree of a robot at different positions and time, and carrying out the searching of a path at a bottom layer, generating a robot initial path based on thermodynamic diagram guidance and updating the thermodynamic diagram; in the high-level search, on the basis of the initial path of the robot and the updated thermodynamic diagram, path set nodes with the highest priority are selected for extension, conflicts in the path set nodes are detected, if conflicts exist, splitting is carried out according to the heat priority, new constraint nodes are generated, and then bottom-level path search is carried out again; and continuously repeating the above steps until all the robots obtain a conflict-free path or reach a preset termination condition. Compared with the prior art, the method has the advantages that conflicts can be accurately predicted, path planning time is shortened, and synchronism and stability in the task execution process are improved.

Description

Technical Field

[0001] This invention relates to the field of robot path planning technology, and in particular to a heatmap-guided multi-agent path planning method for high-density warehouse environments. Background Technology

[0002] Multi-Agent Path Finding (MAPF) is a core problem widely present in complex environments such as smart warehousing, automated logistics, and manufacturing. Its goal is to generate a set of conflict-free paths from a starting point to a target point for multiple robots in a given environment, while optimizing overall performance metrics such as total path cost (Sumof Costs) or maximum completion time (Makespan). In typical warehouse robot systems, a large number of mobile robots need to simultaneously perform tasks such as picking, delivering, and returning within a limited space. The rationality of the paths directly determines the system's throughput efficiency and stability. As the number of robots and task density increase, the probability of conflict increases dramatically, and the complexity of the MAPF problem grows exponentially, proving it to be an NP (Non-deterministic Polynomial) hard problem.

[0003] Among centralized solution methods, Conflict-Based Search (CBS) is a classic algorithm. CBS employs a framework combining a high-level Constraint Tree (CT) and a low-level single-robot shortest path search: the high-level tree detects and decomposes conflicts and generates different constraint nodes; the low-level tree, based on search algorithms such as A*, plans paths for a single robot under given constraints. This method guarantees the optimality of the global solution and performs well in small to medium-scale scenarios. However, as the robot size increases or the task density rises, the number of nodes in the high-level constraint tree expands rapidly, and the cost of repeated searches at the low level also increases dramatically, thus limiting the practical solution efficiency of CBS. To improve the algorithm's practicality, researchers have proposed near-optimal improved versions, such as ECBS (Enhanced CBS), which introduces FOCAL search and relaxation factors. This allows solutions to be found within a bounded suboptimal range, thus significantly reducing search time. Further improvements, such as EECBS (Experience-Enhanced CBS), introduce an explicit estimation search (EES) framework to achieve even more efficient bounded suboptimal solutions.

[0004] Besides the CBS series, academia has proposed various heuristic and learning-driven MAPF methods. For example, some methods guide the search by designing algorithmic heuristic functions and utilizing heuristic cost distributions, enabling paths to avoid potential conflict areas to some extent. Other studies attempt to introduce machine learning and deep reinforcement learning techniques to train models to predict conflict risks or path feasibility, thereby adjusting the search direction in advance and reducing ineffective exploration. Furthermore, there are distributed path planning and local conflict detection mechanisms that allow robots to make online adjustments during task execution, enhancing the algorithm's real-time performance and flexibility. These methods have shown advantages in specific scenarios, but they also place higher demands on the scale of training data, model generalization ability, and communication reliability.

[0005] While existing multi-robot path planning techniques have improved solution efficiency and path quality to some extent, they still have significant shortcomings in high-density, multi-task warehouse environments. First, centralized optimization algorithms, such as CBS, can solve high-quality paths at smaller scales, but with a significant increase in the number of robots and conflict frequency, the number of nodes in the high-level constraint tree expands exponentially, and the amount of redundant computation in the low-level search also increases dramatically, severely limiting the algorithm's actual computational efficiency and scalability. Even improved algorithms such as ECBS and EECBS, while reducing some search overhead through suboptimal relaxation or empirical heuristics, do not model the conflict distribution characteristics in high-density scenarios in their core search process. Therefore, in large-scale scenarios, they still struggle to avoid a large amount of ineffective expansion and redundant computation.

[0006] Secondly, existing experience-enhanced methods primarily rely on historical search information for their heuristic evaluation functions, lacking characterization of dynamic spatiotemporal congestion patterns within the task area. In warehouse robot systems, multiple robots easily form high-conflict zones near narrow passages, intersections, or picking stations, and these zones constantly change with task scheduling and execution. Traditional heuristics struggle to capture the evolutionary trends of these hotspots in a timely manner, resulting in untargeted path selection. Robots may still frequently enter potential congestion zones, triggering repeated conflict detection and splitting processes. While some research based on machine learning or distributed methods can mitigate conflicts or perform local obstacle avoidance to some extent, these methods rely on large amounts of training data, have limited generalization capabilities, and are difficult to adapt to the dynamically changing task requirements in real-world warehouse environments. Furthermore, distributed local conflict detection methods typically focus on short-term avoidance, making it difficult to optimize overall path quality from a global perspective, which can easily lead to increased task execution time or unstable scheduling. Summary of the Invention

[0007] The purpose of this invention is to overcome the shortcomings of the existing technology and provide a heat map-guided multi-agent path planning method for high-density warehousing environments, which can accurately predict conflicts, reduce path planning time, and improve the synchronization and stability during task execution.

[0008] The objective of this invention can be achieved through the following technical solution: a heatmap-guided multi-agent path planning method for high-density warehousing environments, employing a heatmap-guided explicit estimation conflict-based search (HEECBS) algorithm. First, a heatmap in the environment is initialized to record the degree of robot aggregation at different locations and times. Then, during the bottom-level path search, initial robot paths are generated based on the heatmap, and the heatmap is updated. During the high-level search, based on the initial robot paths and the updated heatmap, the highest-priority path set nodes are selected for expansion, and conflicts are detected among the path set nodes. If conflicts exist, the nodes are split according to heat priority, generating new constraint nodes, and then returning to the bottom-level path search. These steps are repeated until all robots obtain a conflict-free path or a preset termination condition is met.

[0009] Furthermore, the heatmap in the initialization environment is specifically initialized based on the task and existing paths as follows: Where t is the time step, For the robot's position, Indicates at time step At time, position Number of visits Represents the set of non-negative integers.

[0010] Furthermore, the underlying path search process includes the following steps: D1. Based on the initial heatmap, initialize the open, closed, and focal tables; D2. Check if the open table is empty. If it is empty, directly generate the robot's initial path and update the heatmap; otherwise, proceed to step D3. D3. Select the path point with the lowest popularity value in the focal table, delete the path point in the open table, add the path point in the closed table, then add adjacent path points to the open table and update the focal table. D4. Determine if the destination has been reached. If yes, generate the robot's initial path and update the heatmap; otherwise, return to step D3.

[0011] Furthermore, the formula for calculating the heat value in step D3 is as follows: in, Indicates the current path length. It is an estimate of the distance to the target point. As a heat regulation factor, This represents the composite heat value of the current waypoint at a given time step, which is used to simultaneously reflect the historical congestion of the waypoint itself and the local load of its surrounding area.

[0012] Furthermore, the composite calorific value is specifically as follows: in, This is the heat decay value. The time decay coefficient, This is the current path planning time; The average heat within the node's neighborhood; and These are the weights for heat decay and average heat, respectively.

[0013] Furthermore, the high-level search process includes the following steps: U1. For the path set nodes output by the underlying planning, calculate the value of each node and select the path set node with the smallest value. The path set node contains the initial path of each robot. U2. Detect whether there is a conflict within the selected path set nodes. If there is a conflict, proceed to step U3. Otherwise, output the currently selected path set nodes as the robot's final path and execute it. U3. Calculate the conflict score, select the conflict with the highest conflict score for processing, add the corresponding constraints, and then return to perform the underlying path search.

[0014] Furthermore, step U1 uses an extended evaluation function to calculate the node evaluation, which includes a path heat penalty term to measure the congestion of the path in time and space.

[0015] Furthermore, the extended evaluation function is specifically as follows: in, It is a traditional EECBS node evaluation function, which includes nodes Costs already incurred and nodes Heuristic cost estimation , This is the heat adjustment coefficient. For nodes Path heat penalty item, For the current node The set of paths for all robots. For all time-space points in the path, It is a 3D heatmap.

[0016] Furthermore, the heat adjustment coefficient The value is dynamically adjusted: the initial stage is set to a lower value. It converges quickly to a feasible solution, and its speed gradually improves as the conflict tree depth increases. This increases the weight of heat penalty terms, forcing a gradual avoidance of high-density areas and improving the stability and diversity of solutions.

[0017] Furthermore, the formula for calculating the conflict score in step U3 is as follows: in, The heat level of the conflict point on the heat map indicates the "sensitivity" of the conflict. This indicates that the earlier the occurrence, the higher the priority; This indicates that the shorter the path, the greater the impact of the conflict on the system; This is an indicator of edge conflict; higher scores are given when edge conflicts are more difficult to resolve. These correspond to various adjustment weights and can be flexibly adjusted under different density scenarios.

[0018] Compared with the prior art, the present invention has the following advantages: This invention proposes a heatmap-guided explicit estimation conflict search algorithm, HEECBS, which is an innovative improvement on the EECBS algorithm framework. It introduces a spatiotemporal heatmap mechanism, using heatmaps to record the degree of robot aggregation at different locations and times. Heatmap information is integrated into both the lower-level path search and higher-level search processes. On one hand, the initial robot path is generated based on the heatmap; on the other hand, the highest-priority path set nodes are selected for expansion based on the heatmap, and conflicts within the path set nodes are detected. If conflicts exist, the nodes are split according to heat priority, generating new constraint nodes, and then returning to the lower-level path search. This achieves a multi-agent path planning scheme that combines global congestion perception and local path optimization capabilities. It enables more efficient and stable path scheduling in complex warehouse environments, significantly reducing planning time and conflict frequency, and improving synchronization and stability during task execution.

[0019] This invention employs an incremental update strategy to construct a heatmap. After each round of path planning, it iterates through all possible paths of the robot's current path. Triples are generated, and the corresponding cell values ​​in the heatmap are incremented. Simultaneously, to ensure the heatmap evolves over time without excessively accumulating historical information, an exponential decay mechanism is introduced to dynamically suppress the influence of distant historical paths. Furthermore, a local spatial smoothing operation is applied to each location to calculate the average heat within its neighborhood, further enhancing the spatial continuity and predictive ability of the heatmap. Finally, a composite heatmap is obtained to guide the search. This composite heatmap simultaneously reflects the historical congestion of a node and the local load of its surrounding area, effectively compensating for the lack of representativeness of single-point heatmaps.

[0020] This invention introduces a heatmap guidance mechanism into the underlying path search process. By dynamically evaluating the popularity of path points, it integrates congestion status into the path selection process, enabling explicit perception and proactive avoidance of local conflict risks. The significance of popularity guidance lies in the fact that when multiple candidate paths have similar costs, it tends to select paths that have historically been less visited and have relatively empty neighborhoods, even if these paths are slightly longer. This mechanism embodies a "risk avoidance" strategy: proactively sacrificing some path costs in exchange for a lower probability of conflict. Furthermore, selecting the path point with the lowest popularity value in the focal table for expansion prioritizes exploring lower-risk alternative paths within the heuristic cost tolerance range, thus achieving a dynamic balance between "search flexibility" and "risk-driven" approaches.

[0021] This invention introduces a heatmap-based high-level scoring mechanism and conflict prioritization strategy during the high-level search process to enhance the strategic nature and goal awareness of the search. By extending the evaluation function and designing a path heat penalty term that is incorporated into the evaluation of path set nodes, the congestion level of path set nodes can be effectively measured, fully reflecting potential conflict risks or the degree of global resource competition. This prioritizes expanding path set nodes with lower heat and more "dispersed" paths, thereby reducing the cost of subsequent conflict splitting. In addition, the introduction of heatmap-driven conflict scoring transforms conflict management into a "task scheduling" problem, with the goal of prioritizing high-cost and high-risk conflicts, thereby improving search efficiency and solution feasibility. Attached Figure Description

[0022] Figure 1 This is a schematic diagram of the method flow of the present invention; Figure 2 This is a schematic diagram of the application process in Example 1; Figure 3 This is a schematic diagram of a heat map; Figure 4 This is an example diagram of logistics warehousing in Example 2. Detailed Implementation

[0023] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments.

[0024] Example 1 like Figure 1 As shown, a heatmap-guided multi-agent path planning method for high-density warehousing environments employs the HEECBS explicit estimation conflict search algorithm guided by heatmaps. First, a heatmap of the environment is initialized, recording the degree of robot aggregation at different locations and times. Then, during the bottom-level path search, initial robot paths are generated based on the heatmap, and the heatmap is updated. During the top-level search, based on the initial robot paths and the updated heatmap, the highest-priority path set nodes are selected for expansion, and conflicts are detected within the path set nodes. If conflicts exist, the nodes are split according to heat priority, generating new constraint nodes, and then returning to the bottom-level path search. These steps are repeated until all robots obtain a conflict-free path or a preset termination condition is met.

[0025] Among them, the HEECBS explicit estimation conflict search algorithm based on heatmap guidance proposed in this scheme has an innovative improvement on the EECBS algorithm framework. The core is to introduce a spatiotemporal heatmap mechanism to quantitatively model the congestion and conflict risks that may be formed by the robot during path execution, and to integrate heatmap information into the high-level and low-level search processes at the same time, thereby achieving the coordinated optimization of global perception and local avoidance.

[0026] In its overall process, HEECBS first initializes a heatmap of the environment to record the degree of robot aggregation at different locations and times. This heatmap is continuously updated as new paths are generated, and incorporates time decay and spatial smoothing strategies to realistically reflect the congestion risk of different areas in future time periods. The heatmap not only provides a basis for the expansion order of higher-level nodes but also directly participates in the generation and selection of lower-level paths.

[0027] This embodiment applies the above-described solution, such as Figure 2 As shown, firstly, a heatmap is initialized based on the task and existing paths; secondly, guided by the heatmap, new paths are generated for the constrained robot at the bottom layer, then the heatmap is updated, and the highest priority path set nodes are selected for expansion in the high-level constraint tree. Then, it is checked whether there are conflicts in the path set nodes. If there are conflicts, the nodes are split according to the heat priority and new constraint nodes are generated. Then, the process returns to the bottom layer path search.

[0028] The specific content includes: Heatmap Mechanism Design and Construction The core idea of ​​the heatmap mechanism is to record the access frequency of each robot path in the spatiotemporal dimension, forming a three-dimensional congestion representation structure in time and space.

[0029] like Figure 3 As shown, the heat map defined in this scheme is as follows: To indicate at time step At time, position The number of times a location is visited. After each successful underlying path planning, the heatmap for each location on the robot's path increases by 1 unit at the corresponding time step. The heatmap is constructed using an incremental update strategy. After each round of path planning, the system iterates through all locations on the robot's current path. Triples are generated, and the corresponding cell values ​​in the heatmap are incremented. Simultaneously, to ensure the heatmap evolves over time without excessively accumulating historical information, an exponential decay mechanism is introduced to dynamically suppress the influence of distant historical paths. If the time step at the current location exceeds a threshold... The heat at that location will decrease in the following manner: in, This is the current path planning time.

[0030] To further enhance the spatial continuity and predictive power of heat maps, this scheme introduces a local spatial smoothing operation for each location and calculates the average heat within its neighborhood: The final composite heat function used to guide the search is obtained as follows: in The experiment is generally set up This composite heat map can simultaneously reflect the historical congestion level of a node and the local load situation of its surrounding area, thus effectively compensating for the lack of representativeness of single-point heat maps.

[0031] The heatmap mechanism designed in this scheme takes into account three modeling trade-offs: spatial resolution—fine-grained heat recording improves path avoidance accuracy; temporal decay—prevents historical paths from "contaminating" the current search trend; and local smoothness—enhances the system's sensitivity and foresight to potential conflict areas.

[0032] II. Heatmap Guidance in Bottom Path Search In HEECBS, the underlying path planning acts as a bridge connecting the global search objective and local navigation actions. Its core task is to generate a feasible trajectory for each robot under current constraints, minimizing conflicts with other robots' paths. To this end, this solution introduces a heatmap guidance mechanism at the underlying level. By dynamically evaluating the heat information of path points, it incorporates congestion status into the path selection process, enabling explicit perception and proactive avoidance of local conflict risks.

[0033] The design extends the heuristic function from the traditional A* form to a three-term fusion structure: in, Indicates the current path length. It is an estimation of the distance to the target point, and This represents the composite heat value of the current waypoint at a given time step. The heat value includes not only the direct access frequency of the location, but also the average congestion level of neighboring locations, capturing the "regional density trend" rather than focusing on a single point.

[0034] The significance of heat-based guidance lies in the fact that when multiple candidate paths have similar costs, the algorithm tends to choose paths that have historically been less visited and have relatively empty neighborhoods, even if these paths are slightly longer. This mechanism embodies a "risk avoidance" strategy: proactively sacrificing some path costs in exchange for a lower probability of conflict. For example, in a typical cross-shaped path scenario, traditional A* search would choose the shortest path with direct intersection, while the heat-based guidance mechanism would proactively seek alternative routes to avoid future high-level conflicts, thereby improving the overall planning stability. This mechanism is particularly suitable for scenarios with high task density and limited space resources, effectively delaying the system from entering a conflict-intensive state.

[0035] Furthermore, this scheme enhances the role of heatmaps in the Focal Search strategy by constructing a set of candidate path points in the Focal List and prioritizing the expansion of path points with the lowest popularity. Essentially, this strategy prioritizes exploring lower-risk alternative paths within the tolerable range of heuristic costs, thereby achieving a dynamic balance between "search flexibility" and "risk-driven" approaches.

[0036] After the paths are generated, the adopted paths are written into the heatmap, and each path is processed accordingly. Nodes update their heatmaps and simultaneously deduct the heatmap from old paths, providing real-time feedback on the environment. This mechanism ensures spatiotemporal consistency, guaranteeing that the heatmap dynamically evolves with the system state, accurately reflecting the current global path distribution. In practical applications, heatmap adjustment factors... These are not fixed parameters and can be adaptively adjusted based on local task density or the number of robots. For example, in situations with excessively dense tasks or frequent congestion, increasing... It can enhance obstacle avoidance capabilities; in situations where tasks are sparse or passages are plentiful, it can reduce the heat penalty ratio and improve path compactness.

[0037] In the underlying path search, this scheme directly incorporates heatmap values ​​into the path search evaluation mechanism. Traditional single-robot search typically only considers distance costs, while this scheme adds heatmap penalties to the evaluation function, enabling the search process to proactively avoid congested areas during path generation. Even if the selected path has a slightly increased distance, it significantly reduces the probability of conflicts with other robots, thereby reducing rework and replanning caused by secondary conflicts.

[0038] III. Heatmap Scoring Mechanism and Conflict Priority Selection in High-Level Search In the Conflict-Based Search framework, the core task of the high-level layer is to progressively split conflicting nodes, continuously constructing new constraint combinations, and thus converging to a globally conflict-free solution. However, traditional EECBS high-level search still appears "blind" or "uniform" in terms of conflict selection and node expansion order, potentially wasting significant computational resources on low-priority conflicts or ineffective splitting paths. Therefore, this scheme introduces a heatmap-based high-level scoring mechanism and a conflict priority strategy to enhance the strategic nature and goal awareness of the high-level search.

[0039] First, to measure the congestion level of a solution at a certain high-level node, a path heat penalty term is defined: in For the current node The set of paths for all robots. For all time-space points in the path, It's a 3D heatmap. It essentially measures the "crowding" of a path in time and space, and can reflect the potential risk of conflict or the degree of global resource competition.

[0040] Based on this, this solution extends the valuation function: in, It is the traditional EECBS node evaluation function. This is the heat adjustment coefficient. This improvement introduces a risk assessment mechanism while retaining the goal of minimizing costs, making the algorithm tend to prioritize expanding nodes with lower heat and more "dispersed" paths, thereby reducing the cost of subsequent conflict and splitting.

[0041] Furthermore, the conflict selection strategy has been fundamentally restructured. While EECBS defaults to random or chronologically ordered conflict selection for splitting, HEECBS introduces a heatmap-driven conflict scoring function. in: The heat level of the conflict point on the heat map indicates the "sensitivity" of the conflict. This indicates that the earlier the occurrence, the higher the priority; This indicates that the shorter the path, the greater the impact of the conflict on the system; This is an indicator of edge conflict; higher scores are given when edge conflicts are more difficult to resolve. These correspond to various adjustment weights and can be flexibly adjusted under different density scenarios.

[0042] This scoring mechanism essentially transforms conflict management into a "task scheduling" problem, aiming to prioritize high-cost, high-risk conflicts to improve search efficiency and solution feasibility. Through an adjustable structure design of the scoring function, the system can adaptively change its behavior based on real-time scenarios. To avoid a rigid scoring mechanism, this solution also introduces a dynamic adjustment mechanism: setting a lower initial value... It converges quickly to a feasible solution, and its speed gradually improves as the conflict tree depth increases. By increasing the weight of the heat penalty term, the system is forced to gradually avoid high-density areas, thereby improving the stability and diversity of the solution.

[0043] In the high-level search process, traditional EECBS node evaluation relies solely on path cost and empirical heuristics, while this scheme adds heatmap information on top of that. That is, the priority of candidate nodes depends not only on path length and estimation error but also on the heat of the regions their paths traverse. If a node contains a large number of paths located in high-heat regions, its expansion priority will be reduced, thereby decreasing ineffective high-risk expansion. Furthermore, this scheme also designs a heat-based conflict priority mechanism. When multiple conflicts exist simultaneously, conflicts occurring in high-heat regions or early in the task execution phase will be prioritized to prevent these high-risk conflicts from causing larger-scale expansions in later stages.

[0044] Example 2 To further verify the applicability of the proposed multi-robot path planning algorithm in complex warehousing environments, this embodiment first models the operational scenario of warehousing robots. For example... Figure 4 As shown, the warehouse environment is abstracted as a two-dimensional discrete grid structure. Fixed facilities such as shelves and walls are designated as impassable areas, while the remaining grid areas serve as mobile areas for robots. A robot is always at a specific grid position and is only allowed to perform one action per time step, including moving in four directions (up, down, left, right) or remaining stationary. Time is discretized, with each time step corresponding to one global scheduling cycle. Regarding the task model, each robot executes tasks according to a fixed process, specifically including three stages: picking, delivering, and returning. That is, the robot first goes to the target shelf to pick up the goods, then delivers the goods to the designated picking station, and finally returns empty to the preset return area to await new task assignment. If an order involves multiple shelf locations, the system will decompose it into multiple sub-tasks, which the robots must complete sequentially. Task allocation employs a strategy combining random allocation and load balancing: new orders are randomly assigned to currently idle robots, and picking stations prioritize those with the fewest queued tasks to achieve overall operational balance.

[0045] Path conflicts are inevitable during the parallel operation of multiple robots. These conflicts mainly fall into two categories: vertex conflicts, where two robots attempt to enter the same grid position at the same time; and edge conflicts, where two robots exchange positions within adjacent time steps. These conflicts are common in warehouse environments with dense aisles and frequent intersections, and if not handled effectively, can lead to localized congestion or even deadlock. Therefore, the goal of path planning is to generate a set of conflict-free feasible paths for all robots while ensuring the task is completed according to the workflow. Simultaneously, the planning process must also consider optimization metrics, including shortening the overall task completion time, reducing the total path length to decrease energy consumption, and controlling algorithm computation time to meet real-time scheduling requirements. Through the above modeling, the constraints and optimization objectives of the system operation can be clearly defined, thus providing a foundation for subsequent experimental setup and algorithm verification.

[0046] In this embodiment, the experimental setup covers warehousing scenarios of different scales. Small-scale scenarios use a 50*50 grid to simulate the operating environment of small to medium-sized warehouses; medium-scale scenarios use a 75*75 grid to reproduce the typical scale of e-commerce transit warehouses; and large-scale scenarios use a 100*100 grid to simulate intelligent warehousing systems with high concurrency and high task density. In each scenario, the number of robots gradually increases, ranging from dozens to hundreds, to examine the algorithm's adaptability and scalability under different scales and task densities. The experimental procedure first initializes task allocation based on the different scales of the warehousing scenarios and the number of robots. Each robot is randomly assigned to a starting point and a target storage location to ensure the diversity and representativeness of the experimental conditions.

[0047] To verify the efficiency advantage of the HEECBS algorithm, we conducted a systematic comparison with the classic CBS and EECBS algorithms. Table 1 shows the comparison data of path planning time (in seconds) under different configurations. The overall trend shows that as the number of robots increases, the computation time of all algorithms increases. This is because frequent conflicts in high-density robot environments lead to a significant increase in the computational load of high-level search nodes and low-level path generation. However, the growth rate of HEECBS's runtime is significantly lower than that of CBS and EECBS, demonstrating its good scalability and stability in high-density, multi-task environments.

[0048] Table 1 In small-scale scenarios (50×50 grid), as the number of robots increases from dozens to over a hundred, the computation time of CBS exhibits a rapid exponential increase, failing to complete the planning task within the time limit with a higher number of robots. While EECBS reduces some redundant searches through the EES mechanism, its runtime still increases significantly with the number of tasks. In contrast, HEECBS consistently maintains the lowest runtime, and its time increase is relatively gradual with the number of robots. This indicates that HEECBS's heatmap-guided mechanism effectively reduces redundant node expansion in high-level conflict selection and low-level path generation, improving computational efficiency. In medium-scale scenarios (75×75 grid), as the number of robots increases, conflicts caused by high-density tasks become more frequent. CBS becomes unable to complete the planning task with a slightly higher number of robots, and the computation time of EECBS increases rapidly, while the growth rate of HEECBS's computation time is significantly lower than other algorithms. This demonstrates that HEECBS can reduce the number of high-level search node expansions and the computational load of low-level paths by prioritizing potentially high-conflict areas and rationally planning robot paths, maintaining good solution efficiency even in the high-density task environment of a medium-sized warehouse. In large-scale scenarios (100×100 grids), with further increases in task density and robot count, CBS fails to complete path planning in most configurations, while EECBS experiences a sharp increase in computation time and even timeouts in some high-density scenarios. In contrast, HEECBS successfully completes planning even with extremely high robot counts, and its total runtime is less than half that of EECBS, demonstrating excellent scalability and robustness. HEECBS's advantage in such high-density, large-scale task environments is mainly attributed to its heatmap-driven conflict prioritization mechanism and low-level path optimization strategy. This not only reduces the expansion of invalid nodes but also significantly reduces redundant conflict searches, thereby improving overall path planning efficiency.

[0049] In summary, the experimental results show that HEECBS maintains a lower runtime compared to CBS and EECBS in small, medium, and large warehouse environments, and its time advantage becomes more pronounced as the task scale and number of robots increase. This fully verifies the efficiency and scalability of HEECBS in multi-robot path planning, making it particularly suitable for high-density, multi-task intelligent warehousing systems.

[0050] To evaluate the performance of the HEECBS algorithm in terms of path quality, we compared HEECBS with the LACAM algorithm, which performs well in large-scale path planning problems, focusing first on the total path execution length (total path length consumed or the sum of Makespans). Table 2 shows the comparison data of path planning length under different configurations. The overall trend shows that the total path length of both algorithms increases with the number of robots. However, the growth rate of HEECBS is generally lower than that of LACAM in various scenarios, indicating that its planned path is closer to the optimal solution in high-density task environments.

[0051] Table 2 In small-scale scenarios, HEECBS's advantages are mainly reflected in its efficient avoidance of local conflicts and reduction of redundant path detours. Although the total path length is not significantly different from LACAM, HEECBS can more rationally arrange robot paths, effectively alleviating congestion in high-conflict areas, thus achieving a slight reduction in path length and improved scheduling efficiency overall. In medium-scale scenarios, HEECBS's advantages become even more apparent. As the number of robots increases, local conflicts and path overlaps become more frequent, and LACAM's path length grows rapidly. HEECBS, however, effectively reduces unnecessary waiting and detours by guiding conflict selection through high-level heatmaps and optimizing low-level paths, ensuring the overall compactness of path planning. With high robot density, HEECBS's cost savings compared to LACAM are significantly greater, demonstrating its superiority in the task-intensive environment of medium-sized warehouses. In large-scale scenarios, path planning challenges are most pronounced in high-density, multi-robot environments. HEECBS still manages to maintain a lower total path length than LACAM, indicating that its algorithm not only has good scalability in large-scale task scheduling but also can continuously optimize path layout and conflict avoidance strategies. Especially under extremely high task density, HEECBS's proactive avoidance of conflicts and intelligent path selection result in a lower overall path length growth rate than LACAM, further highlighting its applicability and stability in complex, high-concurrency warehousing environments.

[0052] Overall, under varying warehouse scales and task densities, HEECBS consistently achieves shorter total path execution lengths compared to LACAM, demonstrating its superior path planning quality in multi-robot collaborative path planning. This fully validates that the HEECBS algorithm can achieve better path quality while maintaining efficiency, thus improving overall scheduling performance.

[0053] Furthermore, this embodiment also evaluates the performance and scheduling stability of the HEECBS algorithm in actual execution time, again using Lacam as the benchmark algorithm for comparison. In the experiment, the actual execution time under different sized warehouse scenarios and different numbers of robots was recorded, and the mean and variance of each experimental group were calculated to analyze the stability of the algorithm in path execution.

[0054] Table 3 Table 3 shows a comparison of actual path execution times under different configurations. From the perspective of average execution time, HEECBS outperforms LACAM in warehouse scenarios of different sizes. In small scenarios (50×50 grid), even with a gradual increase in the number of robots, HEECBS's average execution time remains low, indicating that the algorithm can efficiently complete path execution in small to medium-sized warehouse environments, rationally allocate robot movement order, and reduce unnecessary waiting and conflicts. In medium-sized scenarios (75×75 grid), the local congestion and potential conflicts caused by high-density tasks significantly increase, but HEECBS still maintains a lower average execution time than LACAM. This demonstrates the significant role of the heatmap guidance mechanism in high-level conflict selection and low-level path generation, effectively optimizing robot scheduling and mitigating execution delays caused by task density. In large scenarios (100×100 grid), the complex environment of high concurrency and multiple tasks places higher demands on path planning, yet HEECBS still maintains a relatively good average execution time, showing the algorithm's good scalability and execution efficiency in large-scale, high-density warehouse environments. From the perspective of execution stability, HEECBS's execution time fluctuation is significantly smaller than LACAM's. Analysis of the variance data across the three scenarios reveals that HEECBS maintains low execution time variance across different task sizes and robot densities, demonstrating the predictability and stability of its path planning results. This advantage is primarily attributed to HEECBS's heatmap mechanism, which dynamically identifies potential high-conflict areas, guides the expansion of higher-level nodes and conflict resolution, and optimizes the robot's movement sequence in the lower-level path search. This reduces local congestion and waiting, resulting in a more balanced overall execution process. Furthermore, the PIBT algorithm and re-insertion mechanism within the Lacam algorithm contribute to a relatively strong degree of randomness in the planning results.

[0055] In summary, the HEECBS proposed in this paper not only outperforms LACAM in terms of average execution time, but also demonstrates a significant advantage in execution stability. Its stability advantage becomes even more pronounced as task size and robot density increase, fully demonstrating that HEECBS can achieve efficient and reliable multi-robot path planning and scheduling in complex, high-concurrency warehouse environments.

[0056] Through the two sets of experiments, one set compared the path planning efficiency of HEECBS with CBS and EECBS in warehouse scenarios of different scales. The results showed that HEECBS consistently maintained the lowest computation time in all scenarios, especially in medium and large-scale high-density task environments, where its runtime growth was significantly lower than other algorithms, demonstrating good scalability and robustness. The other set compared HEECBS with the LACAM algorithm in terms of path quality and execution stability. The results showed that the total path execution length of HEECBS was lower than that of LACAM in all scenarios, indicating that the algorithm can reasonably avoid conflicts and optimize path layout. At the same time, the actual execution time of HEECBS not only had a lower average value than LACAM, but its fluctuation range was also significantly reduced, showing higher scheduling balance and predictability.

[0057] In summary, this proposal presents a heatmap-guided multi-agent path planning method aimed at improving the efficiency of multi-robot path planning, optimizing path quality, and enhancing task execution stability in complex warehousing environments. The HEECBS algorithm designed in this scheme improves upon the classic EECBS framework: First, a heatmap scoring mechanism is introduced in the high-level conflict tree search, dynamically adjusting node expansion priorities based on the spatial distribution of potential conflicts and historical experience, allowing the algorithm to prioritize high-risk areas and reduce invalid node expansion; second, heatmap information is combined in the conflict selection and low-level path generation stages to optimize robot movement order and avoidance strategies, reducing redundant path searches and local congestion; finally, an experience-enhancing mechanism is used to update the estimated cost function, achieving adaptive adjustment to high-density task environments, thus balancing planning efficiency and path quality. HEECBS's design innovations are mainly reflected in three aspects: First, the introduction of a heatmap guidance mechanism enables high-level search to dynamically identify potentially conflict-dense areas and prioritize nodes that may cause widespread congestion, thereby significantly reducing redundant node expansion. Second, the combination of conflict selection strategy and low-level path search depth with heatmap information allows the robot path to more rationally avoid congested areas and optimize the overall path length. Third, the experience-enhanced cost estimation mechanism continuously updates the search strategy based on historical search results, achieving adaptive optimization for complex environments. These design innovations not only improve the algorithm's solution efficiency but also directly improve path planning quality and execution stability.

[0058] The HEECBS algorithm described above, through heatmap guidance, conflict priority optimization, and experience enhancement mechanisms, achieves a comprehensive improvement in path planning efficiency, path quality, and execution stability in high-density, multi-task warehousing environments. Its innovative design enables the algorithm to dynamically adapt to complex environments, effectively reducing redundant computation and local congestion. It is not only suitable for task scheduling in small and medium-sized warehouses but also for efficient and stable multi-robot path planning in large-scale, high-concurrency environments, providing solid technical support and application value for the optimized scheduling of intelligent warehousing systems.

[0059] To address the issues of inefficiency, inaccurate conflict prediction, and unstable path execution in existing multi-robot path planning methods in high-density warehousing environments, this solution introduces a dynamic spatiotemporal heatmap mechanism during path planning. This mechanism quantitatively models the degree of robot aggregation at different times and in different areas. Combined with time decay and spatial smoothing strategies, it effectively captures potential congestion areas and high-risk conflict points, providing more forward-looking reference information for global path planning and local avoidance. This solution realizes a multi-robot path planning method that combines global congestion perception and local path optimization capabilities. It can achieve more efficient and stable path scheduling in complex warehousing environments. This method can significantly reduce planning time and conflict frequency, improve synchronization and stability during task execution, and thus provide stronger operational guarantees and higher operational efficiency for intelligent warehousing systems.

Claims

1. A heatmap-guided multi-agent path planning method for high-density warehousing environments, characterized in that, The HEECBS explicit estimation conflict search algorithm guided by heatmap is adopted. First, a heatmap in the environment is initialized, which is used to record the degree of robot aggregation at different locations and times. Then, in the underlying path search process, the initial path of the robot is generated based on the heatmap and the heatmap is updated. During the high-level search, based on the robot's initial path and the updated heatmap, the highest priority path set node is selected for expansion, and conflicts in the path set node are detected. If a conflict exists, the path set node is split according to its heat priority and a new constraint node is generated. Then, the search returns to the lower level path. The above steps are repeated until all robots obtain a conflict-free path or the preset termination condition is met.

2. The heatmap-guided multi-agent path planning method for high-density warehousing environments according to claim 1, characterized in that, The heatmap in the initialization environment is specifically initialized based on the task and existing paths as follows: Where t is the time step, For the robot's position, Indicates at time step At time, position Number of visits Represents the set of non-negative integers.

3. The heatmap-guided multi-agent path planning method for high-density warehousing environments according to claim 2, characterized in that, The underlying path search process includes the following steps: D1. Based on the initial heatmap, initialize the open, closed, and focal tables; D2. Check if the open table is empty. If it is empty, directly generate the robot's initial path and update the heatmap; otherwise, proceed to step D3. D3. Select the path point with the lowest popularity value in the focal table, delete the path point in the open table, add the path point in the closed table, then add adjacent path points to the open table and update the focal table. D4. Determine whether the destination has been reached. If yes, generate the robot's initial path and update the heatmap; otherwise, return to step D3.

4. The heatmap-guided multi-agent path planning method for high-density warehousing environments according to claim 3, characterized in that, The formula for calculating the heat value in step D3 is as follows: in, Indicates the current path length. It is an estimate of the distance to the target point. As a heat regulation factor, This represents the composite heat value of the current waypoint at a given time step, which is used to simultaneously reflect the historical congestion of the waypoint itself and the local load of its surrounding area.

5. The heatmap-guided multi-agent path planning method for high-density warehousing environments according to claim 4, characterized in that, The specific composite heat value is: in, This is the heat decay value. The time decay coefficient, This is the current path planning time; The average heat within the node's neighborhood; and These are the weights for heat decay and average heat, respectively.

6. The heatmap-guided multi-agent path planning method for high-density warehousing environments according to claim 2, characterized in that, The high-level search process includes the following steps: U1. For the path set nodes output by the underlying planning, calculate the value of each node and select the path set node with the smallest value. The path set node contains the initial path of each robot. U2. Detect whether there is a conflict within the selected path set nodes. If there is a conflict, proceed to step U3. Otherwise, output the currently selected path set nodes as the robot's final path and execute it. U3. Calculate the conflict score, select the conflict with the highest conflict score for processing, add the corresponding constraints, and then return to perform the underlying path search.

7. The heatmap-guided multi-agent path planning method for high-density warehousing environments according to claim 6, characterized in that, Step U1 uses an extended evaluation function to calculate the node evaluation. The extended evaluation function includes a path heat penalty term, which is used to measure the congestion of the path in time and space.

8. The heatmap-guided multi-agent path planning method for high-density warehousing environments according to claim 7, characterized in that, The extended evaluation function is specifically as follows: in, It is a traditional EECBS node evaluation function, which includes nodes Costs already incurred and nodes Heuristic cost estimation , This is the heat adjustment coefficient. For nodes Path heat penalty item, For the current node The set of paths for all robots. For all time-space points in the path, It is a 3D heatmap.

9. A heatmap-guided multi-agent path planning method for high-density warehousing environments according to claim 8, characterized in that, The heat regulation coefficient The value is dynamically adjusted: the initial stage is set to a lower value. It converges quickly to a feasible solution, and its speed gradually improves as the conflict tree depth increases. This increases the weight of heat penalty terms, forcing a gradual avoidance of high-density areas and improving the stability and diversity of solutions.

10. A heatmap-guided multi-agent path planning method for high-density warehousing environments according to claim 6, characterized in that, The formula for calculating the conflict score in step U3 is as follows: in, The heat level of the conflict point on the heat map indicates the "sensitivity" of the conflict. This indicates that the earlier the occurrence, the higher the priority; This indicates that the shorter the path, the greater the impact of the conflict on the system; This is an indicator of edge conflict; higher scores are given when edge conflicts are more difficult to resolve. These correspond to various adjustment weights and can be flexibly adjusted under different density scenarios.

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