Particle Swarm Optimization (PSO), Genetic Algorithm (GA) and ACO are classical methods of path search. By comparing the performance of path planning based on different algorithms in raster map, the algorithm more suitable for graph theory is selected. Specifically, the static simulation environment (20 x 20) is designed, and the starting point and ending point of the target are known in the environment. PSO, GA and ACO are used for shortest path planning. The parameters of ACO are set as follows: the number of ants M=10, the maximum number of iterations Nmax=50, information heuristic operator α=1, expectation heuristic factor β=5, pheromone intensity Q=5, pheromone residue operator ρ=0.7. The parameters of PSO are set as follows: particle size S=20, number of iterations Gmax=50, inertia weight operators w=0.5, acceleration operators c1=c1=1, r1=r2=0.3. The parameters of GA are set as follows: population size N⁢P=20, iteration times Gmax=50, crossover rate pc=0.7, mutation rate Pm=0.05.

Figure 1.

Path planning results of different algorithms.

1) The coordinate system is built with the target as the center, and the distance between the initial position of the UAV and the target is the far boundary. Extraction of terrain, wind speed, and visibility within the airspace. Due to the wide range of airspace, an adaptive spatial distribution decomposition of airspace based on obstructive factors is proposed. Among them, the obstacle factor is the environmental factor that causes the UAV's flight direction to be impossible and the heading direction must be changed [28]. In this paper, elevation greater than 5200 m, visibility less than 0.8 km, and wind speed greater than 25 m/s, are considered as obstruction factors. As shown in Figure 3(a), there are two obstacle factors in the airspace, which is divided into three-ring layers.

2) To reduce the evaluation errors among sub-regions, each ring layer is divided into 32, 16, and 8 sub-regions from outer to inner layers. As illustrated in Figure 3(b), the first layer (o-R1) of the ring region is divided into eight sub-regions. The second layer of the ring region (R2-R1) is divided into 16 sub-regions. The third layer (R3-R2) is divided into 32 sub-regions.

Figure 3.

Spatial decomposition based on obstacle ring.

During UAV flight operations, this study primarily considers the impacts of elevation, visibility, and wind speed on reconnaissance mission performance. Through information fusion, we subsequently evaluate the flight selection probability within each sub-region.

(1) Most UAVs adopt low-altitude penetration tactics, so the terrain elevation has a decisive effect on the heading. The lower the ground elevation, the stronger the shielding ability, and the greater the selection possibility Pe.

(2) When the visibility is below 1.22 km, the clarity of vision is poor, which seriously threatens the flight safety of the UAV, and its flight direction selection is limited. When the visibility is between 1.22 km and 6 km, it significantly affects the clarity of vision and the degree of radar attenuation. When the visibility exceeds 6 km, the possibility Pv of choosing the direction is higher.

(3) With the increase in wind speed, the control performance and maneuvering stability of the UAV deteriorate. Therefore, when selecting the flight direction, it often avoids regions with high wind speeds. When the wind speed is less than 1.5 m/s, it does not affect the UAV's flight, and the selection possibility is high. Wind speed starts to impact operational performance in the range of 1.5 m/s to 20 m/s, and the selection possibility decreases as wind speed increases. When the wind speed exceeds 30 m/s, it becomes unsafe to fly, and the selection possibility Pw is low.

Combined with the grading standards of terrain and meteorology, this paper makes a fuzzy representation of environmental factors, as shown in Table 1.

Table 1

Classification of heading selection probability.

Then the calculation formula based on multi-source information fusion of the heading selection probability is shown in Eq.(12), where S(i,j)′, S(i,j) represent the barrier area and total area of the subregion (i,j) respectively.

P c ⁢ s 1 = S ( i , j ) ′ S ( i , j ) ⁢ ( 0.4 × P e + 0.3 × P v + 0.3 × P w )

The course of UAV is described by selecting the sub-region sequence set, c⁢o⁢u⁢r⁢s⁢e={r⁢e⁢g⁢i⁢o⁢n1,⋯,r⁢e⁢g⁢i⁢o⁢nN}.

The quality of feasible solution is evaluated from sequence length, possibility, and tortuosity. The length of the sequence is not only limited by the flight range of UAV, but also affects the flight time; a shorter sequence length results in a shorter flight time, which is crucial for completing the flight task efficiently. A higher mean probability of the sequence reduces the risk to the UAV. The sequence's tortuosity should be as smooth as possible due to flight maneuverability constraints. We propose that the direction sequence quality be evaluated according to Eq.(13).

J c ⁢ o ⁢ u ⁢ r ⁢ s ⁢ e = w P L ⁢ P L ⁢ ( c ⁢ o ⁢ u ⁢ r ⁢ s ⁢ e ) + w P c ⁢ s ⁢ P c ⁢ s ⁢ ( c ⁢ o ⁢ u ⁢ r ⁢ s ⁢ e ) + w P θ ⁢ P θ ⁢ ( c ⁢ o ⁢ u ⁢ r ⁢ s ⁢ e )

where PL⁢(c⁢o⁢u⁢r⁢s⁢e), Pc⁢s⁢(c⁢o⁢u⁢r⁢s⁢e), Pθ⁢(c⁢o⁢u⁢r⁢s⁢e)∈(0,1) represent the length ratio, mean probability, tortuosity of sequence respectively. The calculation formulas are as shown in (14). And wPL, wPc⁢s, wPθ are the corresponding weights.

P L ⁢ ( c ⁢ o ⁢ u ⁢ r ⁢ s ⁢ e ) = L m ⁢ a ⁢ x − L c ⁢ o ⁢ u ⁢ r ⁢ s ⁢ e L m ⁢ a ⁢ x − L m ⁢ i ⁢ n P c ⁢ s ⁢ ( c ⁢ o ⁢ u ⁢ r ⁢ s ⁢ e ) = C ⁢ S c ⁢ o ⁢ u ⁢ r ⁢ s ⁢ e N P θ ⁢ ( c ⁢ o ⁢ u ⁢ r ⁢ s ⁢ e ) = θ m ⁢ a ⁢ x − θ c ⁢ o ⁢ u ⁢ r ⁢ s ⁢ e θ m ⁢ a ⁢ x

where Lc⁢o⁢u⁢r⁢s⁢e represents total length of sequence sets, Lm⁢i⁢n is the straight-line distance from the flight start point to the end point, Lm⁢a⁢x=Lm⁢i⁢n+2⁢π⁢Lm⁢i⁢n. C⁢Sc⁢o⁢u⁢r⁢s⁢e represents total probability of sequence sets, N is the number of sub-regions in the course sequence set. θm⁢a⁢x is a constant, representing the maximum allowable direction tortuosity of UAV.

When the flight environment is complex, ACO is challenged due to the large search space when selecting the next region, leading to extended processing time and subpar real-time performance. To address this issue, an improved ant colony algorithm (IACO) is proposed.

Figure 5.

Schematic of path safety.

(1) The objective function of 3D path planning is similar to that of heading planning, as both have specific requirements for path length and smoothness. The difference lies in the calculation method. Moreover, flight path planning also takes into account the safety of UAVs.

F = ω 1 ⁢ F l + λ 1 ⁢ ω 2 ⁢ F e + λ 2 ⁢ ω 3 ⁢ F s

where, Fl, Fe, Fs are the three cost functions, ω1, ω2, ω3 are the corresponding weight coefficients, λ1, λ2 are the scaling factors. The smaller the value of the objective function, the better the quality of the path.

1) Path length

F l = ∑ i = 1 n ( x i + 1 − x i ) 2 + ( y i + 1 − y i ) 2 + ( z i + 1 − z i ) 2

where n represents the number of track points, and (xi,yi,zi) represents the position of track point Pi.

2) Path smoothness The 3D path design must meet its maximum climb angle and climb rate limits to ensure the curvature of the path and avoid sharp changes in direction and height fluctuations. li represents the vector between two track points. Eqs.(22) and (23) respectively represent the yaw angle φi and the pitch angle ϕi. The cost function of path smoothness can be defined as Eq.(24).

φ i = arccos ⁡ ( l i ⋅ l i + 1 ‖ l i ‖ × ‖ l i + 1 ‖ )

ϕ i = arctan ⁡ ( z i + 1 − z i ( x i + 1 − x i ) 2 + ( y i + 1 − y i ) 2 )

F e = ∑ i = 1 n − 2 φ i + ∑ i = 1 n − 1 ( ϕ i − ϕ i − 1 )

3) Path safety In order to ensure the safe flight of UAVs, a safety-oriented mechanism must be incorporated into the path planning process. As shown in Figure 5, there is an obstacle K in the airspace, whose central coordinate is Ok and radius is Rk. The vertical distance dk between the flight node of the UAV and the obstacle should be greater than the safety distance threshold S=800 m.

F s = { 0 , d k ≥ S S − d k S , R k < d k < S ∞ , d k < R k

Figure 6.

The influence of three main parameters.

(1)Initialization strategy based on Tent Chaos When dealing with complex optimization problems, the original DBO randomly generates a population for initialization. However, this approach may lead to low population diversity, uneven population distribution, and rapid convergence to local optimal solutions. Tent Chaotic mapping offers better uniformity and correlation. Therefore, we utilize it as a new initialization method to enhance the population diversity of DBO. The formula is presented in Eq.(29).

x i + 1 = { x k α , 0 < x k ≤ α ( 1 − x k ) ( 1 − α ) , α < x k ≤ 1

(2)Reverse learning strategy

Since the global search ability of reproduction and stealing behavior decreases with the increase of iterations, a random reverse learning strategy is introduced to enhance the global search ability of these two behaviors.

x r t = l ⁢ b + u ⁢ b − r ⁢ d × x i t x i t = { x i t , f ⁢ ( x i t ) < f ⁢ ( x r t ) x r t , f ⁢ ( x r t ) < f ⁢ ( x i t )

where, xrt is the reverse solution, l⁢b and u⁢b are D-dimensional vectors representing the lower and upper bounds of each dimension, and r⁢d is the D-dimensional random vector of (0,1), xit is the current feasible solution.

Figure 7.

Heading planning results in M1.

The setting of hyperparameters significantly impacts on the performance of evolutionary algorithms. Since there are no theoretical methods to determine the best parameter combination, hyperparameters must be fine-tuned and optimized through experiments to ensure the algorithm can find high-quality solutions efficiently [19]. In the ACO, pheromone and heuristic information play crucial roles in ant selection. Therefore, we focus on three key parameters: pheromone important parameter α, heuristic factor important parameter β, pheromone evaporation coefficient ρ. The other initial parameters are: the total number of ants is 50, the maximum number of iterations is set to 100, and pheromone increase intensity coefficient Q=20.

This paper analyzes the influence of different values of α∈{1,1.5,2,2.5,3}, ρ∈{0.1,0.2,0.3,0.5,0.8}, and β∈{1,3,5,7,9} on the convergence results. Set a set of default values: α=2, β=3, ρ=0.2. When analyzing the values of multiple parameters using the control variable method, one parameter's value is changed at a time while keeping the other parameters at their default values. To enhance the algorithm's robustness, the selected parameter combinations from each group are simulated in 20 random two-dimensional probability maps, and the average convergence values are compared and analyzed. The experimental results are shown in Figure 6.

It can be seen from the results in Figure 5 that when α=2, β=5, ρ=0.3, the average convergence value reaches the maximum value, and the influence of α on the average convergence result fluctuates the most. Therefore, the combination of α=2, β=5, ρ=0.3 is adopted in the subsequent experiments in this paper.

In order to further verify the effectiveness of heading planning based on IACO, the heading sequence is planned in the same 2D probability map (M1 and M2) based on D*, ACO, and GA. Figure 6 and Figure 7 display the planning results in M1 and M2, respectively.

Figure 9.

Convergence curves of three algorithms.

Compared with heading prediction in a 2D map, the algorithm's global search capability and search efficiency are more critical for the precise planning of a 3D path. DBO, Improved Grey Wolf Optimizer (IGWO), and Sparrow Search Algorithm (SSA) are comparisons to verify the optimization performance of AMDBO. Six functions with different optimal characteristics are compared, and their specific details are presented in Table 3. F1 and F2 are single-peak reference functions that are used to evaluate the algorithm's single-objective solving capability. F3-F4 represent multi-modal reference functions, while F5-F6 are mixed reference functions that assess the algorithm's ability to escape local optima.

(1) Performance analysis In order to enhance the reliability of the experimental results, this paper adopts a consistent experimental setting by fixing the population size of each algorithm at 30 and the number of iterations at 500. Thirty independent experiments are performed on each function to obtain the target solution under each run. The average value (AVE), best value (BEST), and standard deviation (STD) are then calculated. These values serve as indicators to evaluate the reliability and stability of the optimization algorithm. Table 4 displays the test results of the four optimization algorithms on the six test functions.

Figure 13.

Heading sequence planning results derived from the multi-source data-fused two-dimensional map.

AMDBO introduces three improvement strategies to enhance the DBO algorithm. By implementing these strategies, three different algorithms derive through individual enhancements to DBO. The algorithms are as follows: 1) DBO1, which incorporates the Tent Chaos Initialization strategy. 2) DBO2, which integrates the reverse learning strategy of breeding and stealing. 3) DBO3, which includes the adaptive mutation strategy. These four algorithms are evaluated on test functions , and the AVE, BEST and STD of each algorithm are calculated after 30 independent runs. The experimental results are presented in Table 5.

In summary, the average optimization accuracy of a single improved strategy is higher than that of DBO. However, its variance is worse than that of DBO in test functions, indicating that the stability of a single improved strategy is insufficient. Compared with DBO, the performance indicators of AMDBO are improved, indicating that the comprehensive improvement of the three strategies is effective.

Taking the endpoint as the center of the circle and the distance between the starting point (-674,156) and the endpoint as the radius, the environmental information such as elevation, visibility, wind speed, and obstruction factors in this vast airspace is simulated, as shown in Figure 11. Then, the UAV trajectory is planned based on the fused multi-source environmental information.

According to the spatial distribution of obstacle factors in Figure 11(d), the airspace is adaptively decomposed into 80 km, 200 km, 310 km, 460 km, 540 km, and 700 km. The environmental situation information in each sub-area is shown in Figure 12. Comparing it with the actual scene grid information in Figure 11(a-c), it can be observed that the decomposition method accurately reflects the changing trend of environmental information while reducing the number of subregions. The results indicate that the adaptive spatial decomposition method effectively addresses the issue of sparse environmental information.

Based on Eq.(12), the 2D probability map is constructed, as shown in Figure 13(a). The IACO is used to search the heading sequence, and the result is shown in Figure 13(b). The flight direction gradually changed from 292.5° to 225°-270°. The average choice possibility of the route is 0.46, and the minimum value is 0.32, indicating that flight safety is relatively reliable. The tortuous degree of the heading sequence is 0.52, and the length is 1075.1 km, which is relatively smooth and short.

The AMBDO is used to plan a 3D path in regions where the heading sequence is located. The results are shown in Figure 14. The length of the 3D path is 4536 km, the average height is 4674 m, and the average tortuosity is 7.9. The 3D path is relatively smooth, and obstructions can be avoided.

In order to verify the advantages of the two-layer path planning model proposed in this paper, the AMDBO is used to plan the path directly in 3D space. The planning results are shown in Figure 15. It is evident from Figure 13 that feasible paths are successfully planned using the three methods. The paths generated by DBO are distributed in regions with elevated terrain. To ensure a safe distance from the ground, the resulting paths are at a higher altitude and exhibit a greater tortuous degree. The other two methods exhibit a similar path trajectory, opting to first navigate towards lower-lying areas.

Figure 16.

Comparison of iterative convergence curves.