A Glowworm Swarm Optimization Algorithm for Uninhabited Combat Air Vehicle Path Planning

2.1 Problem Description

Path planning for UCAV is the design of an optimal flight route to meet certain performance requirements according to the special mission objective and is modeled by the constraints of the terrain, data, threat information, fuel, and time [13]. In this article, the route-planning problem is first transformed into a D-dimensional function optimization problem (Figure 1).

Figure 1.

Coordinates Transformation Relation.

In Figure 1, we transform the original coordinate system into a new coordinate whose horizontal axis is the connecting line from the starting point to the target point, according to the transform expressions shown in Equations (1) and (2), where point (x,y) is the coordinate in the original ground coordinate system O_XY, point (x′, y′) is the coordinate in the new rotating coordinate system O_X′Y′, θ is the rotation angle of the coordinate system, A and B are the start and end points, respectively, |AB→| is the Euclidean distance from the starting point to the target point, and (x₁, y₁) and (x₂, y₂) are the coordinate of the starting point and the target point in the original coordinate system, respectively.

Then we divide the horizontal axis X′ into D equal partitions and then optimize the vertical coordinate Y′ on the vertical line for each node to get a group of points composed of the vertical coordinate of the D points. Obviously, it is easy to get the horizontal abscissas of these points. We can get a path from the start point to the end point by connecting these points together so that the route-planning problem is transformed into a D-dimensional function optimization problem.

2.2 Performance Indicator

A performance indicator of path planning for UCAV is mainly the completion of the mandate of the safety performance indicator and fuel performance indicator, for example, indicators with the least threat and the least fuel.

• Minimum of performance indicator for threat:

  (3)minJt = ∫0Lwtdl,  where Listhelengthofthepath. (3)

• Minimum of performance indicator for fuel:

  (4)minJf = ∫0Lwfdl,   where  Listhelengthofthepath. (4)

• Total performance indicators for UCAV route:

  (5)minJ = kJt + (1 − k)Jf, (5)

where w_t denotes the threat cost for each point on the route, w_f denotes the fuel cost for each point on the path that depends on path length (in this article, w_f ≡ 1), k(∈[0, 1]) is a balanced coefficient between safety performance and fuel performance, whose value is determined by the special-task UCAV performance; for example, if flight safety is of vital importance to the task, then the larger k is chosen, whereas if the speed is critical to the aircraft task, a smaller k is selected. For the same battlefield environments, the total cost value will decrease with the increase in k. In our experiment, we let k = 0.5 (to further clarify, see Section 6.1).

2.3 Threat Cost

When the UCAV is flying along path L_ij, the total threat cost generated by N_t threats is calculated by

To simplify the calculations (as shown in Figure 2), each edge is divided into five equal partitions and the threat cost of every edge is calculated via the five midpoints of the five equal partitions. If the distance from the threat point to the edge is within the threat radius, we can calculate the corresponding threat cost according to

Figure 2.

Calculation for Threat Cost.

where L_ij is the length of the subsegment connecting node i and node j, d0.1, k4 is the distance from the 1/10 point on subsegment L_ij to the k^th threat, t_k is the threat level of the k^th threat. Moreover, it can simply consider the fuel cost w_f to L. As fuel cost is related to flight length, we can consider w_f to L≡, for simplicity, and the fuel cost of each edge can be expressed by w_f,Lij = L_ij.

3.1 Glowworm Distribute Phase

A set of n glowworms are randomly distributed in different locations of the search space. All glowworms carry an equal quantity of luciferin l₀.

3.2 Luciferin Update Phase

The luciferin update phase depends on the objective function value and previous individual luciferin level. The luciferin update rule is

where l_i(t) denotes the luciferin value of glowworm i at the t^th iteration, ρ and γ are the luciferin decay and enhancement factor, respectively, and J(x_i(t)) is the value of the objective function at the location of glowworm i.

3.3 Glowworm Movement Phase

During the movement step, each glowworm, according to a probabilistic mechanism, move toward a neighbor that has more luminosity than its own. For each glowworm i, the probability of moving toward a neighbor j is

where j∈N_i(t), N_i(t) is a set, which can be confirmed by

where d_ij(t) denotes the Euclidean distance between glowworms i and j at the t^th iteration and rdi(t) represents the variable neighborhood range associated with glowworm i at the t^th iteration. Then the model of the glowworm movement is

where x_i(t)∈R^m is the location of glowworm i at the t^th iteration in the m-dimensional real space R^m, ∣∣·∣∣ represents the Euclidean norm operator, and st(> 0) is the step size.

3.4 Neighborhood Range Update Phase

Assuming r₀ as an initial neighborhood domain for all glowworm, the neighborhood domain of each glowworm is given in the following in every generation:

where β is a constant, n_t is a parameter to control the number of neighborhoods, r_s denotes the sensory radius of glowworm, and N_i(t) denotes neighborhoods set. The pseudo-code of GSO is given in Algorithm 1.

4.1 Individual Location Update Model of PGSO

The basic GSO algorithm has shown some weaknesses in global and high-dimension search such as slow convergence rate and low computational accuracy. The slow convergence rate is caused mainly by the individual location update model of the original GSO, which only contains local information. It is obvious that a glowworm individual changes its position only via local information, which slows the global convergence rate. The low computational accuracy is because of the individual updating its location in all dimensions at the same time. Sometimes, the objective function value is likely to deteriorate in a high-dimensional space if independent variables change their value in all dimensions at the same time. Inspired by the PSO, we proposed an individual PGSO location update according to Equations (13) and (14). According to Equation (13), individual location updates relate to both local and global information. In addition, the update dimensions of individual locations decrease gradually along with the increase in generations.

where x_i(t) is the location of individual i at the t^th iteration, x_j(t) is the location of the i’s neighbor that is selected, x_gb(t) is the location of the global optimal individual, c1 and c2 denote the acceleration factors, and rand is a random number.

where m(t) is the update dimensions of location at the t^th iteration, D is the problem dimension, and gen is the maximum evolution generations.

4.2 Parallel Hybrid Mutation

The idea of parallel hybrid mutation is derived from Ref. [7], in which the authors have concluded that this model can be used to improve population diversity. The detailed process of parallel hybrid mutation is given as follows:

1. Solve Equation (15) to set the mutation capability value of each individual,

   (15)mci = 0.05 + 0.45(exp(5(i − 1)/(n − 1)) − 1)exp(5) − 1, (15)

   where n denotes the population size, i denotes the individual serial number, and mc_i denotes the mutation capability value of an individual, which is represented by i. Figure 3 shows an example of the mutation capability value assigned for 40 glowworms. Each glowworm corresponds to a mutation capability value, which ranges from 0.05 to 0.5.

2. Choose the mode of mutation according to Algorithm 2. p_u is the mutation factor that denotes the ratio of the uniform distribution mutation; correspondingly, 1 – p_u is the ratio of the Gaussian distribution mutation. The Gaussian(σ) returns a random number drawn from a Gaussian distribution with a standard deviation σ. Ceil(p) generates the elements of p to the nearest integers greater than or equal to p. Here we adopt linear mutation factor, and the function is given as

Figure 3.

Individual Mutation Capability Curve.

where t denotes the current generation and gen denotes the maximum generation.

4.3 Local Searching Strategy

In Ref. [10], the authors presented a modified GSO algorithm, which was inspired by the glowworms’ mating behavior. Enlightened by this idea, we introduced the strategy of local searching in PGSO. The strategy of local searching is somewhat similar to the above-mentioned idea but is not the same with it. In PGSO, we implement local search in the global optimal individual vicinity of each five generations. For each local search, we implement a search near the global optimal individual. If the objective value of a new position is better than the original position, the global optimal individual moves to the new position and the search is terminated. Local search can be performed four times at most. Algorithm 3 is the local search algorithm.

where x_gb(t) denotes the global optimal position of the t^th generation, st(t) denotes the step size of the t^th iteration, and x_gb(t) denotes the new position. The value of st(t) is calculated via

where st(0) is the initial step size, t is the current number of iterations, gen is the maximum number of iterations, and 10^–4 is the lower bound of the step size, whose value can be found in Reference [15]. The PGSO algorithm pseudo-code is as follows:

5.1 PGSO for Solving UCAV Path-Planning Algorithm

In PGSO, the standard ordinates are inconvenient in directly solving UCAV path planning. To apply PGSO to UCAV path planning, one of the key issues is to transform the original ordinate into rotation ordinate using Equations (1) and (2). Then we divide the horizontal axis X′ into D equal partitions and optimize the vertical coordinate Y′ on the vertical line for each node to get a group of points composed of the vertical coordinate of the D points. We can get a path from the start point to the end point by connecting these points together, so that the route-planning problem is transformed into a D-dimensional function optimization problem. At the beginning, we let every individual denote a feasible solution. The luciferin value of every glowworm is related to the objective function value of its location. If an objective value is smaller, its luciferin is larger. By running PGSO, worms will move to the global optimal location. The detailed PGSO process for UCAV path planning is shown in Algorithm 5.

5.2 Parameters Analysis

To analyze how these two parameters, c1 and c2, influence optimization results, we run our proposed algorithm for the UCAV path-planning problem with c1 = 1.0, 1.1, 1.2, …, 2.0 and c2 = 1.0, 1.1, 1.2, …, 2.0. In the parameter analysis tests, the evolutionary generation, population, and problem dimensions are 200, 30, and 20, respectively. The optimization results are shown in Table 1. Each data in Table 1 is the mean value of the PGSO algorithm independently running 30 times. From Table 1, we can see that the minimum threaten cost value 52.31 is acquired when the parameter combination is c1 = 1.2 and c2 = 1.7. Hence, the parameter value of c1 and c2 are set to 1.2 and 1.7, respectively, on the simulation experiments in this article.

6.1 The Value of the k Setting

To illustrate how the value of k in the model impacts on the total cost, we let our algorithm run 30 times in the same battlefield (see Table 3). Table 2 shows the mean total cost in the same battlefield when the value of k increases from 0.1 to 0.9. It is obvious that the total cost value decreases with the k increases in the same battlefield. To reconcile with Ref. [9], we let k = 0.5 in our experiment, which means that we assume that the importance of safety and fuel is equal.

6.2 Experimental Platform and Parameter Settings

All the experiments were implemented on a PC with an AMD II X4 640 running at 3.0 GHz, 4.00 MB of RAM, and a hard drive of 500G. Our implementation was compiled using MATLAB R2011b (7.1) running under Windows 7. PGSO algorithm parameter setting is ρ = 0.4, β = 0.08, γ = 0.6, n_t = 5, l₀ = 5, st(0) = 0.03, σ = 1, c1 = 1.2, c2 = 1.7.

6.3 Wilcoxon Rank Sum Test

To ensure the effectiveness of PGSO, we adopted the nonparametric Wilcoxon rank sum tests to determine whether the difference is significant or not. We implement the Wilcoxon rank sum tests when the population size is 30, maximum generation is 200, and dimension is 20. Table 4 shows the test results. The test criterion can be found in Equation (19).

If P is less than α, two algorithm experiment data are statistically different. If P is equal or greater than α, two algorithm experiment data are not statistically different. From Table 4, we can observe that the performance of PGSO is statistically different from the other five algorithms.

6.4 Comparing the Worst, Best, and Mean Optimization Results

From Tables 5–7., we see that the performance of PGSO is better than ACO, BBO, ES, PBIL, PSO, and SGA on different evolutionary generations. Compared with FA, the mean and worst values of our algorithm are better; however, the best is not good as FA. The performance of PGSO is worse than MFA on different evolutionary generations.

From Tables 8–10, we understand that PGSO performance is better than 10 other algorithms on dimensions 5 and 10. With the increase in problem dimension, the performance of DE, FA, MFA, and SGA becomes better than the represented algorithm. The performance of the proposed algorithm is better than ACO, BBO, ES, PBIL, and PSO on all test dimensions.

6.5 Comparing the Standard Deviation

To further prove the effectiveness of PGSO, we choose six algorithms with better performance and test their standard deviation. The data can be found in Tables 11 and 12.

Table 11 shows that the standard deviation values of six algorithms on different evolutionary generations. It is obvious that the performance of PGSO is more superior than other algorithms except MFA.

Table 12 shows that the standard deviation values of six algorithms on different dimensions. It is obvious that the performance of PGSO is more superior than the performance of DE, FA, and PSO on all different dimensions and has a more superior performance than SGA with the exception of dimensions 10 and 15. The MFA has the more superior performance than four other algorithms. Figure 4 is the best optimization for the UCAV flying trajectory when problem dimension, population size, and evolutionary generation are 20, 30, and 200, respectively. Figure 5 is the convergence graph. The coordinates of the best optimization path of UCAV are given in Table 13.

Figure 4.

UCAV Flying Trajectory.

Figure 5.

Convergence Curves.