Target position estimation method based on composite whale optimization algorithm

Abstract

The invention belongs to the technical field of positioning, and particularly relates to a target position estimation method based on a composite whale optimization algorithm, and the method comprises the following steps: S1, after a plurality of monitoring stations receive signals transmitted by a target, employing a generalized cross-correlation method to estimate the time delay inequality [delta] t1j between the monitoring stations and a reference monitoring station, then calculating the time when the signal arrives at the remaining monitoring stations according to the delay inequality, converting a plurality of unknown time parameters into parameters only related to the time parameters of the reference monitoring station, calculating the distances r1, r2 and r3 between the monitoring stations and the target point according to the signal propagation speed, and establishing a TDOA-based positioning model by taking the monitoring stations as the circle center and the distances as the radius. According to the method, the bionic intelligent optimization algorithm is integrated into the TDOA positioning algorithm, the TDOA positioning problem is converted into the optimal value searching problem, the whale optimization algorithm is improved by adopting the nonlinear factor, the adaptive inertia weight and the crossover variation strategy, and the algorithm performance is improved.

Description

Technical Field

[0001] The present invention relates to the technical field of positioning, and specifically to a method for estimating the target position based on a composite whale optimization algorithm. Background Technique

[0002] With the rapid development of the Internet of Things and wireless sensor networks, positioning technology has become one of the key technologies in many application fields, such as intelligent transportation, indoor navigation, and drones. According to different technical principles, positioning technologies can be divided into the following types: the direction-of-arrival (AOA) positioning technology that uses azimuth angle information for positioning, the time-difference-of-arrival (TDOA) positioning technology that locates by the time difference of signals arriving at different observation stations, and the frequency-difference-of-arrival (FDOA) positioning technology that locates based on the Doppler frequency shift caused by the relative motion between the target and the observation station. Among them, the positioning method based on time difference has higher positioning accuracy compared to other methods and has been more widely applied.

[0003] The positioning method based on TDOA establishes an overdetermined nonlinear equation system of TDOA according to the time delay of the target arriving at two monitoring stations, and the solution of the equation system is the estimated position of the target. Common solution algorithms include non-iterative methods, such as the least squares method, Fang algorithm, and Chan algorithm. There are also iterative algorithms, such as the Taylor algorithm. There are also some that transform the equation system into a geometric problem, and the above algorithms have obvious defects, such as large positioning errors in a high-noise background and the need for initial position estimation. At the same time, bionic intelligent optimization algorithms are also applied to the positioning problem, such as the whale optimization algorithm, particle swarm optimization algorithm, and grey wolf optimization algorithm.

[0004] For positioning technology, positioning accuracy, positioning time, and anti-interference ability are the key indicators to measure the excellence of a positioning method. Currently, traditional bionic optimization algorithms have problems such as slow iteration speed, easy to fall into local optimal solutions, and large errors in traditional positioning methods. Therefore, we propose a method for estimating the target position based on a composite whale optimization algorithm to solve the above problems. Summary of the Invention

[0005] (1) Technical Problems to be Solved

[0006] Aiming at the deficiencies of the existing technology, the present invention provides a method for estimating the target position based on a composite whale optimization algorithm, which solves the problems raised in the above background technique.

[0007] (2) Technical Solutions

[0008] To achieve the above object, the present invention specifically adopts the following technical solutions:

[0009] A method for estimating the target position based on a composite whale optimization algorithm, comprising the following steps:

[0010] S1: After multiple monitoring stations receive the signals emitted by the target, the generalized cross-correlation method is used to estimate the time delay difference Δt between the monitoring station and the reference monitoring station 1j , and then the time when the signal arrives at the remaining monitoring stations is calculated according to the time delay difference, so that multiple unknown time parameters are converted into only those related to the time parameter of the reference monitoring station, and the distances between the monitoring stations and the target point are calculated according to the signal propagation speed, which are r 1 , r 2 , r 3 respectively. Taking the monitoring station as the center of the circle and the distance as the radius, a TDOA-based positioning model is established;

[0011] S2: After establishing the TDOA positioning model, r 2 , r 3 are represented by r 1 , then the position of the target point is only related to r 1 . Once the distance r 1 of the reference monitoring station is determined, the estimated target position can be determined. Once the optimal distance value is found, the optimal target estimated position can also be determined, that is, the intersection of multiple circles is the optimal target estimated position, and the solution space of the optimal solution is determined according to the geometric interpretation of the mathematical model;

[0012] S3: Use the composite whale optimization algorithm to search for the optimal solution in the solution space to determine the estimated position of the target. First, initialize the whale population, let the population be distributed in the solution space, set the fitness function according to the geometric interpretation of the positioning mathematical model, calculate the fitness function of all current whales, and the position of the whale with the minimum fitness represents the current global optimal position. Compare the minimum fitness with the threshold to determine whether the current global optimal solution meets the iteration end condition. If it meets, output the current optimal position. If it does not meet, update the whale position. After updating the position, recalculate the fitness of all whales until the whale position that meets the end condition is found. At this time, the whale position is the global optimal estimated position.

[0013] Furthermore, in S1, the generalized cross-correlation method is used to calculate the time delay difference Δt between the reference monitoring station and other monitoring stations 1j , then the times when the target reaches the three monitoring stations are: t, t + Δt 12 , t + Δt 13 ; Convert the three unknown parameters into one unknown parameter.

[0014] Further, the positioning equation based on TDOA established in S1 is a circular positioning equation with the monitoring station as the center and the distance as the radius, which is:

[0015]

[0016] where (x, y) is the position of the target, and (x i , y i ) is the position of the monitoring station.

[0017] Further, the specific content of solving the solution space in S2 is: when two circles intersect, the radii of the two circles should satisfy:

[0018]

[0019] r i + r i + Δr ij ≥ R ij

[0020] r i ≥ (R ij - Δr ij ) / 2

[0021] When the target is located within the triangular region formed by three nodes, the maximum radius of the circle should satisfy the formula:

[0022] r i ≤ max(R ij )

[0023] Therefore, the solution space of the reference radius is: R min ≤ r i ≤ R max , where R min = min[(R ij - Δr ij ) / 2], R max = max(R 1j ).

[0024] Further, the specific content of setting the fitness function in S3 is: first, determine the distance of each current whale from the reference monitoring station. Since the position of the whale is known when initializing the whale distribution, the reference radius of each whale is also known, denoted as r 1 ;

[0025] When the reference radius is r 1 , according to the circular positioning equation, we can obtain:

[0026]

[0027] Let

[0028] where R ij = r j - r i , (x i , y i ) are the coordinates of the monitoring station; r 1 is the reference radius;

[0029] Then the solution of the current value can be obtained as:

[0030] Then the fitness function is:

[0031] Furthermore, when comparing the fitness value with the threshold in S3, a suitable threshold T needs to be set. When f(r 1 ) ≤ T, the search can be stopped; the size of the threshold is set according to the computing performance and specific application scenarios. Under the same computing performance, the smaller the threshold, the smaller the positioning error, but the longer the positioning time required.

[0032] Furthermore, the composite whale optimization algorithm adopted in S3 is the multi-strategy improved whale optimization algorithm, which incorporates a non-linear function, particle swarm adaptive inertia weight, and crossover mutation. The specific content is as follows: A non-linear function is used to enhance the global search ability of the whale optimization algorithm. The non-linear function is:

[0033]

[0034] where iter is the current iteration number, and iter max is the maximum iteration number;

[0035] An adaptive inertia weight is adopted to avoid the problem of the whale optimization algorithm falling into local optimum; the updated position formula of the improved whale optimization algorithm is:

[0036] X(t + 1) = δ * X'(t) - A * D'

[0037] X(t + 1) = δ * e bl * cos(2πl) + X'(t)

[0038] X(t + 1) = δ * X rand (t) - A * D'

[0039] where δ is the inertia factor, A and C are coefficient vectors, X rand (t) is the random whale position, D' represents the distance between the i-th individual and the optimal individual, b is a constant, l is a random number between [-1, 1], and different position update methods are selected according to the probability and the value of A;

[0040] The global search ability and population richness of the whale optimization algorithm are enhanced by using crossover and mutation; uniform crossover means randomly selecting a crossover point and then exchanging the parts of two individuals after the crossover point; assume there are two individuals P 1 and P 2 , and their position vectors are respectively where n is the dimension of the vector; randomly generate the crossover point c, then the new individuals after crossover are:

[0041]

[0042] Mutation means randomly changing the values of some dimensions of the individual vector; for each dimension, mutate with a probability; the mutated value is a random value on this dimension, and the calculation formula is:

[0043]

[0044] In the formula, u*b j and l*b j are the upper and lower bounds of the j-th dimension respectively, and rand is a random number uniformly distributed in the interval [0, 1].

[0045] (III) Beneficial effects

[0046] Compared with the prior art, the present invention provides a method for estimating the target position based on a composite whale optimization algorithm, which has the following beneficial effects:

[0047] The present invention integrates the bionic intelligent optimization algorithm into the TDOA positioning algorithm, transforms the TDOA positioning problem into a problem of finding the optimal value, and at the same time improves the performance of the algorithm by using non-linear factors, adaptive inertia weights, and crossover and mutation strategies. Set the fitness function according to the geometric interpretation of the positioning equation. Use the composite whale optimization algorithm to search for the optimal value in the solution space, and the method proposed by the present invention improves the accuracy, time, and robustness of positioning. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] Figure 1 is the flowchart of the method of the present invention;

[0049] Figure 2 is the schematic diagram of the distribution of monitoring stations and targets of the present invention;

[0050] Figure 3 is the schematic diagram of the TDOA positioning model of the present invention;

[0051] Figure 4 is the performance comparison diagram of the composite whale optimization algorithm of the present invention;

[0052] Figure 5 is the positioning error diagram of the present invention. DETAILED DESCRIPTION OF THE INVENTION

[0053] Next, in combination with the accompanying drawings in the embodiments of the present invention, the technical solutions in the embodiments of the present invention will be clearly and completely described. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present invention.

[0054] Embodiment

[0055] As Figures 1-5 shown, a method for estimating a target position based on a composite whale optimization algorithm proposed in an embodiment of the present invention includes the following steps:

[0056] S1: Establish a positioning model based on TDOA. As Figure 2 shown, in a 10m×10m area, there is a target point p(x p , y p ). At the same time, there are 3 monitoring stations, and their positions are S 1 (0, 0), S 2 (10, 0), S 1 (10, 10), where S 1 is the reference node. The target point emits a signal. Since the distances from the target point to the monitoring stations are different, there will be a time difference when the signal arrives at the monitoring stations. This time difference is defined as the time difference of arrival (TDOA), which is obtained by various methods such as generalized cross-correlation. Assume that the obtained time differences are Δt 12 and Δt 13 , which are the time differences between the signal arriving at the second and third monitoring stations and the first monitoring station respectively. Assume that the time when the signal arrives at the reference monitoring station is t, then the times when the signal arrives at the second and third monitoring stations are t + Δt 12 and t + Δt 13 . If the propagation speed of the signal is known as v, then the distances from the target point to the three monitoring stations can be obtained as r 1 , r 2 , r 3 . Then a circular positioning equation based on TDOA can be established. The equation is:

[0057]

[0058] where (x, y) is the position of the target, and (x i , y i ) is the position of the monitoring station;

[0059] S2: After establishing the positioning model based on TDOA, according to the positioning equation and the geometric model, convert the problem of solving the positioning equation into a problem of finding the optimal solution. AsFigure 3 As shown, it is the geometric model of the positioning equation. From the equation and the geometric model, it is not difficult to see that if the optimal reference radius at which three circles intersect at the same point can be found, then the intersection point of the three circles is the estimated position of the target. However, in this mathematical model, r 1 is not known, and the intersection point cannot be directly obtained to determine the position of the target. Combining the above description, the positioning equation can be completely transformed into one that is only related to the reference radius r 1 . Then, if the optimal reference radius r 1 can be found, the optimal estimated position can be known. The reference radius is crucial for determining the target position. The search interval of the reference radius has a direct impact on the performance and accuracy of the positioning algorithm. The size of the search interval determines the search space of the algorithm when looking for the optimal value. If the search interval is too small, the optimal target angle may be isolated outside the search interval, resulting in an inability to obtain the correct estimated value and thus affecting the positioning accuracy. If the search interval is too large, the computational burden will increase and the efficiency of the algorithm will decrease. Therefore, it is necessary to determine the solution space of r 1 . When two circles intersect, the radii of the two circles should satisfy:

[0060]

[0061] r i +r i +Δr ij ≥R ij

[0062] r i ≥(R ij -Δr ij ) / 2

[0063] When the target is located within the triangular region formed by three nodes. If the target is within the node enclosing region, the maximum radius of the circle should satisfy the formula:

[0064] r i ≤max(R ij )

[0065] Therefore, the solution space of the reference radius is: R min ≤r i ≤R max , where R min =min[(R ij -Δr ij ) / 2], R max =max(R 1j );

[0066] S3: After determining the solution space of the reference radius, in order to quickly obtain the optimal point, the Improved Whale Optimization Algorithm (IWOA) is adopted to search for the optimal solution within the solution space. The Improved Whale Optimization Algorithm is the whale optimization algorithm improved by multiple strategies, which incorporates a non-linear function, the inertia weight of the particle swarm algorithm, and the crossover and mutation strategies. WOA is an efficient meta-heuristic search algorithm, which is inspired by the hunting behavior of humpback whales and simulates three main behaviors of humpback whales during hunting: encircling prey, bubble-net attacking, and spiral hunting. These behaviors are abstracted into mathematical models for searching for the optimal solution in the solution space. First, initialize the algorithm parameters, and evenly distribute an appropriate number of whale individuals within the area enclosed by the reference radius solution space. Calculate the fitness value of each whale according to the fitness function, and the magnitude of the fitness value determines the optimality of the solution represented by the position of the whale.

[0067] The specific content of setting the fitness function is as follows: First, determine the reference radius of each current whale. Since the position of the whale is known when initializing the whale distribution, the reference radius of each whale is also known, denoted as r 1 .

[0068] When the reference radius is r 1 , according to the circle positioning equation, we can obtain:

[0069]

[0070] Let

[0071] Then the solution of the current value can be obtained as

[0072] Then the fitness function is:

[0073] Set an appropriate threshold T. When f(r 1 ) ≤ T, the search can be stopped. Calculate the fitness values of all current whales, find the minimum fitness value and compare it with the threshold. If the condition is satisfied, the position of this whale is the target position; otherwise, adjust the position of the whale according to the following method: Ⅰ: Encircle the prey, simulate the circular or spiral path of the humpback whale around the prey, and approach the optimal solution by updating the position of the whale;

[0074] D' = ∣C * X'(t) - X(t)∣

[0075] X(t + 1) = X′(t) - A * D

[0076] In the formula, t is the iteration number, A and C are coefficient vectors, X'(t) is the current optimal position, and X(t) is the current position vector of the whale individual. The calculation of A and C in the formula is as follows:

[0077] A = 2a * r - a

[0078] C = 2 * r

[0079] a decreases linearly from 2 to 0 as the number of iterations increases, and r is a random vector between [0, 1]. In the standard WOA, global search and local optimization are achieved through the linear convergence factor a. A larger convergence factor a has better global search ability, and a smaller a has better local optimization ability. However, in the standard WOA algorithm, the linearly varying convergence factor a cannot fully reflect this process and may lead to problems such as the algorithm falling into local optima and slow convergence speed. In view of this, a non-linear function is introduced, and its expression is:

[0080] where iter is the current number of iterations, and iter max is the maximum number of iterations.

[0081] Ⅱ: Hunting behavior. The WOA algorithm precisely attacks the prey through hunting behavior. The humpback whale will gradually contract to surround the prey and move towards the prey in a spiral. In the algorithm, this behavior is used to adjust the candidate solutions to find the optimal solution. The mathematical model of the hunting behavior can be expressed as:

[0082] X(t + 1) = D' * e bl * cos(2πl) + X'(t)

[0083] D' = |X'(t) - X(t)|

[0084] where D' represents the distance between the i-th individual and the optimal individual, b is a constant, and l is a random number between [-1, 1].[[]]

[0085] To simultaneously simulate the contraction and encirclement mechanism and the spiral update mechanism of the whale, assuming that the probabilities of these two mechanisms being executed are equal, it can be expressed by the following mathematical expression:

[0086]

[0087] Ⅲ: Searching for prey. The WOA algorithm increases the exploration ability through the search behavior. The humpback whale not only surrounds and attacks the prey but also randomly searches for food sources. In the algorithm, the search behavior allows the algorithm to jump out of the local optimum and thus explore new regions in the solution space. The mathematical model of the search behavior can be expressed as:

[0088] X(t + 1) = X rand - A * D'

[0089] D' = |C * X rand - X|

[0090] where X rand is the position of a random whale.

[0091] The adaptive weight mechanism in the particle swarm algorithm allows the algorithm to automatically adjust its inertia weight according to the performance of the particles during the operation. Therefore, after introducing the adaptive weight, the whale optimization algorithm can adjust its search behavior based on the historical performance of each whale, making it approach the optimal solution more effectively. This helps the algorithm avoid premature convergence and falling into local optima, while improving the global search ability. The adaptive inertia weight formula is:

[0092]

[0093] where β is a constant, β = 0.8.

[0094] After introducing the adaptive inertia weight δ, the position update formula is:

[0095] X(t + 1) = δ * X'(t) - A * D'

[0096] X(t + 1) = δ * e bl * cos(2πl) + X'(t)

[0097] X(t + 1) = δ * X rand (t) - A * D'

[0098] Uniform crossover mutation is a commonly used genetic operation that can effectively generate new individuals and increase the diversity of the population. In IWOA, the crossover mutation operation is applied to the position update process of whales to increase the diversity of the population, thereby improving the global search ability of the algorithm and enhancing the population richness.

[0099] Uniform crossover means randomly selecting a crossover point and then swapping the parts of two individuals after the crossover point. Suppose there are two individuals P 1 and P 2 , and their position vectors are respectively where n is the dimension of the vector. Randomly generate the crossover point c, then the new individuals after crossover are:

[0100]

[0101] Mutation means randomly changing the values of some dimensions of the individual vector. For each dimension, mutate with a certain probability. The mutated value is a random value on that dimension, and the calculation formula is:

[0102]

[0103] where u * b j and l * b jThey are the upper and lower bounds of the j-th dimension respectively, and rand is a random number uniformly distributed in the interval [0, 1].

[0104] Update the whale position according to the above method, and then recalculate whether the fitness value of the optimal whale position meets the end condition. If it meets, output the current position as the optimal position; otherwise, return to the above steps to continue the update. In summary, the flowchart of the IWOA execution process proposed by the present invention is as Figure 1 shown.

[0105] The effects of the present invention will be further described below in conjunction with simulation experiments.

[0106] Simulation conditions: The positions of the monitoring stations have been determined, and the actual radius difference plus a 5dB Gaussian white noise error is used as the radius difference measured from the time delay value. 2000 points are randomly generated in a 20m * 20m area, and the position estimation is performed on 500 points according to the above steps. The performance evaluation index is the horizontal positioning error, and the calculation formula is:

[0107]

[0108] where e represents the true position, represents the estimated value of the true position.

[0109] Simulation experiment:

[0110] First, perform the performance test of the composite whale optimization algorithm, and compare it with the standard whale optimization algorithm and the particle swarm optimization (PSO) algorithm. Taking the iteration speed and the fitness value as the evaluation indexes, the experimental results are as Figure 4 shown. It can be clearly seen from the figure that the iteration speed of the composite whale optimization algorithm is significantly faster than that of other algorithms, and the fitness value is lower, indicating that this algorithm has more superior performance than the traditional algorithms. Secondly, perform a simulation experiment on 2000 points, calculate the horizontal positioning error, Figure 5 shows the error bar distribution diagram of each point. It can be seen from the figure that most of the positioning errors are around 0.15m, and the positioning effect is good.

[0111] Finally, it should be noted that the above are only the preferred embodiments of the present invention and are not used to limit the present invention. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing embodiments, or perform equivalent replacements for some of the technical features. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.

Claims

1. A target position estimation method based on a composite whale optimization algorithm, characterized in that: The steps include: S1: After multiple monitoring stations receive the signals transmitted by the target, the generalized cross-correlation method is used to estimate the time delay difference Δt between the monitoring station and the reference monitoring station. 1j , and then calculate the time for the signal to reach the remaining monitoring stations based on the time delay difference, so that multiple unknown time parameters are converted to be related only to the time parameters of the reference monitoring station, and calculate the distance between the monitoring station and the target point according to the signal propagation speed, which are r1, r2, and r3 respectively. With the monitoring station as the center and the distance as the radius, a positioning model based on TDOA is established; S2: After establishing the TDOA positioning model, r2 and r3 are represented by r1, so the position of the target point is only related to r1. Once the distance r1 of the reference monitoring station is determined, the estimated target position can be determined. Once the optimal distance value is found, the optimal target estimated position can also be determined, that is, the intersection of multiple circles is the optimal target estimated position. The solution space of the optimal solution is determined according to the geometric interpretation of the mathematical model; S3: Use the composite whale optimization algorithm to search for the optimal solution in the solution space to determine the estimated position of the target. First, initialize the whale population and distribute the population in the solution space. Set the fitness function according to the geometric interpretation of the positioning mathematical model, and calculate the fitness function of all current whales. The whale position with the minimum fitness value represents the current global optimal position. Compare the minimum fitness value with the threshold to determine whether the current global optimal solution meets the iteration end condition. If it does, output the current optimal position. If not, update the whale position. After updating the position, recalculate the fitness values ​​of all whales until the whale position that meets the end condition is found. At this time, the whale position is the global optimal estimated position.

2. The target position estimation method based on the composite whale optimization algorithm according to claim 1 is characterized in that: In S1, the generalized cross-correlation method is used to calculate the time delay difference Δt between the reference monitoring station and other monitoring stations. 1j , then the time when the target reaches the three monitoring stations is: t, t+Δt 12 ,t+Δt 13 ; Convert three unknown parameters into one unknown parameter.

3. The target position estimation method based on the composite whale optimization algorithm according to claim 1 is characterized in that: The positioning equation based on TDOA established in S1 is a circular positioning equation with the monitoring station as the center and the distance as the radius: Among them, (x, y) is the position of the target, (x i ,y i ) is the location of the monitoring station.

4. The target position estimation method based on the composite whale optimization algorithm according to claim 3 is characterized by: The specific content of solving the solution space in S2 is: when two circles intersect, the radii of the two circles should satisfy: r i +r i +Δr ij ≥R ij r i ≥(R ij -Δr ij ) / 2 When the target is located in the triangular area formed by three nodes, the maximum radius of the circle should satisfy the formula: r i ≤max(R ij ) Therefore, the solution space of the reference radius is: R min ≤r i ≤R max , where R min =min[(R ij -Δr ij ) / 2],R max =max(R 1j ).

5. The target position estimation method based on the composite whale optimization algorithm according to claim 4 is characterized in that: The specific content of setting the fitness function in S3 is: first determine the distance from each whale to the reference monitoring station. Since the position of the whale is already known when the whale distribution is initialized, the reference radius of each whale is also known, which is recorded as r1; When the reference radius is r1, according to the circle positioning equation: make Where R ij =r j -r i ,(x i ,y i ) are the coordinates of the monitoring station; r1 is the reference radius; Then the solution of the current value is: Then the fitness function is:

6. The target position estimation method based on the composite whale optimization algorithm according to claim 1 is characterized in that: When comparing the adaptation value in S3 with the threshold, a suitable threshold T needs to be set. When f(r1)≤T, the search can be stopped. The size of the threshold is set according to the computing performance and the specific application scenario. Under the same computing performance, the smaller the threshold, the smaller the positioning error, but the required positioning time is longer.

7. The target position estimation method based on the composite whale optimization algorithm according to claim 1 is characterized in that: The composite whale optimization algorithm used in S3 is a whale optimization algorithm improved by multiple strategies, which incorporates nonlinear functions, particle swarm algorithm inertia weight and crossover mutation strategy. The specific content of the improvement is: using nonlinear functions to enhance the global search capability of the whale optimization algorithm. The nonlinear function is: Among them, iter current iteration number, iter max is the maximum number of iterations; Adaptive inertia weight is used to avoid the whale optimization algorithm from falling into the local optimal problem; the update position formula of the improved whale optimization algorithm is: X(t+1)=δ*X'(t)-A*D' X(t+1)=δ*e bl *cos(2πl)+X'(t) X(t+1)=δ*X rand (t)-A*D' Among them, δ is the inertia factor, A and C are coefficient vectors, and X rand (t) is the random whale position, D' is the distance between the i-th individual and the optimal individual, b is a constant, l is a random number between [-1,1], and different position update methods are selected according to the probability and A value; Crossover mutation is used to enhance the global search capability and population richness of the whale optimization algorithm; uniform crossover means randomly selecting a crossover point and then exchanging the parts of the two individuals after the crossover point; suppose there are two individuals P1 and P2, and their position vectors are Where n is the dimension of the vector; the crossover point c is randomly generated, and the new individual after crossover is: Mutation refers to randomly changing the values ​​of certain dimensions of individual vectors; for each dimension, mutation is performed with probability; the mutated value is a random value of that dimension, and the calculation formula is: Where u*b j and l*b j are the upper and lower bounds of the j-th dimension, respectively, and rand is a random number uniformly distributed in the interval [0,1].

Images