Mechanical arm trajectory planning method based on improved particle swarm optimization

Abstract

The invention provides an improved particle swarm optimization (PSO) algorithm for mechanical arm trajectory planning, and aims at solving the problems that a traditional PSO algorithm is low in convergence speed and prone to falling into a local optimal solution. Therefore, the inertia weight and the learning factor of the algorithm are dynamically adjusted, so that the inertia weight and the learning factor change along with the increase of the number of iterations, and the global search capability and the convergence speed are enhanced; meanwhile, an elite reverse learning (EL) strategy is introduced, an optimal trajectory planning scheme is selected according to a fitness function, a new search area is explored through reverse particles, and the local search capability is enhanced; and a Gaussian-Cauchy variation (GC) strategy is combined, so that the global search capability is further improved, and a local optimal solution is avoided. In the optimization process, a single-target optimization method is adopted to carry out hierarchical classification on population solutions, and the performance of the algorithm under single-target optimization and constraint conditions is improved. Finally, the trajectory of the mechanical arm is optimized based on the algorithm, and particularly, each time period in a 3-5-3 polynomial interpolation method is finely optimized, so that the time from an initial point to a target point is shortened, a trajectory planning scheme with the optimal time is obtained, and the working efficiency of the mechanical arm is improved.

Description

Technical Field

[0001] The present invention belongs to the fields of industrial robots and computer applications. More precisely, the present invention relates to a manipulator trajectory planning method based on an improved particle swarm algorithm. Background Art

[0002] In recent decades, the continuous innovation of automation technology has promoted the wide application of manipulators in production systems. Especially in industrial assembly lines and production lines, manipulators are highly praised for their high speed, high reliability, and precise task execution capabilities. However, in these efficient industrial environments, how to perform precise trajectory planning for manipulators to ensure their efficient and accurate completion of various tasks has become a key problem to be solved urgently. Therefore, the trajectory planning problem of manipulators has become an important research direction in the field of robotics and automation.

[0003] The trajectory planning time of a manipulator directly affects its efficiency and is crucial for meeting the time limits and requirements of industrial production. To improve the efficiency of the manipulator to meet the actual application needs, researchers have tried to optimize the trajectory planning with time as an indicator. However, the current main method - single-objective optimization based on the Particle Swarm Optimization (PSO) algorithm has two major defects. First, this method fails to comprehensively consider the constraint conditions of each joint of the manipulator, so it cannot accurately reflect the overall comprehensive performance of the manipulator in terms of operation time. Second, the standard PSO algorithm is prone to falling into local optima and difficult to find the optimal time solution, which further reduces the efficiency of the manipulator.

[0004] Specifically, although the Particle Swarm Optimization (PSO) algorithm has various advantages, it also has significant disadvantages. In a function with multiple local extreme points, the algorithm is prone to falling into local optima, resulting in premature convergence and low convergence accuracy. This is mainly because the diversity of particles in the algorithm quickly disappears, and there is a lack of dynamic adjustment of velocity, which may cause the algorithm to stop searching prematurely during the iteration process and thus fail to find the global optimal solution. Therefore, in the trajectory planning problem of manipulators, how to overcome these defects of the PSO algorithm and improve the efficiency and accuracy of trajectory planning is an important research topic at present.

[0005] The Chinese patent publication number is "CN 117301048 A", and the patent name is "A Method for Trajectory Planning of Manipulator Based on Time Optimal". A method for trajectory planning of manipulator based on time optimal is proposed, in which the whale optimization algorithm and quintic polynomial interpolation are combined. By improving and enhancing the whale optimization algorithm, its convergence speed is increased and it is prevented from falling into the local optimal state, ensuring that it can obtain the motion trajectory under the optimal time to improve the working efficiency of the manipulator. Among them, by introducing a piecewise attenuation rate and an inertia weight to avoid the algorithm falling into the local optimum. Although this method can improve the working efficiency of the manipulator and achieve time-optimal trajectory planning, its local search ability is weak, more dependent on the global search strategy, and may be relatively slow in improving the local accuracy. Summary of the Invention

[0006] The technical problem to be solved by the present invention is that the PSO algorithm may face limitations of slow convergence speed and easy to fall into local optimal solutions in applications. A method for optimal time trajectory planning of a manipulator with an improved particle swarm algorithm is provided. This method adjusts the inertia weight and learning factor in the algorithm to dynamically change their numerical values as the number of iterations increases, thereby improving the global search ability of the algorithm, and uses elite reverse learning to optimize it to improve the algorithm efficiency.

[0007] The technical solution for the present invention to solve the technical problem:

[0008] Step 1: In this paper, the manipulator is taken as the research object, and kinematic modeling of the manipulator is carried out based on the classical D-H method. According to the shape characteristics of the manipulator and its D-H parameters, a D-H parameter table is established.

[0009] Step 2: Conduct forward kinematic analysis of the manipulator, and construct a homogeneous transformation matrix for each joint based on the D-H parameter method. By cascading the transformation matrices of each joint, the overall transformation matrix of the end effector is obtained. From this, the position and attitude information of the end effector in the Cartesian space are extracted, the displacement and direction of the end effector relative to the base coordinate system are clarified, and a foundation is laid for subsequent inverse kinematic analysis.

[0010] Step 3: Conduct inverse kinematic analysis of the manipulator. According to the given desired position and attitude of the end effector in the Cartesian space, an inverse kinematic equation is established. Corresponding constraint conditions (such as joint angle range, manipulator workspace limitation, etc.) are introduced, and the inverse kinematic equation is solved by the analytical method to obtain the solutions of each joint angle and determine a series of key points. These joint angle solutions reflect the motion state of the manipulator in the joint space, ensuring that the end effector can achieve the desired motion trajectory and attitude in the Cartesian space.

[0011] Step 4: Perform trajectory planning on the robotic arm using 3-5-3 polynomial interpolation to generate a smooth trajectory curve.

[0012] Step 5: Construct a trajectory planning model based on the improved PSO (Particle Swarm Optimization) algorithm, set key parameters such as the size of the particle swarm, the maximum number of iterations, the initial value and range of the inertia weight, the initial value and range of the learning factor, etc., and introduce an adaptive inertia weight and a dynamic learning factor to improve the global search ability and convergence speed of the algorithm.

[0013] Step 6: Introduce an elite reverse learning strategy to select the optimal trajectory planning scheme based on the fitness function value. Generate reverse particles for each elite particle and explore new search areas by reversely adjusting the position information. At the same time, use the Gaussian-Cauchy mutation strategy to randomly perturb the particle positions to enhance the diversity of the population and the global search ability. If the maximum number of iterations or the fitness value convergence condition is reached, stop the iteration and extract the optimal trajectory planning scheme.

[0014] Step 7: Use the improved particle swarm optimization algorithm to optimize the interpolation time of each segment of the 3-5-3 polynomial, and finally obtain the optimal time of the robotic arm.

[0015] Furthermore, the improved particle swarm algorithm is as follows:

[0016] Introduce an adaptive inertia weight and a dynamic learning factor mechanism into the particle swarm algorithm, add an elite reverse learning strategy, use the Gaussian-Cauchy mutation strategy to randomly perturb the positions of the particles, increase the diversity of the population, and enhance the global search ability. The velocity update formula of the particle swarm algorithm is as follows:

[0017]

[0018] where The velocity of particle i at generation k; The position of particle i at generation k;

[0019] pbest i : The historical best position of particle i; gbest: The global best position; w k : Inertia weight;

[0020] c1, c2: Learning factors; r1, r2: Two random numbers, with values in the range [0,1];

[0021] w k is the inertia weight, obtained by the following formula:

[0022]

[0023] w k is the inertia weight at generation k; w​max is the initial inertia weight, set to 0.9; w min is the final inertia weight; k is the current iteration number; K is the maximum iteration number;

[0024] where c1 and c2 are learning factors, obtained by the following formula:

[0025]

[0026]

[0027] where c1 and c2 are the personal learning factor and the social learning factor; c min and c max are the minimum and maximum values of the learning factor, used to control the range of the learning factor; k is the current iteration number; K is the maximum iteration number; τ is a parameter controlling the adjustment rate, used to control the rate of change of c1 and c2 with the iteration number.

[0028] Compared with the prior art, the present invention has the following beneficial effects:

[0029] 1. The improved particle swarm optimization (PSO) algorithm improves the global search ability and convergence speed by introducing an adaptive inertia weight and a dynamic learning factor, effectively overcoming the local optimum problem of traditional PSO in high-dimensional spaces. Especially when facing complex multi-objective optimization, it can find better solutions.

[0030] 2. The elite reverse learning strategy generates a new search area through reverse particles, avoiding the dilemma that the conventional PSO algorithm is prone to fall into local optimum. At the same time, through the information update of elite particles, the information transmission efficiency in the search process is improved, thus accelerating the convergence process.

[0031] 3. Introducing the Gaussian-Cauchy mutation strategy can effectively increase the diversity of the population through random perturbation of particle positions, prevent the particle swarm from falling into local optimum prematurely, and achieve a balance between global search and local search. BRIEF DESCRIPTION OF THE DRAWINGS

[0032] Figure 1 is the overall flowchart of the present invention;

[0033] Figure 2 is the flowchart of the improved particle swarm algorithm of the present invention;

[0034] Figure 3 is the schematic diagram of the 3-5-3 piecewise polynomial;

[0035] Figure 4 is the schematic diagram of the interpolation path of the 3-5-3 piecewise polynomial;

[0036] Figure 5Comparison Chart of Optimal Fitness between Improved Particle Swarm Optimization Algorithm and Standard Particle Swarm Optimization Algorithm Detailed Implementation Manner

[0037] First, some technical terms related to the present invention will be described below.

[0038] There are two methods for the trajectory planning of a robotic arm: joint space and Cartesian space. Joint space trajectory planning refers to directly designing the trajectory in the joint angle space of the robotic arm. That is, by changing the angles of each joint to control the motion trajectory of the end effector of the robotic arm. Joint space trajectory planning usually needs to consider the constraints between joints, such as joint limits, motion ranges, speed and acceleration limits, etc. Cartesian space trajectory planning refers to designing the trajectory in the Cartesian coordinate system of the end effector (i.e., the position and orientation in space). This method directly specifies the trajectory of the end effector in three-dimensional space and calculates the corresponding joint angles through the inverse kinematics algorithm to achieve the motion of the end effector.

[0039] The Particle Swarm Optimization (PSO) algorithm is an optimization algorithm that simulates the group behavior in nature, proposed by James Kennedy and Russ Eberhart in 1995, inspired by natural phenomena such as bird flocks foraging. The PSO algorithm conducts global optimization search by simulating the information transfer and cooperation among individuals, and has the characteristics of being simple to understand, easy to implement, and high in computational efficiency. Its basic process includes initializing the particle swarm, calculating the fitness value, updating the personal and global optimal solutions, and adjusting the velocity and position of the particles according to these solutions. During the iterative process, the particle swarm adjusts the motion trajectory according to the current global and personal optimal solutions to achieve global exploration of the search space. The algorithm stops when the maximum number of iterations or fitness convergence is reached and outputs the optimal solution. The PSO algorithm is simple and efficient, especially suitable for continuous optimization problems, and can effectively avoid falling into local optima. It has strong self-adaptability, can adapt to different optimization problems, and is widely used in fields such as function optimization, neural network training, control system design, path planning, and feature selection and model optimization in machine learning. However, the performance of PSO is also affected by parameter settings and the initial solution, so it needs to be adjusted and optimized according to specific problems in practical applications. Generally speaking, PSO has become one of the important methods in the optimization field.

[0040] The present invention will be further described below with reference to the accompanying drawings:

[0041] Figure 1 is the flowchart of the present invention, combined with Figure 1 as shown. In this embodiment, a robotic arm trajectory planning method using an improved particle swarm optimization algorithm is adopted:

[0042] 1. Taking the PUMA560 manipulator as an example, kinematic modeling of the manipulator is carried out based on the classical D-H method to determine the geometric parameters and relative position relationships of each link of the manipulator. According to the shape of the manipulator and the D-H parameter coordinate system, the D-H parameter table is listed as follows:

[0043]

[0044] Its limit parameters are as follows:

[0045] joint <![CDATA[θ max > <![CDATA[v max > <![CDATA[a max > <![CDATA[J max > <![CDATA[τ max > 1 π 8 10 30 97.6 2 3π / 4 10 12 40 186.4 3 3π / 4 10 12 40 89.4 4 π 5 8 20 24.2 5 3π / 4 5 8 20 20.1 6 3π / 4 5 8 20 21.3

[0046] The D-H parameter table contains four parameters:

[0047] Joint angle (θ): The rotation angle of the joint coordinate system relative to the previous joint coordinate system.

[0048] Link offset (d): The distance that the origin of the joint coordinate system is translated along the z-axis of the previous joint coordinate system.

[0049] Link length (a): The distance along the x-axis from the origin of the previous joint coordinate system to the origin of the current joint coordinate system.

[0050] Twist angle (α): The angle between the z-axis of the previous joint coordinate system and the z-axis of the current joint coordinate system.

[0051] 2. Conduct forward kinematic analysis of the manipulator and construct the homogeneous transformation matrix of each joint based on the D-H parameter method. According to the chain rule of coordinate transformation, the transformation matrix from coordinate system i-1 to coordinate system i can be written as:

[0052]

[0053] The homogeneous transformation matrices of each adjacent joint angle can be obtained from the homogeneous transformation matrix of the classical DH parameter method as follows:

[0054]

[0055]

[0056]

[0057]

[0058]

[0059]

[0060] By cascading each joint, the total transformation matrix from the base coordinate system to the end effector coordinate system is obtained. As follows:

[0061] T60 = T1 0 ·T2 1 ·T3 2 ·T4 3 ·T5 4 ·T6 5

[0062]

[0063] 3. Use the analytical method to solve the inverse kinematics of the robotic arm:

[0064] During the solution process, each joint angle needs to satisfy the following constraint conditions:

[0065] (1) Joint angle limit: θ i ∈[θ i,min , θ i,max

[0066] (2) Velocity and acceleration limits:

[0067] Target position: x = 0.6, y = 0.4, z = 0.7;

[0068] Attitude: The end remains parallel to the base;

[0069] Then the target matrix is:

[0070]

[0071] Then we can obtain p x = 0.6, p y = 0.4, p z = 0.7;

[0072] Use the D-H parameter table and geometric relationship of the robotic arm to calculate

[0073] θ1 = 0.3805 rad

[0074] θ2 = 0.982 rad

[0075] θ3 = 1.248 rad

[0076] θ4 = 0 rad

[0077] θ5 = 0 rad

[0078] θ6 = π rad

[0079] ​4. When performing 3-5-3 polynomial interpolation, first divide the spatial working trajectory of the robotic arm into three segments, and insert two transition points into the trajectory to form four key nodes: the initial point, transition point 1, transition point 2, and the end point. Through inverse kinematic analysis, determine the initial joint angles and end joint angles of the robotic arm to provide boundary conditions for trajectory interpolation. The general formula for 3-5-3 polynomial interpolation is as follows:

[0080]

[0081]

[0082]

[0083] Let the initial point: (x0, y0, z0) = (0.3, 0.1, 0.4), transition point 1: (x1, y1, z1) = (0.4, 0.2, 0.5), transition point 2: (x2, y2, z2) = (0.5, 0.3, 0.6); the target point: (x3, y3, z3) = (0.6, 0.4, 0.7). After inverse kinematic calculation, we can get:

[0084]

[0085] 5. The present invention proposes a robotic arm trajectory planning method based on an improved particle swarm optimization (PSO) algorithm. First, use the classical D-H method to perform kinematic modeling of the robotic arm and conduct forward and inverse kinematic analysis to obtain the position and attitude information of the end effector. Then, use 3-5-3 polynomial interpolation for smooth trajectory planning and combine the improved PSO algorithm to optimize the trajectory. By setting an adaptive inertia weight, dynamic learning factor, elite reverse learning strategy, and Gaussian-Cauchy mutation strategy, the global search ability and convergence speed are improved, and the time-optimal trajectory of the robotic arm is achieved. This method is applicable to a six-degree-of-freedom robotic arm and can determine the optimal trajectory path of the robotic arm according to user requirements.

[0086] As Figure 2 shown, the present invention proposes the specific process of the improved particle swarm optimization algorithm. Specifically as follows: S1: Construct a trajectory planning model

[0087] Construct a robotic arm trajectory planning model based on the improved PSO algorithm. Set key parameters such as the size of the particle swarm to 50, the maximum number of iterations to 100, the range of the inertia weight to 0.4 - 0.9, the initial value of the learning factor and its range, etc., to provide a basis for the subsequent optimization process.

[0088] S2: Introduce an adaptive inertia weight and a dynamic learning factor

[0089] Introduce an adaptive inertia weight and a dynamic learning factor mechanism into the velocity update formula to optimize the particle swarm search process, improve the global search ability and convergence speed of the algorithm, and thus effectively avoid local optimal solutions. During the velocity update process, the positions of the particles are updated simultaneously, and the fitness value of the particles is calculated after each position update to timely evaluate the search effect and adjust the search direction. The velocity update formula, as well as the inertia weight and learning factor formulas, are as follows:

[0090]

[0091] where The velocity of particle i in generation k; The position of particle i in generation k; pbest i : The historical best position of particle i; gbest: The global best position; w k : Inertia weight; c1, c2: Learning factors; r1, r2: Two random numbers, with a value range of [0,1];

[0092]

[0093] where w k is the inertia weight in generation k; w max is the initial inertia weight, set to 0.9; w min is the final inertia weight; k is the current iteration number; K is the maximum iteration number;

[0094]

[0095]

[0096] where c1 and c2 are the personal learning factor and the social learning factor; c min and c max are the minimum and maximum values of the learning factor, used to control the range of the learning factor; k is the current iteration number; K is the maximum iteration number; τ is a parameter controlling the adjustment rate, used to control the rate of change of c1 and c2 with the iteration number;

[0097] The fitness function formula is as follows:

[0098] fitness(k) = t(k)

[0099] where k is the current iteration number.

[0100] S3: Elite opposition-based learning strategy

[0101] Introduce the elite opposition-based learning strategy to select the optimal trajectory planning scheme through the fitness function. For each elite particle, generate an opposition particle and explore new search areas by adjusting the position of the particle in the opposite direction, thus expanding the search space. When generating the opposition particle, it is necessary to calculate its fitness and ensure that the fitness evaluation meets the design requirements of the objective function. For each opposition particle, compare its fitness with the current optimal solution, and then update the local optimal solution and the global optimal solution. The elite opposition-based learning formula is as follows:

[0102] x reverse,j = L j + U j - x elite,j + ξ·r

[0103]

[0104] L j = min(x j ), U j = max(x j )

[0105] Among them, x reverse refers to the position of the opposition particle, then x reverse,j refers to the position of the opposition particle in the j-th dimension; L j refers to the lower boundary of the j-th dimension; U j refers to the upper boundary of the j-th dimension;

[0106] After generating the opposition particle, it is necessary to calculate its fitness value and compare it with the current optimal solution. If the fitness of the opposition particle is better, then update the local optimal solution and the global optimal solution. Assume that the fitness function is f(x), then for the fitness evaluation of the opposition particle:

[0107] f reverse = f(x reverse )

[0108] If f reverse < f pbest , then update the local optimal solution:

[0109] pbest i = x reverse

[0110] If f reverse < f gbest , then update the global optimal solution:

[0111] gbest = x reverse

[0112] S4: Gaussian-Cauchy mutation strategy

[0113] The Gaussian-Cauchy mutation strategy is used to randomly perturb the positions of the particles, increasing the diversity of the population, enhancing the global search ability, and avoiding falling into local optimal solutions. The formula is as follows:

[0114]

[0115] Where, is the updated particle position. is the current particle position, that is, the particle before mutation; α·σ·N(0,1) is the Gaussian perturbation, and β·γC(0,1) is the Cauchy perturbation; α and β are the weights of the Gaussian distribution and Cauchy distribution perturbations respectively, satisfying α + β = 1.

[0116] S5: Convergence judgment and optimal solution extraction

[0117] Judge whether to end the iteration according to the maximum number of iterations or the convergence condition of the fitness value. The convergence condition is that if the change in the fitness value is less than a certain threshold, it is considered that the algorithm has converged. If the stopping condition is met, stop the iteration and extract the optimal trajectory planning scheme.

[0118] S6: Trajectory optimization and time optimal solution

[0119] Through the improved particle swarm optimization algorithm, further optimize the interpolation time of each segment of the 3-5-3 polynomial, and finally obtain the optimal time-optimal trajectory of the robotic arm.

[0120] By comparing the improved particle swarm algorithm with the standard particle swarm algorithm, the results show that the improved particle swarm algorithm has shown obvious advantages in optimizing the running time of the robotic arm. The elite opposition-based learning (EL) strategy enhances the local search ability of the algorithm and can accelerate the convergence process; while the Gaussian-Cauchy mutation (GC) strategy improves the global search ability and effectively avoids the algorithm falling into local optimal solutions. After adopting these improvement strategies, the cumulative time of the robotic arm from the initial point to the target point is significantly shortened, from 5.0327 seconds of the standard particle swarm algorithm to 4.7623 seconds, a reduction of 5.37%, effectively improving the system performance.

Claims

1. A manipulator trajectory planning method based on an improved particle swarm optimization algorithm, characterized in that The method includes the following steps: (1) Construction of a trajectory planning model; (2) 3-5-3 polynomial interpolation; (3) Introduction of an improved algorithm; (4) Convergence determination and extraction of the optimal solution; (5) Simulation of the robotic arm's motion trajectory.

2. The robotic arm trajectory planning method based on an improved particle swarm algorithm according to claim 1, wherein, In step (1), it is first necessary to determine the geometric parameters of each link of the robotic arm, including link rotation angles, link lengths, link offsets, and joint angles. These parameters are obtained by analyzing the shape of the robotic arm and the relative positional relationship between each joint, and are recorded in accordance with the standard D-H table form. After establishing the kinematic model, the forward kinematic analysis can be performed using this model. The position and orientation of the end effector in the base coordinate system are calculated through the homogeneous transformation matrix. At the same time, through inverse kinematic analysis, based on the target position and orientation of the end effector, the angles or displacements of each joint are deduced.

3. A robotic arm trajectory planning method based on an improved particle swarm optimization algorithm according to claim 1, characterized in that In step (2), after determining the path point positions of the robotic arm's end in the Cartesian coordinate system, through inverse kinematic analysis, these end positions are converted into the corresponding joint angles in the joint space, and then the path points in the joint space are obtained. The 3-5-3 polynomial interpolation method is used for trajectory planning. Specifically, the trajectory is divided into three segments: cubic polynomial interpolation is used at both ends, and quintic polynomial interpolation is used in the middle segment to ensure the continuity of position, velocity, and acceleration. In the endpoint segments, the cubic polynomial interpolation ensures the continuity of the initial and final positions, velocities, and accelerations of the joints. In the middle segment, quintic polynomial interpolation is used to control the acceleration change rate (i.e., jerk) of the joints, thereby reducing jitter during the intermediate process and improving the smoothness of the trajectory. During the interpolation process, appropriate boundary conditions are set, and the polynomial coefficients of each segment are calculated to ensure the smoothness and continuity of the overall trajectory, and finally an optimized joint trajectory is obtained.

4. A robotic arm trajectory planning method based on an improved particle swarm optimization algorithm according to claim 1, characterized in that, In step (3), an improved particle swarm algorithm is introduced to optimize the process of trajectory planning. First, relevant parameters of the particle swarm algorithm are set, including the particle size, maximum number of iterations, inertia weight, learning factors, etc., to provide a basis for optimization. In the velocity update formula, an adaptive inertia weight and a dynamic learning factor mechanism are adopted to enhance the global search ability and convergence speed of the particle swarm, and avoid falling into local optimal solutions. The improved algorithm introduces an elite reverse learning strategy, selects particles with better fitness to generate reverse particles, and further expands the search space through the exploration of reverse particles to enhance the global search ability of the algorithm. Through the Gaussian-Cauchy mutation strategy, the positions of the particles are randomly perturbed to increase the diversity of the population and further avoid the trap of local optimal solutions. Through multiple iterations, the trajectory path of the robotic arm is gradually optimized, and finally the optimal trajectory scheme is obtained. This process can effectively balance global search and local optimization, and provide an accurate, smooth, and time-optimal robotic arm trajectory.

5. A robotic arm trajectory planning method based on an improved particle swarm optimization algorithm according to claim 1, characterized in that In the said step (4), through convergence determination and optimal solution extraction, the optimization process is completed and the optimal time trajectory is obtained. First, a convergence criterion is set. Usually, it is judged whether the stopping condition is satisfied according to the maximum number of iterations or the change of the fitness value of the particle swarm. If the change of the fitness value tends to be stable in several consecutive generations of iterations, or reaches the set convergence threshold, it is considered that the algorithm has converged and further iterations are stopped. After convergence, the global optimal solution is extracted as the optimal time trajectory of the robotic arm. At this time, the optimal solution corresponds to minimizing the total movement time of the robotic arm from the starting point to the ending point while meeting the requirements of trajectory planning. Through this process, the optimization algorithm ensures that the robotic arm can complete the movement in the shortest time when performing tasks, achieving the optimal time performance and providing an efficient trajectory planning scheme.

6. A robotic arm trajectory planning method based on an improved particle swarm optimization algorithm according to claim 1, characterized in that, In the said step (5), a simulation of the robotic arm's motion trajectory is carried out to evaluate the time required for the robotic arm to move from the starting point to the ending point, ensuring the effectiveness of the optimal time trajectory.

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