Swarm Division-Based Aircraft Velocity Obstacle Optimization Considering Low-Carbon Emissions

Abstract

In the pursuit of sustainable aviation, this paper presents an innovative approach that adopts a swarm division strategy to enhance and refine the velocity obstacle (VO) method, guided by a low-carbon principle. A dynamic elliptical protection zone model forms the core of this innovative approach. Specifically, this dynamic elliptical protection zone is created based on the difference in aircraft velocity, and a swarm division strategy is introduced in this process. Initially, aircraft that share the same route and type, and have similar velocities and distances, are grouped into swarms. Then, the characteristics of the swarms, such as mass points, velocities, and protection zones, are recorded. Second, the collision cone (CC) between swarms is established, and planar geometrical analysis is used to determine the optimal relief velocity and heading of aircraft on the low-carbon objective while ensuring a safe interval between aircraft in the swarm during the relief period. Additionally, a swarm control algorithm is utilized to adjust the velocity of the aircraft by a small margin. Finally, simulation experiments are conducted using Python, revealing that the swarm relief efficiency of the enhanced VO method sees a notable increase of over 33%. Concurrently, the need for adjustments decreases by an average of 32.78%, while fuel savings reach as high as 70.18%. The strategy is real-time and operational, significantly reduces the air traffic controller (ATC) workload, improves flight efficiency and safety, and contributes positively to the reduction in carbon emissions, which is beneficial for the environment.

1. Introduction

Civil aviation has become a cornerstone in China’s march toward modernization, fueled by advancements in aviation technology and the dynamics of globalization. In alignment with the global decarbonization movement, the Chinese government has launched the ambitious “Peak Carbon Action Program by 2030”, setting forth clear low-carbon development goals for the aviation sector. In response, the Civil Aviation Administration of China (CAAC) has rolled out the “14th Five-Year Plan for Green Development of Civil Aviation”, establishing eight quantitative targets to guide the industry toward a more sustainable, low-carbon future. Despite these initiatives, transitioning from the current fossil fuel-based aviation kerosene paradigm to a more sustainable model poses significant short-term challenges. The broad adoption of cutting-edge decarbonization technologies remains a goal yet to be fully achieved. Nonetheless, with China’s growing demand for civil aviation and the steady increase in energy consumption and emissions, the industry is urged to take more proactive, long-term measures. Simultaneously, the policy of “flight encryption for greater convenience” requires the civil aviation sector not only to manage the increase in flight frequency and passenger volume but also to address the twin challenges of flight safety and low-carbon development. The growing trend of “high density” and “small spacing” in air traffic underscores the critical role of dynamic route path planning in ensuring flight safety while also pursuing low-carbon objectives [1,2,3]. Path planning in aviation is generally divided into global and local strategies. Global path planning focuses on optimizing flight paths by avoiding known static obstacles and utilizing pre-existing knowledge to find the most efficient route. This method reduces flight distance and time, lowering energy consumption and emissions. On the other hand, local path planning revolves around real-time sensing and analysis, essential for avoiding collisions with unexpected and dynamic aerial obstacles. This requires the implementation of advanced local path planning algorithms, such as the velocity obstacle (VO) method, to ensure real-time obstacle avoidance and the safe navigation of aircraft. Using swarm dynamics and collision cone (CC) in these algorithms is crucial for managing the complexities of modern air traffic [4,5].

In the domain of local path planning for aviation, a diverse array of algorithms has been implemented, each characterized by its unique methodology and specific applications. These encompass the artificial potential field method, the guidance method, machine learning-based obstacle avoidance algorithms, the VO method, and the dynamic window approach [6,7]. The artificial potential field method, functioning through the calculation of gravitational and repulsive forces to navigate around obstacles, faces certain limitations. Its accuracy in reaching target coordinates, maintaining path stability, and ensuring viable flight paths is not optimal, particularly in ensuring precise navigation to the intended destination while maintaining a stable flight trajectory [8,9]. The guidance method, drawing inspiration from missile guidance principles, faces challenges in scenarios with multi-obstacle aircraft intrusions and the need for subsequent flight maneuvers, highlighting the complexities of navigating in densely populated airspaces [10]. Machine learning-based obstacle avoidance algorithms, along with other intelligent algorithms, require significant computational resources and time, showing their best performance in less complex scenarios like global path planning [11,12,13]. The dynamic window approach and the VO method are both designed for real-time path planning in dynamic environments [14,15,16]. The former is based on velocity and acceleration constraints, along with local sensing data, while the latter identifies potential conflicts between vehicles and outlines evasive maneuvers by constructing a CC. The VO method, with its direct incorporation of velocity information, is exceptionally suited for dynamic environments. It applies not only to avoiding obstacles for two or more moving objects but also in multi-entity scenarios [17,18]. These methods have been thoroughly investigated and applied in the field of robotics, proving their versatility and effectiveness. The VO method, in particular, has been widely adopted for obstacle avoidance and trajectory planning in aircraft, unmanned aerial vehicles (UAVs), and other aerial vehicles. Therefore, to cope with the growing trend of “high-density” and “small spacing” swarm flight conditions and meet the government’s low-carbon policy, this study presents an innovative approach that adopts a swarm division strategy to enhance and refine the VO method considering low-carbon emissions.

2. Literature Review

Scholars have studied the VO method from various perspectives. Regarding the improvement of collision avoidance techniques, HyunJae J and Du Zhe compared two algorithms: VO and artificial potential field (APF) [19,20]. Among them, Du Zhe introduced the uncertainty factor of the VO method to further optimize the obstacle avoidance maneuvers of the unmanned surface vessel (USV). The results showed that the VO method outperforms the APF method in terms of ensuring safety and speeding up formation transformations. Pan W proposed a method for detecting and resolving 3D spatial flight conflicts [21]. The method uses the VO method and particle swarm optimization (PSO) to determine the optimal avoidance strategy. By simplifying the three-dimensional problem into two dimensions, significant improvements in computational efficiency are obtained. This enabled the method to find the optimal solution in approximately 0.4 s, making it suitable for real-time conflict resolution. Tan Y C proposed an improvement to the 3D VO algorithm to increase the efficiency of collision avoidance for multi-UAV systems [22]. The improvements included algorithmic corrections, enhanced ability to handle 3D cubic obstacles, and verification through real flight tests. Fangwei L proposed an improved VO method based on an elliptical ship domain to more accurately describe the ship’s kinematic characteristics and achieve collision avoidance [23]. Unlike prior research, this method takes into account the ship domains of both the home and the target ships and determines whether the domains of the two ships overlap using a generalized characteristic polynomial. Then, the VO is created by discretizing the heading angle and calculating the collision speeds at different angles. Consequently, the optimal heading angle is then determined using the velocities on the boundary of the VO. The algorithm was tested in various encounter scenarios and successfully achieved collision avoidance. Jing G proposed an improved reciprocal velocity obstacle method (RVO) to reduce steering angle and path jitter during obstacle avoidance of UAVs to ensure a smoother flight along the target direction [24]. In the realm of dynamic path planning and decision-making, Hongyang Z proposed the optimal collision avoidance point (OCAP) method for USVs, which combines USV dynamics and VO methods to handle multiple moving obstacles in real time and optimize collision avoidance decisions [25]. The OCAP improves the USV’s heading by identifying critical conditions for collision avoidance and determining the optimal avoidance point and velocity adjustment. Also, OCAP demonstrates high efficiency in pathfinding and robustness in environments with high obstacle density. Huarong Z presents a dynamic path-planning method for smart ships that integrates navigation rules and uncertainty and evaluates collision risk using fuzzy logic [26]. This method can adjust path planning for various encounter scenarios, including crossing, overtaking, and head-on situations. The effectiveness of this method was verified using real ship data from the Yangtze River region in Wuhan, China. Mingzhu P developed a multi-UAV collaborative obstacle avoidance algorithm that combines heartbeat information filtering and UDP communication, along with the VO method and B-spline curves to generate smooth paths [27]. Yumin S proposed a novel path planning algorithm, the constrained locking sweeping method and velocity obstacle (CLSM-VO), for USVs that improves search efficiency and path smoothing in complex dynamic marine environments [28]. This algorithm combines a lock-and-scan method (LSM) and a VO method, and the results indicate that the algorithm is effective in generating smooth and safe paths that avoid multiple moving obstacles. Delai X proposed an autonomous collision avoidance navigation method for USVs that combines RVO and deep reinforcement learning (DRL) [29]. Through the improved proximal policy optimization (PPO) algorithm and gated recurrent unit (GRU)-based neural network, achieve effective collision avoidance can be achieved in complex environments containing dynamic and static obstacles. The accuracy and practicality of the autonomous navigation of USVs are supported by simulation experiments, providing a new theoretical and methodological reference. Choi M used the VO method to handle obstacles under limited perception conditions and analyzed sensor data to plan collision-free trajectories, enabling UAVs to successfully avoid obstacle paths [30]. K. Shi proposed a conflict detection and relief method applicable to UAVs sharing airspace with manned aircraft [31]. This methodology involves designing dynamic protected zones using the closest point of approach (CPA) strategy and a conflict risk ranking mechanism and minimizes trajectory deviations using VO. The numerical results demonstrate the effectiveness of the methodology. Overall, these studies not only improve the safety and efficiency of unmanned systems in complex environments but also provide new ideas and technical support for future unmanned system operations.

All of the studies mentioned above focus on improving environmental factors and enhancing the accuracy of path planning. Although significant progress has been made in collision avoidance, path planning, and risk assessment of aircraft and UAVs, there are still fewer studies on the fuel consumption generated by the obstacle avoidance process. Additionally, there are shortcomings such as increased computational complexity and high reliance on training data, resulting in a longer computation time, low efficiency, and limited model generalization ability and application scope. These limitations make it difficult to deal with high-density, multi-target, and diversified aviation environment problems.

To address these issues, this paper introduces an innovative local path-planning method tailored for real-time obstacle avoidance in dynamic swarm obstacle aircraft, with a focus on low-carbon objectives. This algorithm features the creation of a dynamic elliptical protection zone around aircraft with significant speed differences. When aircraft have similar speeds, headings, and proximities, they are grouped into “swarms”. Utilizing this swarm concept, the VO method is optimized and enhanced. The advanced VO method is effective in identifying conflicts both within and between these swarms. A critical aspect of this method is the incorporation of low-carbon objectives for navigating obstacles between swarms. The swarm principle forms the basis for the optimization and enhancement of the VO method, which is then employed to resolve conflicts between swarms. In seeking the optimal relief strategy, which includes adjustments in velocity and heading, the algorithm gives priority to reducing carbon emissions. To this end, the velocities of the aircraft in the swarms are finely adjusted using swarm control algorithms. This arrangement guarantees conflict avoidance among aircraft in the same swarm during the relief phase, thereby facilitating the safe and efficient execution of cruising activities. This novel approach not only overcomes existing limitations in path planning and collision avoidance but also aligns with reducing carbon emissions in aviation.

3. Description of the Problem

Within a designated altitude stratum, as illustrated in Figure 1, the scenario involves airspace intersected by the paths of 10 aircraft, serving as a model for exploring conflict detection and resolution. The airway is defined to be 20 km wide, with a 10 km margin on each side of its central line. In-flight adjustments to original flight plans, driven by unexpected circumstances, can lead to potential conflict risks at $t_1$, especially when aircraft at the same horizontal level meet at the crossing points of these airways.

Aircraft $P_i$, with similar speeds, headings, and distances from one another, are grouped into swarms $S_h$. Different swarms are inscribed with different colored ellipses. In this context, $i$ denotes the index of the aircraft, and $h$ denotes the index of the swarm, where $i=1,2,3,\dots ,k$, $h=1,2,3,\dots ,k$, and $k$ belong to integers. As shown in Figure 2, swarm $S=\left\{S_1,S_2\right\}$, with $S_1=\left\{P_1,P_2,P_3\right\}$ and $S_2=\left\{P_4,P_5\right\}$. The blue ellipse represents swarm $S_1$, and the green ellipse represents swarm $S_2$. Furthermore, the assumption of non-overlapping security ranges between aircraft is satisfied within the swarms as no conflicts are generated within them [32]. After dividing the swarms, by expanding the safety radius of the intruding aircraft, the aircraft is simplified as mass points; the VO cone is constructed between $S_1$ and $S_2$, represented by the red zone; and geometric analyses are performed to decide whether to perform avoidance or not, as well as to compute the obstacle avoidance strategy under low fuel consumption.

In summary, this study presents an aircraft obstacle avoidance model that is based on a variable elliptical protection zone and a cluster division strategy in the context of low carbon consumption. The proposed model aims to improve the flight efficiency and safety of the aircraft. The feasibility and effectiveness of the model are verified through simulation experiments that satisfy the aircraft dynamics model and the actual operation situation.

4. Mathematical Modeling

4.1. Problem Assumptions

Without loss of generality, the following assumptions are introduced to facilitate the model formulation process:

(1) There are flight conflicts in both the horizontal and vertical directions between aircraft, but the emphasis here is focused on conflicts in the horizontal direction. Therefore, the model is simplified to a two-dimensional plane and is not limited to a specific control scenario or flight phase. This approach is also applicable to free-route airspace.

(2) The aircraft flies straight along the front of the detected obstructing aircraft, maintaining its course and speed until instructed to adjust.

(3) The urgency level of a potential conflict for an aircraft pair (two aircraft considered together) is undifferentiated and equal during conflict identification.

(4) During conflict detection and resolution, the aircraft and the obstructing aircraft are represented as ellipses with a safety radius. By increasing the safety radius of the obstructing aircraft, the aircraft can be reduced to a point mass.

4.2. Protection Zone Creation and Analysis

4.2.1. Velocity-Based Dynamic Elliptical Protection Zone

In compliance with air traffic control regulations, aircraft are mandated to maintain a specified safety distance to avert flight conflicts. The conventional safety interval standard, used to define flight protection zones, is insufficient for accurately emulating actual flight status due to the heterogeneity in performance and speed among aircraft types. This discrepancy often leads to substantial inaccuracies in detecting flight conflicts. An elliptical model is employed to represent the protection zone for various aircraft models, as shown in Figure 3.

In Figure 3, the ellipse’s center is designated as $P_i=(x_i,y_i)$, while the velocity is denoted as ${\overrightarrow{V}}_i$, with its magnitude defined as $v_i$. This velocity ${\overrightarrow{V}}_i$ is positioned along the ellipse’s major semiaxis, forming an angle ${\theta }_i$ with the x-axis, where ${\theta }_i\in (0,2\pi )$. The size of the protection zone for an aircraft is ascertained based on the precise safety intervals specific to different models and their respective cruising speeds. Here the minor semiaxis $b_i$ is taken from the exact safety intervals corresponding to different models derived from previous studies [33]. Specifically, the precise protection zones are obtained by building a model for the calculation of safe flight intervals for lateral flights of different aircraft types. This model optimizes the basis for determining aircraft contiguity in the flight conflict network, taking into account the aircraft’s heading, speed, different performances, and states, thus bringing the flight conflict network closer to the TBO (trajectory-based operations) mode of operation. In contrast, the major semiaxis $a_i$ is contingent upon the magnitude of the cruising speeds. A relational variable is represented by $\lambda$. The x-axis represents the horizontal distance, and the y-axis represents the vertical distance. These parameters are quantitatively determined using Equation (1), providing a more nuanced and model-specific approach to defining the size of the protection zone, thereby enhancing the accuracy of flight conflict detection in diverse operational scenarios. Throughout the text, velocity denotes the speed and heading, and speed denotes the magnitude of the velocity.

$$\left\{\begin{array}{l}{\overrightarrow{V}}_i=\left[\begin{array}{l}{\overrightarrow{V}}_{xi}\\ {\overrightarrow{V}}_{yi}\end{array}\right]=\left[\begin{array}{l}{v}_i\cos {\theta }_i\\ v_i\sin {\theta }_i\end{array}\right]\\ a_i=\lambda v_i\end{array}\right.$$

(1)

4.2.2. Dynamic Elliptical Protection Zone for Aircraft Swarms

Definition of Swarms

A clustering model is employed to categorize aircraft $P_i$ that satisfy specific criteria into swarms $S_h$. These criteria encompass a maximum inter-aircraft distance of 10 km, a heading divergence no greater than 5°, and a speed variance not exceeding 0.5 km/min. The implementation of this model involves several key steps: (1) determining aircraft separation by calculating the shortest distance between points on different ellipses; (2) ensuring that all three clustering conditions are concurrently satisfied; and (3) assigning different numbers to each swarm that meets the clustering criteria, as delineated above, to differentiate inter-swarm relationships.

Within each swarm, it is presumed that the safety ranges between aircraft do not overlap, thereby precluding internal conflicts. Postclustering, the aircraft are simplified into point masses by enlarging the safety radius of the intruding aircraft. The VO method then establishes a CC between the swarms, which is visible as a cone-shaped zone. Geometric analysis is then applied to ascertain the necessity of avoidance maneuvers and to compute the extent of the avoidance zone.

To ensure there is no internal conflict within each divided swarm, it is important to avoid overlapping of aircraft (ellipses). Such overlapping is considered a conflict between aircraft. $S$ is a conflict zone, and point $n_k=(x_k,y_k)$ is any point in the conflict zone, $n_k\in S$. For any point $n_k$, it is calculated using Equation (2):

$$n_k=\left\{(x_k,y_k)\in {ℝ}^2:\frac{{\left[\left(x_k-x_i\right)\cos {\theta }_i+\left(y_k-y_i\right)\sin {\theta }_i\right]}^2}{{a_i}^2}+\frac{{\left[\left(x_i-x_k\right)\sin {\theta }_i+\left(y_k-y_i\right)\cos {\theta }_i\right]}^2}{{b_i}^2}\le 1\right\}$$

(2)

In this model, $(x_i,y_i)$ represents the coordinates of the target aircraft’s ellipse center, while $(x_k,y_k)$ corresponds to the coordinates on the obstacle aircraft’s protection zone, defined by an ellipse. Here ${ℝ}^2$ is used to denote a plane that consists of all possible ordered pairs of $(x_k,y_k)$, where both $x_k$ and $y_k$ are real numbers. This plane is referred to as the “two-dimensional real number space”.

Parameterized Description

The swarm’s protection zone is delineated by an elliptical profile, termed the “swarm ellipse”. This ellipse is defined by the swarm’s collective center of mass, velocity, and the size of the protection zone, all of which are contingent on the position, velocity, and protection zone size of each aircraft within the swarm. The swarm’s ellipse angle on the x-axis indicates the aircraft’s collective velocity direction, and the optimal center of mass is determined by the midpoint of the rectangle formed by all ellipse boundary points. As the angle difference between the aircraft in the swarm is small, it is possible to employ the mean value of each aircraft coordinate as the swarm’s center of mass. This center of mass is denoted as $P_{Ni}=(x_{Ni},y_{Ni})$, and $n$ represents the number of aircraft, which can be calculated using Equation (3):

$$\left\{\begin{array}{l}{x}_{Ni}=\frac{{\displaystyle \sum _{i=1}^n{x}_i}}{n}\\ y_{Ni}=\frac{{\displaystyle \sum _{i=1}^n{y}_i}}{n}\end{array}\right.$$

(3)

The swarm’s velocity ${\overrightarrow{V}}_{Ni}$ is divided into speed direction and magnitude, where the angle ${\theta }_{Ni}$ between the speed direction and the positive half-axis of the x-axis represents the angle formed by the aircraft’s combined velocity and the positive half-axis of the x-axis. The speed magnitude $v_{Ni}$ is obtained from the average speed of each aircraft. ${\overrightarrow{V}}_{Nxi}$ and ${\overrightarrow{V}}_{Nyi}$ are the horizontal and vertical components of the combined velocity of the aircraft. Its calculation method is given by Equation (4):

$$\left\{\begin{array}{l}{v}_{Ni}=\frac{{\displaystyle \sum _{i=1}^n{v}_i}}{n}\\ {\theta }_{Ni}=\frac{{\overrightarrow{V}}_{Nyi}}{{\overrightarrow{V}}_{Nxi}}={\displaystyle \sum _{i=1}^n{\overrightarrow{V}}_i}\\ {\overrightarrow{V}}_{Ni}=\left[\begin{array}{l}{v}_{Ni}\cos {\theta }_{Ni}\\ v_{Ni}\sin {\theta }_{Ni}\end{array}\right]\end{array}\right.$$

(4)

The protection zone size of the swarm ellipse is determined through an optimization model. Solving the model involves three steps: (1) The objective function is to minimize the area of the swarm ellipse. (2) Traverse the points on each ellipse and substitute each point into the swarm ellipse formula to satisfy the constraint condition of less than or equal to 1, ensuring that each point is inside the swarm ellipse. (3) To generate a swarm ellipse that contains all ellipses and has the smallest area, take the center of mass as the center of the swarm and ${\theta }_{Ni}$ as the angle between the swarm velocity ${\overrightarrow{V}}_{Ni}$ and the x-axis. This process will yield values for the major semiaxis $a_{Ni}$ and minor semiaxis $b_{Ni}$ that satisfy the predefined conditions. The details can be seen in Algorithm 1 as follows:

Algorithm 1: Swarm elliptic optimization solution algorithm

Initialization and Imports: Reads $x_i$ (aircraft horizontal coordinates), $y_i$ (aircraft vertical coordinates), $a_i$ (size of the protection zone in the direction of the velocity of the aircraft), $b_i$ (size of the protection zone of the aircraft perpendicular to the direction of velocity), ${\theta }_i$ (horizontal angle between the direction of the aircraft’s velocity and the positive half-axis of the x-axis), $v_i$ (aircraft speed magnitude) from an Excel file into a DataFrame. Step 1. Calculate Ellipse Points: Function get_ellipse_points ($x_i$, $y_i$, $a_i$, $b_i$, ${\theta }_i$) Create an array of points points for angle from 0 to 2π Calculate and add points on the ellipse to points Return points Step 2. Calculate Average Speed Direction: Function compute_${\theta }_{Ni}$ (data) Initialize the integration variables for velocities for each data point in data Calculate the sum of velocities Calculate and return the average velocity angel ${\theta }_{Ni}$ Step 3. Draw Ellipses and Velocity: Function draw_ellipse_and_velocity (data, $x_{Ni}$, $y_{Ni}$, $a_{Ni}$, $b_{Ni}$, ${\theta }_{Ni}$) Create the figure and axes for each ellipse data in data Draw the ellipse and velocities Annotate the ellipse number Calculate and draw the combined velocity Draw and annotate the optimized ellipse Display the figure and return the average speed Step 4. Execute Optimization to Obtain the Optimal Ellipse: Function main_optimization (ellipse_points, $x_{Ni}$, $y_{Ni}$, ${\theta }_{Ni}$) Set the optimization goal to minimize the area of the ellipse Create constraints to ensure all points are within the ellipse Solve the optimization problem using the minimize function Return the optimal ellipse parameters and area Step 5. Extract Data from Excel and Calculate Ellipse Center: Create an empty data list for each row in the Excel DataFrame Extract ellipse parameters Call the get_ellipse_points function Accumulate to calculate center coordinates Calculate the average position of the ellipse center Step 6. Call Optimization Function to Get Optimal Parameters: Call the main_optimization function Store the optimal parameters and area Step 7. Draw Results and Calculate Average Speed: Call the draw_ellipse_and_velocity function Print result information Step 8. Calculate Speed Differences: Function velocity_difference (data, $v_{Ni}$, ${\theta }_{Ni}$) Create a different list for each ellipse in the data Calculate the speed and angle differences Add to the difference list Return the difference list Output Final Results: Call the speed_difference function for each difference in the difference list

Print $x_{Ni}$ (horizontal coordinates of swarm aircraft), $y_{Ni}$ (vertical coordinates of swarm aircraft), $a_{Ni}$ (the size of the protection zone in the direction of the velocity of the swarm aircraft), $b_{Ni}$ (the size of the protection zone perpendicular to the direction of velocity for a swarm aircraft), ${\theta }_{Ni}$ (horizontal angle between the direction of the swarm aircraft’s velocity and the positive half-axis of the x-axis), $v_{Ni}$ (swarm aircraft speed magnitude)

4.3. Conflict Sensing Technology

4.3.1. Traditional Velocity Obstacle Conflict Perception Model

Principle of Collision between Aircraft Pairs

The traditional VO method aims to determine potential flight conflicts between aircraft within the encounter geometric space based on their current trajectory, including position and velocity, without considering other factors. However, the process of relief can be seen as the point mass avoiding the VO expansion ellipse. The aircraft’s velocity space generates a conical obstacle zone. If the relative velocity of the aircraft is outside the conical zone, it will not collide with other aircraft in the future. This method enables the selection of the appropriate speed and heading of the aircraft while avoiding conflicts [20].

In distributed multi-aircraft motion coordination, each aircraft treats other aircraft as motion-obstructed aircraft. Let $a_i$ be the forward radiation distance of the aircraft and $b_i$ be the lateral radiation distance. The dynamic obstacle aircraft has a forward radiation distance of $a_j$ and a lateral radiation distance of $b_j$. At time $t_1$, two aircraft are being monitored: the first has a position marked as $P_i$ with a corresponding velocity ${\overrightarrow{V}}_i$, and for the second, an obstacle, is positioned at $P_j$ with velocity vector ${\overrightarrow{V}}_j$. Their speeds are given by the magnitudes $v_i$ and $v_j$, with their directions indicated by yaw angles ${\theta }_i$ and ${\theta }_j$, respectively. The spatial and directional information, termed posture, for the first is captured by the tuple $(P_i,{\overrightarrow{V}}_i)$ and for the second by $(P_j,{\overrightarrow{V}}_j)$. In the whole text, $i$ and $j$ denote indices used to distinguish between different aircraft, and $i\ne j$, $i=1,2,3,\dots ,k$, $j=1,2,3,\dots ,k$, and $k$ are integers. The above variables can be calculated by Equation (5):

$$\left\{\begin{array}{l}{P}_i=(x_i,y_i)\\ P_j=(x_j,y_j)\\ {\overrightarrow{V}}_i=\left[\begin{array}{l}{\overrightarrow{V}}_{xi}\\ {\overrightarrow{V}}_{yi}\end{array}\right]=\left[\begin{array}{l}{v}_i\cos {\theta }_i\\ v_i\sin {\theta }_i\end{array}\right]\\ {\overrightarrow{V}}_j=\left[\begin{array}{l}{\overrightarrow{V}}_{xj}\\ {\overrightarrow{V}}_{yj}\end{array}\right]=\left[\begin{array}{l}{v}_j\cos {\theta }_j\\ v_j\sin {\theta }_j\end{array}\right]\end{array}\right.$$

(5)

Figure 4 shows an ellipse with center $P_j$, major semiaxis $a_{ij}$, and minor semiaxis $b_{ij}$. A cone tangent to the ellipse $P_j$ at point $P_i$ and to rays $l_1$ and $l_2$ can be formed. ${\overrightarrow{V}}_{ij}$ represents the relative velocity, and $l_m$ is the ray in the direction of vector ${\overrightarrow{V}}_{ij}$. The above variables can be calculated by Equation (6):

$$\left\{\begin{array}{l}{a}_{ij}=a_i+a_j\\ b_{ij}=b_i+b_j\\ {\overrightarrow{V}}_{ij}={\overrightarrow{V}}_i-{\overrightarrow{V}}_j\end{array}\right.$$

(6)

If the end of vector ${\overrightarrow{V}}_{ij}$ falls within the CC, the dynamic obstacle aircraft poses a threat to the aircraft. To define the collision zone set (collision cone, CC) in the velocity space, as shown in Equation (7),

$$CC=\left\{{\overrightarrow{V}}_{ij}\mid l_m\cap \odot P_j\ne \varnothing \right\}$$

(7)

where $\odot P_j$ represents the expanded safety protection zone of the obstacle aircraft, which is an elliptical zone.

Determining whether there is a flight conflict with an obstructing aircraft is usually more intuitive when using the aircraft’s velocity, ${\overrightarrow{V}}_i$. Figure 5 shows that $P_i$ and $\odot P_j$ are translated by distance $\parallel {\overrightarrow{V}}_j\Delta t\parallel$ in the direction of vector ${\overrightarrow{V}}_j$ to obtain $Q_i=(x_i^{\prime },y_i^{\prime })$, $Q_j=(x_j^{\prime },y_j^{\prime })$, and the unit time interval is $\Delta t$. The above variables can be calculated by Equation (8):

$$\left\{\begin{array}{l}{Q}_i=P_i+{\overrightarrow{V}}_j\Delta t\\ Q_j=P_j+{\overrightarrow{V}}_j\Delta t\end{array}\right.$$

(8)

The zone resulting from the translation of $CC$ by ${\overrightarrow{V}}_j$ is referred to as the new collision cone ($NCC$), as shown in Equation (9):

$$NCC=CC\oplus {\overrightarrow{V}}_j$$

(9)

where the vector addition operation is denoted by $\oplus$ in Minkowski space, which takes into account the structure of space–time [34]. The Minkowski vector sum operation is a four-dimensional spatial vector operation used in relativity theory that combines time and space dimensions to describe space–time relationships at high speeds.

Principles of Collisions between Multi-Aircraft

This method aims to calculate the corresponding $NC{C}_i$ for the conflict between the aircraft and the multi-dynamic obstacle aircraft. If the aircraft’s velocity ${\overrightarrow{V}}_i$ falls within the multiple new collision cone ($MNCC$), a flight conflict with the multi-obstacle aircraft occurs. Otherwise, there is no conflict. Figure 6 illustrates a scenario in which one aircraft conflicts with two obstacle aircraft simultaneously. The aircraft’s velocity, ${\overrightarrow{V}}_i$, is within the overlapping zone of $NC{C}_1$ and $NC{C}_2$, and the conflict zone can be expressed by Equation (10):

$$A_{MNCC}=\underset{i=1}{\overset{N}{\cup }}{A}_{{NCC}_i}$$

(10)

4.3.2. Swarm-Oriented Dynamic VO Method Conflict Awareness Model

Collaborative Control within Swarms

To ensure stable coordination within the swarm during obstacle avoidance and avoid conflicts or deviations between aircraft, a swarm control algorithm is used [35]. The algorithm follows the principle of alignment: each aircraft calculates the direction of the combined velocity of all aircraft and the size of the average speed to obtain the swarm velocity and then attempts to align with this velocity. The method is feasible for practical application as the aircraft only requires a small fine adjustment within the acceptable range of navigation accuracy error. Equation (11) provides the heading for the adjustment:

$${\overrightarrow{V}}_{Ni}=\left[\begin{array}{l}{v}_{Ni}\cos {\theta }_{Ni}\\ v_{Ni}\sin {\theta }_{Ni}\end{array}\right]$$

(11)

Principle of Collision between Swarm Pairs

This section shows the improvement of the traditional VO method based on aircraft swarms, where a CC is constructed on the swarm ellipse to facilitate conflict identification between swarms. To facilitate explanation, the relevant information is defined first. Assume that at time $t_1$, the forward and lateral radiation distances of the aircraft swarm are $a_{Ni}$ and $b_{Ni}$, respectively; the forward and lateral radiation distances of a dynamic threatening obstacle aircraft are $a_{Nj}$ and $b_{Nj}$, respectively. Meanwhile, the positions of the aircraft swarm and the obstacle aircraft swarm are $P_{Ni}$ and $P_{Nj}$, of which the velocities are ${\overrightarrow{V}}_{Ni}$ and ${\overrightarrow{V}}_{Nj}$, the magnitudes of the speeds are $v_{Ni}$ and $v_{Nj}$, and the yaw angles are ${\theta }_{Ni}$ and ${\theta }_{Nj}$, respectively.

Based on the above settings, then the points $Q_{Ni}=(x_{Ni}^{\prime },y_{Ni}^{\prime })$ and $Q_{Nj}=(x_{Nj}^{\prime },y_{Nj}^{\prime })$ can be obtained by translating the distances $\parallel {\overrightarrow{V}}_{Nj}\Delta t\parallel$ from $P_{Ni}$ and $\odot P_{Nj}$ along the direction of the velocity ${\overrightarrow{V}}_{Nj}$. The improved VO method of the aircraft swarm situation relationship is shown in Figure 7. In Figure 7, a bule ellipse is obtained with $Q_{Nj}$ as the center, $a_{Nij}$ as the major semiaxis, and $b_{Nij}$ as the minor semiaxis. In addition, let $Q_{Ni}$ be the vertex as a cone tangent to the ellipse $Q_{Nj}$, as shown in Figure 7. The above variables can be calculated by Equation (12):

$$\left\{\begin{array}{l}{a}_{Nij}=a_{Ni}+a_{Nj}\\ b_{Nij}=b_{Ni}+b_{Nj}\end{array}\right.$$

(12)

4.4. Conflict Resolution Strategies for Swarms

4.4.1. Optimized Reachability Analysis for Heading and Speed

The solution to the relief strategy depends on the magnitude of the speed and heading of the velocity ${\overrightarrow{V}}_{Ni}$ of the aircraft swarm when it reaches the boundary of the $NCC$. Given the inherent performance constraints of the aircraft, the range of its maneuverability is inherently bounded. The angle formed between the aircraft’s velocity, denoted as ${\overrightarrow{V}}_{Ni}$, and the horizontal aircraft is defined as ${\theta }_{Ni}$. Equation (13) delineates the permissible spectrum for the modification of the aircraft’s relief velocity, ${\overrightarrow{V}}_{Ni}^{\prime }$, and its relief heading, ${\theta }_{Ni}^{\prime }$, over a unit time interval $\Delta t$.

$$\left\{\begin{array}{l}{\left|{\overrightarrow{V}}_{Ni}\right|}_{\min }\le \left|{\overrightarrow{V}}_{Ni}^{\prime }\right|\le {\left|{\overrightarrow{V}}_{Ni}\right|}_{\max }\\ 0\le \left|\Delta {\theta }_{Ni}\right|\le \left|{\theta }_{Ni}^{\prime }-{\theta }_{Ni}\right|\le \Delta {\theta }_{Ni\max }\end{array}\right.$$

(13)

4.4.2. Analysis of Swarm Application in Conflict Resolution

Figure 8 shows a scenario where two swarms clash in the future. For the problem of resolving inter-swarm conflicts among aircraft, the improved VO method dictates that the velocity of aircraft, ${\overrightarrow{V}}_{Ni}$, must be adjusted ${\overrightarrow{V}}_{Ni}$ to fall outside the $NCC$. The boundary parameters of the $NCC$ are determined through geometric analysis, and the resulting relief strategy is the one that minimizes changes in speed and heading.

When the aircraft swarm establishes a CC with the obstacle aircraft swarm, the velocity ${\overrightarrow{V}}_{Nj}$ of the obstacle aircraft swarm is translated. The center of mass of the aircraft swarm is $Q_{Ni}$, and the center of mass of the obstacle aircraft swarm is $Q_{Nj}$. The protection zone (ellipse) of the inflated obstacle aircraft swarm is $\odot Q_{Nj}$, hereinafter referred to as “inflated swarm ellipse”, with the tangent point $P_{\tau }$ and tangent lines $l_3$ and $l_4$, where $Q_{Ni}=(x_{Ni}^{\prime },y_{Ni}^{\prime })$, $Q_{Nj}=(x_{Nj}^{\prime },y_{Nj}^{\prime })$, and $P_{\tau }=\left(x_{\tau },y_{\tau }\right)$. The major semiaxis of the inflated swarm ellipse $\odot Q_{Nj}$ is $a_{Nij}$, while the minor semiaxis is $b_{Nij}$.

First, take the center of the ellipse as $Q_{Nj}$ and the major and minor semiaxes as $a_{Nij}$ and $b_{Nij}$, respectively. This establishes the standard equation of $\odot Q_{Nj}$. Next, derive the derivative of $x$, and then bring the tangent point $P_{\tau }$ into the standard equation after deriving the derivative to obtain the derivative value at the tangent point; second, the slope $k_{l_3}$ is obtained through points $P_{\tau }$ and $Q_{Nj}$. This is equated with the derivative value at the tangent point $P_{\tau }$, obtained in the previous step. Then, the equation involving $x_{\tau }$ and $y_{\tau }$ is derived; third, bring the tangent point $P_{\tau }$ into the standard equation of the expanding swarm ellipse $\odot Q_{Nj}$, then solve the equation of $x_{\tau }$ and $y_{\tau }$, then solve $x_{\tau }$, and then solve $y_{\tau }$ and $k_{l_3}$ in turn; finally, obtain the function $y_{l_3}$ of the line where $l_3$ is located from the point $Q_{Ni}$ and the slope $k_{l_3}$, i.e., solve the function $y_{l_3}$ by using the point-slope equation. The calculation of the above variables is given by Equation (14):

$$\left\{\begin{array}{l}\frac{{\left[\left(x-x_{Nj}^{\prime }\right)\cos {\theta }_{Nj}+\left(y-y_{Nj}^{\prime }\right)\sin {\theta }_{Nj}\right]}^2}{{a_{Nij}}^2}+\frac{{\left[\left(x_{Nj}^{\prime }-x\right)\sin {\theta }_{Nj}+\left(y-y_{Nj}^{\prime }\right)\cos {\theta }_{Nj}\right]}^2}{{b_{Nij}}^2}=1\\ {\frac{\partial y}{\partial x}}_{(x_{\tau },y_{\tau })}=\frac{\frac{2\cos {\theta }_{Nj}\left[\left(x_{\tau }-x_{Nj}^{\prime }\right)\cos {\theta }_{Nj}+\left(y_{\tau }-y_{Nj}^{\prime }\right)\sin {\theta }_{Nj}\right]}{{a_{Nij}}^2}-\frac{2\sin {\theta }_{Nj}\left[\left(x_{Nj}^{\prime }-x_{\tau }\right)\sin {\theta }_{Nj}+\left(y_{\tau }-y_{Nj}^{\prime }\right)\cos {\theta }_{Nj}\right]}{{b_{Nij}}^2}}{\frac{2\sin {\theta }_{Nj}\left[\left(x_{\tau }-x_{Nj}^{\prime }\right)\cos {\theta }_{Nj}+\left(y_{\tau }-y_{Nj}^{\prime }\right)\sin {\theta }_{Nj}\right]}{{a_{Nij}}^2}+\frac{2\cos {\theta }_{Nj}\left[\left(x_{Nj}^{\prime }-x_{\tau }\right)\sin {\theta }_{Nj}+\left(y_{\tau }-y_{Nj}^{\prime }\right)\cos {\theta }_{Nj}\right]}{{b_{Nij}}^2}}\\ k_{l_3}=\frac{(y_{Nj}^{\prime }-y_{\tau })}{(x_{Nj}^{\prime }-x_{\tau })}\\ k_{l_3}={\frac{\partial y}{\partial x}}_{(x_{\tau },y_{\tau })}\\ \frac{{\left[\left(x_{\tau }-x_{Nj}^{\prime }\right)\cos {\theta }_{Nj}+\left(y_{\tau }-y_{Nj}^{\prime }\right)\sin {\theta }_{Nj}\right]}^2}{{a_{Nij}}^2}+\frac{{\left[\left(x_{Nj}^{\prime }-x_{\tau }\right)\sin {\theta }_{Nj}+\left(y_{\tau }-y_{Nj}^{\prime }\right)\cos {\theta }_{Nj}\right]}^2}{{b_{Nij}}^2}=1\\ y_{l_3}=k_{l_3}(x-x_{Ni}^{\prime })+y_{Ni}^{\prime }\end{array}\right.$$

(14)

In Figure 8a, when obtaining the relief velocity $v_{Ni}^{\prime }$, without changing the heading ${\theta }_{Ni}$, combine the straight line $y_{{\overrightarrow{V}}_{Ni}^{\prime }}$ where ${\overrightarrow{V}}_{Ni}^{\prime }$ is located in conjunction with the tangent equation $y_{l_3}$ to obtain the point of intersection $M=(x_m,y_m)$. Then, through the point $M$ with respect to $P_{Ni}$, obtain the horizontal and vertical components as $d{x}_{v_{Ni}^{\prime }}$ and $d{y}_{v_{Ni}^{\prime }}$, respectively. Finally, calculate the value of $v_{Ni}^{\prime }$, which is given by Equation (15):

$$\left\{\begin{array}{l}d{x}_{v_{Ni}^{\prime }}=x_m-x_{Ni}/d{y}_{v_{Ni}^{\prime }}=y_m-y_{Ni}\\ k_{{\overrightarrow{V}}_{Ni}^{\prime }}=\frac{d{y}_{v_{Ni}^{\prime }}}{d{x}_{v_{Ni}^{\prime }}}\\ y_{{\overrightarrow{V}}_{Ni}^{\prime }}=k_{{\overrightarrow{V}}_{Ni}^{\prime }}(x-x_{Ni})+y_{Ni}\\ v_{Ni}^{\prime }=\frac{d{y}_{v_{Ni}^{\prime }}}{\sin {\theta }_{Ni}}\end{array}\right.$$

(15)

In a parallel approach to the boundary $l_4$ of the $NCC$ zone, a congruent methodology is employed to ascertain the relief velocity $v_{Ni}^{″}$.

As shown in Figure 8b, when solving the relief heading ${\theta }_{Ni}^{\prime }$, the velocity magnitude of $v_{Ni}$ is maintained. Vector ${\overrightarrow{V}}_{Ni}^{\prime }$ and tangent $y_{l_3}$ intersect at point $G$, and the horizontal and vertical components of ${\overrightarrow{V}}_{Ni}^{\prime }$ concerning $P_{Ni}(x_{Ni},y_{Ni})$ are $d{x}_{{\theta }_{Ni}^{\prime }}$ and $d{y}_{{\theta }_{Ni}^{\prime }}$. The point $G=(x_{Ni}+d{x}_{{\theta }_{Ni}^{\prime }},y_{Ni}+d{y}_{{\theta }_{Ni}^{\prime }})$ is brought into the equation of $y_{l_3}$, and ${\theta }_{Ni}^{\prime }$ is solved, which is given by Equation (16):

$$\left\{\begin{array}{l}\sin {\theta }_{Ni}^{\prime }=\frac{d{y}_{{\theta }_{Ni}^{\prime }}}{v_{Ni}}/\cos {\theta }_{Ni}^{\prime }=\frac{d{x}_{{\theta }_{Ni}^{\prime }}}{v_{Ni}}\\ \left(y_{Ni}+d{y}_{{\theta }_{Ni}^{\prime }}\right)=k_{l_3}\left[\left(x_{Ni}+d{x}_{{\theta }_{Ni}^{\prime }}\right)-x_{Ni}^{\prime }\right]+y_{Ni}^{\prime }\\ {\theta }_{Ni}^{\prime }=arctan(\frac{d{y}_{{\theta }_{Ni}^{\prime }}}{d{x}_{{\theta }_{Ni}^{\prime }}})\end{array}\right.$$

(16)

In a parallel approach to the boundary $l_4$ of the $NCC$ zone, a congruent methodology is employed to ascertain the relief heading ${\theta }_{Ni}^{″}$.

4.4.3. Analysis of Multi-Swarm Applications in Conflict Resolution

The $MNCC$ boundary delineates a critical threshold for aerial conflict between swarms of aircraft and obstacle aircraft. When an aircraft’s velocity, ${\overrightarrow{V}}_{Ni}$, intersects the $MNCC$’s overlapping zone, it signifies an impending conflict with dynamically moving obstacle aircraft. Conflict aversion necessitates positioning the aircraft’s velocity, either ${\overrightarrow{V}}_{Ni}^{\prime }$ or ${\overrightarrow{V}}_{Ni}^{″}$, outside the $MNCC$’s domain, as depicted in Figure 9.

In Figure 9a, the shaded area represents the adjustable range of speed and heading per unit time and is defined as “E”. The shaded area in Figure 9b illustrates the area of $MNCC$, denoted as “F”. Meanwhile, the shaded area in Figure 9c highlights the relief range, which is the area beyond the collision zone within the accessible range. This area, defined as $C_E(E\cap F)$, is essentially the complement of the intersection between “E” and “F”.

The computation of the boundary of the $MNCC$ zone’s parameter is similar to that of a single $NCC$ zone and will not be repeated here.

4.4.4. Decision Optimization Models with Low-Carbon Objectives

An optimal conflict resolution strategy centered on low-carbon emissions is proposed, prioritizing minimal fuel consumption and CO₂ emissions in aircraft. The strategy identifies the most fuel-efficient course of action within a given speed or heading intervals for conflict resolution. Heading adjustments adhere to a minimum change tangent strategy, maintaining the aircraft’s original course and minimizing unnecessary trajectory deviations. Speed adjustments involve an analysis and comparison of fuel efficiency at various speeds to ascertain the most energy-efficient value. This approach integrates heading and speed adjustments, effectively resolving potential conflicts between aircraft while aligning with sustainable development goals in the air transport industry. The Base of Aircraft Data (BADA), encompassing performance and operational process parameters for 294 aircraft types, serves as a foundational resource. A relationship equation, derived from BADA’s reference model, correlates speed per unit time with fuel consumption [36]. It can be calculated by Equations (17) and (18):

Objective function:

$$\min C_t(t)=c_1\left[1+\frac{v_{Ni}^{\prime }(t)}{c_2}\right]\left[\alpha v_{Ni}{(t)}^2+\frac{\beta }{v_{Ni}^{\prime }{(t)}^2}+m\frac{\mathrm{d}{v}_{Ni}(t)}{\mathrm{d}t}\right]$$

(17)

Constraints:

$${\left|v_{Ni}\right|}_{\min }\le \left|v_{Ni}^{\prime }\right|\le {\left|v_{Ni}\right|}_{\max }$$

(18)

where $m$ represents the mass of the aircraft; $c_1$ and $c_2$ represent the first and second thrust specific fuel consumption coefficients, respectively; $\alpha$ and $\beta$ represent the first and second fuel flow coefficients, respectively; $v_{Ni}(t)$ represents the speed at time t; $v_{Ni}^{\prime }(t)$ represents the decision speed at time t; and $C_t(t)$ represents the fuel consumption per unit of time at the decision speed. Please note that $c_1$, $c_2$, $\alpha$, and $\beta$ are dependent on the type of aircraft.

5. Simulation Experiment and Analysis

A method for mitigating conflicts in aircraft swarms is developed, enhancing the VO method. The approach is bifurcated into a preprocessing stage and a solution stage, as delineated in Figure 10. The process involves extracting and processing aircraft data using a clustering model, fine-tuning the velocity within a swarm using a swarm control algorithm, and establishing an advanced VO model between swarms for conflict detection. The optimal strategy for reducing carbon emissions in terms of speed and heading is then calculated using the minimize function in the SciPy library for the optimization model.

In the field of aircraft simulation, commonly used software includes MATLAB/Simulink(R2021b) for designing systems and control algorithms, X-Plane and Flightgear for flight dynamics and environmental simulation, and Ansys Fluent for aerodynamic analysis. Additionally, the Modelica language is commonly used for multi-domain system modeling, and the Ada language is used for aeronautical software development due to its high reliability. This paper assesses the reliability of conflict detection algorithms through Python, which is used to compute and visualize the designed simulation experiments.

5.1. Verification and Analysis of Dynamic Elliptic Protection Zones

Table 1. Aircraft position information at different safety intervals.

Exp. Grp.	IS	AT	(xi, yi)	ai	bi	θi	vi	(xj, yj)	aj	bj	θj	vj
1	exact	A320 and A380	(70, 60)	4.08	3.18	50	13.6	(50, 90)	4.5	3	5	15
	traditional		(70, 60)	5	5	50	13.6	(50, 90)	5	5	5	15
2	exact	B737-800 and DA40	(270, 260)	4.2	3.01	135	14	(250, 300)	1.35	0.93	180	4.5
	traditional		(270, 260)	5	5	135	14	(250, 300)	5	5	180	4.5
3	exact	A380 and B737-800	(65, 70)	4.5	3.18	20	15	(100, 120)	4.2	3.01	280	14
	traditional		(65, 70)	5	5	20	15	(100, 120)	5	5	280	14

Note: IS means interval standard; AT means aircraft type; the unit of (xi, yi), ai, bi, (xj, yj), aj, and bj is km; the unit of θi and θj is °; the unit of vi and vj is km/min.

presents the initial position, heading, speed, and protection zone dimensions for three pairs of different aircraft types in a cross-conflict scenario. An accurate inter-aircraft protection zone distance strategy is employed to define the minor semiaxis of the protection zone. The major semiaxis length is determined by the velocity magnitude, ensuring it surpasses the short half-axis while not exceeding the conventional 15 km interval, for which $\lambda$ is set at 0.3. Utilizing these parameters, the improved VO method is applied to compute the necessary relieving speed and heading [33].

Table 2. Conventional interval and dynamic elliptical protection zone relief strategies.

Exp. Grp.	IS	vi′	Vi″	θi′	θi″	Min V	Max V	Min Heading	Max Heading
1	exact	11.36	16.69	12.37	79.49	2.24	3.09	37.63	29.49
	traditional	10.65	17.34	11.30	88.28	2.95	3.74	38.70	38.28
2	exact	9.90	18.34	127.26	138.75	4.10	4.34	7.74	3.75
	traditional	8.40	36.14	121.80	143.82	5.63	22.11	13.20	8.82
3	exact	12.97	23.93	−0.07	26.60	2.03	8.93	20.07	6.60
	traditional	11.63	26.37	−3.80	31.56	3.37	11.37	23.80	11.56

Note: the unit of vi′ and vi″ is km/min; the unit of θi′ and θi″ is °; Min V and Max V mean minimum and maximum relief speed differentials, respectively; Min Heading and Max Heading mean minimum and maximum adjustments in the relief heading differentials, respectively.

displays the results for experimental group 1, compared with the traditional method, using the proposed method (exact) indicating a reduction of 24.07% (2.95 − 2.24 = 0.71 km/min) and 17.38% (3.74 − 3.09 = 0.65 km/min) in the minimum and maximum relief speed differentials, respectively. Additionally, the minimum and maximum adjustments in the relief heading differentials were reduced by 2.76% (38.70 − 37.63 = 1.07°) and 22.96% (38.28 − 29.49 = 8.79°), respectively. The data from experimental groups 2 and 3 exhibited comparable optimization effects. Experimental group 2 showed a significant 80.37% reduction in the maximum relief speed difference, while experimental group 3 showed a 39.76% reduction in the minimum relief speed difference. Regarding heading adjustments, experimental group 2 achieved a maximum percentage reduction of 57.48%, while experimental group 3 achieved a maximum percentage reduction of 42.91%. There was an average reduction of 32.78 percent change in the three experimental groups. The data demonstrated that speed-based variable protection zones were effective in reducing the need to adjust heading and speed in both large and small aircraft, which was consistent across aircraft types. Therefore, dynamic protection zones based on speed not only optimize the relief strategy and reduce the need for adjustments but also meet the requirements of fine-grained management, improving the safety and efficiency of aircraft operations.

5.2. Validation and Analysis of Clustering Model

Data 1 of

Table 3. Position information for four different types of aircraft (data 1).

Aircraft	(xi, yi) (km)	ai (km)	bi (km)	θi (°)	vi (km/min)
1	(55, 73)	4.05	3	2	13.5
2	(20, 78)	4.47	3.18	0	14.9
3	(80, 54)	4.17	3.01	60	13.9
4	(80, 44)	4.17	3.01	62	13.8
5	(50, 80)	4.08	3	5	13.60
6	(45, 73)	4.065	3	6	13.55
7	(15, 70)	4.05	3	10	13.5
8	(110, 75)	4.44	3.18	10	14.8
9	(80, 100)	1.35	0.93	60	4.5
10	(65, 73)	4.02	3	4	13.4

delineates positional data for four randomly selected aircraft types. Utilizing the aircraft clustering model, ten aircraft were grouped with parameters set to a maximum separation of 10 km, a maximum heading divergence of 5°, and a maximum velocity differential of 0.5 km/min. The outcomes of this exercise are depicted in Figure 11, where the ten aircraft are categorized into six distinct swarms. Notably, no conflicts were identified within these swarms. For ease of differentiation, each swarm has its own distinct color. The swarms are designated as $S=\left\{S_1,S_2,S_3,S_4,S_5,S_6\right\}$, where $S_1=\left\{P_1,P_5,P_6,P_{10}\right\}$, $S_2=\left\{P_3,P_4\right\}$, $S_3=\left\{P_2\right\}$, $S_4=\left\{P_7\right\}$, $S_5=\left\{P_8\right\}$, and $S_6=\left\{P_9\right\}$.

The swarm optimization ellipse model illustrates the mass points, velocities, and dimensions of the protection zone for swarm $S_1$ (represented by red aircraft) and swarm $S_2$ (depicted by yellow aircraft). Figure 12 and Figure 13 display the ellipse information for swarms $S_1$ and $S_2$, respectively. For swarm $S_1$, the ellipse’s area met the specified conditions at 453.998 km², with a major semiaxis of 16.55 km, a minor semiaxis of 8.73 km, and an angle of 4.253° between the major semiaxis and the x-axis. The speed magnitude was 13 km/min. The ellipse’s center was positioned at coordinates (53.75, 74.75). For swarm $S_2$, the ellipse’s parameters fulfilled the condition with a major semiaxis of 10.125 km, a minor semiaxis of 6.6625 km, and an angle of 59.999° between the major semiaxis and the x-axis. The speed magnitude here was 13.85 km/min, with the center of the ellipse located at coordinates (80.0, 49.0).

Table 4. The differences between the velocities of the aircraft and the swarm.

Swarm	Aircraft	Angular Diff. (°)	Velocity Diff. (km/min)
S1	1	2.25	0.01
S1	5	0.75	0.09
S1	6	1.75	0.04
S1	10	0.25	0.11
S2	3	1	0.05
S2	4	1	0.05

indicates that the velocity and angular differences between the aircraft and its respective swarm were calculated. The speed difference ranged between [0–0.2] km/min, and the angular divergence was within [0–3] degrees. The adjustments in each aircraft’s velocity remained within the bounds of navigational accuracy, suggesting that micro-adjustments to the velocities are a viable approach. By fine-tuning these vectors, a new collective velocity for each swarm can be derived, demonstrating an effective method for dynamic swarm management.

When there are “m” aircraft in the aircraft swarm and “n” aircraft in the obstacle aircraft swarm, the traditional VO method requires m × n conflict judgments. In the case of the obstacle aircraft swarm $S_1$ with four aircraft and the aircraft swarm $S_2$ with two aircraft, a total of eight conflict judgments are necessary. The computational efficiency of the swarm improvement VO method is improved by 87.5% compared with the traditional VO method, as $S_1$ and $S_2$ only need to go through one conflict judgment. To demonstrate the advantage of the improved VO method with larger swarms, the position information of 10 aircraft of four different types is re-given in data 2 of

Table 5. Position information for four different types of aircraft (data 2).

Aircraft	(xi, yi) (km)	ai (km)	bi (km)	θi (°)	vi (km/min)
1	(55, 73)	4.05	3	2	13.5
2	(51, 82)	4.09	3.18	0	13.63
3	(80, 54)	4.17	3.01	60	13.9
4	(80, 44)	4.14	3.01	62	13.8
5	(60, 80)	4.08	3	5	13.60
6	(46, 69)	4.065	3	6	13.55
7	(20, 75)	4.05	3	10	13.5
8	(46, 76)	4.04	3.18	7	13.45
9	(80, 90)	1.35	0.93	60	4.5
10	(65, 73)	4.02	3	4	13.4

and swarmed into $S=\left\{S_7,S_8,S_9,S_{10}\right\}$, which resulted in the findings shown in Figure 14 with swarms $S_7=\left\{P_1,P_2,P_5,P_6,P_8,P_{10}\right\}$, $S_8=\left\{P_3,P_4\right\}$, $S_9=\left\{P_7\right\}$, and $S_{10}=\left\{P_9\right\}$. There will be a conflict between swarm $S_7$ (red aircraft) and swarm $S_8$ (green aircraft) in the future. The six aircraft in the obstacle aircraft swarm $S_7$ and two aircraft in the aircraft swarm $S_8$ will need to go through 12 conflict judgments. This is 91.67% more efficient than the traditional method. The efficiency of optimization is calculated by subtracting the number of original judgments from the number of optimized judgments, dividing the difference by the number of original judgments, and multiplying the result by 100. The optimization of the improved VO method depends mainly on the number of aircraft included in the swarm, rather than the total number of aircraft. Current research has shown excellent computational efficiency on a limited number of aircraft models. To cope with broader and more complex aircraft classification in the future, intelligent cluster classification methods and modern multi-core parallel computing can be further studied to reduce computing time.

5.3. Validation and Analysis of the Swarm Conflict Resolution Model

The ensuing section validates the reliability of this approach by analyzing aircraft conflict resolution scenarios in swarm $S_1$ and swarm $S_2$. The simulation depicted in Figure 15 reveals that the aircraft pair encountered the closest conflict at T = 1.35 min. Figure 15a–d show that after 1.35 min, aircraft $P_3$ conflicted with $P_{10}$ but safely navigated past $P_1$, $P_5$, and $P_6$. Likewise, Figure 15e–h show that aircraft $P_4$ encountered conflicts at different times: it conflicted with $P_1$ at 2.40 min, with $P_5$ at 3.10 min, and with $P_{10}$ at 2.00 min. Importantly, during these sequences, it managed to navigate past aircraft $P_6$ without any conflict. Within a brief span of 5 min, four conflicts arose between two swarms of aircraft, indicating a situation of high urgency.

From the CC formed by the Improved VO Method, it is clear that $S_1$ and $S_2$ will collide at some point in the future.

Table 6. Strategy for aircraft deconstruction.

Swarms	T (min)	${\mathit{v}}_{\mathit{N}\mathit{i}}^{{\displaystyle \prime }}$ (km/min)	${\mathit{v}}_{\mathit{N}\mathit{i}}^{{\displaystyle ″}}$ (km/min)	${\mathit{\theta }}_{\mathit{N}\mathit{i}}^{{\displaystyle \prime }}$ (°)	${\mathit{\theta }}_{\mathit{N}\mathit{i}}^{{\displaystyle ″}}$ (°)
S1 and S2	1.35	4.93	27.58	12.84	143.24

delineates the relief speed and heading derived from geometric analysis. The maximum heading adjustment is set at 120°, with a minimum speed adjustment of 1 km/min and a maximum of 30 km/min. This range is determined in conjunction with the maximal unit-time adjustment for an A320, establishing the boundary for relief measures. The aircraft relief strategy in the improved VO method involves selecting a zone outside the $NCC$ boundary.

In pursuit of an optimal relief strategy focused on minimizing carbon dioxide emissions, a comprehensive analysis of the Airbus A320 was conducted. Key parameters were meticulously selected to accurately represent the aircraft’s performance. These included a mass (m) of 24,000 kg, a cruising speed ($v_{Ni}$) of 13.85 km/min, and an acceleration per unit time ($d{v}_{Ni}/dt$) of 0.35 km/min². Additionally, the study incorporated specific coefficients: $c_1$ at 1.202 × 10⁻⁷ and $c_2$ at 0.365, alongside constants $\alpha$ and $\beta$, valued at 2300 and 7,979,854, respectively. Figure 16 vividly illustrates the intricate relationship between the aircraft’s flight time, speed, and fuel consumption, providing critical insights into the efficiency of the A320 under varying operational conditions.

Figure 17 delineates the achievable range of the unwinding strategy, coupled with an in-depth analysis of the fuel consumption rate $v_{Ni}$ at various speeds. The examined speeds and their corresponding fuel consumptions were 1 km/min (3.82 kg/km), 4.92 km/min (1.49 kg/km), 27.58 km/min (4.93 kg/km), and 30 km/min (5.34 kg/km). Intriguingly, the graphical representation reveals a pivotal finding: the most fuel-efficient speed was 4.2 km/min, at which the fuel consumption was minimized to 1.47 kg/km. The lowest point of the curve is indicated by the star in the figure. By subtracting the minimum fuel consumption of 1.47 kg/km from the fuel consumption corresponding to the speeds at the boundary of the reachable speed range, dividing it by the respective fuel consumption, and multiplying it by 100, a maximum saving of 72.47% ((5.34 − 1.47)/5.34 × 100%) was achieved, with a minimum saving of 1.34% ((1.49 − 1.47)/1.49 × 100%).

As can be seen in Figure 18a, the velocity $v_{Ni}$ of swarm $S_2$ falls into the $NCC$ zone. The reliability of this method is corroborated by employing a relief speed $v_{Ni}^{\prime }$ of 4.2 kg/km and a relief heading ${\theta }_{Ni}^{\prime }$ of 12°. As depicted in Figure 18b,c, the velocity ${\overrightarrow{V}}_{Ni}$ of swarm $S_2$ does not intersect with the $NCC$, signifying a future non-conflict scenario between the two swarms. In Figure 18d, a simulation step size of 0.05 was used, and over an 8 min simulation, a T = 1.35 min conflict scenario was observed between swarms $S_1$ and $S_2$. The introduction of swarm pairs (two swarms per pair) using the VO method resulted in a conflict duration consistent with the most recent conflict time of the aircraft pair, a notable improvement over the traditional VO method. This technique supports the effectiveness of the aircraft conflict identification that this novel approach introduced. The simulation diagram marks the conflict time of the swarm pair in red, with the trajectory darkening as the proximity between the two swarms increases. Figure 18e,f confirm the absence of conflict during the flight of the two swarms. Figure 18g shows the distance between the swarm pairs diminishing to zero over 2.35 min. Conversely, in Figure 18h,i, the distance between swarm pairs decreases and then increases, yet never falls below zero. To preclude collisions in swarms of multi-dynamic obstacle aircraft, a conflict resolution strategy is derived by selecting zones outside multiple $NCC$. This strategy, although based on the same principle, is not further elaborated upon here.

When dealing with a greater number of swarms, the optimized VO method demonstrates increased efficacy. This is of considerable practical importance for dynamic route planning as it reduces operational costs for airlines and the workload for controllers, while concurrently decreasing fuel consumption and exhaust gas emissions. It is crucial to note that this improvement is objective and grounded in empirical evidence.

6. Conclusions

To enhance the current real-time aircraft obstacle avoidance strategies, this paper refines the traditional VO method by integrating the dynamic ellipsoid model with swarm intelligence concepts, while also considering low-carbon emissions. Specifically, the swarm control algorithm is used to optimize inter-aircraft obstacle avoidance and minimize the risk of aircraft collisions while maintaining swarm integrity. The method accurately sets the protection zone, reducing the need for speed and heading adjustments by an average of 32.78% during obstacle avoidance compared with traditional aircraft intervals. In multi-swarm conflict scenarios, this strategy can quickly formulate a resolution, significantly reducing the number of conflict detections by the aircraft. This improves computational efficiency by over 87.5% and reduces the frequency of conflict resolution. In addition, by adjusting the speed to reduce fuel consumption, it is possible to achieve fuel savings of up to 72.47%, which not only reduces fuel consumption and emissions but also helps to reduce operating costs. Also, the feature of the proposed method has great potential in the field of swarm formation flying for future urban air traffic.

The reliability and practicality of the new model have been confirmed through in-depth analysis of existing methods and Python simulation experiments. Despite the positive results, aircraft flight dynamics have not been adequately considered in CO₂ simulation studies, which play an important role in accurately estimating the relationship between fuel consumption and speed. Future research will focus on this by actively exploring and using open-source simulation platforms such as BlueSky and AirTrafficSim for more accurate flight dynamics simulations [37,38], to improve the accuracy and applicability of the models. This work makes an important contribution to the improvement of aircraft obstacle avoidance strategies, which is an important reference for the development of a low-carbon and efficient air transport system and is expected to promote the sustainable development of the aviation industry.