Mechanical arm trajectory planning method based on COEHHO algorithm and quintic polynomial interpolation method

Abstract

The invention discloses a mechanical arm trajectory planning method based on a COEHHO algorithm and a quintic polynomial interpolation method, and belongs to the technical field of mechanical arm trajectory planning. By fusing Circle chaotic mapping, the diversity of populations in the initial stage can be enhanced, so that the populations are uniformly distributed, the global search capability is enhanced, and the difficulty of falling into local optimum is avoided; through an elite reverse learning strategy, the convergence speed of the algorithm is improved; finally, a nonlinear escape energy updating strategy is introduced, and the search precision of a globally optimal solution is improved; and finally, combining a COEHHO algorithm with a quintic polynomial interpolation method, and comparing with an existing trajectory planning method through a simulation experiment to obtain a continuous and smooth trajectory planning curve with optimal time.

Description

Technical Field

[0001] The present invention relates to the technical field of robot arm trajectory planning, and in particular to a robot arm trajectory planning method based on a COEHHO algorithm and a quintic polynomial interpolation method. Background Art

[0002] As a key link in the motion control of the robot arm, trajectory planning directly affects the efficiency, accuracy and safety of its task execution. In the trajectory planning theory of industrial robots, according to the differences in the motion characteristics of the task, trajectory planning is usually divided into point-to-point (PTP) trajectory planning (see Figure 1 ) and Continuous Path (CP) trajectory planning (see Figure 2 ) Among them, PTP planning only needs to ensure that the end effector is accurately positioned between discrete path points, and there is no specific constraint on the motion trajectory between points; while CP planning requires the robot arm to strictly follow the preset continuous space path, and to meet the continuity requirements of position, posture and high-order motion parameters during the motion process.

[0003] The movement path of the robot in space can be divided into joint space path planning and Cartesian space path planning. Compared with the Cartesian space trajectory planning method, the joint space trajectory planning takes the rotation joints of the robot as the direct control object, and describes the motion trajectory by establishing a functional relationship between the joint angular displacement and time. This method has two major technical advantages: first, by avoiding the frequent conversion between the end posture coordinates and the joint angle coordinates, it effectively reduces the time delay caused by the kinematic positive solution calculation, and significantly improves the real-time performance of the trajectory generation algorithm; second, while eliminating the cumulative effect of the coordinate system conversion error, it can directly constrain and optimize the motion characteristics of each joint, thereby ensuring the trajectory tracking accuracy of the end effector. Figure 3 This is the basic flow chart of joint space trajectory planning. Commonly used joint space trajectory planning methods include polynomial interpolation method, spline curve interpolation method, etc.

[0004] Swarm Intelligence (SI) algorithm is an optimization algorithm developed based on the behavior of biological groups such as ants, bees, and whales in nature. These biological groups show significant decentralized and decentralized self-organization characteristics at the collective level. Their individuals can efficiently solve complex problems such as dynamic optimization problems, constrained optimization problems, uncertain environment optimization problems, and multi-objective optimization problems through interaction mechanisms. In recent years, with the rapid development of artificial intelligence technology, swarm intelligence optimization algorithms have emerged and made significant progress. Among them, a group of algorithms represented by sparrow search algorithm, particle swarm optimization algorithm, genetic algorithm, etc. have been widely used in the field of optimization and solution due to their strong self-learning, adaptability, self-organization and other intelligent characteristics, as well as simple algorithm structure and fast convergence speed. These algorithms not only provide effective tools and technical support for solving complex optimization problems, but also open up new directions for the research of artificial intelligence, which have important theoretical significance and practical application value.

[0005] The process characteristics analysis of the welding robot arm shows that its operation task requires the end effector to complete continuous movement along the weld trajectory and must accurately pass through the intermediate path point sequence determined by the welding process parameters, which places strict requirements on the parameters of the robot arm during operation. Therefore, a robot arm trajectory planning method based on the COEHHO algorithm and the quintic polynomial interpolation method is proposed. Summary of the invention

[0006] The technical problem to be solved by the present invention is: how to generate an optimal feasible path from the starting point to the target point while ensuring the smoothness of the motion process and optimizing energy consumption under the premise of satisfying the kinematic and dynamic constraints of the robot arm, and provide a robot arm trajectory planning method based on the COEHHO algorithm and the quintic polynomial interpolation method.

[0007] The present invention solves the above technical problems through the following technical solutions, and the present invention comprises the following steps:

[0008] S1: Initialize parameters, set population size and maximum number of iterations;

[0009] S2: Initialize the Harris hawk population using Circle chaos map;

[0010] S3: Take the objective function as the fitness function, calculate the fitness value of each individual, and regard the individual with the best fitness as prey;

[0011] S4: Use the elite reverse learning strategy to select elite individuals to generate a reverse population, that is, to obtain a new population;

[0012] S5: Constructing the nonlinear convergence factor E 1 , used to dynamically adjust the prey’s escape energy;

[0013] S6: According to the prey escape energy E 1 The position update strategy is determined by the prey escape probability r, the position of Harris hawks in the new population under different predation strategies is calculated, and the global optimal position is updated with the number of iterations, where the prey escape energy E 1 is the nonlinear convergence factor E 1 ;

[0014] S7: Generate individual behavior paths based on Harris Hawk location points;

[0015] S8: Optimizing the generated path using a quintic polynomial interpolation method;

[0016] S9: If the maximum number of iterations is reached, the optimal fitness value is output and the process ends, thereby obtaining the optimal path. Otherwise, return to step S2 to continue iterating.

[0017] Furthermore, in step S2, the iterative formula of the Circle chaotic map is:

[0018]

[0019] Among them, x k ∈[0,1), represents the value of the kth iteration; a∈[0,1), represents the offset of the control sequence; b∈R + , which means controlling the strength of the sine term; mod 1 means mapping the result to the interval [0,1).

[0020] Furthermore, in step S3, the objective function is to minimize the total time for the robot to complete the entire operation process, and the total time T is expressed as:

[0021]

[0022] Among them, t 0 is the starting time; t m is the time when the target posture is reached, m-1 represents the number of segments of the motion trajectory of each joint of the robot arm, and h i Represents the time interval of the i-th trajectory.

[0023] Furthermore, in step S4, the elite reverse learning strategy generates a reverse solution for the elite individuals in the current group and uses the position information of the elite individuals to guide the search direction, wherein the top 10% of individuals are selected from the individuals with the best fitness to form elite individuals;

[0024] Assume that the position of an elite individual in the d-dimensional space is P elite =(x 1 ,x 2 ,...x d), its reverse solution P opposite The generation formula is:

[0025] P opposite,j =α 1 ·(LB j +UB j )-x j

[0026] Among them, LB j and UB j are the upper and lower bounds of the j-th dimension variable, α 1 It is a dynamic adjustment coefficient, with an initial value of 1 and gradually decreasing with iterations.

[0027] Furthermore, in step S5, the prey escape energy E 1 The definition is as follows:

[0028]

[0029] Among them, α and β are adjustment parameters used to control the prey escape energy E 1 The decay rate and nonlinearity of , t is the current iteration number, and T is the maximum iteration number.

[0030] Furthermore, in step S6, the location update strategy is specifically as follows:

[0031] If |E 1 |≥1, enter the search phase to update the position;

[0032] If 0.5≤|E 1 |<1 and r≥0.5, the soft siege strategy is used to update the position;

[0033] If |E 1 |<0.5 and r≥0.5, use the hard siege strategy to update the position;

[0034] If 0.5≤|E 1 |<1 and r<0.5, a soft siege strategy with raid is used to update the position;

[0035] If |E 1 |<0.5 and r<0.5, use the hard siege strategy with raid to update the position.

[0036] Furthermore, the mathematical model of the search phase is:

[0037]

[0038] Where q is the probability threshold, t is the number of iterations; P(t) is the current position of the eagle; P(t+1) is the position of the eagle after the next iteration; P rand(t) is the randomly selected eagle position; P prey (t) is the position of the prey; LB and UB represent the upper and lower bounds of the solution space; r 1 、r 2 、r 3 、r 4 ∈[0,1], indicating a uniformly distributed random number; P mean (t) represents the average position of the population.

[0039] Furthermore, the mathematical model of the soft siege strategy is:

[0040] P(t+1)=ΔP(t)-E 1 ·|JP prey (t)-P(t)|

[0041] ΔP(t)=P prey (t)-P(t)

[0042] J=2(1-r 5 )

[0043] Among them, P(t) represents the current position of the eagle; P(t+1) represents the position of the eagle after the next iteration; P prey (t) is the position of the prey; ΔP(t) is the position difference between the individual and the prey; J is the jumping intensity of the prey; r 5 ∈[0,1] is a random number;

[0044] The mathematical model of the hard siege strategy is:

[0045] P(t+1)=P prey (t)-E 1 |ΔP(t)|

[0046] The mathematical model of the soft siege strategy with raid is:

[0047] P(t+1)=P prey (t)-E 1 ·|JP prey (t)-P(t)|+Levy(D)

[0048] Where D represents the dimension of the solution space, Levy() is the Levy flight function;

[0049] The mathematical model of the hard siege strategy with raid is:

[0050] P(t+1)=P prey (t)-E 1 ·|JP prey (t)-P mean (t)|

[0051] Among them, P mean represents the average position of the population.

[0052] Furthermore, in step S8, the specific process of optimizing the generated path using the quintic polynomial interpolation method is as follows:

[0053] S81: Taking the angular velocity, angular acceleration and angular jerk of the robot arm joint at the starting point and the end point as constraint conditions, a corresponding fifth-order polynomial is established;

[0054] S82: Solving the coefficients of the quintic polynomial to obtain the path function, thereby optimizing the generated path.

[0055] Compared with the prior art, the present invention has the following advantages: the robot arm trajectory planning method based on the COEHHO algorithm and the quintic polynomial interpolation method, integrating the Circle chaotic map can enhance the diversity of the population in the initial stage, make it evenly distributed, enhance the global search capability, and avoid the dilemma of falling into the local optimum; then through the elite reverse learning strategy, the convergence speed of the algorithm is improved; finally, the nonlinear escape energy update strategy is introduced to improve the search accuracy of the global optimal solution; finally, the COEHHO algorithm is combined with the quintic polynomial interpolation method, and compared with the existing trajectory planning method through simulation experiments, a continuous, smooth, and time-optimal trajectory planning curve is obtained. BRIEF DESCRIPTION OF THE DRAWINGS

[0056] Figure 1 It is a schematic diagram of the PTP trajectory planning process in the prior art;

[0057] Figure 2 It is a schematic diagram of the CP trajectory planning process in the prior art;

[0058] Figure 3 It is a schematic diagram of the joint space trajectory planning process in the prior art;

[0059] Figure 4 It is a trajectory simulation result diagram under the cubic polynomial interpolation method in an embodiment of the present invention;

[0060] Figure 5 1 is a trajectory simulation result diagram under the fifth-order polynomial interpolation method in an embodiment of the present invention;

[0061] Figure 6 is a graph showing a comparison of convergence factors in an embodiment of the present invention;

[0062] Figure 7 It is a schematic diagram of the improved Harris Eagle optimization algorithm flow in an embodiment of the present invention;

[0063] Figure 8 is an iterative curve diagram of the average fitness of the population in the embodiment of the present invention, where (a) is based on F1 The average fitness iteration curve of the population, (b) is based on F 2 The iteration curve of the average fitness of the population;

[0064] Fig. 9 is a particle distribution diagram in an embodiment of the present invention, wherein (a) is a random distribution and (b) is a Circle distribution. DETAILED DESCRIPTION

[0065] The following is a detailed description of an embodiment of the present invention. This embodiment is implemented on the premise of the technical solution of the present invention, and a detailed implementation method and a specific operation process are given, but the protection scope of the present invention is not limited to the following embodiment.

[0066] The design process of the robot arm trajectory planning method based on the COEHHO algorithm and the quintic polynomial interpolation method in this embodiment is as follows:

[0067] 1. Comparative study of polynomial interpolation methods

[0068] Polynomial interpolation methods in robot trajectory planning are mainly classified according to the kinematic constraint dimension. According to the order of the polynomial, cubic and quintic polynomials are typical representatives.

[0069] (1) Cubic polynomial interpolation method

[0070] When the robot motion accuracy is not high, the trajectory planning usually adopts the cubic polynomial interpolation method, which uses the speed, starting point, and end point angles as constraints. Under the premise of knowing the initial posture and expected posture of the robot arm, the joint variables of the posture can be solved. The determination of the angle of each joint in the cubic polynomial interpolation process is expressed as:

[0071]

[0072] Among them, a 0 、a 1 、a 2 、a 3 is the unknown coefficient, t∈[t 0 ,t f ], t 0 is the initial time, t f is the end time;

[0073] Connect two adjacent points as the starting point and the end point, and regard them as a PTP trajectory planning. 0 and θ f Denote the initial and desired angles, v 0 and v f Denote the starting and ending speeds, t 0 and t fare the start and end times. List the constraint formulas:

[0074]

[0075] For the convenience of calculation, t 0 Set it to 0, and combine equations (1) and (2) to solve the polynomial coefficients:

[0076]

[0077] Substituting the coefficients obtained from equation (3) into equation (1), the trajectory equation can be obtained.

[0078] After obtaining the trajectory equation, given the parameters, the robot trajectory planning using cubic polynomial interpolation was simulated on MATLAB software. In this process, the position, velocity, and acceleration of joint three were analyzed. In order to simulate the real scene of the robot during welding, the running time was set to 10s, the initial and terminal velocities were set to 0, and the joint angles of joint three at the five selected path points were shown in Table 1. The trajectory simulation results are shown in Figure 4 shown.

[0079] Table 1 Joint angles of joint 3 at the five selected path points

[0080] Path sequence number Waypoint 1 Waypoint 2 Waypoint Three Waypoint Four Waypoint Five Angle / (°) 0 50 150 100 0

[0081] (2) Quintic polynomial interpolation method

[0082] The fifth-order polynomial interpolation method describes the trajectory of the angle of the robot arm joint changing with time through a fifth-order polynomial function, which usually includes six constraints: the position of the starting point and the end point, the speed and the acceleration. Let the time be t and the joint angle be θ(t), then its expression is:

[0083] θ(t)=a 0 +a 1 t+a 2 t 2 +a 3 t 3 +a 4 t 4 +a 5 t 5 (6)

[0084] Assume that the initial time t 0 = 0, position θ(0) = θ 0 ,speed Acceleration End time t = T, position θ(T) = θ f ,speed Acceleration After substituting the boundary conditions, six linear equations are obtained:

[0085]

[0086] By solving the linear equations, determine a 3 、a 4 、a 5 , we can get the complete quintic polynomial, and thus the trajectory equation.

[0087] After the solution is completed, in order to compare with the cubic polynomial interpolation method, the same parameters are used to simulate it, and the trajectory simulation results are as follows Figure 5 shown.

[0088] By comparing the simulation results of the cubic polynomial interpolation method and the quintic polynomial interpolation method in trajectory planning, it can be analyzed that the quintic polynomial interpolation method has significant advantages in trajectory planning performance. Specifically, the trajectory curve generated by this method shows higher smoothness in terms of joint angle and speed; its angular acceleration curve remains continuous at the path points, avoiding mutations; in addition, at the starting and ending points of the movement, the acceleration value is significantly reduced, which effectively reduces the impact effect during the operation of the robot arm and can effectively suppress the jitter of the robot arm, thereby ensuring the accuracy of the welding position and improving the welding quality. However, there are also problems such as slow calculation process and long data processing time resulting in insufficient response. Therefore, using only the quintic polynomial interpolation method for trajectory planning of the robot arm cannot meet the requirements, and other conditions need to be added for optimization.

[0089] 2. Harris Eagle Optimization Algorithm and Its Improvement

[0090] In the previous section, the present invention conducted a comparative study on the related issues of trajectory planning of the robot arm using polynomial interpolation method in joint space. The research results show that it is difficult to meet the practical application requirements of the present invention by relying solely on the interpolation method. In view of the significant advantages of swarm intelligence algorithms in optimization and a large number of precedents of use in robotics technology, this section will focus on the Harris Hawks Optimization (HHO) algorithm that has emerged in recent years. On this basis, the present invention proposes a COEHHO algorithm (improved Harris Hawk Optimization algorithm) to improve the performance of robot arm trajectory planning. Through systematic simulation experiments, the COEHHO algorithm is compared and analyzed with other swarm intelligence algorithms to verify its effectiveness and superiority.

[0091] 2.1 Harris Eagle Optimization Algorithm

[0092] The Harris Hawk Optimization Algorithm was proposed by Heidari, Mirjalili and others in 2019. It was inspired by the searching, tracking, encircling, and consuming behaviors exhibited by Harris Hawks during predation. From the perspective of biological bionics, the dynamic division of labor and cooperation mechanism exhibited by hawks during predation is abstracted into a universal optimization framework, especially its unique collaborative foraging behavior (including multi-directional encirclement, dynamic tracking, and adaptive attack strategies), which provides a novel solution for solving complex optimization problems. At present, some scholars have applied the HHO algorithm to related optimization problems. In the established mathematical model, HHO iteratively updates the position of the solution to simulate the dynamic adjustment and strategy optimization of hawks during predation. The specific process can be divided into the following stages:

[0093] (1) Tracking and Exploration Behavior

[0094] During the search phase, Harris's hawks do not gather together, but roost in high places (such as treetops or rock peaks) as individuals. They continuously monitor the environment through vision until prey appears, and dynamically choose the monitoring strategy based on the location of the prey and other companions. Specifically, the algorithm balances the two roosting strategies through the probability threshold q. When q<0.5, individuals tend to roost in local areas near population members to ensure that they maintain a moderate distance when attacking; when q≥0.5, a random dispersion strategy is adopted to select high sites for detection in the global range. The corresponding calculation formula is as follows:

[0095]

[0096] Where t represents the number of iterations; P(t) represents the current position of the eagle; P(t+1) represents the position of the eagle after the next iteration; P rand (t) is the randomly selected eagle position; P prey (t) is the position of the prey; LB and UB represent the upper and lower bounds of the solution space; r 1 、r 2 、r 3 、r 4 ∈[0,1], indicating a uniformly distributed random number; P mean (t) represents the average position of the population, and its expression is:

[0097]

[0098] Among them, P i (t) represents the position of each eagle in iteration t, and N represents the total number of eagles.

[0099] (2) Transition stage

[0100] As the prey evades capture, its energy E decreases with each iteration, driving the algorithm from exploration to exploitation. The energy of the prey can be simulated as:

[0101]

[0102] Where T represents the maximum number of iterations; E 0 represents the randomly generated initial energy, simulating the uncertainty of the initial physical strength of the prey, E 0 ∈[-1,1];

[0103] The energy E decays linearly with the number of iterations t, and eventually approaches -2E 0 When |E|≥1, the prey is physically strong and triggers a global search; when |E|<1, the prey is physically weak and enters local development.

[0104] (3) Development stage

[0105] According to the research on Harris Hawk's hunting behavior, it can dynamically adjust and display various hunting strategies when facing different avoidance behaviors of prey. The HHO algorithm simulates this phenomenon and summarizes it into four different strategies.

[0106] ①Soft Surround

[0107] When the prey escape probability r≥0.5 and |E|≥0.5, it means that the prey has medium physical strength and tries different behaviors to escape capture. In this case, the Harris Hawk will gradually surround the prey, consume its physical strength, and suddenly attack the prey after it is tired, thus capturing the prey. Its mathematical model is:

[0108] P(t+1)=ΔP(t)-E·|JP prey (t)-P(t)| (11)

[0109] ΔP(t)=P prey (t)-P(t) (12)

[0110] J=2(1-r 5 ) (13)

[0111] Where ΔP(t) represents the position difference between the individual and the prey; J represents the jumping intensity of the prey; r 5 ∈[0,1] is a random number.

[0112] ②Hard Surround

[0113] When the prey escape probability r≥0.5 and |E|<0.5, it means that the prey is about to run out of energy and is in a state of exhaustion, and the possibility of escape is extremely low. At this time, the Harris Hawk will approach the prey and kill it directly. The corresponding mathematical model is:

[0114] P(t+1)=Pprey (t)-E·|ΔP(t)| (14)

[0115] ③Soft Surround with Dive

[0116] When the prey escape probability r<0.5 and E≥0.5, it means that the prey has enough energy to escape. To prevent the prey from escaping, the Harris Hawk will adopt a more rigorous soft encirclement strategy to consume the prey's physical strength until the prey's physical strength is exhausted and then hunt it. This capture strategy is called a soft siege strategy with surprise attack. Among them, the Levy flight function is used to realize the soft siege of the prey, and the corresponding mathematical model is:

[0117] P(t+1)=P prey (t)-E·|JP prey (t)-P(t)|+Levy(D) (15)

[0118] Where D represents the dimension of the solution space.

[0119] The Levy flight concept introduced is used to simulate the movement trajectory of prey in the stage of avoiding pursuit, which is consistent with the habits of most animals in nature. The Levy flight function expression is as follows:

[0120]

[0121] Among them, β is taken as a constant 1.5; u, v∈(0,1) are random variables, σ is a normalization constant, and Γ is a gamma function.

[0122] ④Hard Surround with Dive

[0123] When the prey escape probability r<0.5 and |E|<0.5, it means that the prey has run out of energy and cannot escape. At this time, the Harris Hawk will launch a hard siege, gradually narrowing the distance between it and the prey, and surround and capture the prey. The mathematical model is:

[0124] P(t+1)=P prey (t)-E·|JP prey (t)-P mean (t)| (17)

[0125] Although the Harris Eagle has the advantages of low dependence on parameters and the ability to take into account both global and local search, it also has defects such as weak adaptability to dynamic environments, slow convergence in high-dimensional environments, and easy to fall into local optimality.

[0126] 2.2 Improved Harris Hawk Optimization Algorithm

[0127] (1) Circle Chaos Map

[0128] Chaotic mapping has the characteristics of ergodicity and initial value sensitivity. It can generate an evenly distributed and diverse initial population. At the same time, it provides chaotic disturbances for the local development stage to prevent the algorithm from falling into the local optimum. It is often used for algorithm optimization and improvement. Commonly used chaotic mapping can be divided into SPM chaotic mapping, Logistic chaotic mapping, Tent chaotic mapping, Circle chaotic mapping, etc. according to their functions. In view of the problems of uneven distribution and lack of diversity in the initial stage of HHO, the population is generated by random methods. By comparison, the present invention selects Circle chaotic mapping to optimize the population initialization process, expand its search space, and increase the diversity and randomness of the population. The iterative formula of Circle chaotic mapping is:

[0129]

[0130] Among them, x k ∈[0,1), represents the value of the kth iteration; a∈[0,1), represents the offset (bias parameter) of the control sequence; b∈R + , which means controlling the intensity of the sine term (nonlinear parameter); mod 1 means mapping the result to the interval [0,1).

[0131] To verify the performance optimization after adding the chaotic mapping strategy, the parameters are initialized, a = 0.5, b = 0.2, and x 0 = 0.1, and the number of particles generated in the dimension [0, 10000] is compared. The results are as follows Fig. 9 As shown in the figure, the particle distribution generated by introducing Circle chaotic mapping is more uniform than that randomly generated by the HHO algorithm, which can effectively enhance the diversity of the initial stage of the algorithm and further improve the convergence speed and global search ability of the algorithm.

[0132] (2) Elite Reverse Learning Strategy

[0133] In the HHO algorithm, the iterative update of the Harris Hawk population individuals is randomly selected, with large uncertainty, low iteration efficiency, and the quality after iteration cannot be guaranteed. There is also the problem of being easily trapped in the local optimum during the search process. Therefore, the elite reverse learning strategy is introduced to optimize the algorithm. The principle is to generate a reverse solution for the elite individuals in the current group (select the top 10% of individuals from the individuals with the best fitness to form elite individuals), and use the position information of the elite individuals to guide the search direction, thereby improving the algorithm's search ability and convergence speed. Assume that the position of an elite individual in the d-dimensional space is P elite =(x 1 ,x 2 ,…x d ), its reverse solution Popposite The generation formula is:

[0134] P opposite,j =α·(LB j +UB j )-x j (19)

[0135] Among them, LB j and UB j are the upper and lower bounds of the j-th dimension variable, and α is the dynamic adjustment coefficient (the initial value is 1, which gradually decreases with iteration).

[0136] (3) Nonlinear escape energy update strategy

[0137] In the Harris Hawk optimization algorithm, the energy factor E (convergence factor) is a key parameter for the dynamic balance between the global search and local search of the control algorithm. The traditional HHO algorithm uses a linear decreasing method to update the energy factor E = 2 → 0, and its mathematical expression is:

[0138]

[0139] Among them, t is the current iteration number and T is the maximum iteration number.

[0140] However, the linear decreasing mechanism can easily lead to insufficient global search capabilities in the later stages of the algorithm, excessive bias towards local development, and difficulty in balancing the synergy between the exploration and development stages, thereby increasing the risk of falling into local optimality. To this end, the present invention proposes an energy factor update strategy based on a nonlinear decreasing mechanism. The improved energy factor introduces a dynamic adjustment mechanism to slow down the energy decay rate in the later stages of algorithm iteration, retain a certain global search capability, and improve local development accuracy. The improved formula is defined as follows:

[0141]

[0142] Among them, α and β are adjustment parameters used to control the attenuation rate and nonlinearity of the energy factor.

[0143] Compared with the traditional linear method, the improved strategy maintains a higher energy value in the early stage of iteration to strengthen the global search, and retains moderate exploration ability by gently decreasing it in the later stage of iteration, effectively coordinating the contradiction between the global optimization and local convergence of the algorithm, such as Figure 6 Experiments show that this strategy can enhance the algorithm's exploration efficiency of complex solution spaces, suppress premature convergence, and thus improve the search accuracy and stability of the global optimal solution.

[0144] The improved Harris Eagle optimization algorithm flow chart is as follows: Figure 7 As shown, the specific implementation steps are as follows:

[0145] Step 1: Input algorithm-related parameters, including population size N and maximum number of iterations T max ;

[0146] Step 2: Initialize the Harris hawk population using Circle chaos mapping;

[0147] Step 3: Calculate the fitness value of each individual and regard the individual with the best fitness as prey;

[0148] Step 4: Construct the nonlinear convergence factor E 1 , used to dynamically adjust the escape energy;

[0149] Step 5: For each individual, according to the prey escape energy E 1 and the prey escape probability r to determine the position update strategy;

[0150] If |E 1 |≥1, enter the search phase to update the position;

[0151] If 0.5≤|E 1 |<1 and r≥0.5, the soft siege strategy is used to update the position;

[0152] If |E 1 |<0.5 and r≥0.5, use the hard siege strategy to update the position;

[0153] If 0.5≤|E 1 |<1 and r<0.5, a soft siege strategy with raid is used to update the position;

[0154] If |E 1 |<0.5 and r<0.5, use the hard siege strategy with raid to update the position;

[0155] Step 6: Determine whether the current number of iterations t is equal to the maximum number of iterations T max If they are equal, output the optimal solution; otherwise, return to step 5 to continue iterating.

[0156] 2.3. Simulation analysis of optimization algorithm

[0157] In order to verify the convergence performance and optimization ability of the proposed improved Harris Hawk Optimization Algorithm (COEHHO), the Dung Beetle Optimization Algorithm (DBO), the Whale Optimization Algorithm (WOA), the Harris Hawk Optimization Algorithm (HHO) and the improved algorithm (COEHHO) are selected for comparative experiments. The experiment is based on the standard test function in Table 2, where the unimodal function F 1 Used to evaluate the convergence ability of the algorithm, the multi-peak function F 2 Used to test the global search capability of the algorithm. Through simulation experiments, the convergence speed and convergence accuracy of each algorithm are intuitively compared to verify the optimization effect of the COEHHO algorithm.

[0158] Table 2 Test function table

[0159]

[0160] The experiment sets the population size N to 30 and the maximum number of iterations to 1000. 1 and F 2 Perform 20 independent tests and obtain Figure 8 The average fitness iteration curve of the population is shown.

[0161] like Figure 8 As shown in (a), in F 1 In the test function, the fitness values ​​of SSA and WOA always approach 1, while the fitness value of COEHHO continues to decrease with the iteration process and approaches 0 infinitely, which shows that COEHHO is significantly better than the mainstream swarm intelligent optimization algorithm in terms of convergence speed and accuracy, and the improvement is 38.6% compared with the traditional HHO algorithm. It is worth noting that the traditional HHO algorithm has a good performance in F 2 Although the global optimum can be quickly located within 50 iterations in the test ( Figure 8 (b) in the figure, but its fitness value enters steady-state convergence too early, revealing that the algorithm has the inherent defects of low convergence accuracy and easy to fall into local optimum. To overcome this limitation, the present invention constructs an initial population uniformly distributed in high-dimensional space through Circle chaotic mapping, which improves the population diversity by 21.4%. At the same time, it adopts the fusion elite reverse learning strategy and nonlinear escape energy update strategy. The improved COEHHO algorithm is 2 The test showed multi-peak optimization characteristics, and its fitness curve showed a continuous step-like decline (including 3 significant inflection points), proving that the algorithm has the characteristics of dynamically jumping out of the local optimum and continuously approaching the global optimal solution. Experimental data show that the final convergence accuracy of COEHHO is 2-3 orders of magnitude higher than that of HHO, SSA, and WOA, respectively, verifying the effectiveness of the proposed improvement strategy.

[0162] 2.4 Mathematical model of time-optimal trajectory planning

[0163] (1) Objective function

[0164] In the robot operation task, it is assumed that the end effector needs to pass through m path points (including the starting point and the target point) when running from the starting position to the target position. Through the inverse kinematics solution, these m path points are mapped to the joint space to obtain the m joint nodes corresponding to each joint of the robot arm. Based on these joint nodes, the motion trajectory of each joint of the robot arm can be divided into m-1 segments. Let the time interval of the i-th trajectory be h i =t i+1 -t i(i=0,1,…m-1), where t i is the time when the robot moves to the i-th joint node. The total time T for the robot to complete the entire operation process can be expressed as:

[0165]

[0166] Where: t 0 is the starting time; t m is the time when the target position is reached.

[0167] (2) Design variables

[0168] In formula (21), since the time interval h during the operation of the robot arm i are independent and adjustable, so h i As an optimization design variable, it is reasonable and feasible. This variable selection can not only fully reflect the timing characteristics of the robot arm's motion process, but also facilitate the precise control and optimization of the motion trajectory.

[0169] (3) Constraints

[0170] In the trajectory planning of the welding robot, the initial and final positions of the target welding trajectory are known, and its operation time mainly depends on the kinematic parameters of each joint of the robot, including angular velocity, angular acceleration and angular jerk. In order to optimize the operation time, the maximum value of the above motion parameters needs to be used as a constraint condition, which is defined as follows:

[0171] Angular velocity limit:

[0172]

[0173] Angular acceleration limit:

[0174]

[0175] Angular jerk limit:

[0176]

[0177] Time-optimal constraints:

[0178] T≥T min

[0179] In the formula: i = 1, 2,…, m, m is the degree of freedom of the robot arm.

[0180] According to the user manual, the constraints of the angular velocity, angular acceleration, and angular jerk of each joint of the RM-65 robot arm are shown in Table 3:

[0181] Table 3 Constraints of angular velocity, angular acceleration, and angular jerk of each joint of the RM-65 manipulator

[0182]

[0183] (4) Select path points

[0184] According to the requirements of welding operations, the present invention designs three sets of trajectory optimization experiments. Based on the path point joint angle values ​​shown in Table 4, the following three motion paths are defined: ①P1→P2→P3→P4; ②P1→P3→P4→P2; ③P2→P1→P4→P3. Taking path ① as an example, its motion process can be decomposed into three consecutive stages: P1→P2, P2→P3 and P3→P4. The running time of each stage is determined by the maximum motion time of the six joints, and the total path time is the sum of the time required for each stage.

[0185] Table 4 Joint angle values ​​corresponding to path points

[0186]

[0187] 2.5 Experimental Process

[0188] Now we use the RM-65 robotic arm, the quintic polynomial and the COEHHO algorithm to do the time optimal trajectory planning experiment. The specific process is as follows:

[0189] Step 1: Initialize parameters, set the population size N = 60, and the maximum number of iterations T = 600;

[0190] Step 2: Initialize the Harris hawk population using Circle chaos map;

[0191] Step 3: Take the objective function as the fitness function, calculate the fitness value of each individual, and regard the individual with the best fitness as prey;

[0192] Step 4: Use the elite reverse learning strategy to select elite individuals to generate a reverse population, that is, to obtain a new population;

[0193] Step 5: Construct the nonlinear convergence factor E 1 , used to dynamically adjust the prey’s escape energy;

[0194] Step 6: Escape energy E according to the prey 1 The position update strategy is determined by the prey escape probability r, the position of Harris hawks in the new population under different predation strategies is calculated, and the global optimal position is updated with the number of iterations, where the prey escape energy E 1 is the nonlinear convergence factor E 1 ;

[0195] Step 7: Generate individual behavior paths based on Harris Hawk location points;

[0196] Step 8: Optimize the generated path using the quintic polynomial interpolation method;

[0197] Step 9: If the maximum number of iterations is reached, the optimal fitness value is output and the process ends, thereby obtaining the optimal path. Otherwise, return to step S2 to continue iterating.

[0198] 2.6 Simulation Experiment

[0199] Based on the experimental design, the present invention conducts a comparative experiment on robot trajectory planning under the following three conditions:

[0200] Experiment 1: Only the fifth-order polynomial interpolation method was used;

[0201] Experiment 2: Quintic polynomial interpolation combined with HHO algorithm;

[0202] Experiment 3: Quintic polynomial interpolation method combined with COEHHO algorithm.

[0203] During the experiment, the total time for the robot to pass through the three paths (the paths defined in part (4) of Section 2.4) and the running time of each joint in each stage were recorded. To ensure the reliability of the evidence, the experiment was repeated 30 times for each path and the average value was taken. The experimental results are shown in Tables 5 to 7. The data with the longest average time among the six joints in each stage are bolded, and the value "0" indicates that the joint is stationary at this stage.

[0204] Table 5 Experimental results of Experiment 1

[0205]

[0206] Table 6 Experimental results of Experiment 2

[0207]

[0208] Table 7 Experimental results of Experiment 3

[0209]

[0210]

[0211] According to the comparison of experimental data, the trajectory planning scheme using the quintic polynomial interpolation method combined with the COEHHO algorithm has a significant improvement in time performance and achieves the purpose of time optimization. The specific performance is shown in Table 8:

[0212] Table 8 Comparison of experimental data of each group

[0213]

[0214] Although the embodiments of the present invention have been shown and described above, it is to be understood that the above embodiments are exemplary and are not to be construed as limitations of the present invention. A person skilled in the art may change, modify, replace and vary the above embodiments within the scope of the present invention.

Claims

1. A robot arm trajectory planning method based on the COEHHO algorithm and the quintic polynomial interpolation method, characterized in that: The following steps are involved: S1: Initialize parameters, set population size and maximum number of iterations; S2: Initialize the Harris hawk population using Circle chaos map; S3: Take the objective function as the fitness function, calculate the fitness value of each individual, and regard the individual with the best fitness as prey; S4: Use the elite reverse learning strategy to select elite individuals to generate a reverse population, that is, to obtain a new population; S5: construct a nonlinear convergence factor E1 to dynamically adjust the prey escape energy; S6: Determine the position update strategy according to the prey escape energy E1 and the prey escape probability r, calculate the position of Harris's hawk in the new population under different predation strategies, and update the global optimal position with the number of iterations, where the prey escape energy E1 is the nonlinear convergence factor E1; S7: Generate individual behavior paths based on Harris Hawk location points; S8: Optimizing the generated path using a quintic polynomial interpolation method; S9: If the maximum number of iterations is reached, the optimal fitness value is output and the process ends, thereby obtaining the optimal path. Otherwise, return to step S2 to continue iterating.

2. The robot arm trajectory planning method based on the COEHHO algorithm and the quintic polynomial interpolation method according to claim 1, characterized in that: In step S2, the iterative formula of Circle chaos mapping is: Among them, x k ∈[0,1), represents the value of the kth iteration; a∈[0,1), represents the offset of the control sequence; b∈R + , which means controlling the strength of the sine term; mod 1 means mapping the result to the interval [0,1).

3. The robot arm trajectory planning method based on COEHHO algorithm and quintic polynomial interpolation method according to claim 2, characterized in that: In step S3, the objective function is to minimize the total time for the robot to complete the entire operation process, and the total time T is expressed as: Among them, t0 is the starting time; t m is the time when the target posture is reached, m-1 represents the number of segments of the motion trajectory of each joint of the robot arm, and h i Represents the time interval of the i-th trajectory.

4. The robot arm trajectory planning method based on the COEHHO algorithm and the quintic polynomial interpolation method according to claim 2, characterized in that: In step S4, the elite reverse learning strategy generates a reverse solution for the elite individuals in the current group and uses the position information of the elite individuals to guide the search direction, wherein the top 10% of individuals are selected from the individuals with the best fitness to form elite individuals; Assume that the position of an elite individual in the d-dimensional space is P elite =(x1,x2,…x d ), its reverse solution P opposite The generation formula is: P opposite,j =α1·(LB j +UB j )-x j Among them, LB j and UB j are the upper and lower bounds of the j-th dimension variable, α1 is the dynamic adjustment coefficient, the initial value is 1, and it gradually decreases with iteration.

5. The robot arm trajectory planning method based on COEHHO algorithm and quintic polynomial interpolation method according to claim 4 is characterized in that: In step S5, the prey escape energy E1 is defined as follows: Among them, α and β are adjustment parameters used to control the attenuation rate and nonlinearity of the prey escape energy E1, t ​​is the current iteration number, and T is the maximum iteration number.

6. The robot arm trajectory planning method based on COEHHO algorithm and quintic polynomial interpolation method according to claim 1 or 5, characterized in that: In step S6, the location update strategy is as follows: If |E1|≥1, enter the search phase to update the position; If 0.5≤|E1|<1 and r≥0.5, the soft siege strategy is used to update the position; If |E1|<0.5 and r≥0.5, the hard siege strategy is used to update the position; If 0.5≤|E1|<1 and r<0.5, the soft siege strategy with surprise attack is used to update the position; If |E1|<0.5 and r<0.5, use the hard siege strategy with raid to update the position.

7. The robot arm trajectory planning method based on COEHHO algorithm and quintic polynomial interpolation method according to claim 6, characterized in that: The mathematical model of the search phase is: Where q is the probability threshold, t is the number of iterations; P(t) is the current position of the eagle; P(t+1) is the position of the eagle after the next iteration; P rand (t) is the randomly selected eagle position; P prey (t) is the position of the prey; LB and UB represent the upper and lower bounds of the solution space; r1, r2, r3, r4∈[0,1] represent uniformly distributed random numbers; P mean (t) represents the average position of the population.

8. The robot arm trajectory planning method based on COEHHO algorithm and quintic polynomial interpolation method according to claim 6, characterized in that: The mathematical model of the soft siege strategy is: P(t+1)=ΔP(t)-E1·|JP prey (t)-P(t)| ΔP(t)=P prey (t)-P(t) J＝2(1-r5) Among them, P(t) represents the current position of the eagle; P(t+1) represents the position of the eagle after the next iteration; P prey (t) is the position of the prey; ΔP(t) represents the position difference between the individual and the prey; J represents the jumping intensity of the prey; r5∈[0,1] is a random number; The mathematical model of the hard siege strategy is: P(t+1)=P prey (t)-E1·|ΔP(t)|; The mathematical model of the soft siege strategy with raid is: P(t+1)=P prey (t)-E1·|JP prey (t)-P(t)|+Levy(D) Where D represents the dimension of the solution space, Levy() is the Levy flight function; The mathematical model of the hard siege strategy with raid is: P(t+1)=P prey (t)-E1·|JP prey (t)-P mean (t)| Among them, P mean represents the average position of the population.

9. The robot arm trajectory planning method based on COEHHO algorithm and quintic polynomial interpolation method according to claim 1, characterized in that: In step S8, the specific process of optimizing the generated path using the quintic polynomial interpolation method is as follows: S81: Taking the angular velocity, angular acceleration and angular jerk of the robot arm joint at the starting point and the end point as constraint conditions, a corresponding fifth-order polynomial is established; S82: Solving the coefficients of the quintic polynomial to obtain the path function, thereby optimizing the generated path.

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