Mobile Robot Path Planning Method Based on Particle Swarm and Differential Evolution Algorithms

Abstract

The present invention discloses a path planning method for a mobile robot based on particle swarm and differential evolution algorithms, comprising the following steps: inputting a map model; initializing path parameters and particle swarm parameters; designing a path planning objective function, which consists of a path length function and a penalty function; improving the traditional particle swarm algorithm by introducing enterprise governance ideas, adding an adaptive adjustment weight and an acceleration coefficient, optimizing and updating the velocity V and the position X, generating a better elite population as the initial population of the improved differential evolution algorithm, combining the scaling factor F and the crossover probability factor CR of the adaptive parameters, performing mutation operations, crossover operations, and selection operations, realizing the iterative optimization of the particle swarm, and updating the Adm of the particle swarm, outputting an optimized path and performing smoothing processing. The present invention can effectively improve the path planning ability of the mobile robot in a static complex environment model, improve the planning efficiency, and generate a collision-free path with the shortest path length and smoothness to reach the target point.

Description

Technical Field

[0001] The present invention relates to the field of robot motion planning, and particularly to a path planning method for a mobile robot based on particle swarm and differential evolution algorithms. Background Art

[0002] Path planning is one of the important research directions in mobile robot technology. Path planning of a mobile robot refers to planning a collision-free path that meets certain conditions (usually the optimal one) to reach the target point in a static or dynamic environment. Good mobile robot path planning technology can be applied to robots to explore harsh environments that humans cannot reach; replace humans in high-risk rescues such as fire fighting; help visually impaired people with path guidance; and can also be applied to the field of intelligent warehousing to improve the efficiency of material transportation and reduce human and material resources.

[0003] In the prior art, heuristic algorithms are used to optimize path planning problems, including particle swarm algorithm (PSO), artificial bee colony algorithm (ABC), genetic algorithm (GA), grey wolf algorithm (GWO), ant colony algorithm (ACO), differential evolution algorithm (DE), etc. Among them, Jianfang Lian et al. used chaotic adaptive particles to optimize the particle swarm algorithm and adaptively adjust the parameters. Faiza Gul et al. used a grey wolf algorithm-particle swarm algorithm hybrid optimization algorithm to solve the problem of finding the minimum path. Chang-Feng Chen et al. proposed a hybrid algorithm based on ABC algorithm and PSO algorithm, namely PSO-ABC algorithm. However, the improved algorithms still have limitations such as low convergence accuracy and easy premature convergence, resulting in poor path planning efficiency. Summary of the Invention

[0004] The object of the present invention is to overcome the above-mentioned drawbacks and propose a path planning method for a mobile robot based on particle swarm and differential evolution algorithms, which can effectively improve the path planning ability of the mobile robot in a static complex environment model, improve the planning efficiency, and generate a collision-free path with the shortest path length and smoothness to reach the target point.

[0005] The path planning method for a mobile robot based on particle swarm and differential evolution algorithms of the present invention, wherein: the method comprises the following steps:

[0006] Step 1, map construction: obtain the position information of the mobile robot itself and the coordinate information of the target point, and at the same time obtain obstacle information for map construction to form a map model and obtain environmental information;

[0007] Step 2, initialize path parameters: including the starting point Start of the mobile robot, the end coordinate Goal, path nodes, the maximum number of iterations T, and the acceleration factor c i (i = 1, 2, 3, 4);

[0008] Step 3. Randomly initialize the particle swarm parameters: including the dimension D, the number of population particles N, the position X, and the velocity V;

[0009] Step 4. Design the path planning objective function: According to the requirements of path planning, design evaluation indicators, construct the objective function, calculate the path fitness value fit, and determine the maximum fitness value fit max and the minimum fitness value fit min ;

[0010] Step 5. Improve the traditional particle swarm algorithm: Improve the traditional particle swarm algorithm by introducing the enterprise governance idea, adding an adaptive adjustment weight and acceleration coefficient; Based on the calculated path fitness value, calculate the individual best position, local best position, global best position, and elect the manager particle Adm, so as to optimize and update the velocity V and position X, and generate a better elite population;

[0011] The formulas for optimizing and updating the velocity and position are as follows:

[0012]

[0013]

[0014] 2) Inertia factor: ω * ·V i t

[0015] The first term of the velocity update formula (1) is the inertia factor, which is the product of the adaptive inertia weight ω * and the velocity V of the i-th particle in the t-th generation, where the concept of the inertia weight is the same as that of the traditional particle swarm principle; i t

[0016] 2) Individual best position:

[0017] The second term of formula (1) calculates the distance between the current position of particle i at the t-th generation and the individual best position P of this particle i t , multiplied by the acceleration factor c1 and the random number r1 ∈ [0, 1];

[0018] 3) Local best position:

[0019] The third term in expression (1) calculates the current position of particle i at the t-th generation and the best position of this particle in the peer group area The distance is multiplied by the acceleration factor c2 and a random number r2 ∈ [0, 1];

[0020] 4) Global best position:

[0021] The fourth term in expression (1) calculates the current position of particle i when iterating to the t-th generation The distance from the global best position Adm of the particle swarm t is multiplied by the acceleration c3 and a random number r3 ∈ [0, 1]; where Adm t is the position of the manager particle in the t-th generation, generated based on the voting concept;

[0022] Let the initial value of the number of votes Opvote cast for the operation right particle Operator be The number of votes Owvote cast for the owner particle Owner is Furthermore, it causes a deviation in the number of votes, where The voting mechanism is implemented through the roulette wheel algorithm, and each particle votes, is a random number within the range [0, 1], and the selection of the manager particle Adm is as follows:

[0023]

[0024] Among them, represents the position of particle i in dimension d when iterating to the t-th generation as the manager particle;

[0025] 5) The position of particle i in the (t + 1)-th generation in expression (2) is equal to the position of particle i in the t-th generation plus the velocity multiplied by the acceleration factor C4 and a random number r4 ∈ [0, 1] The sum;

[0026] Step Six: Improve the differential evolution algorithm: Adjust the two parameters in the traditional differential evolution algorithm, the scaling factor F and the crossover probability factor CR, to adaptive parameters to improve the convergence performance and iteration accuracy of the algorithm;

[0027] Step Seven: Use the elite population obtained in Step Five as the initial population of the improved differential evolution algorithm, combine the scaling factor F and the crossover probability factor CR of the adaptive parameters, perform mutation operations, crossover operations, and selection operations to achieve the iterative optimization of the particle swarm, and update the Adm of the particle swarm

[0028] Step Eight: Determine whether the fitness value fit is less than the minimum fitness value f min , that is, fit < fit min : If so, end and output the optimized path; otherwise, return to Step Four;

[0029] Step Nine: Smooth the output optimized path.

[0030] The above mobile robot path planning method based on particle swarm and differential evolution algorithms, wherein: the design evaluation indexes in the above Step Four include safety level and path length.

[0031] The above mobile robot path planning method based on particle swarm and differential evolution algorithms, wherein: the objective function in the above Step Four is constructed based on the evaluation indexes and consists of two parts: a path length function and a penalty function; set the starting point coordinates as Start(x0,y0), and the target point coordinates as Goal(x n+1 ,y n+1 ), and each particle in the particle swarm represents a set of node coordinates H = {Start, (x1,y1), (x2,y2),..., (x n ,y n ), Goal} that a path passes through;

[0032] 3) Path length function

[0033] The path length function f1 is used to calculate the path length of the mobile robot from the starting point Start to the target point Goal, and can be expressed as follows:

[0034]

[0035] 4) Penalty function

[0036] The more times the path intersects with obstacles, the greater the danger level of the generated path. Use the danger level to set the penalty function to penalize the path nodes that intersect with obstacles;

[0037] The obstacles are represented by circles, denoted as C k , with the center as O k , k is the number of obstacles, and set the obstacle radius as the safety threshold, denoted as R = {r0, r1,..., r k}, to obtain a collision-free path, it is necessary to ensure that the distance between the path nodes and the connection line of the path nodes and the obstacles is greater than the safety threshold;

[0038] Take m mid-nodes on the connection line between two adjacent path nodes, use the path mid-nodes to judge whether the route where they are located intersects with obstacles, and calculate the Euclidean distance Dis between each mid-node and the path node and the center of the obstacle k (k = 1, 2,..., t), then the penalty degree Risk between node i and node i-1 (i = 0, 1,..., n+1) is calculated according to the following formula:

[0039]

[0040]

[0041] where r k is the k-th safety threshold set by the obstacle radius; for the penalty degree, two values of 0 and 1 are set. When the distance between the middle node and the path node and the obstacle is less than the safety threshold, the node is penalized and the penalty degree is set to 1. When the distance is greater than the safety threshold, the penalty degree is set to 0. Given the weight coefficient η, which represents the influence degree of the penalty degree on the path node, the penalty function expression is as follows:

[0042]

[0043] where n is the number of path nodes; m is the number of middle nodes;

[0044] Therefore, the objective function, i.e., the fitness function, is the following expression:

[0045] fit = f1 + f2

[0046] In the above mobile robot path planning method based on the particle swarm and differential evolution algorithms, in the global best position of the fifth step: in order to better control the influence of the particle swarm on the global best position Adm t on the particle swarm, a trigonometric function is introduced, and an adaptive parameter β is proposed, which makes the algorithm converge faster and the accuracy higher. The formula is as follows:

[0047]

[0048] where β min and β max are the maximum and minimum values of the parameter β; t is the number of iterations, and T is the set maximum number of iterations.

[0049] In the above mobile robot path planning method based on the particle swarm and differential evolution algorithms, in the global best position of the fifth step: the voting mechanism is implemented by the roulette wheel algorithm. Each particle votes, and the votes are accumulated and updated to facilitate recording the influence of the Operator and the Owner. The cumulative update formula is:

[0050]

[0051] where, is a random number in the range of [0, 1], N is the number of population particles, M represents the number of votes cast by the particle in the given dimension d to the corresponding candidate leader Operator or Owner, and the total number of support votes after accumulation can intuitively indicate the elected final Adm; for the convenience of further operation, after each iteration is completed, the voting needs to be normalized:

[0052]

[0053]

[0054] Among them,

[0055] For the above mobile robot path planning method based on particle swarm and differential evolution algorithms, where: in step five, boundary processing is performed on the particle velocity If it exceeds the velocity boundary, it is immediately absorbed:

[0056]

[0057] Where is the component of the velocity of the t-th generation in dimension d, v min and v max are the minimum and maximum values of the velocity respectively.

[0058] For the above mobile robot path planning method based on particle swarm and differential evolution algorithms, where: the self-adaptive optimization formula of the scaling factor F in step six is as follows:

[0059]

[0060] In the formula, F i represents the scaling factor of the i-th particle vector in the population; at the t-th iteration, three individuals x p1 (t), x p2 (t), x p3 (t) are randomly selected from the particle population, and p1≠p2≠p3≠i, fit(x p1 (t)), fit(x p2 (t)), fit(x p3 (t)) represent the fitness values of the three individuals x p1 (t), x p2 (t), x p3 (t) respectively.

[0061] For the above mobile robot path planning method based on particle swarm and differential evolution algorithms, where: the self-adaptive optimization formula of the crossover probability factor CR in step six is as follows:

[0062]

[0063] In the formula, CR i is the mutation probability factor corresponding to the i-th particle vector in the population; CR min , CR max are the minimum and maximum values of the crossover probability factor CR respectively; fit(x i ) is the fitness value of the position vector x i , fitmin , fitness max For the minimum and maximum values of fitness, fitness mean is the average value of fitness.

[0064] The above path planning method for a mobile robot based on particle swarm and differential evolution algorithms, wherein: in step nine, the output optimized path is smoothed, and cubic spline interpolation is used for calculation to obtain a path including path nodes, interpolation points, and start and end points.

[0065] Compared with the prior art, the present invention has obvious beneficial effects. As can be seen from the above solutions, the proposed adaptive parameter β and acceleration coefficient are used to improve the convergence speed when updating the global best position; the idea of enterprise governance is introduced to optimize the particle swarm algorithm and improve the performance of the algorithm; for the traditional DE algorithm, the scaling factor F and crossover probability factor CR are adaptively optimized, enabling the algorithm to adaptively control the search accuracy and the degree of mutation, thereby improving the optimization accuracy of the algorithm; the improved IDE algorithm is used to optimize the global optimal position of the IPSO algorithm, preventing the IPSO algorithm from falling into local optimal solutions while improving the optimization ability of the algorithm; a new objective function applied to path planning is proposed, which consists of a path length function and a penalty function, simplifying the path planning problem of the mobile robot into an objective function optimization problem; the cubic spline interpolation method is used to smooth the generated path to prevent the path from having sharp points. In summary, the present invention can effectively improve the path planning ability of the mobile robot in a static complex environment model, improve the planning efficiency, and generate a collision-free path with the shortest and smoothest path length to reach the target point.

[0066] The following further illustrates the beneficial effects of the present invention through specific embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0067] Figure 1 is a flow chart of the present invention;

[0068] Figure 2 is a comparison chart of the path planning results of the present invention and the comparative algorithm. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0069] The following, in conjunction with the accompanying drawings and preferred embodiments, details the specific embodiments, features, and effects of a path planning method for a mobile robot based on particle swarm and differential evolution algorithms proposed according to the present invention.

[0070] Refer to Figure 1 , the path planning method for a mobile robot based on particle swarm and differential evolution algorithms of the present invention, wherein: the method includes the following steps:

[0071] Step 1. Map construction: Obtain the position information of the mobile robot itself and the coordinate information of the target point. At the same time, obtain the obstacle information to construct a map, form a map model, and obtain the environmental information;

[0072] Step 2. Initialize the path parameters: including the starting point Start of the mobile robot, the end coordinate Goal, the path nodes, the maximum number of iterations T, and the acceleration factor c i (i = 1, 2, 3, 4);

[0073] Step 3. Randomly initialize the particle swarm parameters: including the dimension D, the number of population particles N, the position X, and the velocity V;

[0074] Step 4. Design the path planning objective function: According to the requirements of path planning, design evaluation indicators, construct the objective function, calculate the path fitness value fit, and determine the maximum fitness value fit max and the minimum fitness value fit min ;

[0075] The evaluation indicators include the safety level and the path length; the objective function is constructed based on the evaluation indicators and consists of two parts: the path length function and the penalty function; set the starting point coordinates as Start(x0, y0), the target point coordinates as Goal(x n+1 , y n+1 ), and each particle in the particle swarm represents a set of node coordinates H = {Start, (x1, y1), (x2, y2),..., (x n , y n ), Goal} passed by a path;

[0076] 5) Path length function

[0077] The path length function f1 is used to calculate the path length of the mobile robot from the starting point Start to the target point Goal and can be expressed as follows:

[0078]

[0079] 6) Penalty function

[0080] The more times the path intersects with the obstacles, the greater the danger level of the generated path. Use the danger level to set the penalty function to penalize the path nodes that intersect with the obstacles;

[0081] The obstacles are represented by circles, denoted as C k , with the center as O k , k is the number of obstacles, and the obstacle radius is set to the safety threshold, denoted as R = {r0, r1,..., r k}, to obtain a collision-free path, it is necessary to ensure that the distance between the path nodes and the connecting lines of the path nodes and the obstacles is greater than the safety threshold;

[0082] Take m mid-nodes on the connecting line between two adjacent path nodes, and use the path mid-nodes to determine whether the route intersects with the obstacles, and calculate the Euclidean distance Dis between each mid-node and the center of the obstacle and the path node k (k = 1, 2, …, t), then the penalty degree Risk between node i and node i - 1 (i = 0, 1, …, n + 1) is calculated according to the following formula:

[0083]

[0084]

[0085] Among them, r k is the kth safety threshold set by the obstacle radius; for the penalty degree, two values of 0 and 1 are set. When the distance between the mid-node and the path node and the obstacle is less than the safety threshold, the node is penalized and the penalty degree is set to 1. When the distance is greater than the safety threshold, the penalty degree is set to 0. Given the weight coefficient η, which represents the influence degree of the penalty degree on the path node, the penalty function expression is as follows:

[0086]

[0087] Among them, n is the number of path nodes; m is the number of mid-nodes;

[0088] Therefore, the objective function, that is, the fitness function, is the following expression:

[0089] fit = f1 + f2

[0090] Step Five: Improve the traditional particle swarm optimization algorithm: Improve the traditional particle swarm optimization algorithm by introducing the enterprise governance idea, adding an adaptive adjustment weight and acceleration coefficient; based on the calculated path fitness value, calculate the individual best position, local best position, global best position, and elect the manager particle Adm, so as to optimize and update the speed V and position X, and generate a better elite population;

[0091] The formulas for optimizing and updating the speed and position are as follows:

[0092]

[0093]

[0094] 3) Inertia factor: ω * ·V i t

[0095] The first term of the velocity update formula (1) is the inertia factor, which is the adaptive inertia weight ω * and the velocity V of the i-th particle in the t-th generation i t The concept of the inertia weight is the same as that of the traditional particle swarm principle;

[0096] 2) Individual best position:

[0097] The second term of formula (1) calculates the distance between the current position of particle i at the t-th generation and the individual best position P of this particle i t multiplied by the acceleration factor c1 and a random number r1 ∈ [0, 1];

[0098] 3) Local best position:

[0099] The third term in expression (1) calculates the distance between the current position of particle i at the t-th generation and the best position of this particle within the peer group area multiplied by the acceleration factor c2 and a random number r2 ∈ [0, 1];

[0100] 4) Global best position:

[0101] The fourth term in expression (1) calculates the distance between the current position of particle i at the t-th generation and the global best position Adm of the particle swarm t multiplied by the acceleration c3 and a random number r3 ∈ [0, 1]; where Adm t is the position of the manager particle in the t-th generation, which is generated based on the voting concept;

[0102] The global best position Adm of the particle swarm t , in order to better control the influence of Adm t on the particle swarm, a trigonometric function is introduced, and an adaptive parameter β is proposed to accelerate the convergence speed and improve the accuracy of the algorithm. The formula is as follows:

[0103]

[0104] where β min and β max are the maximum and minimum values of the parameter β; t is the number of iterations, and T is the set maximum number of iterations;

[0105] Adm is generated by the administrator based on the voting concept. Particles will follow the Adm elected by voting. As the iteration progresses, the leadership of the operator particle Operator gradually increases, that is, the number of votes Opvote cast for the operator particle Operator increases, and the influence of the owner is relatively reduced, that is, the number of votes Owvote cast for the owner particle Owner decreases, and the convergence speed increases. However, if the operator particle cannot lead the particles to the optimum, the owner particle Owner needs to take over the management right. At this time, even if the number of votes Owvote is small, the dominance of the owner particle Owner needs to be increased. Therefore, a manager regulation factor is introduced.

[0106] Since the initial population is randomly generated, a tendency is artificially imposed first, and the initial voting ranges of the two candidate manager particles are asymmetrically processed. The asymmetric range is controlled within [0,1] through standardization. Therefore, in the initial iteration, the particles will choose the operator particle (Operator) as the Adm, and the owner (Owner) as the candidate manager particle that plays a supervisory role.

[0107] Let the initial value of the number of votes Opvote cast for the operator particle Operator be The number of votes Owvote cast for the owner particle Owner is Furthermore, a deviation in the number of votes is caused, where The voting mechanism is implemented through the roulette wheel algorithm. Each particle votes. is a random number within the range [0,1]. The selection of the manager particle Adm is as follows:

[0108]

[0109] Among them, represents the position of particle i in dimension d at the t-th generation as the manager particle;

[0110] The voting mechanism is implemented through the roulette wheel algorithm. Each particle votes, and the number of votes is accumulated and updated to facilitate recording the influence of Operator and Owner. The accumulation and update formula is:

[0111]

[0112] Among them, is a random number within the range [0,1], N is the number of population particles, M represents the number of votes cast by particle i in dimension d for the corresponding candidate leader Operator or Owner, and the total number of support votes after accumulation can intuitively indicate the elected final Adm; for the convenience of further calculation, after each iteration is completed, the voting needs to be standardized:

[0113]

[0114]

[0115] Among them,

[0116] 5) The position of particle i in the (t + 1)-th generation in expression (2) is equal to the position of particle i in the t-th generation and the sum of the velocity multiplied by the acceleration factor C4 and the random number r4 ∈ [0, 1]; and;

[0117] For the boundary treatment of the particle velocity if it exceeds the velocity boundary, it is immediately absorbed:

[0118]

[0119] where is the component of the velocity of the t-th generation in dimension d, v min and v max are the minimum and maximum values of the velocity respectively.

[0120] Step Six: Improve the differential evolution algorithm: Adjust the two parameters in the traditional differential evolution algorithm, the scaling factor F and the crossover probability factor CR, to adaptive parameters to improve the convergence performance and iteration accuracy of the algorithm;

[0121] 1) The adaptive optimization formula for the scaling factor F is as follows:

[0122]

[0123] In the formula, F i represents the scaling factor of the i-th particle vector in the population; at the t-th iteration, three individuals x p1 (t), x p2 (t), x p3 (t) are randomly selected from the particle population, and p1 ≠ p2 ≠ p3 ≠ i, and fit(x p1 (t)), fit(x p2 (t)), fit(x p3 (t)) represent the fitness values of the three individuals x p1 (t), x p2 (t), x p3 respectively;

[0124] 2) The adaptive optimization formula for the crossover probability factor CR is as follows:

[0125]

[0126] In the formula, CR i is the mutation probability factor corresponding to the i-th particle vector of the population; CR min , CR max are the minimum and maximum values of the crossover probability factor CR respectively; fit(x i ) is the fitness value of the position vector x i , fit min , fit max are the minimum and maximum values of the fitness, and fit mean is the average fitness value;

[0127] Step Seven: Use the elite population obtained in Step Five as the initial population of the improved differential evolution algorithm, and combine the scaling factor F and the crossover probability factor CR of the adaptive parameters to perform mutation operation, crossover operation, and selection operation to achieve iterative optimization of the particle swarm and update the Adm of the particle swarm

[0128] Step Eight: Determine whether the fitness value fit is less than the minimum fitness value f min , that is, fit < fit min : If so, end and output the optimized path; otherwise, return to Step Four.

[0129] Step Thirteen: Smooth the generated path.

[0130] For the smoothing process of the generated path, cubic spline interpolation method is used for calculation to obtain a path including path nodes, interpolation points, and start and end points. That is, the generated path is represented by H = {Start, (x1, y1), (x2, y2),..., (x n , y n ), Goal}, that is, there are n + 2 path nodes including the start and end points. Cubic spline interpolation method is used for calculation in the intervals (x0, x1,..., x n+2 ) and (y0, y1,..., y n+2 ), and the path formed by connecting the path nodes, interpolation points, and start and end points is the required path.

[0131] Among them:

[0132] 1 Traditional PSO algorithm and DE algorithm

[0133] 1.1 Basic principle of traditional PSO algorithm

[0134] The PSO algorithm sets a bird predation scenario: the bird flock randomly searches for food within a certain area. It can be determined that there is only one piece of food in this search range, and its location is unknown. However, the birds know the distance between their current location and the food. The best way to solve the bird predation problem is to search the surrounding area of the bird that is closest to the food in the current flock. The particle swarm algorithm abstracts the birds into massless particles and transforms the optimization problem into finding specific particles in the search space. Each particle is a solution to the optimization problem. Each particle has two attributes: position and velocity, which control the movement of the particle in the search space, making the particle move towards the optimal solution direction, thereby optimizing the problem.

[0135] In the target search space with dimension D, the number of particles in the population is set to N, and the position of particle i is X i =(x i1 ,x i2 ,…,x iD ), i = 1, 2,..., N; the velocity of particle i is V i =(v i1 ,v i2 ,...,v iD ), i = 1, 2,..., N; the individual best position experienced by particle i is P i =(p i1 ,p i2 ,...,p iD ); the global best position obtained after judging the entire particle swarm is G = (g1, g2,…, g D ). The particle velocity and position update formulas are as follows:

[0136]

[0137]

[0138] In the formula, ω is the inertia weight, which is used to suppress the inertial velocity of the previous iteration, thereby affecting the convergence accuracy, and further improving the parameter to an adaptive parameter, enabling the algorithm to perform adaptive adjustment during the iteration process. That is, as the number of iterations increases, ω decreases. This weight enables the algorithm to have a relatively large search speed at the beginning of the iteration, improving the search ability and ensuring the operation efficiency; at the end of the iteration, the search speed decreases, improving the search accuracy. The specific formula is as follows:

[0139]

[0140] where ω max and ω min are the maximum and minimum values of ω respectively, t is the number of iterations, and T is the set maximum number of iterations.

[0141] r1 and r2 are random numbers within the range of [0, 1], aiming to increase the search range of the algorithm; both c1 and c2 are learning factors, which are values greater than or equal to zero. Among them, c1 is the cognitive factor, that is, the cognitive ability of the individual particle itself, controlling the changes within the local range of the individual particle itself. c2 is the social factor, referring to the influence ability at the social level, that is, controlling the influence of the optimal particle in the particle swarm on the entire region. The two learning factors work together to promote the particles to continuously approach the optimization direction.

[0142] P i t is the individual optimal position of the i-th particle at the t-th iteration of the algorithm; G t is the global optimal position of the particles at the t-th iteration of the algorithm; Let the fitness function of the algorithm be fit, then the expressions for the individual optimal position and the global optimal position are as follows:

[0143]

[0144]

[0145] 1.2 Basic Principles of Traditional DE Algorithm

[0146] The principle of the DE algorithm is similar to that of the genetic algorithm, and also includes three operations: "mutation", "crossover", and "selection". However, there are differences in the mutation and crossover operations. The genetic algorithm directly compares the parent generation and the offspring generation and selects the individual with higher fitness, adhering to the principle of "survival of the fittest". The DE algorithm introduces a differential vector for mutation, with a more concise structure and higher efficiency.

[0147] The specific implementation process of DE is as follows:

[0148] 1) Establish an initial population and initialize parameters

[0149] In the solution space {x j_min , x j_max}, j = 1, 2,..., D, randomly and uniformly generate N individuals. The individuals are candidate solution vectors, and the dimension is set to D. The expression for the i-th candidate solution vector in the initial population is as follows:

[0150] X i (0) = (x i1 (0), x i2 (0),..., x iD (0)), i = 1, 2,..., N

[0151] The initialization formula for the value of the j-th dimension of the i-th individual is as follows:

[0152] x ij (0) = x j_min + rand(0, 1)·(xj_max -x j_min )

[0153] 2) Mutation Operation

[0154] The DE algorithm is based on the parent individual x i (t), i = 1, 2, …, M, randomly selects an individual for differential mutation. The mutation strategy is expressed as "DE / offspring generation method / number of groups for differential / crossover method". Generally, there are the following four differential methods for generating the mutation vector:

[0155] (1) DE / rand / 1:

[0156] h i (t) = x p1 (t) + F·(x p2 (t) - x p3 (t)) (1 - 6)

[0157] (2) DE / rand / 2:

[0158] h i (t) = x p1 (t) + F·(x p2 (t) - x p3 (t)) + F·(x p4 (t) - x p5 (t))(1 - 7)

[0159] (3) DE / best / 1:

[0160] h i (t) = x best (t) + F·(x p2 (t) - x p3 (t))(1 - 8)

[0161] (4) DE / current-to-best / 1:

[0162] h i (t) = x i (t) + F·(x best (t) - x i (t)) + F·(x p1 (t) - x p2 (t))(1 - 9)

[0163] Among them, h i (t) is the generated mutation vector; x p1 (t), x p2 (t), x p3 (t), x p4 (t), xp5 (t) is the solution vector numbered p1, p2, …, p5 in the t-th generation of the population. The numbers are randomly selected and pairwise different from i; x best (t) is the optimal individual in the t-th generation of the population; F is the scaling factor, which is used to appropriately scale the difference vector, and its value range is generally controlled in (0, 1.2].

[0164] The present invention selects the DE / rand / 1 strategy for mutation operation. That is, at the t-th iteration, three individuals x p1 (t), x p2 (t), x p3 (t) are randomly selected from the population, and p1≠p2≠p3≠i. The generated mutation vector h i (t) is shown in Equation (1-6). 3) Crossover operation:

[0165] The parent vectors x i (t), i = 1, 2, …, M and the mutation vector h i (t) are hybridized with a crossover probability CR to generate new individual vectors, which become the trial vectors v ij (t). The j-th (j = 1, 2, …, D) dimension in the trial vector is selected from the parent and mutation vectors according to CR, and the formula is as follows:

[0166]

[0167] In the formula: rand is a uniform random number in the range of [0, 1]; CR is the crossover probability factor, CR ∈ [0, 1]; j rand is a random positive integer in the range of [0, 1], so that at least one component is produced by the mutation vector, thereby ensuring the generation of a new vector.

[0168] 4) Selection operation

[0169] The vector v i (t + 1) generated after the crossover and mutation operations is compared with the parent vector x i (t), and the vector with a better fitness value (fit(*)) is retained. The formula is as follows:

[0170]

[0171] 2 Improved Particle Swarm Optimization (IPSO)

[0172] The traditional PSO algorithm incorporates a leadership concept: an optimal particle is selected from the particle swarm to represent the global optimum G, which is the only leader guiding the particle swarm to the optimal position. However, the single leadership concept has its limitations and cannot ensure that the direction guided by the leader is always correct, leading to premature convergence and falling into a local optimum. To address this issue, the present invention incorporates the concept of corporate governance into particle swarm optimization and optimizes its parameters.

[0173] In economics, the concept of corporate governance refers to an enterprise having two rights, namely, ownership and management rights. Excellent owners and operators are managers with strong influence. The owner confers the right to manage the company on the operator, leading the company's employees to develop in a better direction. If the company's profits are not good under the management of the operator, the owner will step in to make decisions and deprive the operator of the management right, and the two rights will balance each other to promote the development of the enterprise. Based on the above concept, the present invention improves the algorithm. The particle temporarily occupying the management position is called the administrator particle (Adm). The operator (Operator) particle and the owner (Owner) particle are potential administrator particles, and the final Adm particle is selected through a voting mechanism. The Owner particle usually does not occupy the leadership position first. It mainly plays a role in supervising and controlling the overall situation, checking whether the Operator particle has been leading the particle swarm towards the optimal position. In addition, research shows that a good working atmosphere and corporate culture can improve employees' work efficiency, and employees' behaviors will influence each other, that is, peers or within a small group will influence each other. Based on this phenomenon, the concept of a peer group is proposed. Adjacent particles are regarded as a peer group, and their individual best positions influence each other, and the local optimum value (Lbest) is selected within the peer group.

[0174] The present invention improves the traditional PSO algorithm by introducing the concept of corporate governance and adding adaptive parameters and acceleration coefficients. For the update of the particle position in two consecutive iterations, the specific update (Update) formulas for velocity and position are as follows:

[0175]

[0176]

[0177] (1) ω * ·V i t

[0178] The first term of the velocity update formula (1 - 12) is the product of the adaptive inertia weight ω * and the velocity V at the previous iteration t i t of. Among them, ω * is the adaptive inertia weight proposed by the present invention based on the original inertia weight ω, adding a trigonometric function. The improved ω* The convergence rate can be controlled more accurately, and the specific expression is as follows:

[0179]

[0180] where ω * max and ω * min represent the maximum value and the minimum value of ω * respectively.

[0181] (2)

[0182] The second term of formula (1 - 12) calculates the distance between the current position of particle i at the t-th generation of iteration and the individual best position P of this particle, and multiplies it by the acceleration factor c1 and the random number r1 ∈ [0, 1]. i t

[0183] (3)

[0184] The third term in expression (1 - 12) calculates the distance between the current position of particle i at the t-th generation of iteration and the best position of this particle within the peer group area, and multiplies it by the acceleration factor c2 and the random number r2 ∈ [0, 1].

[0185] (4)

[0186] The fourth term in expression (1 - 12) calculates the distance between the current position X i t of particle i at the t-th generation of iteration and the global best position of the particle swarm, that is, Adm t , and multiplies it by the adaptive parameter β, acceleration c3, random number r3 ∈ [0, 1] and supervision factor ψ. Among them, in order to better control the influence of Adm t on the particle swarm, the present invention introduces a trigonometric function and proposes an adaptive parameter β to accelerate the convergence rate of the algorithm; ψ represents the supervision factor of the administrator particle. The formula for the adaptive parameter β is as follows:

[0187]

[0188] ​Adm is generated based on the voting concept. Particles will follow the Adm elected by voting. As the iteration progresses, the leadership of the Operator gradually increases, that is, the number of votes for the Operator (Opvote) increases, and the number of votes for the Owner (Owvote) decreases. Consequently, the influence of the Owner weakens, and the convergence speed increases. However, if the Operator cannot lead the particles to the optimum, the Owner needs to take control. At this time, even if the number of Owvote is small, the influence of the Owner should be enhanced. Therefore, a supervision factor ψ is introduced, where

[0189]

[0190] Since the initial population is randomly generated and particles do not have the ability to select a better leader, a tendency is artificially imposed first. The initial voting ranges of the two candidate leader particles are processed asymmetrically, and the asymmetric range is controlled within [0,1] through standardization. Therefore, in the initial iteration, particles will choose the Operator as the Adm, and the Owner as the second choice to play a supervisory role. Set the initial value of Opvote to Owvote is This further causes a deviation in the number of votes, where

[0191] The voting mechanism is implemented through the roulette wheel algorithm, and each particle votes is a random number within the range of [0,1]. The selection of the Adm is as follows:

[0192]

[0193] Accumulate and update the number of votes to facilitate recording the influence of the Operator and the Owner:

[0194]

[0195] M represents the number of votes a particle casts for the corresponding candidate leader in a given dimension. After accumulation, it is the total number of supporting votes, which can intuitively show which leader can be elected as the final Adm. For the convenience of further calculation, after each iteration is completed, the voting needs to be normalized:

[0196]

[0197]

[0198]

[0199] voteAdm Refers to the standardized number of votes obtained by specific leaders (Operator and Owner). Factor is expressed as follows:

[0200] Case 1:

[0201] Case 2:

[0202] The update expressions for Operator and Owner are as follows:

[0203]

[0204] Operator d = gbest d

[0205] where the parameters φ and rand d are uniformly distributed random numbers in the range [0, 1], pro = 1 / N, d ∈ {1, 2, …, D}, and the Owner specific dimension is selected between [X min , X max , and the other parts are from Operator.

[0206] (5) Improved position update formula

[0207] The position of the particle in the (t + 1)-th generation in expression (1 - 13) is equal to the historical position and the sum of the velocity V multiplied by the acceleration factor C4 and the random number r4 ∈ [0, 1]. i t+1 Sum.

[0208] In addition, Boundary processing is performed: Since the particles search in the solution space at velocity v in the algorithm, if the particle velocity exceeds the range and flies out of the solution space, it will affect the algorithm operation. In the present invention, the particle velocity is processed, and if it exceeds the velocity boundary, it is immediately absorbed.

[0209]

[0210] where, where is the component of the velocity in dimension d at the t-th generation, v min and v max are the minimum and maximum values of the velocity, respectively.

[0211] 3 Improved differential evolution algorithm (IDE)

[0212] The DE algorithm has a simple structure and a fast convergence speed. This algorithm contains two parameters, namely, the scaling factor F and the crossover probability factor CR. In the standard DE algorithm, these two parameters are fixed values. If the parameters are improved to adaptive parameters, the convergence performance and iteration accuracy of the algorithm can be improved.

[0213] 3) Adaptive optimization of the scaling factor F

[0214] F can control the degree of mutation. When the value of F is relatively large, that is, the degree of mutation is relatively large, the search range of the algorithm can be expanded, which is conducive to global search. However, in the later stage of iteration, premature convergence may occur. When the value of F is small, that is, the degree of mutation is small, it is beneficial to perform local search, thereby improving the search accuracy. However, it is easy to fall into the local optimal solution.

[0215] Therefore, the present invention makes an adaptive improvement to F, and the improved adaptive parameters are as follows:

[0216]

[0217] In the formula, F i represents the scaling factor of the i-th vector in the population, that is, each individual is analyzed, and a specific scaling factor suitable for this individual is selected. fit(x p1 (t)), fit(x p2 (t)), fit(x p3 (t)) respectively represent the fitness values of the vectors x p1 , x p2 , x p3 . F min , F max are the minimum and maximum values of the scaling factor. If the fitness values of x p2 and x p3 differ greatly, then the value of F i is reduced, thereby reducing the search range and improving the search accuracy; conversely, if the fitness values of the two vectors are similar, then the value of F i is increased, thereby preventing it from falling into the local optimal solution and increasing the degree of mutation.

[0218] 4) Adaptive optimization of the crossover probability factor CR

[0219] To improve the convergence speed of the algorithm, the crossover probability factor is improved. CR is a factor that affects the crossover degree between the parent vector and the mutant vector. If CR is too large, the crossover degree increases, but it may cause the individuals with better fitness to be damaged due to excessive mutation; if CR is too small, the degree of crossover and mutation is insufficient, and it may fall into local convergence, reducing the search efficiency and the mutation process is slow. In the present invention, the fitness of a specified individual is compared with the average fitness of the population. If it is smaller than the average value, it indicates that the individual is relatively excellent, and then its crossover and mutation degree is reduced; if it is larger than the average value, it indicates that the individual still needs further optimization, and then its mutation degree is increased to promote the iterative search for the optimal individual. The specific expression is as follows:

[0220]

[0221] In the formula, CR i is the mutation probability factor corresponding to the i-th vector of the population; CR min , CR max are the minimum and maximum values of CR respectively; fit(x i ) is the fitness value of the vector x i , fit min , fit max are the minimum and maximum values obtained after comparing the fitness values of each vector in the population, and fit mean is the average fitness value.

[0222] 4 Improved Particle Swarm Optimization - Differential Evolution Hybrid Algorithm (IPSO - IDE)

[0223] 4.1 Principle of IPSO - IDE Hybrid Algorithm

[0224] To improve the optimization ability of the PSO algorithm, the present invention combines the governance of human society and the concept of voting, and introduces an adaptive factor for optimization to improve the convergence speed. However, since the principle of the algorithm is to continuously iterate to update the velocity and position of the particles, and approach the optimal position through a simple motion mode, it makes the particles easily affect each other, that is, the inherent property of the algorithm causes the algorithm to still easily converge at a non - optimal position during optimization. The DE algorithm compares the parent vector with the target vector generated after mutation and crossover in the early stage, and preferentially retains the better one, making it have a very high convergence speed, which can be used to optimize the algorithm.

[0225] In summary, to improve the accuracy of the algorithm, while improving the traditional PSO algorithm and DE algorithm, the two are fused. The limitations generated by the inherent properties of the PSO algorithm are "broken" by the DE algorithm, and a new PSO-DE hybrid optimization mechanism is proposed. The idea of "mutual benefit and win-win", that is, the partnership, is introduced. After the partner optimizes its own capabilities, it will in turn provide greater benefits to the partner. Based on this idea, the improved IPSO-IDE algorithm is proposed. The IPSO algorithm and the IDE algorithm are in a cooperation mode. The DE algorithm is used to optimize the leader of the PSO algorithm, making the leader's position closer to the optimal position and improving its ability to guide the particles to the optimal position. In this way, the performance of the iteratively updated population is more superior, and this population is called the "elite population". The elite population is used to train the differential evolution algorithm to obtain better results.

[0226] 4.2 Implementation Process of IPSO-IDE Hybrid Algorithm

[0227] The implementation steps of the improved IPSO-IDE hybrid algorithm are as follows:

[0228] Step1: Initialize parameters, including acceleration factor c i (i = 1, 2, 3, 4), support votes Opvote, opposition votes Owvote, etc.;

[0229] Step2: Randomly initialize the particle swarm, including dimension D, number of population particles N, position X, velocity V, etc.;

[0230] Step3: Calculate the fitness value fit of the particles according to the set objective function;

[0231] Step4: Based on the fitness value fit, calculate the individual best position P i t , local best position global best position, and elect Adm according to expression (1-15);

[0232] Step5: Update the position X and velocity V according to the improved formulas (1-12)(1-13) to generate a better elite population;

[0233] Step6: Process the boundary using expression (1-16);

[0234] Step7: Use the elite population as the initial population of the IDE algorithm, combine the adaptive parameters (1-17)(1-18), and perform "high-intensity" iterative optimization using formulas (1-6)(1-10)(1-11);

[0235] Step8: Apply the result optimized by the IDE algorithm to update the global optimal position of the particle swarm;

[0236] Step 9: Determine whether the output result meets the termination condition: If yes, end and output the final result; if no, return to Step 2. The pseudo-code of the IPSO-IDE algorithm is as follows:

[0237]

[0238]

[0239] 5 Experimental Analysis

[0240] Compare and analyze the path planning results of the algorithm (IPSO-IDE) of the present invention with the traditional PSO algorithm, DE algorithm, and ABC algorithm. The scenario is a mixed map of square and circular obstacles, with a total of 10 obstacles. The domain values of x and y are [0, 10]. The square and "X" represent the starting point and the ending point respectively. Since the obstacles are relatively dense, the number of path nodes is selected as 5. The population size of all algorithms is 15, the maximum number of iterations is 100, and each algorithm runs 20 times. Figure 2 Shows the best paths of several algorithms. Table 3 lists the average value (Mean), best fitness value (Best), worst fitness value (Worst), and standard deviation (Std) of the 20 running results. It can be seen that the IPSO-IDE algorithm has better path optimization ability compared with the traditional heuristic algorithms, and the standard deviation value Std is smaller, indicating better stability.

[0241] Table 1 Path Planning Results of Each Algorithm

[0242]

[0243] The above are only the preferred embodiments of the present invention, and do not impose any form of limitation on the present invention. Any simple modification, equivalent change, and modification made to the above embodiments based on the technical essence of the present invention without departing from the content of the technical solution of the present invention still fall within the scope of the technical solution of the present invention.

Claims

1. A path planning method for a mobile robot based on particle swarm and differential evolution algorithms, characterized in that: The method includes the following steps: Step 1, Map construction: Obtain the location information of the mobile robot itself and the coordinate information of the target point. At the same time, obtain obstacle information for map construction to form a map model and obtain environmental information; Step 2, Initialize path parameters: Include the starting point Start of the mobile robot, the end coordinate Goal, path nodes, the maximum number of iterations T, and the acceleration factor; Step 3, Randomly initialize particle swarm parameters: Include the dimension D, the number of population particles N, the position X, and the velocity V; Step 4. Design the path planning objective function: According to the requirements of path planning, design evaluation indicators, construct the objective function, calculate the path fitness value fit, and determine the maximum fitness value fit max and the minimum fitness value fit min ; Step 5, Improve the traditional particle swarm algorithm: Improve the traditional particle swarm algorithm by introducing enterprise governance ideas, adding adaptive adjustment weights and acceleration coefficients; Based on the calculated path fitness value, calculate the individual best position, local best position, global best position, and elect the manager particle Adm, so as to optimize and update the velocity V and position X, and generate a better elite population; The formulas for optimizing and updating the velocity and position are as follows: 1) Inertial factor: ω * ·V i t The first term of the velocity update formula (1) is the inertia factor, which is the adaptive inertia weight ω * and the velocity V of the i-th particle in the t-th generation i t The concept of the inertia weight is the same as that of the traditional particle swarm principle; 2) Individual best position: The second term of formula (1) calculates the current position of the i-th particle when iterated to the t-th generation and the personal best position P of this particle i t The distance between them is multiplied by the first acceleration factor c1 and a random number r1 ∈ [0, 1]; 3) Local optimal position: The third term in the expression (1) calculates the current position of the $i$-th particle when it iterates to the $t$-th generation and the best position of the particle within the peer group area The distance is multiplied by the second acceleration factor $c_2$ and a random number $r_2\in[0,1]$; 4) Global best position: The fourth term in expression (1) calculates the current position of the $i$-th particle when it iterates to the $t$-th generation from the global best position Adm of the particle swarm t , multiplied by the adaptive parameter $\beta$, the third acceleration $c_3$, the random number $r_3\in[0,1]$, and the supervision factor $\psi$; where Adm t is the position of the manager particle in the $t$-th generation, generated based on the voting concept Since the initial population is randomly generated and the particles do not have the ability to select a better leader, the initial voting ranges of the two candidate leader particles are asymmetrically processed, and the asymmetric range is controlled within [0, 1] through normalization. Therefore, in the initial iteration, the particles will select the operator as the position of the management particle Adm, and the owner as the second choice to play a supervisory role. Let the initial value of the number of votes Opvote cast for the operator particle Operator be The number of votes Owvote cast for the owner particle Owner is This further causes a deviation in the number of votes, where The voting mechanism is implemented through the roulette wheel algorithm, and each particle votes. is a random number within the range of [0, 1]. The selection of the management particle position Adm is as follows: Among them, represents the position of the i-th particle in the given dimension d at the t-th generation as the leader particle; To better control the global best position Adm of the particle swarm t To address the impact on the particle swarm, trigonometric functions are introduced, and an adaptive parameter β is proposed, which accelerates the convergence speed of the algorithm and improves the accuracy. The formula is as follows: where β min and β max are the maximum and minimum values of the parameter β; t is the number of iterations, and T is the set maximum number of iterations; The voting mechanism is implemented by the roulette algorithm. Each particle votes, and the votes are accumulated and updated to facilitate recording the influence of the Operator and Owner. The cumulative update formula is: Among them, is a random number within the range of [0, 1], N is the number of population particles, M represents the number of votes cast by a particle in a given dimension d for the corresponding candidate leader Operator or Owner, and the total number of support votes after accumulation can intuitively indicate the elected final Adm; for the convenience of further operations, after each iteration, the voting needs to be normalized: Among them, 5) The position of the $i$-th particle in the $(t + 1)$-th generation in expression (2) is equal to the position of the $i$-th particle in the $t$-th generation and the sum of the velocity $V$ multiplied by the fourth acceleration factor $C4$ and a random number $r4\in[0,1]$ i t+1 ; Step 6, Improve the differential evolution algorithm: Adjust the two parameters in the traditional differential evolution algorithm, the scaling factor F and the crossover probability factor CR, to adaptive parameters to improve the convergence performance and iteration accuracy of the algorithm; Step 7, Use the elite population obtained in Step 5 as the initial population of the improved differential evolution algorithm, combine the scaling factor F and the crossover probability factor CR of the adaptive parameters, perform mutation operations, crossover operations, and selection operations to achieve iterative optimization of the particle swarm, and update the Adm of the particle swarm Step Eight: Determine whether the fitness value fit is less than the minimum fitness value f min , that is, fit < f min : If so, end and output the optimized path; otherwise, return to Step Four; Step 9, Smooth the output optimized path.

2. The path planning method for a mobile robot based on particle swarm and differential evolution algorithms according to claim 1, characterized in that: The design evaluation indicators in Step 4 include the safety level and the path length.

3. The path planning method for a mobile robot based on particle swarm and differential evolution algorithms according to claim 1 or 2, characterized in that: The objective function in step 4 is constructed based on the evaluation index and consists of two parts: a path length function and a penalty function. Set the starting point coordinates as Start(x0,y0) and the target point coordinates as Goal(x n+1 ,y n+1 ). Each particle in the particle swarm represents a set of node coordinates H = {Start, (x1,y1), (x2,y2),...,(x n ,y n ), Goal} passed by a path; 1) Path length function The path length function f1 is used to calculate the path length of the mobile robot from the starting point Start to the target point Goal, and can be expressed as follows: 2) Penalty function The more times the path intersects with obstacles, the greater the danger of the generated path. Use the danger level to set the penalty function to penalize the path nodes that intersect with obstacles; The obstacle is represented by a circle, denoted as C k , with the center at O k , where k is the number of obstacles. The radius of the obstacle is set as the safety threshold R = {r0, r1, …, r k}. To obtain a collision-free path, it is necessary to ensure that the distance between the path nodes and the line connecting the path nodes and the obstacle is greater than the safety threshold; Take m mid-nodes on the line connecting two adjacent path nodes, and use the path mid-nodes to determine whether the route where they are located intersects with obstacles. Calculate the Euclidean distance Dis between the center of each mid-node and path node and the obstacle k , where k = 1, 2, …, t, then the penalty degree Risk between nodes is calculated according to the following formula: where r k is the k-th safety threshold set by the obstacle radius; for the penalty degree, two values of 0 and 1 are set. When the distance between the middle node and the path node and the obstacle is less than the safety threshold, the node is penalized and the penalty degree is set to 1. When the distance is greater than the safety threshold, the penalty degree is set to 0. Given that the weight coefficient is η, which represents the influence degree of the penalty degree on the path node, the penalty function expression is as follows: Among them, n is the number of path nodes; m is the number of middle nodes; Therefore, the objective function, that is, the fitness function, is the following expression: fit = f1 + f2.

4. The mobile robot path planning method based on particle swarm and differential evolution algorithms according to claim 1, characterized in that: In step five, the particle velocity is processed at the boundary. If it exceeds the velocity boundary, it is immediately absorbed: where is the component of the velocity of the t-th generation in dimension d, v min and v max are the minimum and maximum values of the velocity, respectively.

5. The mobile robot path planning method based on particle swarm and differential evolution algorithms according to claim 1, characterized in that: The adaptive optimization formula for the scaling factor F in Step 6 is as follows: Wherein, F i represents the scaling factor of the i-th particle vector of the population; at the t-th iteration, three individuals x p1 (t), x p2 (t), x p3 (t) are randomly selected from the particle population, and p1≠p2≠p3≠i, and fit(x p1 (t)), fit(x p2 (t)), fit(x p3 (t)) represent the fitness values of the three individuals x p1 (t), x p2 (t), x p3 (t), respectively.

6. The mobile robot path planning method based on particle swarm and differential evolution algorithms according to claim 1, characterized in that: The adaptive optimization formula for the crossover probability factor CR in Step 6 is as follows: In the formula, C R i is the mutation probability factor corresponding to the i-th particle vector of the population; CR min , CR max are respectively the minimum and maximum values of the crossover probability factor CR; fit(x i ) is the fitness value of the position vector x i , fit min , fit max are respectively the minimum and maximum values of the fitness, and fit mean is the average fitness value.

7. The mobile robot path planning method based on particle swarm and differential evolution algorithms according to claim 1, characterized in that: In Step 9, the output optimized path is smoothed, and the cubic spline interpolation method is used for calculation to obtain a path including path nodes, interpolation points, and the starting point and the target point.

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