Four-Dimensional Trajectory Planning Algorithm for Fixed-Wing Aircraft Formation Based on Improved Hunter—Prey Optimization

Abstract

The aircraft four-dimensional trajectory planning is an important technology for multiple aircraft to achieve cooperation. However, the current four-dimensional trajectory planning technology is mainly used for civil aviation and helicopters and is difficult to meet the requirements of fixed-wing aircraft. This paper proposed a four-dimensional trajectory planning algorithm for a fixed-wing aircraft formation, considering the speed range, turning radius and maximum overload. The improved tau-J strategy (ITJS) is used to generate the four-dimensional trajectory of the aircraft. This strategy is a bio-inspired trajectory planning algorithm that can generate a four-dimensional trajectory with continuous acceleration. Furthermore, the improved hunter–prey optimization (IHPO) algorithm is used to optimize the trajectory to make the generated trajectory meet the constraints and speed up the algorithm convergence. This algorithm improves the updated strategy and initialization strategy based on the hunter–prey optimization (HPO) algorithm, which prevents the algorithm from falling into local optima. The results of the benchmark test function show that the optimization result of the algorithm is improved by more than 10% compared with the original HPO algorithm. The simulation results show that the proposed algorithm jumps out of local optima and generates a trajectory that meets the constraints.

1. Introduction

Fixed-wing aircraft with higher speed have been playing an important role in modern warfare and emergency response [1]. However, the above environment is highly dangerous and unstable. A single aircraft often faces great challenges when performing tasks. However, aircraft formations can enhance aircraft flexibility and reduce the risk of mission failure [2]. Four-dimensional trajectory planning is a key technology for the aircraft formation and collaboration. It is composed of traditional three-dimensional (latitude, longitude and altitude) trajectory planning and a required-time-of-arrival (RTA) constraint, which requires the aircraft to arrive at required locations at required times [3]. Therefore, the four-dimensional trajectory planning of aircraft is an important technology to realize the aircraft formation and improve the efficiency of the aircraft formation’s task execution. It plays an important role in realizing obstacle avoidance, situation awareness and precision strike of the aircraft formation.

Many researchers have performed a great deal of work on four-dimensional trajectory planning for aircraft. By investigating the historical trajectory data of civil aircraft and comprehensively considering constraints such as aircraft flight performance, fuel consumption and atmospheric conditions, a multi-objective four-dimensional trajectory collaborative optimization algorithm based on the non-dominated sorting genetic algorithm and simulated annealing algorithm was proposed [4]. Comparing the trajectory obtained by this algorithm with the historical trajectory data, the fuel consumption is reduced by 4.5%. It can also meet the demand of real-time ballistic calculation. To keep the balance between the demand and capacity in the four-dimensional trajectory planning task, a hybrid optimization strategy was proposed [5]. The method combines a time-window-based sequential decision-making architecture, a heuristic strategy (greedy strategy) and an optimization algorithm to achieve fast trajectory planning for large-scale flights. By constructing a matrix related to flight conflicts, the nonlinear model is converted into a linear model in the optimization model based on continuous time, which greatly improves the speed of model solving. The results show that this method can effectively reduce flight delay. Lucas et al. used the genetic algorithm, non-dominated sorting genetic algorithm II (NSGA-II) [6], to solve a multi-objective optimization problem for glider mission planning that includes multiple parameters such as longitude, latitude, depth and time. Through this method, multiple Pareto optimal solutions can be quickly found in the case of a given fixed obstacle. At the same time, this method has low requirements for computer performance and can be effectively used in practical tasks. Based on the A* algorithm, Wu et al. [7] proposed a multi-step four-dimensional trajectory planning search algorithm multi-step A* (MSA*). The trajectory generated by the algorithm consists of a set of linear trajectories and takes into account many decision criteria and constraints such as wind, fuel, power constraints, turning radius and other flight requirements. The algorithm can meet the needs of offline and online trajectory planning, and the average calculation time is one-fourth of the vector-neighborhood-based A*. However, the above algorithms require historical data and cannot cope with dynamic changes. Fixed-wing aircraft have complex constraints, and their trajectories change frequently; therefore, these methods cannot meet the requirements of four-dimensional trajectory planning for a fixed-wing aircraft formation in high-dynamic environments.

Transforming the four-dimensional trajectory planning task for an aircraft formation into a mathematical optimization problem is an effective way to generate the aircraft four-dimensional trajectory. At present, the algorithms for transforming the aircraft trajectory planning problems into mathematical optimization problems are mainly based on curve fitting algorithms [8,9,10,11], graph search based algorithms [12,13,14,15]. However, the above algorithm can only generate the trajectory but cannot control the speed of the aircraft. Additional speed control strategy is required to generate the four-dimensional trajectory. This leads to the complexity of the aircraft four-dimensional trajectory algorithm. To avoid the above situation, this paper uses a biologically inspired trajectory planning algorithm called the tau theory. This method is widely used in various tasks such as trajectory planning and motion planning [16,17,18,19,20]. It can generate multiple trajectories with consistent time and has the advantages of continuous smooth velocity curve and simple optimization parameters. This makes the tau theory suitable for four-dimensional trajectory planning for fixed-wing aircraft. In this paper, the fixed-wing aircraft use the ITJS to generate the trajectory.

The parameter of the tau-J guidance strategy influences the quality of the generated trajectory. To enable the application of the tau-J guidance strategy to all kinds of complex situations, it is necessary to optimize the tau-J guidance strategy. HPO [21] is an optimization algorithm that mimics the behavior of the predator and the prey. In nature, the predator chases the prey, and the prey runs to the safest place. The HPO algorithm treats the safest place as the optimal location. Therefore, all the predators run to the optimal position, and the algorithm converges gradually. The algorithm has been verified on several test functions and engineering problems. The results show that the HPO algorithm has the advantages of the fast convergence speed and strong optimization ability. However, the HPO algorithm also has the disadvantages of falling into local optima and uneven search scope. Therefore, this paper proposes an improved HPO algorithm (IHPO) based on the HPO algorithm. The feasibility and efficiency of IHPO is verified using test functions.

To solve the problems existing in the current four-dimensional trajectory planning algorithm for fixed-wing aircraft, in this paper, the ITJS algorithm is used to generate the four-dimensional trajectory, and the proposed IHPO algorithm is used to optimize the trajectory. In addition, to verify the effectiveness of the IHPO algorithm, it is validated using the benchmark test function. The structure of this paper is as follows: the second chapter describes the problem and establishes a dynamic model of fixed-wing aircraft. The third chapter introduces the ITJS and HPO algorithms and proposes the IHPO algorithm. The IHPO is compared with other optimization algorithms to prove its feasibility and efficiency. In the fourth chapter, a four-dimensional trajectory planning algorithm for a fixed-wing aircraft formation is proposed, and the validity of the algorithm is verified. The fifth chapter summarizes the IHPO algorithm and the four-dimensional trajectory planning algorithm for fixed-wing aircraft and suggests future work.

2. Background

2.1. Problem Statement

The problem is as follows: a formation composed of two fixed-wing aircraft passes through the navigation point in turn. The distance between the two aircraft is 3 km. The leader is Aircraft 1. The slave is Aircraft 2, which is located on the right side of the leader machine. The formation of the aircraft is shown in Figure 1.

The leader and the slave need to generate their own four-dimensional trajectory according to the given navigation point. The four-dimensional trajectory is required to reach the given navigation point simultaneously. The aircraft only maintain the formation near each navigation point. The trajectory needs to meet the constraints of fixed-wing aircraft. The formation of aircraft flying in a straight line is often difficult to conflict. Therefore, to verify the performance of the algorithm, the selection of navigation points takes turning situation into account. Taking the first navigation point as the origin of the coordinates, the coordinates of the navigation points of the leader are shown in

Table 1. Navigation points of the fixed-wing aircraft formation.

Index of Navigation Point	Coordinates/Unit
1	(0, 8.09, 0)/km
2	(50, 8.1, 50)/km
3	(70, 8.1, 100)/km
4	(100, 8.08, 150)/km
5	(170, 8.1, 50)/km
6	(180, 8.09, −50)/km
7	(170, 8.1, −100)/km

.

The top view of the navigation points is shown in Figure 2.

In addition, the speed, acceleration and turning radius of aircraft are within a certain range. They are shown in

Table 2. Constraints of the aircraft.

Constraint	Range/Unit
Speed Range	[180, 230]/m/s
Acceleration Range	[0, 5]/m/s2
Safety Gap	>1000/m
Turning Radius	>9680/m

. The speed is kept within a certain range. If the speed is too low, the aircraft cannot provide enough lift and overload, and the aircraft’s flight performance will suffer. If the speed is too high, aircraft resistance will increase significantly, the fuel consumption rate will also greatly increase. Due to engine limitations, aircraft acceleration should be less than the given value. The gap between the aircraft is larger than the given value to avoid collision.

The task process of this study is shown in Figure 3.

2.2. Fixed-Wing Aircraft Dynamic Model

To simulate and check the generated trajectory, the dynamic model of the fixed-wing aircraft is established as follows:

$$\{\begin{array}{c} m\frac{dV}{dt}=P\cos \alpha \cos \beta -X-mg\sin \theta \hfill \\ \hspace{0.17em}mV\frac{d\theta }{dt}=P\left(\sin \alpha \cos {\gamma }_v+\cos \alpha \sin \beta \cos {\gamma }_v\right)+\hfill \\ Y\cos {\gamma }_v-Z\sin {\gamma }_v-mg\cos \theta \hfill \\ -mV\frac{d{\psi }_v}{dt}=P(\sin \alpha \sin {\gamma }_v-\cos \alpha \sin \beta \cos {\gamma }_v)+\hfill \\ Y\sin {\gamma }_v+Z\cos {\gamma }_v\hfill \\ M_x=J_x\frac{d{\omega }_x}{dt}-J_{xy}\frac{d{\omega }_y}{dt}+\left(J_x-J_y\right){\omega }_z{\omega }_y+J_{xy}{\omega }_z{\omega }_x\hfill \\ M_y=J_y\frac{d{\omega }_y}{dt}-J_{xy}\frac{d{\omega }_x}{dt}+\left(J_x-J_z\right){\omega }_z{\omega }_x+J_{xy}{\omega }_z{\omega }_y\hfill \\ \hspace{0.17em} M_z=J_z\frac{d{\omega }_z}{dt}+\left(J_y-J_z\right){\omega }_x{\omega }_y+J_{xy}\left({\omega }_y^2{\omega }_x^2\right)\hfill \\ \frac{dx}{dt}=V\cos \theta \cos {\psi }_v\hfill \\ \frac{dy}{dt}=V\sin \theta \hfill \\ \frac{dz}{dt}=-V\cos \theta \sin {\psi }_v\hfill \\ \hspace{0.17em} \frac{d\vartheta }{dt}={\omega }_y\sin \gamma +{\omega }_z\cos \gamma \hfill \\ \hspace{0.17em} \frac{d\psi }{dt}=\frac{1}{\cos \vartheta }\left({\omega }_y\cos \gamma -{\omega }_z\sin \gamma \right)\hfill \\ \hspace{0.17em} \frac{d\gamma }{dt}={\omega }_x-\tan \vartheta \left({\omega }_y\cos \gamma -{\omega }_z\sin \gamma \right)\hfill \end{array}$$

(1)

The meaning of some variables in Equation (1) is shown in the

Table 3. Table of variable meanings.

Variable	Meaning	Variable	Meaning
$V$	Velocity	$\rho$	Density of Air
$m$	Mass	$\alpha$	Attack Angle
$P$	Thrust	$\beta$	Sideslip Angle
$X,Y,Z$	Drag Force, Lift Force and Lateral Force	$M_x,M_y,M_z$	Torque
$x,y,z$	Position	$J_x,J_y,J_z$	Moment of Inertia
$\vartheta$	Pitch Angle	${\omega }_x,{\omega }_y,{\omega }_z$	Angular Velocity
$\theta$	Ballistic Inclination	$\psi$	Yaw Angle
${\psi }_v$	Ballistic Declination	$\gamma$	Roll Angle

.

3. Four-Dimensional Trajectory Planning Algorithm for Fixed-Wing Formation

3.1. Tau Theory and Improved Tau-J Guidance Strategic

The tau theory is a widely used bio-inspired trajectory planning algorithm. It is a bio-inspired algorithm proposed by Lee et al. [22]. Researchers use time-to-collision (TTC) to describe the movement of organisms and define TTC as the tau variable. TTC refers to an estimate of arrival time when an actor approaches the target. To solve other kinds of problems, Lee et al. extended this type of process into the generalized tau theory. The definition of the tau variable in the generalized tau theory is as follows:

$$\tau \left(t\right)=\frac{M\left(t\right)}{\dot{M}\left(t\right)}$$

(2)

where $M\left(t\right)$ is the distance between the target and the actor, and $\dot{M}\left(t\right)$ is the speed of the actor.

In the process of real movement, it is often necessary for two actors to reach their designated positions at the same time. Based on this, the tau-coupling strategy was proposed [23]. The tau variables of two actors ${\tau }_1\left(t\right),{\tau }_2\left(t\right)$ satisfy the following expression:

$${\tau }_2\left(t\right)=k{\tau }_1\left(t\right)$$

(3)

where $k$ is the rate of $\left(0,0.5\right)$. Based on the tau-coupling strategy, Zhang et al. [24] proposed the tau-J guidance strategy, using constant jerk motion $J\left(t\right)$ as the coupled motion. The tau variable expression of this motion is as follows:

$$\{\begin{array}{l}J\left(t\right)=\frac{1}{6}j\left(T^3-t^3\right)\\ {\tau }_J\left(t\right)=\frac{J\left(t\right)}{\dot{J}\left(t\right)}=\frac{1}{3}\left(t-\frac{T^3}{t^2}\right)\end{array}$$

(4)

where $j$ is the constant jerk, $T$ is the expected arrival time, and $t$ is the current time. The expression of the movement coupled with the jerk motion is as follows:

$$\{\begin{array}{l}M(t)=\frac{M\left(0\right)}{T^{3/k}}{\left(T^3-t^3\right)}^{1/k}\hfill \\ \dot{M}(t)=\frac{-3M\left(0\right)t^2}{k{T}^{3/k}}{\left(T^3-t^3\right)}^{1/k-1}\hfill \\ \ddot{M}(t)=\frac{3M\left(0\right)t}{k{T}^{3/k}}\left(\frac{3-k}{k}{t}^3-2T^3\right){\left(T^3-t^3\right)}^{1/k-2}\hfill \end{array}$$

(5)

where $k\in \left(0, 0.5\right)$. This strategy is called the tau-J guidance strategy. The tau-j guidance strategy solves the problem of acceleration discontinuity in other tau guidance strategies [24,25] and reduces the errors of tau guidance strategy in practical application. However, the start and end velocity and acceleration of tau-J guidance strategy are always zero. It is impossible for fixed-wing aircraft.

To avoid zero velocity and zero acceleration, the tau-J guidance strategy needs to be improved. The improved method is similar to that described in the literature [25]. The improved tau variable expression of coupled motion is as follows:

$$\{\begin{array}{l}{J}_p\left(t\right)=-\frac{1}{6}j{t}^3+\frac{1}{2}{a}_J{t}^2+V_{J}t+J_0\\ {\dot{J}}_p\left(t\right)=-\frac{1}{2}j{t}^2+a_{J}t+V_J\\ {\ddot{J}}_p\left(t\right)=-jt+a_J\end{array}$$

(6)

where $J_0$ is the initial position, $V_J$ is the initial velocity, $a_J$ is the initial acceleration, and $j$ is the constant jerk. Based on the tau-coupling strategy, the coupled motion strategy can be obtained as follows:

$$\{\begin{array}{l}X(t)=X_T+{\dot{X}}_T(t-T)+\frac{1}{2}{\ddot{X}}_T{\left(t-T\right)}^2-\frac{{\rho }_0}{J_0^{1/k}}{J}_p^{1/k}\hfill \\ \dot{X}(t)={\dot{X}}_T+{\ddot{X}}_T\left(t-T\right)-\frac{{\rho }_0}{k{J}_0^{1/k}}{\dot{J}}_p{J}_p^{1/k-1}\hfill \\ \ddot{X}\left(t\right)={\ddot{X}}_T-\frac{{\rho }_0}{k{J}_0{}^{1/k}}{J}^{1/k-2}\left[\left(\frac{1-k}{k}\right){\dot{J}}_p{}^2+{\ddot{J}}_p{J}_p\right]\hfill \end{array}$$

(7)

where $X_T,{\dot{X}}_T,{\ddot{X}}_T$ are the target position, target velocity and target acceleration, respectively. $J_p,{\dot{J}}_p,{\ddot{J}}_p$ are the coupled motion, their expression is equivalent to Equation (6). The expression for ${\rho }_0$ is as follows:

$${\rho }_0=X_T-X_0-{\dot{X}}_{T}T-\frac{1}{2}{\ddot{X}}_T{T}^2$$

(8)

Let $t=0$ and put the coupled motion back into Equation (7), then we can obtain the following:

$$\{\begin{array}{l}{V}_J=\frac{k\left({\dot{X}}_T-{\ddot{X}}_{T}T-{\dot{X}}_0\right)J_0}{{\rho }_0}\\ a_J=\frac{\left({\ddot{X}}_T-{\ddot{X}}_0\right)k{J}_0}{{\rho }_0}+\frac{\left(k-1\right){\left({\dot{X}}_T-{\ddot{X}}_{T}T-{\dot{X}}_0\right)}^{2}k{J}_0}{{\rho }_0^2}\end{array}$$

(9)

We assume the following:

$$\{\begin{array}{l}{V}_{DJ}=\frac{V_J}{J_0}=\frac{k\left({\dot{X}}_T-{\ddot{X}}_{T}T-{\dot{X}}_0\right)}{{\rho }_0}\\ a_{DJ}=\frac{a_J}{J_0}=\frac{\left({\ddot{X}}_T-{\ddot{X}}_0\right)k}{{\rho }_0}+\frac{\left(k-1\right){\left({\dot{X}}_T-{\ddot{X}}_{T}T-{\dot{X}}_0\right)}^{2}k}{{\rho }_0^2}\end{array}$$

(10)

By substituting Equation (9) into the first formula of Equation (6), we can obtain the following:

$$J_0=\frac{j{T}^3}{\left(6+3a_{DJ}{T}^2+6V_{DJ}T\right)}$$

(11)

Therefore, parameters $J_0,V_J,a_J$ of the coupled motion can be given the initial values $X\left(0\right),\dot{X}\left(0\right),\ddot{X}\left(0\right)$. This strategy is called the ITJS.

The aircraft trajectory can be described by using the ITJS on the three axes X, Y and Z, respectively. The four-dimensional flight trajectory planning algorithm for the fixed-wing aircraft is as Equation (12).

$$\{\begin{array}{l}X(t)=X_T+{\dot{X}}_T(t-T)+\frac{1}{2}{\ddot{X}}_T{\left(t-T\right)}^2-\frac{{\rho }_{x0}}{J_{x0}^{1/k_x}}{J}_{xp}^{1/k_x}\\ Y(t)=Y_T+{\dot{Y}}_T(t-T)+\frac{1}{2}{\ddot{Y}}_T{\left(t-T\right)}^2-\frac{{\rho }_{y0}}{J_{y0}^{1/k_y}}{J}_{yp}^{1/k_y}\\ Z(t)=Z_T+{\dot{Z}}_T(t-T)+\frac{1}{2}{\ddot{Z}}_T{\left(t-T\right)}^2-\frac{{\rho }_{z0}}{J_{z0}^{1/k_z}}{J}_{zp}^{1/k_z}\end{array}$$

(12)

3.2. Original HPO Algorithm

The ITJS simplifies the aircraft trajectory into a mathematical problem determined by parameters $k_x,k_y,k_z$ and expected arrival time $T$. Therefore, the aircraft four-dimensional flight trajectory can be optimized to obtain the optimal trajectory.

The HPO algorithm is a new type of population-based optimization algorithm proposed by Naruei in 2022 [21]. The algorithm is created by mimicking the behavior of the predator and the prey during the hunt in nature. During the hunt, the hunter chases the prey. The prey consistently adjusts its position to the safest spot in the entire population. The authors consider the safest location as the optimal location. The HPO algorithm has fast convergence speed and strong optimization ability, which is suitable for optimizing the four-dimensional trajectory.

The HPO algorithm for population initialization is as follows:

$$x=rand\left(\dim ,1\right).\ast \left(ub-lb\right)+lb$$

(13)

where $x$ denotes the position of the hunter or the prey. It also represents the position of the search agents in the field of the optimization algorithm. Value $x$ is a random vector with all elements in the range $(ub,lb)$. Value $\dim$ denotes the dimension of $x$. Values $ub,lb$ represent the upper and lower ranges of the search areas in each dimension of $x$, respectively. Value $rand(\dim ,1)$ represents a random vector with all elements in the range (0, 1). The hunter’s position update algorithm is as follows:

$$x(t+1)=x(t)+0.5\left[\left(2CZ{\overrightarrow{P}}_{pos}-x(t)+2(1-C)Z\overrightarrow{\mu }-x(t)\right)\right]$$

(14)

where $x(t+1),x(t)$ are the positions of the agents of the next time and the current time, respectively; $P_{pos}$ denotes the position of the prey’ $Z$ is an adaptive parameter. The adaptive parameter’s expression is as follows:

$$\begin{array}{l}IND=({\overrightarrow{r}}_1<C)\\ Z=r_2\otimes IND+{\overrightarrow{r}}_3\otimes (~IND)\end{array}$$

(15)

where $r_2$ is the random number of $\left(0,1\right)$, $r_1,r_3$ are random vectors in dimension $\dim$ of $\left(0,1\right)$. $\otimes$ is the dot product. The value of $Z$ continues to change as the number of iterations changes. $C$ is an adaptive parameter used to balance wide-area searches with careful searches of specific areas. When $C$ is larger, the particles search the results in a larger search range, and the search is more random; when $C$ is smaller, the particles search the area more carefully.

The expression of $C$ is shown in Equation (16). As the number of iterations increases, the value of $C$ decreases continuously, and the search agents search the area more carefully.

$$C=1-it\left(\frac{0.98}{i{t}_{\max }}\right)$$

(16)

where $\mathit{it},{\mathit{it}}_{\max }$ represent the current iterations and the maximum iterations, respectively. Value $\mu$ represents the average position of all search agents, and its equation is as follows:

$$\overrightarrow{\mu }=\frac{1}{n}{\displaystyle \sum _{i=1}^n\overrightarrow{x}}$$

(17)

The distance between each search agent and $\mu$ is as follows:

$$D_{euc(i)}={\left({\displaystyle \sum _{j=1}^d{\left(x_{i,j}-{\mu }_{i,j}\right)}^2}\right)}^{\frac{1}{2}}$$

(18)

The search agents that have the farthest distance from the average distance are regarded as the location of the prey (${\overrightarrow{P}}_{pos}$):

$${\overrightarrow{P}}_{pos}={\overrightarrow{x}}_{i\max }$$

(19)

where $i\max$ is the index of the maximum value of $D_{euc}$. However, the above selection strategy reduces the population search efficiency and leads to slow convergence. Meanwhile, according to the convergence scenario, the hunter gradually catches up with the prey as time goes by. Therefore, to solve the problem of slow convergence, the prey selection strategy is as follows:

$$\begin{array}{l}{\overrightarrow{P}}_{pos}={\overrightarrow{x}}_i|i is sorted Deuc\left(kbest\right)\\ kbest=round\left(C\ast N\right)\end{array}$$

(20)

where $N$ is the number of search agents. This expression indicates that ${\overrightarrow{P}}_{pos}$ is the i-th closest to the average position.

During the hunt, the prey moves away from its current location to improve the probability of survival; the prey’s position is updated by the following equation:

$$x(t+1)={\overrightarrow{T}}_{pos}+CZ\cos \left(2\pi r_4\right)\times \left({\overrightarrow{T}}_{pos}-x(t)\right)$$

(21)

where ${\overrightarrow{T}}_{pos}$ is the global optimal position, and $r_4$ is a random number of $\left(-1,1\right)$. When the prey is attacked, it runs to the safest location, i.e., near the global optimal location, to avoid the attack.

Combining Equation (14) and Equation (21), the updated equation of search agents is as follows:

$$x(t+1)=\{\begin{array}{c}x(t)+0.5\left[\left(2CZ{\overrightarrow{P}}_{pos}-x(t)+2(1-C)Z\overrightarrow{\mu }-x(t)\right)\right],r_5<\beta \left(a\right)\hfill \\ {\overrightarrow{T}}_{pos}+CZ\cos \left(2\pi r_4\right)\times \left({\overrightarrow{T}}_{pos}-x(t)\right),r_5>\beta \left(b\right)\hfill \end{array}$$

(22)

where $r_5$ is the random number of $\left(0,1\right)$, and $\beta$ is the adjustment parameter set to 0.1. When $r_5<\beta$, the search agent is treated as the hunter, and the position is updated using Equation (22a). When $r_5>\beta$, the search agent is treated as the prey, and the position is updated using Equation (22b).

However, the HPO algorithm only uses the global optimal position for iteration when performing iterations. As the number of iteration rounds increases, more search agents concentrate on searching for the optimal value in a specific region, which affects the convergence speed of the HPO algorithm and also makes the HPO algorithm more likely to fall into local optima. In addition, the HPO algorithm also has an uneven search area, which leads to the instability of convergence speed of the HPO algorithm.

3.3. IHPO Algorithm

To solve the above problems, the improved HPO algorithm is proposed in this section. The improvement strategy is as follows:

3.3.1. Good Point Set

The HPO algorithm initializes the search agents’ position vector using a random method. The search agents’ position vector generated by this method often cannot fully express all the features in the entire space, resulting in unstable convergence speed. Therefore, it is better to use the good point set strategy to make the search agents evenly distributed in space.

The good point set is constructed in an $s$-dimensional space with $n$ points distributed as follows: $p_n(k)=\left\{\left(\left\{r_1^{(n)}\cdot k\right\},\left\{r_2^{(n)}\cdot k\right\},\dots ,\left\{r_s^{(n)}\cdot k\right\}\right),1\le k\le n\right\}$. The expression for r is as follows: $r=\{2\cos (2\pi j/p),1\le j\le s\}$. Value $p$ is the smallest prime number that satisfies the condition. The point sets generated by the good point set and randomly generated point set methods in a space with $s=2,n=300$ are shown in Figure 4.

Applying the good point set method to the HPO, the initialization method of the IHPO algorithm can be obtained as follows:

$$x_j=\left(u{b}_j-l{b}_j\right)\cdot \left\{r_j\cdot k\right\}+l{b}_j$$

(23)

where $j$ denotes the $j$ component value of $x$ The initial position vector generated by the good point set method has a more evenly distributed and stable rate of convergence compared to the random vector [26].

3.3.2. Improved Prey Position Update Algorithm

The HPO algorithm updates the prey’s position using the globally optimal position. The HPO can be trapped in local optima, which affects the convergence speed of the algorithm. Therefore, the local optimum position is introduced for updating the prey position. The update algorithm for the improved prey position is as follows:

$$\begin{array}{c}{x}_i(t+1)={\overrightarrow{T}}_{pos}+CZ\cos \left(2\pi r_4\right)\times \hfill \\ \left(rate\ast {\overrightarrow{T}}_{pos}+\left(1-rate\right)\ast P_{Best}{{}_{\left(i\right)}}_{pos}-x_i(t)\right)\hfill \end{array}$$

(24)

where $i$ denotes the local optima of the i-th search agent, and $P_{Best}{}_{\left(i\right)}$ denotes the local optima of the i-th search agents. Value $rate$ is the scale factor of (0,1). Depending on the scale between $P_{Best(i)}$ and $T_{pos}$, it is calculated as follows:

$$rate=2\ast \left(1-\frac{T_{pos}}{\left(T_{pos}+P_{Best(i)}\right)}\right)-1$$

(25)

It should be noted that in this paper, the goal of the optimization is to minimize the output result, so when $T_{pos}$ is smaller, $rate$ is larger; the prey prefers to update its position using the global optima. Combining it with Equation (22), the update equation for the search agents of the IHPO algorithm is as follows:

$$x(t+1)=\{\begin{array}{c}x(t)+0.5\left[\left(2CZ{\overrightarrow{P}}_{pos}-x(t)+2(1-C)Z\overrightarrow{\mu }-x(t)\right)\right],r_5<\beta \left(a\right)\hfill \\ {\overrightarrow{T}}_{pos}+CZ\cos \left(2\pi r_4\right)\times \hfill \\ \left(rate\ast {\overrightarrow{T}}_{pos}+\left(1-rate\right)\ast P_{Best}{{}_{\left(i\right)}}_{pos}-x_i(t)\right),r_5>\beta \left(b\right)\hfill \end{array}$$

(26)

The flowchart of the improved IHPO algorithm is shown in Figure 5.

3.4. IHPO Performance Evaluation and Analysis of the Results

In this section, 13 benchmark test functions are selected to verify the effectiveness of the IHPO algorithm. The 13 benchmark test functions are widely used for testing various types of optimization functions [27,28]. The IHPO algorithm is compared with the (i) original HPO algorithm [21], (ii) PSO [29], (iii) ALO [30], (iv) GWO [31], (v) TSA [32] and (vi) WOA [33].

The expressions of benchmark functions are shown in

Table 4. Unimodal test functions.

Function	Dim	Range	${\mathit{f}}_{\mathbf{min}}$
$F_1\left(x\right)={\displaystyle {\displaystyle \sum }_{i=1}^n}{x}_i^2$	30	$\left[-100,100\right]$	0
$F_2\left(x\right)={\displaystyle {\displaystyle \sum }_{i=1}^n}\left|x_i\right|+{\displaystyle {\displaystyle \prod }_{i=1}^n}\left|x_i\right|$	30	$\left[-10,10\right]$	0
$F_3\left(x\right)={\displaystyle {\displaystyle \sum }_{i=1}^n}{\left({\displaystyle {\displaystyle \sum }_{j-1}^i}{x}_j\right)}^2$	30	$\left[-100,100\right]$	0
$F_4\left(x\right)=max\left\{\left|x_i\right|,1\le i\le n\right\}$	30	$\left[-100,100\right]$	0
$F_5\left(x\right)={\displaystyle {\displaystyle \sum }_{i=1}^{n-1}}\left[100{\left(x_{i+1}-x_i^2\right)}^2+{\left(x_i-1\right)}^2\right]$	30	$\left[-30,30\right]$	0
$F_6\left(x\right)={\displaystyle {\displaystyle \sum }_{i=1}^n}{\left(\left[x_i+0.5\right]\right)}^2$	30	$\left[-100,100\right]$	0
$F_7\left(x\right)=max\left\{\left|x_i\right|,1\le i\le n\right\}$	30	$\left[-1.28,1.28\right]$	0

and

Table 5. Multimodal test functions.

Function	Dim	Range	${\mathit{f}}_{\mathbf{min}}$
$F_8\left(x\right)={\displaystyle {\displaystyle \sum }_{i=1}^n}-x_i\sin \left(\sqrt{\left|x_i\right|}\right)$	30	$\left[-500,500\right]$	−418.9825 * 5
$F_9\left(x\right)={\displaystyle {\displaystyle \sum }_{i=1}^n}\left[x_i^2-10\cos \left(2\pi x_i\right)+10\right]$	30	$\left[-5.12,5.12\right]$	0
$\begin{array}{c}{F}_{10}\left(x\right)=-20\exp \left(-0.2\sqrt{\frac{1}{n}{\displaystyle {\displaystyle \sum }_{i=1}^n}{x}_i^2}\right)-\hfill \\ \exp \left(\frac{1}{n}{\displaystyle {\displaystyle \sum }_{i=1}^n}\cos \left(2\pi x_i\right)\right)+20+e\hfill \end{array}$	30	$\left[-32,32\right]$	0
$F_{11}\left(x\right)=\frac{1}{4000}{\displaystyle {\displaystyle \sum }_{i=1}^n}{x}_i^2-{\displaystyle {\displaystyle \prod }_{i=1}^n}\cos \left(\frac{x_i}{\sqrt{i}}\right)+1$	30	$\left[-600,600\right]$	0
$\begin{array}{c}{F}_{12}\left(x\right)=\frac{\pi }{n}\{10\sin \left(\pi y_1\right)+{\left(y_n-1\right)}^2+\hfill \\ {\displaystyle {\displaystyle \sum }_{i=1}^{n-1}}{\left(y_i-1\right)}^2\left[1+10{\sin }^2\left(\pi y_{i+1}\right)\right]\}+{\displaystyle {\displaystyle \sum }_{i=1}^n}u\left(x_i,10,100,4\right)\hfill \\ +{\displaystyle {\displaystyle \sum }_{i=1}^n}u\left(x_i,10,100,4\right)y_i=1+\frac{x_i+1}{4}\hfill \\ u\left(x_i,a,k,m\right)=\left\{\begin{array}{cc}k{\left(x_i-a\right)}^m& x_i>a\\ 0& -a<x_i<a\\ k{\left(-x_i-a\right)}^m& x_i<-a\end{array}\right\}\end{array}$	30	$\left[-50,50\right]$	0
$\begin{array}{l}{F}_{13}\left(x\right)=0.1\{{\sin }^2\left(3\pi x_1\right)+{\displaystyle {\displaystyle \sum }_{i=1}^n}{\left(x_i-1\right)}^2\left[1+{\sin }^2\left(3\pi x_i+1\right)\right]\\ +{\left(x_n-1\right)}^2\left[1+{\sin }^2\left(2\pi x_n\right)\right]\}+{\displaystyle {\displaystyle \sum }_{i=1}^n}u\left(x_i,5,100,4\right)\end{array}$	30	$\left[-50,50\right]$	0

. Functions F1~F7 are unimodal test functions. Unimodal test functions have only global optimal results and are suitable for detecting the convergence speed of the optimization algorithm. Functions F8~F13 are multimodal test functions. Multimodal test function has multiple local optimum solutions, which is suitable for detecting the ability of algorithms to jump out of the local optima. In the tables below, “Dim” represents the dimension of the test function, “Range” represents the value range of the test function’s independent variable, and “$f_{\min }$” is the optimal value. To simplify the equation, n is used to represent dimensions in the equations in Table 4 and Table 5. All the test function dimensions are set to 30, the number of search agents is 50, and the maximum number of iterations is 500. To obtain statistically significant data, each test function is solved 30 times.

The above benchmark functions were tested using the IHPO algorithm and the other algorithms. Some of the convergence curves are shown in Figure 6, where the horizontal axis is the number of iterations, and the vertical axis is the average value of 30 rounds. Each plot is a semi-logarithmic convergence plot.

Table 6. Comparison results for benchmarking functions.

Function		PSO	ALO	GWO	TSA	WOA	HPO	IHPO
F1	Min	0.735959	2.89 × 10−5	1.45 × 10−38	4.63 × 10−26	3.81 × 10−96	0	0
	Max	3.696292	0.000406	9.46 × 10−36	8.42 × 10−23	8.09 × 10−85	0	0
	Avg	1.538743	0.000136	1.17 × 10−36	9.47 × 10−24	5.52 × 10−86	0	0
	Std	0.726456	8.48 × 10−5	2.35 × 10−36	1.9 × 10−23	1.77 × 10−85	0	0
F2	Min	0.505629	4.102615	1.04 × 10−22	2.74 × 10−16	1.75 × 10−59	0	0
	Max	7.334057	136.2959	4.43 × 10−21	3.32 × 10−14	1.2 × 10−52	3.9 × 10−299	0
	Avg	2.750852	50.7212	9.41 × 10−22	5.8 × 10−15	6.87 × 10−54	2.3 × 10−300	0
	Std	2.089441	46.71624	8.99 × 10−22	6.34 × 10−15	2.41 × 10−53	0	0
F3	Min	4.150569	392.902	1.19 × 10−10	1.02 × 10−10	11,329.37	0	0
	Max	23.75996	3033.444	1.61 × 10−6	5.59 × 10−5	67,139.76	0	0
	Avg	12.059	1615.409	1.12 × 10−7	2.07 × 10−6	28,537.78	0	0
	Std	4.94186	721.8693	2.9 × 10−7	1 × 10−5	12,785.49	0	0
F4	Min	0.946614	4.622906	1.17 × 10−9	0.001164	0.000627	7.8 × 10−272	7.1 × 10−303
	Max	15.51543	21.56396	4.65 × 10−7	0.447515	82.44428	2.5 × 10−253	3.3 × 10−285
	Avg	6.289478	11.73729	6 × 10−8	0.089326	36.16453	8.4 × 10−255	1.3 × 10−286
	Std	4.338685	3.38342	8.91 × 10−8	0.103862	28.23141	0	0
F5	Min	34.1883	5.082163	25.42878	27.12241	26.91943	20.14847	19.52495
	Max	301.9513	1373.745	28.73775	28.90275	28.53038	24.16472	21.17915
	Avg	115.3122	128.4232	26.78848	28.43383	27.42457	21.41817	20.4386
	Std	73.63469	245.4885	0.732867	0.584376	0.370858	1.004628	0.370193
F6	Min	0.824377	2.96 × 10−5	9.95 × 10−7	0.122444	0.001233	3.32 × 10−18	2.78 × 10−17
	Max	4.085392	0.000213	0.030333	0.211506	0.026571	0.009999	2.64 × 10−14
	Avg	1.938282	0.000106	0.016579	0.160994	0.009973	0.000333	2.21 × 10−15
	Std	0.899895	5.2× 10−5	0.010054	0.024823	0.006268	0.001795	4.95 × 10−15
F7	Min	0.014302	0.010791	0.000434	0.00074	6.04× 10−5	7.23 × 10−6	1.28 × 10−6
	Max	0.141476	0.137476	0.002289	0.007175	0.014234	0.000598	0.000809
	Avg	0.049371	0.052769	0.000929	0.003716	0.002815	0.000136	0.000136
	Std	0.031664	0.026242	0.000426	0.001681	0.00308	0.000155	0.000175
F8	Min	−10,905.3	−12,569.4	−7974.12	−7345.03	−12,569.2	−8462.99	−8917.65
	Max	−6582.45	−5417.67	−5625.65	−4860.38	−8123.81	−6369.35	−7461.27
	Avg	−8464.46	−5787.51	−6777.26	−6039.8	−11274.7	−7415.14	−8040.41
	Std	980.7863	1318.79	609.3698	570.5786	1410.565	704.8123	464.25
F9	Min	33.34937	39.79837	0	106.0835	0	0	0
	Max	150.6205	141.2836	23.79284	280.0721	0	0	0
	Avg	66.63293	76.64964	9.230506	178.1704	0	0	0
	Std	26.21836	23.67553	7.766249	38.44448	0	0	0
F10	Min	6.880052	0.931343	3.29 × 10−14	1.93 × 10−13	8.88 × 10−16	8.88 × 10−16	8.88 × 10−16
	Max	12.24613	3.885776	4.35 × 10−14	3.660763	7.99 × 10−15	8.88 × 10−16	8.88 × 10−16
	Avg	10.20611	2.397031	3.91 × 10−14	2.314967	4.91 × 10−15	8.88 × 10−16	8.88 × 10−16
	Std	1.451209	0.668457	2.54 × 10−15	1.408125	2.38 × 10−15	0	0
F11	Min	0.560561	0.004095	0	0	0	0	0
	Max	0.982951	0.061275	0.037284	0.030828	0.096388	0	0
	Avg	0.848833	0.023623	0.005102	0.006036	0.003213	0	0
	Std	0.098769	0.014485	0.010013	0.008379	0.017302	0	0
F12	Min	2.978176	4.427664	0.012233	2.841487	0.001574	2.45 × 10−17	0
	Max	18.29589	23.63953	0.052492	17.95252	0.159504	1.08 × 10−11	0
	Avg	9.247929	10.54308	0.031104	8.39487	0.010748	4.32 × 10−13	0
	Std	3.983268	4.666686	0.010859	3.410731	0.027921	1.95 × 10−12	0
F13	Min	15.50733	5.6 × 10−5	0.095756	1.679004	0.029248	0.064783	0.109919
	Max	61.65713	48.31666	0.688728	5.453582	0.488421	2.86538	2.796923
	Avg	41.12046	7.057643	0.381324	2.906343	0.20174	1.310092	0.791166
	Std	12.39199	13.0496	0.177102	0.755225	0.134676	0.99245	0.769479

Bold indicates the optimal result under this test function.

shows the minimum value, the maximum value, the average value and the variance of the results of different optimization algorithms verified using different test functions. The bolded data represent the best results under this benchmark function. It can be seen from the results that in all unimodal test functions, the convergence speed of the IHPO and HPO algorithms is much faster than that of the other algorithms. The convergence speed and results of the IHPO algorithm are greatly improved compared with the HPO algorithm. In the multimodal test function, the IHPO algorithm performs best on F9, F10, F11 and F12. On F8 and F13, although the results of the IHPO are not as good as those of WOA, the convergence speed and the final results are improved compared with the original HPO algorithm. The IHPO is also more stable than the HPO and WOA. Moreover, the stability of the IHPO algorithm is improved compared with HPO algorithm in most test functions.

3.5. Implementation of Four-Dimensional Trajectory Planning Algorithm

To optimize the trajectory, an optimization function needs to be designed. Considering the constraints and optimization objectives of the aircraft, the optimization function is given as follows:

$$\begin{array}{c}minimize:\\ J={\displaystyle \sum _{i=1}^2{l}_i}\\ \mathrm{subject} \mathrm{to}:\\ Dist\left(1,2\right)>1000\\ \mathrm{variable} \mathrm{range}: \\ v_{\min }<v_i<v_{\max },\\ a_i<a_{\max },\left(i=1,2\right)\end{array}$$

(27)

where the optimization objective $J$ is used to find the minimum length of the aircraft formation; $l_i$ denotes the trajectory length of the i-th aircraft; $Dist\left(1,2\right)$ denotes the relative distance between the first and second aircraft; $a_i,v_i$ denote the acceleration and speed of the i-th aircraft; $a_{\max },v_{\min },v_{\max }$ denote the maximum acceleration, minimum speed and maximum speed of the aircraft, respectively. From Equation (27), the cost function is given as follows:

$$\begin{array}{c}J={\omega }_1\ast {\displaystyle \sum _{i=1}^2{l}_i}+{\omega }_2\ast \left(\min \left(0,Dist\left(1,2\right)-1000\right)\right)\\ {\omega }_3\ast {\displaystyle \sum _{i=1}^{2}std\left(v_i\right)}+{\omega }_4\ast {\displaystyle \sum _{i=1}^{2}std\left(a_i\right)}+{\omega }_5\ast T\end{array}$$

(28)

where ${\omega }_1,{\omega }_2,{\omega }_3,{\omega }_4,{\omega }_5,{\omega }_6$ represent the weight. Value $\min \left(0,Dist\left(1,2\right)-1000\right)$ represents the penalty term of relative distance. When relative distance is less than 1000 m, the penalty gradually increases. Values $std\left(v_i\right),std\left(a_i\right)$ are the standard deviation of speed and acceleration of the i-th aircraft. Their purpose is to limit the range of speed and acceleration. $T$ is the expected arrival time.

The ITJS generates the trajectory between two navigation points, and the IHPO optimizes the generated trajectory by Equation (27). The steps of the four-dimensional trajectory planning algorithm are as follows:

• The aircraft receives a set of navigation points and selects navigation points 1 and 2 as the start and end points, respectively.
• The aircraft uses the ITJS to generate the trajectory.
• The IHPO algorithm is used to optimize the parameter of the ITJS until the end condition is satisfied.
• The aircraft updates the start and end navigation points and goes back to step two until the last navigation point becomes the end point.

The flowchart of the algorithm is shown in Figure 7.

4. Simulation and Results

This section describes simulations performed to verify the effectiveness of the proposed four-dimensional trajectory planning algorithm. All simulations were tested in the Matlab2022b v9.13 environment on a PC with CPU i5-12500H 2.50GHz. The simulation results are as follows.

A three-dimensional trajectory of the aircraft is shown in Figure 8. The markers are the navigation points. The trajectory is smooth and passes through the designated navigation points in sequence. The arrival time of the aircraft is the same. This proves that the ITJS can generate the four-dimensional trajectory of the aircraft. The arrival time of the aircraft formation is shown in

Table 7. Arrival time of the aircraft formation.

Index of Navigation Points	Time/s
1	0
2	385
3	645
4	987.46716
5	1639.85352
6	2156.48321
7	2417.27828

. It should be noted that the time to reach the second navigation point and the third navigation point is the given time. The time to reach the other navigation points is optimized by the IHPO algorithm. The time setting is flexible.

The gap in the aircraft formation is shown in Figure 9. The results show that the distance is always greater than 2 km, which exceeds the minimum safety distance. The velocity and acceleration of the aircraft formation are shown in Figure 10 and Figure 11. They are all within the required range. The whole simulation time is 16.929 s, which is a short planning time and can meet the demand of the online trajectory planning.

At present, a variety of algorithms have been applied to four-dimensional track planning. This paper compares the proposed algorithm with the improved tau-G-PSO (ITG-PSO) algorithm [34], improved tau-H-PSO (ITH-PSO) algorithm [25], MPC algorithm [35] and unoptimized results. Under the same simulation conditions, the simulation results are shown as follows. The arrival time setting of unoptimized results is the same as in Table 7; the coupling rate is set to 0.3 [36]. The first and second arrival times of the improved tau-G-PSO algorithm and improved tau-H-PSO algorithm are set as in Table 7. The time to reach the other navigation points is optimized by the PSO algorithm. The expected arrival time and the expected trajectory of the MPC algorithm are calculated as in [35].

Figure 12 shows the convergence speed of each algorithm. Due to the difference in the cost functions, all the costs are normalized to present the results clearly. It can be seen from the results that the proposed algorithm has the fastest convergence speed, and the MPC algorithm has the slowest convergence rate.

The simulation results of the improved tau-G-PSO algorithm, improved tau-H-PSO algorithm, MPC algorithm, unoptimized results and the proposed algorithm are shown in

Table 8. Results of different optimization algorithms.

Optimization Algorithm	Trajectory Lengths/m	${\mathit{v}}_{\mathbf{min}}$/m/s	${\mathit{v}}_{\mathbf{max}}$/m/s	${\mathit{a}}_{\mathbf{max}}$/m/s	Simulation Time/s
Proposed algorithm	947,812	183.5	223.4	3.875	16.929
ITG-PSO	939,359	180.0	219.9	13.793	15.051
ITH-PSO	947,355	182.4	219.9	8.714	15.012
MPC	924,506	180	240	1	55.497
None	964,298	128.3	244.5	2.736	8.139

. It can be seen from the results that the velocity generated by the improved tau-G-PSO algorithm and improved tau-H-PSO algorithm meet the given constraints. However, their maximum acceleration exceeds the constraint condition, and the acceleration discontinuity exists at the navigation point (Figure 13). This is because the acceleration is not controlled by the two algorithms. This leads to serious errors in the application of the algorithm.

The expected trajectory designed in the MPC algorithm is directed from the starting point to the end point, so the generated trajectory length is minimal. The speed and acceleration of the MPC algorithm are also within the given range. However, the control parameter of the MPC algorithm is discontinuous (Figure 14), which greatly limits the application of the MPC algorithm. Moreover, the MPC algorithm needs to find the optimal control amount in each step, which greatly increases the calculation time of the MPC algorithm. Therefore, the application of the MPC algorithm is limited.

Through the above experiments, it can be proved that the proposed algorithm can solve the problem of the four-dimensional trajectory planning for a fixed-wing aircraft formation. The position, velocity and acceleration of the generated four-dimensional trajectory are continuous and smooth, which is necessary for the application of the algorithm to real problems. The flight time of each flight trajectory of the two aircraft is consistent and can be generated by the algorithm or set in advance. The flexible time adjustment option can extend the application scope of the algorithm. Compared to the other algorithms, can IHPO can avoid falling into local optima to generate the trajectory that satisfies the constraints. The time consumed is within a short range, which makes the online trajectory planning possible.

5. Conclusions

This paper proposes a four-dimensional trajectory planning algorithm for fixed-wing aircraft in complex environment. In this algorithm, the bio-inspired ITJS algorithm is used to generate the four-dimensional trajectory, and the IHPO algorithm is used to optimize the generated four-dimensional trajectory to meet the constraints.

The IHPO algorithm is proposed based on the HPO algorithm. It is compared with the other algorithms using benchmark functions, and the comparison results show that the IHPO algorithm has obvious improvements compared with the original HPO algorithm and the other algorithms. The final simulation results also show that, compared with the other optimization algorithms, the IHPO algorithm can find better results under restrictive conditions.

The calculation time of the four-dimensional trajectory planning algorithm for fixed-wing aircraft proposed in this paper only depends on the number of navigation points and the optimization times. It is not affected by the length of the flight trajectory. Compared with the flight time of the aircraft, the calculation time of this algorithm only takes 16.929 s. The short calculation time can make the algorithm meet the requirements of the online long-range trajectory generation for fixed-wing aircraft.

This paper also introduces the tau theory and the ITJS algorithm. The ITJS transforms the four-dimensional trajectory planning algorithm into a mathematical optimization problem. The ITJS solves the case in which the start and end velocity of tau-J guidance strategy is always zero. Moreover, compared with the improved tau-G and improved tau-H guidance strategies [25,37], the ITJS acceleration is continuous. It is more in line with actual requirements and reduces the error of the aircraft trajectory execution.

In summary, the proposed four-dimensional flight path planning algorithm can meet the requirements of fixed-wing aircraft in real world: the short calculation time and flexible time adjustment strategy make the algorithm applicable to various complex scenarios; the speed and acceleration within a given range enable the aircraft to correctly execute the given command.

However, the proposed algorithm also has some shortcomings. It can be seen from the results that the length of the trajectory generated by the proposed algorithm is longer than other algorithms. This may affect the performance of the generated trajectory and consume more computing resources. Moreover, in the real world, the acceleration of the aircraft is usually generated by the maneuvering of the aircraft and the engine, which are usually continuous and delayed. However, the jerk of the proposed algorithm is discontinuous, which may lead to errors in some cases. In the next stage, the IHPO algorithm will be further improved to enhance the optimization performance of the algorithm and speed up the optimization. An algorithm for jerk continuity is also necessary to be proposed to solve the problem of jerk discontinuity.