2.1 Assumptions

Figure 1 illustrates the pursuit-evasion dynamics of the TAD engagement. The attacker (A) aims to reach the target (T) while avoiding interception by the defender (D). Both A and D are constrained by their maximum lateral accelerations, which dictate their minimum turning radii. These radii define the tightest possible Dubins paths they can follow.

Figure 1.

Engagement geometry.

In this Cartesian inertial coordinate system, the entities involved are denoted as follows: $T\left( {{x_T},{y_T}} \right)$ for the target, $\;A\left( {{x_A},{y_A}} \right)$ for the attacker, and $D\left( {{x_D},{y_D}} \right)$ for the defender, each with their corresponding velocities ( ${v_T}$ , ${v_A}$ , ${v_D}$ ), maximum accelerations ( ${a_T}$ , ${a_A}$ , ${a_D}$ ), and the heading angles ( ${\gamma _T}$ , ${\gamma _A}$ , ${\gamma _D}$ ).

Within a time interval Δt, the reachable sets of A and D are bounded by their right and left reachable set frontier (RSF). These frontier represent the farthest points each agent can reach while executing maximum-effort right (RS) or left (LS) turns. The intersection of these RSFs forms the equal-time line (ETL), which marks collision points where A and D arrive simultaneously. The virtual target (M), defined as the closest point on the ETL to T, determines the Nash equilibrium interception strategy.

To simplify the theoretical analysis, the following assumptions are made:

1. (1) Target, attacker and defender are assumed to be mass points.

2. (2) Gravity and air resistance’s impacts are disregarded.

3. (3) The speed of target T is significantly slower than that of the attackers and defenders.

4. (4) Given the limited fuel capacity of the attacker, it is impractical to consider multiple engagement rounds in aerial combat scenarios. Thus, the heading angle of the attacker can be constrained within practical bounds based on mission-specific requirements.

5. (5) Each individual’s speed remains constant.

6. (6) The missile autopilot control system is theoretically flawless, excluding any time delays.

7. (7) The attacker or defender can accurately acquire critical state variables (including their own) from all participating agents in the game-theoretic interaction, such as real-time coordinates, maximum acceleration and heading angles.

8. (8) The damage caused to the target being dependent solely on the terminal miss distance of the attacker.

2.2 Equations of motion

According to Hypothesis 3, the velocity and acceleration of target T can be disregarded when juxtaposed with the high-speed flying attacker and defender. Consequently, our attention will be directed exclusively towards the motion and constraints of the defender (D) and the attacker (A), both moving at a constant speed.

The equations governing the motion and constraints for the defender (D) can be articulated as follows:

(1) \begin{align}\left\{ {\begin{array}{*{20}{c}}{{{\dot x}_D} = {v_D}\cos {\gamma _D}}\\{{{\dot y}_D} = {v_D}\sin {\gamma _D}}\\{{{\dot \gamma }_D} = {a_D}/{v_D}}\end{array}} \right.\end{align}

(2) \begin{align}\left| {{a_D}} \right| \le a_D^{\max }\end{align}

Similarly, the motion equations and constraints of attacker A are expressed as:

(3) \begin{align}\left\{ {\begin{array}{*{20}{c}}{{{\dot x}_A} = {v_A}\cos {\gamma _A}}\\{{{\dot y}_A} = {v_A}\sin {\gamma _A}}\\{{{\dot \gamma }_A} = {a_A}/{v_A}}\end{array}} \right.\end{align}

(4) \begin{align}\left| {{a_A}} \right| \le a_A^{\max }\end{align}

In this context, ${v_D}$ and ${v_A}$ represent the flight velocity, while ${\rm{\;}}{\gamma _D}$ and ${\gamma _A}$ denote the heading angles. Additionally, ${a_D}$ and ${a_A}$ refer to the lateral acceleration of the defender and attacker, with their maximum values constrained by $a_D^{\max }$ and $a_A^{\max }$ . The minimum turning radius can be determined as follows:

(5) \begin{align}{r_D} = v_D^2/a_D^{\max }\end{align}

(6) \begin{align}{r_A} = v_A^2/a_A^{\max }\end{align}

When consistently applying the maximum reverse accelerations $a_D^{\max }$ and $a_A^{\max }$ to the defender and attacker, tangential circles with radii ${r_D}$ and ${r_A}$ are formed, denoted as circles ${D_L}$ , ${D_R}$ , ${A_L}$ and ${A_R}$ . These tangential circles are illustrated in Fig. 1.

2.3 Reachable set frontier (RSF) and equal-time line (ETL)

In accordance with Dubins path theory, the reachable set frontier for both the defender and attacker within the time interval [t, t+ $\Delta$ t] can be delineated through an infinite set of path endpoints belonging to {RS, LS}. Stated differently, the curve that minimises the time to reach a specific point must necessarily fall withsin the set {RS, LS}. The axially symmetric nature of the reachable set frontier under RS and LS paths, centred on the initial point with the velocity vector as the axis of symmetry, leads to the subdivision of the reachable set frontier into two components: the right reachable set frontier $RS{F_R}$ and the left reachable set frontier $RS{F_L}$ , which are illustrated in Fig. 1.

Assuming the initial coordinates of defender is $\left( {{x_{D0}},{y_{D0}}} \right)$ and the heading angle is ${\gamma _D}$ , then any point $RS{F_R}\left( {{x_{DR}},{y_{DR}}} \right)$ on the reachable set frontier curve in the right half-plane satisfies:

(7) \begin{align}\begin{array}{c}{{x_{DR}} = {{\tilde x}_{DR}}\sin \left( {{\gamma _D}} \right) - {{\tilde y}_{DR}}\cos \left( {{\gamma _D}} \right) + {x_{D0}}}\\{{y_{DR}} = {{\tilde x}_{DR}}\cos \left( {{\gamma _D}} \right) + {{\tilde y}_{DR}}\sin \left( {{\gamma _D}} \right) + {y_{D0}}}\end{array}\end{align}

where,

(8) \begin{align}\begin{array}{l}{{\tilde x}_{DR}} = \left( {{v_D}\Delta t - {r_D}{\vartheta _D}} \right) \times \sin {\vartheta _D} + {r_D} - {r_D}\cos {\vartheta _D}\\{{\tilde y}_{DR}} = \left( {{v_D}\Delta t - {r_D}{\vartheta _D}} \right) \times \cos {\vartheta _D} + {r_D}\sin {\vartheta _D}\end{array}\end{align}

Any point $RS{F_L}\left( {{x_{DL}},{y_{DL}}} \right)$ on the reachable set frontier curve in the left half-plane satisfies:

(9) \begin{align}\begin{array}{c}{{x_{DL}} = {{\tilde x}_{DL}}\sin \left( {{\gamma _D}} \right) - {{\tilde y}_{DL}}\cos \left( {{\gamma _D}} \right) + {x_{D0}}}\\{{y_{DL}} = {{\tilde x}_{DL}}\cos \left( {{\gamma _D}} \right) + {{\tilde y}_{DL}}\sin \left( {{\gamma _D}} \right) + {y_{D0}}}\end{array}\end{align}

where

(10) \begin{align}\begin{array}{l}{{\tilde x}_{DL}} = \left( {{v_D}\Delta t + {r_D}{\vartheta _D}} \right) \times \sin {\vartheta _D} - {r_D} + {r_D}\cos {\vartheta _D}\\{{\tilde y}_{DL}} = \left( {{v_D}\Delta t + {r_D}{\vartheta _D}} \right) \times \cos {\vartheta _D} - {r_D}\sin {\vartheta _D}\end{array}\end{align}

where ${r_D} = {v_D}^2/a_D^{\max }$ , is the radius of the trajectory circle. The change in heading angle ${\vartheta _D}$ represents the angular change from time t to t + Δt, defined as ${\vartheta _D}{\rm{ = \;}}{\gamma _D}(t + \Delta t) - {\gamma _D}(t)$ . This parameter is constrained within the range [ $ - \vartheta _D^{\max }$ , $\vartheta _D^{\max }$ ], with positive values denoting right turns and negative values indicating left turns. Moreover, $\vartheta _D^{\max } = \min \left[ {\vartheta _D^*,\vartheta _D^{**}} \right]$ depending on the flight duration $\Delta$ t. Specifically:

When $\Delta t \le 2\pi {r_D}/{v_D}$ , $\vartheta _D^{\max } = \vartheta _D^* = {v_D}\Delta t/{r_D}$ .

When $\Delta t \gt 2\pi {r_D}/{v_D}$ , $\vartheta _D^{\max } = \vartheta _D^{**}|{v_D}\Delta t - {r_D}(\vartheta _D^{**}/2) = - \tan (\vartheta _D^{**}/2)$ .

where $\vartheta _D^{**}$ representing the angle reached when the right and left asymptotes are closed to intersect.

Similarly, for attacker, any point $RS{F_A}\left( {{x_{AR}},{y_{AR}}} \right)$ on the reachable set frontier curve in the right half-plane satisfies:

(11) \begin{align}\begin{array}{*{20}{c}}{{x_{AR}} = {{\tilde x}_{AR}}\sin \left( {{\gamma _A}} \right) - {{\tilde y}_{AR}}\cos \left( {{\gamma _A}} \right) + {x_{A0}}}\\{{y_{AR}} = {{\tilde x}_{AR}}\cos \left( {{\gamma _A}} \right) + {{\tilde y}_{AR}}\sin \left( {{\gamma _A}} \right) + {y_{A0}}}\end{array}\end{align}

where

(12) \begin{align}\begin{array}{l}{{\tilde x}_{AR}} = \left( {{v_A}\Delta t - {r_A}{\vartheta _A}} \right) \times \sin {\vartheta _A} + {r_A} - {r_D}\cos {\vartheta _A}\\{{\tilde y}_{AR}} = \left( {{v_A}\Delta t - {r_A}{\vartheta _A}} \right) \times \cos {\vartheta _A} + {r_A}\sin {\vartheta _A}\end{array}\end{align}

Any point $RS{F_L}\left( {{x_{AL}},{y_{AL}}} \right)$ on the reachable set frontier curve in the left half-plane satisfies:

(13) \begin{align}\begin{array}{*{20}{c}}{{x_{AL}} = {{\tilde x}_{AL}}\sin \left( {{\gamma _A}} \right) - {{\tilde y}_{AL}}\cos \left( {{\gamma _A}} \right) + {x_{A0}}}\\{{y_{AL}} = {{\tilde x}_{AL}}\cos \left( {{\gamma _A}} \right) + {{\tilde y}_{AL}}\sin \left( {{\gamma _A}} \right) + {y_{A0}}}\end{array}\end{align}

where

(14) \begin{align}\begin{array}{l}{{\tilde x}_{AL}} = \left( {{v_A}\Delta t + {r_A}{\vartheta _A}} \right) \times \sin {\vartheta _A} - {r_A} + {r_A}\cos {\vartheta _A}\\{{\tilde y}_{AL}} = \left( {{v_A}\Delta t + {r_A}{\vartheta _A}} \right) \times \cos {\vartheta _A} - {r_A}\sin {\vartheta _A}\end{array}\end{align}

The ETL is defined as the locus of points where the defender and attacker simultaneously arrive when following relaxed Dubins paths, as illustrated in Fig. 1. The ETL between the defender and the attacker can be determined when the necessary conditions (the initial coordinates, initial velocities and maximum lateral accelerations of both attacker and defender) are known; however, it cannot be explicitly quantified in relation to time or changes in heading angles.

The points $\left( {{x_{ETL}},{y_{ETL}}} \right)$ on the ETL are formed by the initial intersection of the RSF between the defender and attacker in the temporal dimension. According to formulas (7), (9), (11) and (13), there are four possible cases of ETL, namely, intersection of defender ’s $RS{F_R}$ and attacker ’s $RS{F_R}$ ; intersection of defender ’s $RS{F_R}$ and attacker ’s $RS{F_L}$ ; intersection of defender ’s $RS{F_L}$ and attacker ’s $RS{F_L}$ ; intersection of defender ’s $RS{F_L}$ and attacker ’s $RS{F_R}$ . The following formula can be used to represent it.

(15) \begin{align}\left( {{x_{ETL}},{y_{ETL}}} \right) = f\left( {{v_D},{v_A},a_D^{\max },a_A^{\max },{\gamma _D},{\gamma _A},{x_{D0}},{y_{D0}},{x_{A0}},{y_{A0}}} \right)\end{align}

2.4 Target-defender-attacker differential game problem

In this section, we first derive the system’s terminal states through Dubins path analysis of differential games under optimal solutions. Using these terminal states as the objective function, the optimal manoeuvering strategies for attackers/interceptors obtained via inverse derivation are demonstrated to form a Nash equilibrium solution for the differential game problem.

The differential game problem is first formulated. In accordance with Assumption (8) and differential game theory, the flight strategies of the defender and attacker depend solely on the attacker’s terminal miss distance from the target. Defenders anticipate that attackers will adopt the most unfavourable strategy for defenders and formulate its own defense strategy based on this assumption, aiming for the most advantageous position. Attackers follow a similar approach. This can be expressed using the following equation:

(16) \begin{align}\left\{ {\begin{array}{*{20}{c}}{S_D^* = \begin{array}{*{20}{c}}{\arg \max }\\{{S_D},S_A^*}\end{array}(d)}\\[2pt]{S_A^* = \begin{array}{*{20}{c}}{\arg \min }\\{S_D^*,{S_A}}\end{array}(d)}\end{array}} \right.\end{align}

In the above equations, ${S_D}$ refers to the motion strategy of defender D. Similarly, ${S_A}$ represents the motion strategy of attacker A, and the variable d indicates the distance between the point of collision (where attacker A and defender D intersect) and the designated target T.

The conventional method for solving differential games involves representing the objective function as a function of control variables through a certain linear transformation. This approach then leads to a feedback form for the control variables by optimising the objective function. The advantages of this method include obtaining analytical solutions and fast computation, but it also has the disadvantage of potential precision loss during the transformation process, leading to inaccurate predictions. We now analyse this problem from a geometric perspective and the terminal state of the system is then predicted.

Firstly, if the attacker can reach the target earlier than the defender, it indicates that the target lies between the attacker and the ETL (e.g. point T’ in Fig. 1). In this case, the attacker’s optimal manoeuvering strategy follows a Dubins path with the target as its endpoint. This also implies that the defender has already failed to intercept the attacker from the initial state, unless the attacker deviates from its time-optimal path. The criterion for determining defender failure is given by:

(17) \begin{align}\Delta {t_{AT}} \le \Delta {t_{DT}}\end{align}

where $\Delta {t_{AT}}$ and $\Delta {t_{DT}}$ denote the minimum-time trajectories of the attacker and defender to the target, respectively.

Since this condition is typically evaluated at the initial stage, the target’s coordinates can be substituted into the left-hand side expressions of Equations (7), (9), (11) and (13). Swarm intelligence algorithms such as Particle Swarm Optimization (PSO), Bird Flocking Algorithm or Artificial Bee Colony (ABC) can be employed to estimate the minimum time. For faster computation, the method presented in Section 3.2 may also be used. As this problem can be regarded as a special case of the subsequent discussion, detailed elaboration is omitted here.

In all other cases, based on the defender’s RSF theory mentioned above, the optimal manoeuvering strategy for the defender must result in a Dubins path targeting a specific point on the RSF (points inside the RSF are suboptimal since shorter paths always exist to reach them). Moreover, since the defender can only intercept the attacker by estimating the attacker’s state information, such as the heading angle, velocity, coordinates and acceleration, and given that the defender solely considers pursuit dynamics, this scenario constitutes a norm-type differential game problem, for which the NDG guidance law is optimal. For the attacker, given the defender’s most disadvantageous manoeuvering strategy, their most favourable manoeuvering strategy is to reach the target vicinity as quickly as possible before being intercepted by the defender, thereby minimising the terminal miss distance. Similarly, such a path is also necessarily a Dubins path.

In summary, the terminal state point predicted by the optimal solution of Equation (16) must lie on the curve formed by the intersection of the two RSFs, referred to as the ETL. Furthermore, since the attacker’s objective is to minimise the terminal distance to the target T, the problem can be reformulated as finding the point on the ETL closest to T (denoted as Point M in Fig. 1), which corresponds to the interception point predicted under the optimal solution of Equation (16).

To better describe the predicted terminal point M, we use the change in heading angles ${\vartheta _D}$ and ${\vartheta _A}$ as control variables. Specifically, based on the current heading angle value, the corresponding equations are selected from (7)/(9) and (11)/(13), forming four combinatorial pairs: (7)&(11), (7)&(13), (9)&(11), and (9)&(13). By coupling the selected equations and algebraically eliminating the temporal variable $\Delta$ t, the relationship between parameters ${\vartheta _D}$ and ${\vartheta _A}$ at the ETL endpoint can be derived as follows:

(18) \begin{align}{\psi _{ETL}}\left( {{\vartheta _D},{\vartheta _A}} \right) = 0\end{align}

where ${\vartheta _D} \in \left( {\Delta {\vartheta _{D\min }},\Delta {\vartheta _{D\max }}} \right)$ , $\;{\vartheta _A} \in \left( {\Delta {\vartheta _{A\min }},\Delta {\vartheta _{A\max }}} \right)$ .The detailed formulation of this equation is provided in Appendix A.

Finally, by solving for the terminal state, the same optimal solution as in Equation (16) is obtained, which corresponds to the Nash equilibrium solution. Based on the above analysis, we have obtained an equivalent constraint optimisation problem model from the perspective of plane geometry, which is equivalent to Equation (16):

(19) \begin{align}&{\rm{minimize\, }} {} J = {\left[ {{x_T} - {x_{ETL}}\left( {{\vartheta _D},{\vartheta _A}} \right)} \right]^2} + {\left[ {{y_T} - {y_{ETL}}\left( {{\vartheta _D},{\vartheta _A}} \right)} \right]^2}\nonumber\\&{\rm{subject\, to\, }} {} {\psi _{ETL}}\left( {{\vartheta _D},{\vartheta _A}} \right) = 0\nonumber\\ &\qquad {} {\vartheta _D} \in \left( {\Delta {\vartheta _{Dmin}},\Delta {\vartheta _{Dmax}}} \right)\nonumber\\ &\qquad {} {\vartheta _A} \in \left( {\Delta {\vartheta _{Amin}},\Delta {\vartheta _{Amax}}} \right)\end{align}

Using the virtual target as the NDG tracking objective enhances conceptual clarity while enabling existing guidance design methodologies.

Due to the complexity of this model as a non-convex optimisation problem, it is challenging to transform it into a convex optimisation model or use pattern search methods for optimisation.

3.1 The datasets used for training and testing in neural network

The variables required to solve this model are the target coordinate ${\rm{\;}}\left( {{x_T},{y_T}} \right)$ , defender coordinate $\left( {{x_D},{y_D}} \right)$ , attacker coordinate $\left( {{x_A},{y_A}} \right)$ , defender velocity ${v_D}$ , attacker velocity ${v_A}$ , maximum defender acceleration $a_D^{\max }$ , maximum attacker acceleration $a_A^{\max }$ , defender flight path angle ${\gamma _D}$ and attacker flight path angle ${\gamma _A}$ . The outputs are the Nash equilibrium solution $\;\left( {\vartheta _D^*,\vartheta _A^*} \right)$ , the virtual targets coordinate ${\rm{\;}}\left( {{x_v},{y_v}} \right)$ and terminal miss distance d obtained from the Nash equilibrium.

1. (1) The random enumeration approach for generating input data involves the following steps:

   • • Parameter Range Definition: The bounds of each input variable (e.g. coordinates, speeds, heading angles) are predefined based on mission-specific constraints (e.g. ${x_D} \in $ [0, 4000], ${a_A} \in $ [2g, 5g]).

   • • Random Sampling: Within these bounds, a set of input parameters is stochastically generated to cover diverse engagement scenarios.

2. (2) PSO-based optimisation:

The PSO algorithm is relatively mature and exhibits good adaptability when dealing with nonlinear and non-convex optimisation problems.

Each sampled input set is fed into the PSO algorithm, By considering a set of change in heading angle $\left( {{\vartheta _D},{\vartheta _A}} \right)$ as basic particles and using the negated terminal miss distance (d) as the fitness function, it is possible to iteratively obtain the Nash equilibrium solution $\left( {\vartheta _D^*,\vartheta _A^*} \right)$ based on the aforementioned input variables. Subsequently, virtual target ${\rm{\;}}\left( {{x_v},{y_v}} \right)$ and terminal miss distance d can be derived.

1. (3) Dataset construction:

The resulting pairs of inputs (sampled parameters) and outputs (optimised solutions) form the training dataset for the neural network proxy model.

This approach ensures broad coverage of the state space while leveraging PSO to approximate globally optimal solutions.

However, the PSO algorithm may still converge to local optima even after adopting mitigation strategies, such as adjusting initial particle distributions, tuning learning factors and performing multiple iterations with outlier filtering, resulting in inherent biases in the training dataset.

3.2 Establishing a virtual target proxy model based on BP neural network

Neural network based on the error back propagation (BP) algorithm can effectively approximate a nonlinear function with arbitrary precision by storing quantitative information distribution among individual neurons.

According to the hypothesis, the target coordinate $\left( {{x_T},{y_T}} \right)$ , defender speed ${v_D}$ , attacker speed ${v_A}$ , maximum defender acceleration $a_D^{\max }$ and maximum attacker acceleration $a_A^{\max }$ can be considered constant values. Therefore, for a specific performance of the defender and attacker, the neural network’s proxy model does not need to include the aforementioned variables as input variables. Instead, the other six variables that change over time should be included in the input of the neural network proxy model. These variables are defender coordinate $\left( {{x_D},{y_D}} \right)$ , attacker coordinate $\left( {{x_A},{y_A}} \right)$ , defender flight path angle ${\gamma _D}$ and attacker flight path angle ${\gamma _A}$ .

The proposed back propagation neural network architecture consists of two hidden layers, each containing 10 neurons, designed through empirical optimisation during preliminary trials. The input layer accepts six real-time state variables: $X = \left[ {{x_D},{y_D},{x_A},{y_A},{\gamma _D},{\gamma _A}} \right]$ . The output layer generates two parameters: $Y = \left[ {{x_v},{y_v}} \right]$ .

The structure of the BP neural network is shown in Fig. 2.

Figure 2.

Architecture diagram of the BP neural network.

The back propagation neural network employs a gradient descent training algorithm [Reference Rumelhart, Hinton and Williams38] (TRAINGD) to mitigate premature convergence risks associated with rapid descent methods like Levenberg-Marquardt. The network architecture utilises the hyperbolic tangent sigmoid activation function in its two hidden layers to model nonlinear relationships, while the output layer adopts a linear activation function for regression tasks. Training is executed with a learning rate of 0.01, a momentum coefficient of 0.9. The mean squared error (MSE) serves as the primary performance metric.

(20) \begin{align}e = \frac{1}{{3N}}\sum\limits_{i = 1}^N {[\! \mathop {(\! \mathop x\nolimits_v^* \mathop { - x}\nolimits_v \!)}\nolimits^2 + \mathop {(\! \mathop y\nolimits_v^* \mathop { - y}\nolimits_v \!)}\nolimits^2 + \mathop {(\! \mathop d\nolimits^* - d)}\nolimits^2 } ]\end{align}

In the equation, $x_v^*$ , $y_v^*$ , ${d^*}$ represents the value obtained after estimating the trained model, ${x_v}$ , ${y_v}$ , d represents the actual value, and N represents the number of training iterations for a maximum of 100,000 epochs, with early stopping triggered if validation loss plateaus for 100 consecutive iterations.

The training dataset comprises 100,000 samples generated via PSO, covering predefined operational ranges for coordinates ( ${x_T},{y_T},{x_D},{y_D},{x_A},{y_A} \in \left[ {0,4000} \right]$ ) and speeds ( ${v_D},{v_A} \in \left[ {300,{\rm{ 5}}00} \right]$ ), accelerations ( ${a_D},{a_A} \in \left[ {2g,{\rm{ }}5g} \right]$ ), the heading angles ( ${\varphi _D} \in \left[ {{\rm{0}},{\rm{ }}\pi {\rm{/2}}} \right]$ and ${\varphi _A} \in \left[ {\pi ,{\rm{ 3}}\pi {\rm{/2}}} \right]$ ). Input data is normalised to zero mean and unit variance prior to training. The dataset is partitioned into training (80%), and testing (20%) subsets. Post-training, a relaxation factor (λ = 1) is applied to filter anomalous outputs, and a Holt linear trend model is integrated to compensate for sequential prediction outliers, as defined in Equations (21)–(24).

3.3 Eliminating abnormal outputs and using the Holt linear trend model for prediction

After BP neural network training, there were significant deviations between the actual output values and predicted values at certain points. In order to avoid these anomalous data affecting the guidance system, the corresponding predictions should be removed and supplemented. As the defender or attacker’s manoeuver is continuous, their virtual targets would also be continuous. Hence, a slack factor is introduced to limit the validity threshold of the output values and remove any exceptional predictions.

(21) \begin{align}\sqrt {\mathop {(\! \mathop x\nolimits_v^n - \mathop x\nolimits_v^{n - 1} \!)}\nolimits^2 + \mathop {(\! \mathop y\nolimits_v^n - \mathop y\nolimits_v^{n - 1} \!)}\nolimits^2 } \le \lambda (\! \mathop t\nolimits_n - \mathop t\nolimits_{n - 1} \!)\end{align}

The determination of the relaxation factor λ can be considered as a variant of the L2 regularization approach [Reference Hastie, Tibshirani and Friedman39]. We employ a conventional five-fold cross-validation strategy with candidate values λ = {0.001, 0.01, 0.1, 1, 10}. Given the role of λ in eliminating anomalous data post-training, the selection process is integrated with the subsequent Holt linear trend model and evaluated on a time-dependent training dataset involving interactions between an attacker using DP-NNDG and a defender applying the NDP guidance law. Notably, excessive λ values may fail to suppress outliers effectively, while insufficient λ risks removing valid data points.

To supplement the excluded output portion, the Holt linear trend model [Reference Levine, Stephan, Krehbiel and Berenson40] is employed to predict the current output using the preceding outputs with higher credibility:

(22) \begin{align}\mathop {x}\nolimits_n = \mathop l\nolimits_{n - 1} (x) + h\mathop b\nolimits_{n - 1} (x)\end{align}

(23) \begin{align}\mathop l\nolimits_{n - 1} (x) = \alpha \mathop x\nolimits_{n - 1} + (1 - \alpha )(\! \mathop l\nolimits_{n - 2} + \mathop b\nolimits_{n - 2} \!)\end{align}

(24) \begin{align} b_{n-1}(x)=\beta(l_{n-1}(x)-l_{n-2}(2))+(1-\beta)b_{n-2}(x) \end{align}

Equation (22) represents the forecast equation, while Equation (23) represents the level equation, which denotes the estimated level. Equation (24) represents the trend equation, which denotes the estimated trend. α is the smoothing parameter for the level, β is the smoothing parameter for the trend and n represents the number of iterations.

3.4 Differential game guidance law and chattering suppression design with a virtual target as the objective

After obtaining the virtual target $({x_v},{y_v})$ using the BP neural network, another problem to be solved is how to reach the target following the Dubins path. One solution is to use the following differential game guidance law:

(25) \begin{align} a_{i}= \left\{\begin{array}{c} a^{\max}_{i} \bullet sign(ZEM_{iT}),\,t_{go \gt 0}\\0,\,t_{go}\leq 0 \end{array}\right.\end{align}

In this case, i represents either the defender or the attacker, and $ZE{M_{iT}}$ represents the zero effort miss of individual i to the virtual target. However, when an individual deviates towards the virtual target, its acceleration command will continuously change direction near ZEM = 0, resulting in system chattering issues. To avoid this problem, a saturation function method is used to modify the aforementioned guidance command:

(26) \begin{align} a_{i}= \left\{\begin{array}{c} a^{\max}_{i} \bullet sign(ZEM_{iT}), \,ZEM_{iT}\gt\Delta\\[3pt]a^{\max}_{i}\bullet\frac {ZEM_{iT}}\Delta, \,ZEM_{iT}\leq\Delta\end{array}\right.\end{align}

The modified guidance command is continuous, and therefore the control system is also continuous.

Table 1.

Initial state parameters

Table 2.

Computational efficiency comparison of PSO and BP neural network

This approach integrates neural network methodologies to swiftly predict results, as previously discussed.

4.1 Initial state

The simulation’s initial state parameters are presented in Table 1.

4.2 Training and exception elimination

To create a training dataset, one m sets of input and output data were randomly generated. After three m training iterations, a back propagation (BP) neural network model was obtained. Multiple sets of simulation results are obtained by applying the trained neural network model and combining with the anomaly elimination technique of Holt linear trend model described in Section 3.3.

1. (1) To intuitively demonstrate the computational efficiency advantage of the BP neural network method over the PSO algorithm, we randomly generated 1,000 sets of basic state parameters within the bounds defined in Table 1. For each method, the average computation time and its variance were recorded. As summarised in Table 2, the results show that the BP neural network achieves approximately a 12-fold improvement in computational speed compared to the PSO algorithm, while also exhibiting significantly greater stability.

4.3 The predictive impact of the DP-NNDG guidance law

1. (1) The DP-NNDG trajectory in four scenarios

Based on the initial state parameters specified in Case 1 of Table 1, the DP-NNDG guidance law is applied to generate trajectories corresponding to four defensive outcomes: head-on interception, head-on (defender failed), tail chase and side-on engagement, as illustrated in Fig. 3.

1. (2) The DP-NNDG in an ideal condition

Figure 3.

Case 1: Trajectory simulation results under four different scenarios. (a) is head on(attacker failed), (b) is also head on (defender failed), (c) is tail chase and (d) is side on.

Assuming both the attacker and the defender have identical maximum accelerations (2g), and employ DP-NNDG as the guidance law in TAD three-body game, the simulation results are depicted in Fig. 3. On one hand, throughout this game scenario, as both the attacker and defender adhere to the DP-NNDG guidance law, their flight trajectories generally align with the initial predictive outcomes, with no significant deviations observed during the intermediate stages. On the other hand, the predicted outcomes of the virtual targets during the game process are derived from DP-NNDG, theoretically ensuring consistent virtual target results. Therefore, the distribution of virtual targets reflects the predictive accuracy of DP-NNDG under these conditions.

The virtual targets obtained by DP-NNDG are concentrated within a 1 cubic meter range (c). From an engineering perspective, such prediction accuracy is already sufficient.

However, as observed in Fig. 4(c), the distribution of virtual targets exhibits an approximately linear pattern except for divergence at the endpoints. This phenomenon can be attributed to two primary factors: (1) the application of the DP-NNDG guidance law yields suboptimal control performance with tolerable deviations; (2) the discrete nature of individual flight trajectory simulations introduces cumulative errors, ultimately resulting in the behaviour demonstrated in Fig. 4(c).

1. (3) DP-NNDG considering a first-order autopilot

Figure 4.

Case 2: Without considering the delay of the autopilot. Attacker: DP-NNDG ( $a_A^{\max } = 2g$ ); Defender: DP-NNDG ( $a_D^{\max } = 2g$ ). (a) is trajectory, (b) is acceleration and (c) is virtual target.

More practically speaking, a first-order autopilot is considered for both attacker and defender [Reference Nobahari and Nasrollahi41].

(27) \begin{align} {\dot a}_A & = - 1\big/{\mathop \tau \nolimits_A } + 1\big/\mathop\tau \nolimits_A \mathop u_A \nonumber\\{\dot a}_D & = - 1\big/ \mathop \tau \nolimits_D a_D + 1\big/\mathop \tau \nolimits_D \mathop u_D \end{align}

In this study, $\mathop \tau \nolimits_A $ and $\mathop \tau \nolimits_D $ represent the temporal coefficients for the attacker and defender respectively, while $\mathop u\nolimits_A $ and $\mathop u\nolimits_D $ denote the command inputs of the attacker’s and defender’s autonomous driving systems. It is assumed that both parties have the same maximum acceleration (2g) and employ DP-NNDG as the guidance law in the TAD three-body game, as depicted in Fig. 5. Due to the inertia of the autonomous driving systems, neither party can manoeuver according to the results predicted by the DP-NNDG method, leading to discrepancies between subsequent and preceding state predictions. Nevertheless, the predicted outcomes appear to exhibit temporal regularity, with the movement of virtual targets on the plane approximating a straight line (d). As the final state approaches, the magnitude of predictive variations gradually diminishes until they converge. This phenomenon arises because minor perturbations among the attacker, target and defender within a large spatial range can result in significant changes in terminal states, whereas the reverse is not easily achieved.

1. (4) Comparison of DP-NNDG and NDG Performance

Figure 5.

Case 3: Considering the delay of the autopilot. Attacker: DP-NNDG ( $a_A^{\max } = 2g$ ); Defender: DP-NNDG ( $a_D^{\max } = 2g$ ). (a) is trajectory, (b) is acceleration, (c) is acceleration command and (d) is virtual target.

NDG transforms the three-body pursuit-evasion problem of TAD into two sets of chase-escape problems, AT and DA, for resolution, with the challenge lying in determining the attacker’s manoeuvering strategy. Currently, a widely adopted approach involves determining the attacker’s manoeuvering strategy through a weighted combination of two sets of coefficients. For Case 4 simulations: (1) both groups of pursuit and evasion coefficients for the attacker are assumed to be equal; (2) a first-order autopilot is considered for both attacker and defender; (3) the simulation results presented in Fig. 6 demonstrate a terminal miss distance of 95.89 meters between the attacker and target.

Figure 6.

Case 4: Considering the delay of the autopilot. Attacker: NDG ( $a_A^{\max } = 4g$ ); Defender: NDG ( $a_D^{\max } = 5g$ ). (a) is trajectory, (b) is acceleration and (c) is acceleration command.

Case 5 adopts the attacker guidance strategy of DP-NNDG from Case 4 while keeping other conditions unchanged. Simulation results are shown in Fig. 7, indicating a terminal miss distance of 77.75 meters. Comparing Case 4 and Case 5, the implementation of DP-NNDG in Case 5 significantly reduces the attacker’s terminal miss distance by 18.14 meters. This demonstrates that DP-NNDG outperforms NDG notably in this scenario.

Figure 7.

Case 5: Considering the delay of the autopilot. Attacker: DP-NNDG ( $a_A^{\max } = 4g$ ); Defender: NDG ( $a_D^{\max } = 5g$ ). (a) is trajectory, (b) is acceleration, (c) is acceleration command and (d) is virtual target.

In Case 6, based on Case 5, the guidance strategy of the defender was also set to DP-NNDG, with all other conditions remaining identical to Case 5. The simulation results, depicted in Fig. 8, show a terminal miss distance of 85.40 meters. Comparing Case 6 with Case 5, the adoption of DP-NNDG guidance strategy by the defender resulted in an increase of 7.65 meters in the attacker’s terminal miss distance. This indicates a clear performance advantage of DP-NNDG over NDG in this scenario.