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:
where
is the distance between the target and the actor, and
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
satisfy the following expression:
where
is the rate of
. Based on the tau-coupling strategy, Zhang et al. [
24] proposed the tau-J guidance strategy, using constant jerk motion
as the coupled motion. The tau variable expression of this motion is as follows:
where
is the constant jerk,
is the expected arrival time, and
is the current time. The expression of the movement coupled with the jerk motion is as follows:
where
. 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:
where
is the initial position,
is the initial velocity,
is the initial acceleration, and
is the constant jerk. Based on the tau-coupling strategy, the coupled motion strategy can be obtained as follows:
where
are the target position, target velocity and target acceleration, respectively.
are the coupled motion, their expression is equivalent to Equation (6). The expression for
is as follows:
Let
and put the coupled motion back into Equation (7), then we can obtain the following:
By substituting Equation (9) into the first formula of Equation (6), we can obtain the following:
Therefore, parameters of the coupled motion can be given the initial values . 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).
3.2. Original HPO Algorithm
The ITJS simplifies the aircraft trajectory into a mathematical problem determined by parameters and expected arrival time . 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:
where
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
is a random vector with all elements in the range
. Value
denotes the dimension of
. Values
represent the upper and lower ranges of the search areas in each dimension of
, respectively. Value
represents a random vector with all elements in the range (0, 1). The hunter’s position update algorithm is as follows:
where
are the positions of the agents of the next time and the current time, respectively;
denotes the position of the prey’
is an adaptive parameter. The adaptive parameter’s expression is as follows:
where
is the random number of
,
are random vectors in dimension
of
.
is the dot product. The value of
continues to change as the number of iterations changes.
is an adaptive parameter used to balance wide-area searches with careful searches of specific areas. When
is larger, the particles search the results in a larger search range, and the search is more random; when
is smaller, the particles search the area more carefully.
The expression of
is shown in Equation (16). As the number of iterations increases, the value of
decreases continuously, and the search agents search the area more carefully.
where
represent the current iterations and the maximum iterations, respectively. Value
represents the average position of all search agents, and its equation is as follows:
The distance between each search agent and
is as follows:
The search agents that have the farthest distance from the average distance are regarded as the location of the prey (
):
where
is the index of the maximum value of
. 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:
where
is the number of search agents. This expression indicates that
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:
where
is the global optimal position, and
is a random number of
. 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:
where
is the random number of
, and
is the adjustment parameter set to 0.1. When
, the search agent is treated as the hunter, and the position is updated using Equation (22a). When
, 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.
3.4. 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 and
Table 5. 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 “
” 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 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:
where the optimization objective
is used to find the minimum length of the aircraft formation;
denotes the trajectory length of the
i-th aircraft;
denotes the relative distance between the first and second aircraft;
denote the acceleration and speed of the
i-th aircraft;
denote the maximum acceleration, minimum speed and maximum speed of the aircraft, respectively. From Equation (27), the cost function is given as follows:
where
represent the weight. Value
represents the penalty term of relative distance. When relative distance is less than 1000 m, the penalty gradually increases. Values
are the standard deviation of speed and acceleration of the
i-th aircraft. Their purpose is to limit the range of speed and acceleration.
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.