Nomenclature

${\sigma _{{\textrm{CT}}}}$

cross-track standard deviation

${\sigma _{{\textrm{AT}}}}$

along-track standard deviation

${R_0}$ , ${R_1}$

the lengths of the long axes of the confidence ellipses

${D_{\textrm{H}}}$

horizontal separation

${D_{\textrm{V}}}$

vertical separation

${t_{\textrm{h}}}$

the advance detection time

${t_{\textrm{i}}}$

the time interval for conflict detection

$Co{n_{{\textrm{ij}}}}$

the degree of urgency between conflict aircraft $i$ and conflict aircraft $j$

${t_{{\textrm{lose}}}}$

the moment of loss of safe separation between aircraft $i$ and aircraft $j$

${{\textrm{S}}_1}$

sorting sequence

${{\textrm{S}}_2}$

execution instruction sequence

${{\textrm{S}}_{{\textrm{EXE}}}}$

whether to execute the command sequence

${N_{\textrm{n}}}$

number of aircraft

${N_{\textrm{t}}}$

number of discrete time periods

${N_{\textrm{d}}}$

specific information about each aircraft at a given moment in time

${O_{\textrm{t}}}$

the observation of the agent

${S_{\textrm{t}}}$

the state of the corresponding rectangular airspace

$Info_{\textrm{i}}^T$

the specific flight information of the $i$ th aircraft in the framed airspace at time $T$

$cmd$

a specific command executed by the aircraft

${T_{\textrm{w}}}$

the waiting time for the execution of the action

${T_{\textrm{a}}}$

an upper limit on the execution time of an instruction

$\theta $

the overall network parameter

$p(\tau )$

the distribution of trajectories

Abbreviations

MDP

Markov decision processes

DRL

deep reinforcement learning

ATCOs

air traffic controllers

PBN

performance-based navigation

CD&R

conflict detection and resolution

TCS

tactical conflict solver

SGD

stochastic gradient descent

ACKTR

Actor-Critic Kronecker-factored Trust Region

ATOSS

Air Traffic Operation Simulation System

BADA

base of aircraft data

2.0 Previous related works

In the early stages of flight conflict resolution research, conventional methodologies predominantly included mathematical planning, optimal control and swarm intelligence methods, etc. Mathematical planning methods are typically fast and scalable [Reference Pelegrín and D’Ambrosio3–Reference Liu and Xiao5], but they are often simplified and set a series of assumptions in the modeling process, potentially limiting their practical applicability. Optimal control strategies generate conflict resolution programs typically defined as trajectories comprising sequential position points rather than the manoeuver instructions required by air traffic controllers [Reference Tang, Chen and Li6, Reference Liu, Liang, Ma and Liu7]. Swarm intelligence methods [Reference Pei, Guanlong, Haotian, Yang and Cunbao8, Reference Sui, Zhang and Rey9] cannot guarantee the global optimal solution, and the solution speed in complex dynamic scenes makes it difficult to meet the requirements of practical applications.

In recent years, reinforcement learning-like intelligent methods have been introduced into the domain of flight conflict resolution, yielding promising results. Two-aircraft flight conflicts represent a common scenario in air traffic control. One study [Reference Pham, Tran, Goh, Alam and Duong10] utilised a deep deterministic policy gradient (DDPG) model to resolve a conflict between two aircraft in a continuum-type action space. The model achieved a success rate of approximately 87% under various levels of uncertainty. Literature [Reference Tran, Pham, Goh, Alam and Duong11] considered real-world ATCOs’ experience while training a single agent to resolve a two-aircraft conflict, making the given instructions more acceptable to the ATCOs and increasing anthropomorphism. Literature [Reference Pham, Tran, Alam, Duong and Delahaye12] trained a single-agent model to resolve conflicts between two aircraft in high-density and high-uncertainty traffic flow environments, with the proposed DDPG-2S algorithm demonstrating strong performance. Literature [Reference Sui, Ma and Wei13] extended the conflict resolution environment into three dimensions, trained a model to address two-aircraft conflicts while accounting for uncertainty and successfully combined it with a deep reinforcement learning algorithm to achieve notable success rates.

Multi-aircraft flight conflicts, characterised by spatio-temporal correlation, present significant challenges in conflict resolution due to unstable environments, limited scalability and the lack of effective coordination mechanisms. Efforts to address multi-aircraft conflicts typically rely on multi-agent deep reinforcement learning approaches that incorporate implicit coordination mechanisms. These strategies leverage the concept of stochastic games within multi-agent reinforcement learning algorithms to train agents in cooperative behaviours. Sui et al. [Reference Sui, Xu and Zhang14] solved multi-aircraft conflicts based on the Independent Deep Q Network (IDQN) algorithm, which adopts a downwards-compatible approach to solve the problem of a variable number of conflicting aircraft and reduces the solution time to the level of 0.011s. Ramon et al. [Reference Dalmau and Allard15] combined message-passing neural networks with multi-agent reinforcement learning, enhancing agents’ coordination by exchanging critical information for optimal joint action. Lai et al. [Reference Lai, Cai, Liu and Yang16] employed the multi-agent reinforcement learning algorithm (Multi-Agent Deep Deterministic Policy Gradient) MADDPG to resolve conflicts by diverting flight paths and achieved reasonable success rates on a fixed number of aircraft. Wei’s team [Reference Brittain and Wei17] developed the Distributed Reinforcement Learning for Interval Maintenance D2MAV framework, integrating LSTM to enhance scalability and minimise separation loss for homogeneous agent aerial vehicles. The team further developed the distributed multi-agent autonomous separation assurance (MAASA) framework [Reference Brittain and Wei18], which can predict intruding aircraft’s movement intentions and perform better than the D2MAV-A framework. Ralvi et al. [Reference Isufaj, Aranega Sebastia and Angel Piera19] utilised the MADDPG algorithm for conflict resolution and considered conditions such as time, fuel and airspace complexity while achieving a 93% success rate. Huang et al. [Reference Huang, Petrunin and Tsourdos20] introduced a multi-agent asynchronous advantage actor-critic (MAA3C) framework and used mask recurrent neural networks (RNN) to solve the scalability problem, which provides a solution idea for conflict management at the strategic level. Chen et al. [Reference Chen, Hu, Yang, Xu and Xie21] proposed a general multi-agent approach combined with adaptive manoeuver strategies. The article considered trajectory uncertainty and scalability of the number of aircraft, but the experimental environment was only on a two-dimensional scale. George et al. [Reference Papadopoulos, Bastas, Vouros, Crook, Andrienko, Andrienko and Cordero22] advanced conflict resolution strategies by employing an augmented graph-convolutional reinforcement learning method in a multi-agent environment, featuring a centralised training-decentralised execution training scheme. Despite the potential of multi-agent deep reinforcement learning approaches in enhancing information exchange and collaboration among homogeneous or heterogeneous agents, care must be taken to avoid secondary conflicts stemming from poor coordination. Furthermore, managing the scalability of multi-agent systems remains intricate, with challenges such as dimensional explosion emerging as the number of agents increases.

In contrast to the implicit coordination mechanism inherent in multi-agent reinforcement learning, an explicit coordination mechanism at the macro level external to the agents could offer additional benefits. This approach involves training a single-agent resolution model and subsequently integrating it with an explicit coordination mechanism to facilitate conflict resolution in multi-aircraft flight scenarios. By doing so, aircraft in conflict can be managed sequentially without internal agent coordination. For the training of single-agent models, the literature [Reference Sheng, Egorov and Kochenderfer23] has addressed two- and multi-aircraft conflicts using single-agent systems, developing various conflict resolution frameworks for comparative analysis. Literature [Reference Zhao and Liu24] leveraged the PPO algorithm to train conflict resolution strategies, combining a priori domain knowledge with advanced data analysis.

For the study of the explicit coordination mechanism, Zhang et al. [Reference Zhang, Yu, Zhang, Wang and Yu25] from the perspective of low-altitude rescue aircraft operation, regarded each aircraft as an independent agent, and each agent determined the priority according to the task and state it performed and subsequently made the conflict resolution strategy according to the conflict resolution order respectively. Wang et al. [Reference Wang26] solved the multi-aircraft flight conflict problem based on satisfactory game theory, in which a set of priority ranking mechanisms is proposed to characterise the importance of different aircraft, and the aircraft selects the most satisfactory resolution strategy based on its priority and the preference of others. Wu et al. [Reference Wu, Yang, Bi, Wen, Li and Ghenaiet27] established a flight conflict network based on flight conflict relationships between aircraft using aircraft as nodes. The mechanism of nodes and connecting edges ensures that critical aircraft are prioritised for deployment and high-risk conflicts are prioritised for resolution. Florence et al. [Reference Ho, Geraldes, Gonçalves, Rigault, Sportich, Kubo, Cavazza and Prendinger28] proposed two decentralised coordination approaches to cater to the decentralised requirements in the Multi-Agent Path Finding (MAPF) problem. One method involved prioritising agents according to their cost functions, with lower-priority agents using paths generated by higher-priority agents as constraints to compute subsequent paths. By determining the resolution order, the explicit coordination mechanism coordinates conflicting aircraft in a multi-aircraft flight conflict. This dramatically simplifies the cooperation between aircraft and enables the orderly handling of different numbers of conflicting aircraft.

In order to compare the intelligent conflict resolution methods related to reinforcement learning, Table 1 can be obtained by combing the conflict types, theoretical methods, success rates, scalability, uncertainty, application scenarios, etc. Comparative analyses show that intelligent multi-aircraft flight conflict resolution methods that satisfy all the above requirements still need further exploration. The novel multi-aircraft conflict resolution approach delineated in this study effectively addresses scalability concerns, expands the experimental domain to three dimensions and attains a commendable success rate in densely trafficked environments, all while accommodating aircraft performance constraints.

Table 1. Reinforcement learning based flight conflict resolution literature

Note: 1. This success rate is the conflict resolution rate at low uncertainty levels; 2. This success rate was achieved for a low-density traffic flow with an aircraft count of three; 3. This success rate targets the environment of airspace demand capacity balancing issues in the strategic phase as well as conflict separation; 4. This success rate was achieved in a specific experimental environment and with aircraft less than ten.; 5. The authors claim that this is a micro-average result.

The remainder of this article is organised as follows: Section Section 3 provides a detailed discussion on the concept of multi-aircraft conflict and associated uncertainties. Section 4 describes the logical framework for conflict detection and resolution, Section 5 models the MDP for multi-aircraft conflict resolution, Section 6 introduces the ACKTR algorithm in order to train the agents, Section 7 conducts simulation experiments to validate the feasibility of the proposed scheme, Section 8 discusses the strengths and weaknesses of the model, and finally, Section 9 draws a conclusion and proposes directions for future research.

3.0 Problem statement

This section focuses on the problem statement, including the definition of a multi-aircraft flight conflict scenario and the concept of horizontal uncertainty.

3.1 Multi-aircraft flight conflict scenarios

A multi-aircraft flight conflict is a situation in which the prescribed separation criteria is violated between multiple neighbouring aircraft within a relatively short period, which can also be viewed as a combination of multiple two-aircraft flight conflicts. However, due to the potential interconnections between these individual conflicts, it is not reasonable to simply decompose a multi-aircraft flight conflict into multiple two-aircraft flight conflicts and then resolve them as a two-aircraft conflict.

Figure 1 shows a four-aircraft conflict scenario with spatio-temporal correlation, where the conflict detection mechanism successively detects that during time ${{\Delta }}t$ , the conflict between aircraft A and aircraft B occurs at time ${t_1}$ , the conflict between aircraft A and aircraft C occurs at time ${t_2}$ , and the conflict between aircraft C and aircraft D occurs at time ${t_{\textrm{3}}}$ . The spatio-temporal correlation here is reflected in the fact that the occurrence times of ${t_1}$ , ${t_2}$ , and ${t_{\textrm{3}}}$ are all during the time $\Delta t$ , and the midpoints of the conflict lines between aircraft A and aircraft B, aircraft A and aircraft C, and aircraft C and aircraft D are all in the delineated conflict airspace. Spatio-temporal relevance is also reflected in the fact that an aircraft may be involved in multiple two-aircraft conflicts, which requires the aircraft to consider more constraints when performing conflict resolution manoeuvers and to try to resolve as many conflicts as possible with a single manoeuver. Typically, the airspace consists of conflicting aircraft within flight conflicts, as well as non-conflicting environmental aircraft.

Figure 1. Schematic of the flight conflict scenario.

3.2 Uncertainty of horizontal position

Due to navigation errors, flight technique errors, imperfections in the prediction system, and other factors, there may be an error between the aircraft’s actual position and the prediction results, and this uncertainty must be considered when resolving conflicts. In this study, we address the problem of uncertainty of flight trajectories in the horizontal plane by referring to the model of Lauderdale [Reference Lauderdale29]and measuring the position uncertainty of an aircraft using a probabilistic type of method that relies on a Gaussian representation of the position error. A plane rectangular coordinate system is established with the aircraft heading as the x-axis positive direction and the predicted aircraft position as the origin, as shown in Fig. 2.

Figure 2. Uncertainty in the actual position of aircraft in the horizontal plane.

Let the actual position of the aircraft be $(X,Y)$ , then the two-dimensional random variable $(X,Y)$ obeys the following two-dimensional Gaussian distribution:

(1) \begin{align}{\textrm{Pos}}{(A)_{{\textrm{(x,y)}}}}\sim N(\mu (t,\psi ),{\sum {(t,\psi )} ^2})\end{align}

where $N( \cdot )$ is the normal distribution with mean $\mu $ and standard deviation $\sum {} $ , $t$ is the time in the future, and $\psi $ is the aircraft heading. By decomposing the normal distribution using principal axes and assuming that there is no deviation from the mean, the equation can be transformed into the following form:

(2) \begin{align}{\textrm{Pos}}{(A)_{{\textrm{(x,y)}}}} \sim N\begin{pmatrix}\begin{pmatrix}0 \\0\\\end{pmatrix}\begin{pmatrix}{\sigma _{{\textrm{AT}}}^{\textrm{2}}}\;\;\;\; {}& 0\\0\;\;\;\; & {\sigma _{{\textrm{CT}}}^{\textrm{2}}} {}0\\\end{pmatrix}\end{pmatrix}\end{align}

where ${\sigma _{{\textrm{CT}}}}$ is the cross-track standard deviation, ${\sigma _{{\textrm{AT}}}}$ is the along-track standard deviation. ${\sigma _{{\textrm{CT}}}}$ is a constant. Because the cross-track position is controlled by the flight management system, the error does not increase over time. ${\sigma _{{\textrm{AT}}}}$ is a linear function of time, The along-track-position position error accumulates over time and is satisfied:

(3) \begin{align}{\sigma _{{\textrm{AT}}}} = {\sigma _{{\textrm{A}}{{\textrm{T}}_{\textrm{0}}}}} + \alpha t\end{align}

where $\alpha $ describes how the along-track uncertainty grows with prediction time $t$ and has units of nautical miles per minute.

Combined with the above definition of uncertainty, the conflict detection criteria for two-aircraft in this paper are described as follows: as shown in Fig. 2, the uncertainty in defining the aircraft’s position is expressed as the expansion of the aircraft from the standard protection zone. Depending on the different confidence levels, the confidence ellipse of the 2D Gaussian distribution can be determined. A circle is then made with the ellipse’s long axis as the radius, assuming that the aircraft’s actual position does not extend beyond this circle. Given that aircraft can autonomously adjust their heading, the radius of their flight path circle may expand over time. However, in a performance-based navigation (PBN) operational setting, such as under the assumption of the RNAV1 navigation specification, this radius is constrained to be at most one nautical mile. As shown in Fig. 3, where ${R_0}$ and ${R_1}$ , etc., are the lengths of the long axes of the confidence ellipses of the two aircraft, and the blue ellipse is the confidence ellipse, a flight conflict is considered to have occurred between the two aircraft if ${D_{\textrm{H}}} \le 10\,{\textrm{km}}$ and ${D_{\textrm{V}}} \le {\textrm{300}}\,{\textrm{m}}$ ; otherwise, there is no conflict.

Figure 3. Conflict detection based on horizontal uncertainty.

Figure 4. Conflict detect and resolution mechanism.

4.0 Flight conflict detection and resolution framework

This section introduces the framework of conflict detection and resolution, including the mechanism of conflict detection and resolution, the overall ATCOs decision-making framework, and finally our proposed priority-based ranking mechanism.

4.1 Conflict detection and resolution mechanisms in the continuous time domain

A well-developed conflict detection mechanism is essential for the early detection of real-time flight conflicts. This paper extends conflict detection along the temporal direction to detect multi-aircraft conflicts with spatio-temporal correlation. In the continuous time domain, assuming conflict detection starts at time T, Fig. 4 illustrates the conflict detection and resolution process. By utilising the trajectory speculation module for aircraft trajectory speculation in the airspace, conflicts occurring in $[T + {t_{\textrm{h}}},T + {t_{\textrm{h}}} + \Delta t]$ can be detected. Here, $\Delta t$ corresponds to $\Delta t$ presented in Section 3.1, ${t_{\textrm{h}}}$ is the advance detection time, ${t_{\textrm{i}}}$ is the time interval for conflict detection. At this point, after a certain amount of time of calculation, the instruction to resolve the conflict can be given in $\Delta T$ time, and the conflicting aircraft executes the instruction after receiving it. After an interval of time ${t_{\textrm{i}}}$ , the conflict detection mechanism continues with the subsequent detection, where ${t_{\textrm{i}}}$ is usually set to be slightly larger than $\Delta t$ , conflicts between the next start of conflict detection and the end of the previous conflict detection will be detected in subsequent detection intervals. This conflict detection mechanism on the continuous time domain can realise the comprehensive detection and resolution of flight conflicts in the airspace, which guarantees flight safety in the airspace.

Figure 5. The framework of tactical conflict solver.

4.3 Priority-based ranking mechanism

Two primary approaches exist for resolving conflicts involving multiple flights: simultaneous resolution and sequential resolution. Simultaneous resolution entails coordinating conflicting aircraft to take simultaneous actions to address the flight conflict, often achieved through a multi-agent reinforcement learning framework. However, this approach faces challenges in coordinating actions among the conflicting aircraft, making it difficult to rectify the situation if the conflict persists after post-simultaneous resolution attempts. In such cases, aircraft that have already manoeuvered may require readjustment, which does not align with current air traffic control rule. This paper adopts the sequential resolution method, whereby conflicts between multiple flights are resolved by gradually deploying conflicting aircraft in a specific order, effectively addressing scalability concerns by sequentially resolving flight conflicts.

In practice, ATCOs typically prioritise flight conflict aircraft according to air traffic control rules. High-urgency aircraft are usually deployed first to resolve an imminent conflict, while low-urgency aircraft are deployed afterward, considering their previous deployments. The degree of urgency between two aircraft involved in a flight conflict is defined as the inverse of the time difference between the moment conflict detection begins and the moment of loss of safe separation. The higher the degree of urgency, the more the conflict must be dealt with immediately.

Define the set of conflicting aircraft in a multi-aircraft flight conflict scenario:

(4) \begin{align}\mathcal{C} = \left\{ {Acf{t_1},Acf{t_2}, \ldots, Acf{t_{\textrm{n}}}} \right\}\end{align}

where ${\textrm{n}}$ is the number of conflicting aircraft in a multi-aircraft conflict.

According to Section 3.1, the two-aircraft flight conflict matrix involved in a multi-aircraft flight conflict scenario is defined as:

(5) \begin{align}{\bf{P}} = \left[\begin{array}{c@{\quad}c@{\quad}c}{Co{n_{{\textrm{11}}}}} & {}\cdots {}& {Co{n_{{\textrm{1n}}}}} \\ \vdots & {}\ddots {}& \vdots \\ {Co{n_{{\textrm{n1}}}}} & {}\cdots & {}{Co{n_{{\textrm{nn}}}}} \end{array}\right] \end{align}

where $Co{n_{{\textrm{ij}}}}$ is the degree of urgency between conflict aircraft $i$ and conflict aircraft $j$ :

(6) \begin{align}Co{n_{{\textrm{ij}}}} = \left\{\begin{array}{l} {1/{\left( {{t_{{\textrm{lose}}}} - T} \right),\;conflict\;between\;aircraft\;i\;and\;aircraft\;j}} \\[5pt] 0,\,no\;conflict\;between\;aircraft\;i\;and\;aircraft\;j\;or\;i\;=\;j \end{array} \right.\end{align}

where ${t_{{\textrm{lose}}}}$ is the moment of loss of safe separation between aircraft $i$ and aircraft $j$ , $T$ is the moment to start conflict detection.

Table 2. Priority-based ranking mechanism for multi-aircraft flight conflict resolution

Define the urgency of the $k{\textrm{th}}$ conflicting aircraft as:

(7) \begin{align}{U_{\textrm{k}}} = \sum\limits_{j = 1}^n {Co{n_{{\textrm{kj}}}}}, k = 1 \ldots n\end{align}

After calculating the urgency of each conflicting aircraft, a set of ranked sequences is obtained by arranging them in descending order. Define three sequences based on this: 1. Sorting Sequence ${{\textrm{S}}_1}$ : Storaging conflicting aircraft that have not yet executed a pre-command for conflict resolution; 2. Execution Instruction Sequence ${{\textrm{S}}_2}$ : Storaging conflicting aircraft that have executed a conflict resolution pre-command; 3. Whether to Execute the Command Sequence ${{\textrm{S}}_{{\textrm{EXE}}}}$ : False and True are filled into the queue depending on whether or not the conflict resolution pre-command is executed. The specific flow of conflict resolution according to the order of conflict resolution is shown in Table 2.

Figure 6 gives a specific example. According to the initial sequence ${{\textrm{S}}_1}$ , the conflicting aircraft A should execute the command in priority, and the conflict resolution model is used to generate a pre-command for A. However, after the trajectory prediction module predicts that the pre-command of aircraft A leads to a conflict with the environmental aircraft and does not resolve all of two-aircraft conflict involving aircraft A, the pre-command of aircraft A is withdrawn, and the order of A is moved to the end of the sequence. Currently, the conflicting aircraft D is scheduled to generate pre-command first. The trajectory prediction module determines that aircraft D’s pre-command resolves all conflicts it is involved in and does not create new conflicts, then aircraft D executes the pre-command, moves aircraft D out of the ${{\textrm{S}}_1}$ and into the ${{\textrm{S}}_2}$ , and updates the urgency matrix, recalculates the urgency for each of the conflicting aircraft, and sorts them. The foremost conflicting aircraft A in ${{\textrm{S}}_1}$ is selected to generate the resolution pre-command. However, aircraft A’s pre-command only resolves the conflict between A-C and not between A-B, so it is still necessary to withdraw aircraft A’s pre-command and place aircraft A at the end of ${{\textrm{S}}_1}$ . Next, the conflict aircraft B at the top of ${{\textrm{S}}_1}$ is selected, and the instructions generated using the conflict resolution model can resolve the conflict between A-B and no new conflict occurs, moving B out of the ${{\textrm{S}}_1}$ sequence and into the ${{\textrm{S}}_2}$ sequence and updating the ${{\textrm{S}}_1}$ ordering. At this time, after selecting the conflict aircraft A again to generate and execute the command, the conflicts in the airspace are all resolved, satisfying CONDITION α, and this multi-aircraft flight conflict scenario is resolved successfully. This design of the conflict resolution sequence is dynamically changing with the resolution process, which aligns with the actual situation of ATCOs’ resolution in real life. At the same time, this approach has the flexibility to adjust the resolution order with the airspace situation automatically and can successfully train the agent’s resolution ability in TCS.

Figure 6. Schematic diagram of the conflict resolution process for a multi-aircraft flight.

5.0 Flight conflict resolution model based on markov decision-making process

The flight conflict resolution process is Markovian and, therefore, modeled based on MDP. MDP needs to define tuples with five elements $\langle S,A,{T_{\textrm{r}}},R,\gamma \rangle $ , where $S$ is the state space, $A$ is the action space, ${T_{\textrm{r}}}$ is the state transition function, $R$ is the reward space, $\gamma $ is the discount factor.

At each time step, the environment is in a specific state $s \in S$ , and the agent performs the action $a \in A$ according to the policy. The environment then moves to the next state and gives agent a rewarding feedback $r(s,a)$ . ${s}'$ follows a state transfer probability function that satisfies the principle of Markovianity $p({s'}|s,a)$ . The ultimate goal of MDP modeling is to find a policy that specifies the agent’s actions at each step and maximises the expectation of cumulative discount rewards $E[\sum\nolimits_{t = 0}^{{T_0}} {{\gamma ^t}r({s^t},{a^t})} ]$ over time. Where ${T_0}$ is the time range of the MDP, $0 \le \gamma \le 1$ is the discount factor, ${s^t}$ and ${a^t}$ are the state and action at moment t. $\pi $ represents strategy, the stochastic strategy can be expressed as $\pi (s):S \to A$ , giving the probability of action in each state.

The optimal strategy for an MDP can be solved using reinforcement learning (RL), which defines two value functions to evaluate strategies with discounted long-term rewards. The first function, known as the state value function, measures the expected discounted rewards of starting from a state and following the strategy $\pi $ after that:

(8) \begin{align}{V^\pi }(s) = {{\textrm{E}}_{{a^t},{s^{t + 1}}, \ldots }}[\sum\limits_{\tau = 0}^{{T_0}} {{\gamma ^\tau }} {r^{t + \tau + 1}}|{s^t} = s,\pi ]\end{align}

The other value function is called the action-value function (Q-function) and is defined as:

(9) \begin{align}{Q^\pi }(s,a) = {{\textrm{E}}_{{s^{t + 1}},{a^{t + 1}}, \ldots }}[\sum\limits_{\tau = 0}^{{T_0}} {{\gamma ^\tau }} {r^{t + \tau + 1}}|{s^t} = s,{a^t} = a,\pi ]\end{align}

It measures the expected discounted reward for taking action $a$ in specific state $s$ and adopting strategy $\pi $ .

The Bellman equation for the Q-function is:

(10) \begin{align}{Q^\pi }({s^t},{a^t}) = \sum\limits_{{s^{t + 1}}} {p({s^{t + 1}}|{s^t},{a^t})} \times \left[ {{r^{t + 1}} + \gamma \sum\limits_{{a^{t + 1}}} {\pi ({a^{t + 1}}|{s^{t + 1}}){Q^\pi }({s^{t + 1}},{a^{t + 1}})} } \right]\end{align}

This paper uses the model-free approach, and the transfer function ${T_{\textrm{r}}}$ is derived using the trajectory prediction module. The discount factor $\gamma $ is fixed, the state space, action space and reward function must be explicitly defined.

(1) Observation space:

The state space reflects the state of the target airspace at a particular moment in time and is a dynamic space. With each time increment, the agent derives its observations by examining the state space, resulting in observations that are either subsets of the state space or processed versions. The evolution of the state space is determined by the trajectory prediction module. The observation space of agent in this paper is an N-dimensional vector, $N$ is defined as:

(11) \begin{align}N = {N_{\textrm{n}}} \times {N_{\textrm{t}}} \times {N_{\textrm{d}}}\end{align}

where ${N_{\textrm{n}}}$ is the number of aircraft, including conflict and environmental aircraft. ${N_{\textrm{t}}}$ is the number of discrete time periods, ${N_{\textrm{d}}}$ is specific information about each aircraft at a given moment in time.

Define the midpoint of a two-aircraft conflict as the two-aircraft conflict point, then the conflict point in a multi-aircraft conflict is calculated as the average of the multiple conflict points derived from the respective two-aircraft conflicts. A 200 km × 200 km × 6000 m rectangular airspace is constructed centred on this value to describe the current observational range of the agent. Define the observation of the agent at the moment $t$ to be ${O_{\textrm{t}}}$ , the state of the corresponding rectangular airspace is ${S_{\textrm{t}}}$ , ${O_{\textrm{t}}}$ can be defined as:

(12) \begin{align}{O_{\textrm{t}}} = \left( {{S_{\textrm{t}}},{S_{{\textrm{t + 180}}}},{S_{{\textrm{t + 360}}}},{S_{{\textrm{t + 540}}}},{S_{{\textrm{t + 720}}}}} \right)\end{align}

where ${O_t}$ included the state of the rectangular airspace from moment $t$ to moment $t + 720$ in time intervals of 180s, corresponding to above ${N_{\textrm{t}}}$ . Where ${S_{\textrm{t}}}$ includes the information of 30 aircraft in the airspace at time $t$ . If there are less than 30 aircraft in the airspace, the remaining element is set to 0, and the number of aircraft corresponds to the size of ${N_{\textrm{n}}}$ above:

(13) \begin{align}{S_{\textrm{T}}} = \left( {Info_1^T,Info_2^T, \cdots, Info_{30}^T} \right),{\textrm{ }}T = t,t + 180,t + 360,t + 540,t + 720\end{align}

where $Info_{\textrm{i}}^T$ is the specific flight information of the $i$ th aircraft in the delimited airspace at time $T$ , containing:

(14) \begin{align}Info_{\textrm{i}}^T = \left( {x_{\textrm{i}}^T,y_{\textrm{i}}^T,z_{\textrm{i}}^T,vspd_{\textrm{i}}^T,hspd_{\textrm{i}}^T,h_{\textrm{i}}^T,type_{\textrm{i}}^T,ur_{\textrm{i}}^T} \right)\end{align}

where the number of elements corresponds to ${N_{\textrm{d}}}$ . The meanings are, respectively, longitude, latitude, altitude, rate of climb, horizontal speed, heading, type of aircraft (denoted by serial number), and the length of the long axis of the confidence ellipse of the $i$ th aircraft at the time $T$ . Here $i$ is obtained by ordering the Euclidean distances to the aircraft executing the conflict resolution command, setting the aircraft executing the conflict resolution command as the first aircraft, the closest aircraft to that aircraft as the second aircraft, and so on. Furthermore, the data within the observation space were normalised.

(2) Action space:

The action space is the set of instructions that can be taken by a conflicting aircraft to execute a conflict resolution command, and the action space can be defined as:

(15) \begin{align}A = \{ cmd,{T_{\textrm{w}}}|{T_{\textrm{w}}} = 20x \cdot {\textrm{seconds, }}x \in \left[ {0,{T_a}} \right]\} \end{align}

where $cmd$ represents a specific command executed by the aircraft, containing altitude adjustment, speed adjustment and heading adjustment. This paper addresses discrete actions. ${T_{\textrm{w}}}$ represents the waiting time for the execution of the action. In practice, the executed instructions are usually not executed immediately but at the right time, so the delay in executing the conflict resolution action should be considered and discretised in units of 20s. ${T_{\textrm{a}}}$ is an upper limit on the execution time of an instruction. We specify that the instruction must be executed before a conflict occurs. In addition, the heading conflict resolution manoeuver considered in this paper is dog-leg, the schematic diagram of which is shown in Fig. 7. The angle of departure $\alpha $ is used as a variable and takes the value of 30 or 60 degrees, the angle of return $\beta $ is set to 30 degrees, and the side of departure is specified as 8 nautical miles. The $cmd$ contains a collection of actions as shown in Table 3.

Figure 7. Schematic diagram of the dog-leg manoeuver.

(3) Reward function:

The agent’s goal is to maximise long-term rewards and update the parameters of the neural network based on instantaneous rewards. The setting of the reward function directly affects conflict resolution. To effectively address the multi-aircraft conflict scenario outlined in this study, three essential conditions must be met: (i) the agent’s actions must adhere to air traffic control regulations, ensuring the avoidance of unreasonable actions; (ii) the manoeuvers performed by the agent must not create new conflicts and, simultaneously, not conflict with the environmental aircraft; and (iii) all conflicting aircraft work to resolve conflicts related to themselves first so that the entire multi-aircraft scenario endeavors to satisfy CONDITIONα. We set the reward function as follows:

(16) \begin{align}R = {R_1} + {R_2} + {R_3}\end{align}

where ${R_1}$ indicates whether the conflict resolution manoeuver performed by the aircraft complies with the ATC regulations and a positive reward is given if it does. Otherwise, a negative reward is given. The ATC regulations here mainly mean that the relevant manoeuvers to be performed by the aircraft are under the aircraft’s performance limits. In case of a manoeuver that does not comply with the ATC regulations, the manoeuver will not be performed while giving a negative reward. ${R_1}$ specifically expressed as:

(17) \begin{align}{R_1} = \left\{ \begin{array}{l} + \phi, \,if{ \textrm{action is valid }} \\[5pt] - \phi, \,else\end{array} \right.\end{align}

${R_2}$ aims at the conflicting aircraft doing the manoeuver and denotes the number of two-aircraft conflicts involving that aircraft remaining after that aircraft has done the manoeuver, which is $conflict\_num$ . Each two-aircraft conflict gives a negative feedback value of $\lambda $ .

(18) \begin{align}{R_2} = conflict\_num \times \lambda \end{align}

The ${R_3}$ directly reflect whether the conflicting aircraft doing the action resolves all associated conflicts., defined as:

(19) \begin{align}{R_3} = \left\{\begin{array}{l} + \varphi, \, if{\textrm{ solved successfully}}\\[5pt] - \varphi, \,if\,{\textrm{ solved failed}}\end{array} \right.\end{align}

(4)Termination state:

In this study, we define the conflict resolution process within a multi-aircraft scenario as a ‘round’. Two termination states, or end-of-round conditions, are identified: the round ends after all conflicting aircraft in the training phase have followed their resolution instructions in the order of resolution, at which time resolution is judged successful if CONDITIONα is satisfied; otherwise, resolution fails; before the last conflicting aircraft executes the instruction, the multi-aircraft conflict environment has already satisfied CONDITIONα, and then it is judged that the resolution is successful.

Table 3. Table of commands

6.0 Acktr-based algorithm

In general, the application of deep reinforcement learning to solve MDP models exhibits favourable outcomes in terms of both performance and computational efficiency. While many deep reinforcement learning techniques rely on a simplified version of stochastic gradient descent (SGD) to train neural networks, SGD may prove inefficient in exploring the weight space, often necessitating several days for convergence in control tasks. In this investigation, we employ the actor-critic using the ACKTR algorithm [Reference Wu, Mansimov, Liao, Grosse and Ba30], which leverages natural gradient descent principles. By utilising the Kronecker factor approximation of the natural policy gradient, ACKTR enables the effective inversion of the gradient’s covariance matrix, enhancing computational efficiency.

The ACKTR algorithm is rooted in the actor-critic paradigm. Conventionally, the actor network is responsible for training the actor neural network, while the critic network focuses on training the critic neural network. The output of the actor network is characterised by a distribution, facilitating the straightforward definition of its Fisher information matrix. However, in contrast, the output of a standard critic network is a scalar rather than a distribution, rendering the definition of its Fisher information matrix unfeasible.

The solution given by the ACKTR algorithm is that the actor network shares with the critic network a fully connected network with four hidden layers, as shown in Fig. 8 where the output of the critic network is defined to be a Gaussian distribution $p\left( {v\left| {{S_t}} \right.} \right)\sim \mathcal{N}\left( {v;V\left( {{S_t};{\bf{w}}} \right),{\sigma ^2}} \right)$ , which in practice $\sigma $ can be simply set to 1. Finally, the respective fully connected layers are connected to obtain the policy and value, respectively, in order to update the model using the method of natural gradient descent, ACKTR defines the output of the overall network as $p\left( {a,v\left| s \right.} \right) = \pi \left( {a\left| s \right.} \right)p\left( {v\left| s \right.} \right)$ by assuming independence of the policy and the value distribution. The proposed neural network structure can be viewed as a whole and the overall network is updated using a loss function that synchronises the update of the critic and actor.

Figure 8. Neural network architecture for agent training.

The ACKTR algorithm uses a multi-threaded approach to train the agents. In each thread, the agent interacts with its environment, and the central model performs data aggregation and parameter updates. Let there be a total of $l$ threads, each thread interacts $k$ steps in each multi-threaded interaction, and collects a collection of empirical data, and $\left| N \right| = k \times l$ .

The update formula for the overall network parameter $\theta $ is:

(20) \begin{align}\theta \leftarrow \theta - \eta {{\bf{F}}^{ - 1}}{\nabla _\theta }{\mathcal{J}}\left( \theta \right)\end{align}

where ${\bf{F}} = {{\mathbb E}_{p\left( \tau \right)}}\left[ {\nabla \ln p\left( {a,v\left| s \right.} \right)\nabla \ln p{{\left( {a,v\left| s \right.} \right)}^{\textrm{T}}}} \right]$ , $p(\tau )$ is the distribution of trajectories, given by $ {\prod\limits_{t = 0}^T {\pi \left( {{A_t}\left| {{S_t}} \right.} \right)p\left( {{S_{t + 1}}\left| {{S_t},{A_t}} \right.} \right)} } $ , ${\mathcal{J}}$ is the function to be optimised.

The loss function is defined as shown in the equations (21)–(24):

(21) \begin{align}pgLoss = {1 \over {\left| N \right|}}\sum\limits_{n \in N} {\left( {{R_{t + 1}} + \gamma V\left( {{S_{t + 1}};\theta } \right) - V\left( {{S_t}} \right)} \right) \times \left( { - \ln \pi ({A_t}\left| {{S_t}} \right.;{\bf{\theta }})} \right)} \end{align}

(22) \begin{align}vfLoss = {1 \over 2}\sum\limits_{n \in N} {{{\left( {V({S_t};\theta ) - {R_{t + 1}}} \right)}^2}} \end{align}

(23) \begin{align}entropy = \sum\limits_{a \in {\mathcal{A}}}^{} {\pi \left( {a\left| {{S_t}} \right.;{\bf{\theta }}} \right)} \ln \pi \left( {a\left| {{S_t}} \right.;{\bf{\theta }}} \right)\end{align}

(24) \begin{align}{\mathcal{L}} = pgLoss - e\_coef \times entropy + v\_coef \times vfLoss\end{align}

where $e\_coef$ and $v\_coef$ are constant coefficients.

Table 4 shows the specific flow of the ACKTR algorithm combined with the conflict resolution environment. It is worth noting that for the ACKTR algorithm using multiple threads, many conflicting training samples should be generated for the agent to learn. Where ${\eta _k}$ for the update parameter inside the table is set to $\min ({\eta _{\max }},\sqrt {{{2\delta } \over {\Delta \theta _k^{^{\textrm{T}}}{{{\hat{\bf F}}}_k}\Delta {\theta _k}}}} )$ [Reference Ba, Grosse and Martens31] ${\eta _{\max }}$ is the learning rate and $\delta $ is the trust region, each piece in chunked diagonal matrix ${{\hat{\bf F}}_k}$ corresponds to the Fisher information matrix for each layer of the overall network: ${\hat{\bf F}}_k^i = {X_i} \otimes {Z_i}_{}$ . Let the input to its $i{\textrm{th}}$ layer $\left( {0 \le i \le m} \right)$ be ${x_i}$ , and the output before activation be ${z_i}$ .

Table 4. ACKTR algorithm flow for training the TCS agent

7.0 Simulation experiments

1. Airspace scenario setting:

As shown in Fig. 9, the airspace around HFE, the busiest route point in China in 2018, is selected as the simulation scenario in this paper. The source of airspace, waypoint coordinates, route structure, airport coordinates and other data is the 2018 Aeronautical Information Publication (AIP). A $400\,{\textrm{km}} \times 400\,{\textrm{km}} \times 6\,{\textrm{km}}$ rectangular airspace (i.e. airspace A) is generated in the scenario as a specific airspace for traffic flow simulation; a $200\,{\textrm{km}} \times 200\,{\textrm{km}} \times {\textrm{6}}\,{\textrm{km}}$ rectangular airspace (i.e. airspace B) is delineated therein, and multi-aircraft conflicts occurring in this airspace are recorded, thus generating multi-aircraft conflict scenario samples. Traffic flow data was generated from flight plans flying over the airspace on 1 June 2018, derived from the 2018 National Aeronautical Information Publication (NAIP).

2. Experimental environment:

The Air Traffic Operation Simulation System (ATOSS) serves as the designated deep reinforcement learning environment for training agents in this study. This environment facilitates the generation of multi-aircraft flight conflict scenarios, trajectory prediction, conflict detection, and state space transfer. ATOSS is independently developed by the New Generation Intelligent ATC Laboratory of Nanjing University of Aeronautics and Astronautics (NUAA). It integrates with the airspace database and incorporates a simulation motion engine based on the base of aircraft data (BADA) [32] to simulate the aircraft’s operation posture. BADA includes the performance data sheets of various aircraft types, enabling the evaluation of whether resolution manoeuvers align with flight performance criteria.

The hardware environment for the experiment is : HP Z8-G4 workstation (configured with one Intel Xeon(R) Gold 6242 CPU and 64GB RAM). Software environment: IntelliJ Pycharm as IDE, Python language, deep learning framework-Tensorflow 1.14.0.

3. Experimental process

Randomly load the different flight plans passing through the specified airspace in the flight plan database. For a multi-flight conflict in conflict airspace B, determine whether it meets the definition of spatio-temporal correlation described in the previous section. Upon meeting the criteria for spatio-temporal correlation, a conflict scenario sample is generated. In this study, we generate 7,000 conflict scenario samples, select the first 6,000 samples to form the training set, and the remaining 1,000 samples to form the test set.

Figure 9. Airspace structure.

The ACKTR algorithm is used to train the conflict resolution model for the training set samples. Here, the agent’s ability to resolve conflicts is trained through a well-designed reward function, where a negative reward is given if the aircraft does not resolve all the conflicts associated with it after taking the action solved by the algorithm, and a positive reward is given if it successfully resolves conflicts.

The trained conflict resolution model is validated on a test set, and the resolution is performed by a priority-based ranking mechanism combined with the conflict resolution model. The effect of the number of aircraft in the airspace on the success rate is investigated on the basis of a given airspace environment. During testing, conflict resolution metrics such as resolution time and distribution of resolution actions are meticulously recorded and analysed.

4. Experimental parameters:

The algorithm of deep reinforcement learning contains many parameters, including the hyperparameters in the algorithm, the parameters in the action space and the reward function. The setting of the parameters will affect the training effect of the agent to a certain extent. The parameters of the ACKTR algorithm selected after repeated experiments in this paper are shown in Table 5. Where the number of threads is the number of threads trained simultaneously by ACKTR. The specific values of the other parameters are shown in Table 6, and their values were determined based on several comparative experimental analyses.

Table 5. Hyper-parameter value of ACKTR

Table 6. Other-parameter value

5. Results Analysis:

Randomly loading 20–30 aircraft, including conflict aircraft as well as environmental aircraft, the training environment was constructed in such a way as to train the agent’s ability to resolve conflicts under challenging scenarios. These aircraft are flown along actual flight paths in the ATOSS simulation system, an approach that trains the conflict resolution competence of aircraft from simulated situations.

After 6,000,000 steps, the loss value and reward value changes during training are shown in Fig. 10. The loss value decreases rapidly in the first 2,000,000 steps and converges gradually. The reward value gradually increases and finally converges around 9. This trend signifies a satisfactory training outcome for the model.

Figure 10. Loss and reward.

As shown in Fig. 11, after less than 150 h of training, the ACTKR algorithm can achieve a 94% success rate with low variance and high stability on the training set. The DQN algorithm, on the other hand, is difficult to deal with such more complex task environments, and its success rate is more volatile. Notably, the success rate pertains to the proportion of successfully resolved multi-aircraft conflict scenarios relative to the total number of training samples.

Various factors influence the success rate of aircraft in conflict resolution. One significant determinant is the environment within which the aircraft operate. The most direct influence is the number of aircraft in the airspace, which is a dynamic indicator of the airspace. A higher aircraft density leads to a more constrained solution space for the agent, thus intensifying the complexity of conflict resolution tasks. In addition, static metrics of the airspace can affect the difficulty of conflict resolution, including the number of routes in the airspace, the number of waypoints and the structural complexity of the airspace. This study analyses how the number of aircraft in the airspace affects the success rate based on fixed airspace. The effect of weather factors on success rate is not considered for the time being.

The number of aircraft in the airspace can be adjusted manually, and the variation of success rate with the number of aircraft in the airspace is investigated for certain airspace static metrics and initial horizontal speeds. The model obtained from the training phase was tested on 1,000 test set samples. The two-aircraft success rate is defined as the number of successfully resolved two-aircraft conflicts as a proportion of the total number of two-aircraft conflicts, and the multi-aircraft flight success rate is defined as before, i.e. as the proportion of successfully resolved multi-aircraft samples to the total number of samples. To study the relationship between the number of aircraft in the airspace and the success rate, test results were obtained as shown in Fig. 12, where 20–30 aircraft in the airspace were the same as the number of aircraft in training, and the other scenarios were when the number of environmental aircraft was reduced accordingly. The results show that the obtained conflict resolution model can effectively solve multi-aircraft flight conflicts and is scalable. Reducing the number of aircraft in the airspace will be conducive to successfully resolving multi-aircraft flight conflicts.

Figure 11. Success rate of DQN and ACKTR on the same training set.

Figure 12. Success rates for different numbers of aircraft in the airspace.

In Section 4.3 of this study, a prioritisation mechanism is delineated, and the effectiveness of the mechanism needs to be proved by further comparative experiments. The efficacy of this proposed mechanism is assessed by integrating it with two different ranking approaches: a priority-based dynamic ranking mechanism and a random ranking mechanism. The multi- and two-aircraft success rates on 1,000 test sets are shown in Fig. 13. The results prove that after coordinating conflicting aircraft through the dynamic ranking mechanism, the success rate is significantly improved compared with the random order, and the resolution effect is satisfactory.

Figure 13. Success rate of dynamic order and random order on the same training set.

Corresponding to the number of aircraft in the airspace, the statistical distribution of the initial horizontal mean speed of aircraft in the airspace is shown in Fig. 14. The average initial horizontal speed of aircraft in the airspace also belongs to the dynamic indicators of the airspace. The aircraft’s operability is highly correlated with the horizontal speed, and the larger the average speed, the more difficult it will be to resolve the conflict samples. So, the magnitude of the aircraft’s horizontal speed must also be considered when calculating the success rate.

Figure 14. Mean horizontal speed distribution of aircraft in airspace.

Under the condition that the number of aircraft in the airspace is 20-30, the statistics of the conflicting aircraft selecting the conflict resolution command are shown in Fig. 15, and the TCS agents are more inclined to select the altitude adjustment manoeuver after 6,000,000 steps of training. In 3D space, altitude adjustment is a timely and effective conflict resolution command. Agents think that speed and heading adjustments are less effective than altitude adjustments, so they have fewer choices for speed and heading adjustments.

Figure 15. Distribution of conflict resolution actions.

The results obtained after counting the number of conflict resolution instructions for a sample of 1,000 test sets are shown in Table 7. Over half of the multi-aircraft conflicts can be resolved with only one or two commands, which is the favoured resolution method for ATCOs and pilots. For more complex samples of multi-aircraft conflicts, more than two aircraft is required to execute commands for resolution.

After counting the test set samples, the vast majority of the computation time for TCS to give resolution suggestions is within 2 s, which aligns with the time requirement for conflict resolution. While traditional methods, such as genetic algorithms, to solve the multi-aircraft conflict problem may require tens of seconds of solution time, which could be less conducive to timely conflict resolution. Overall, the model is fast to solve on the test set, the solution is reasonably effective, and it is consistent with the actual air traffic control situation.

Table 7. Minimum resolution action distribution