Multi-target mechanical arm joint trajectory optimization method based on S-shaped speed curve

Abstract

The invention discloses a multi-target mechanical arm joint trajectory optimization method based on an S-shaped speed curve. The method comprises the following steps that a mathematical model of the seven-section type S-shaped speed curve is established; setting an initial position and a target position of each joint and joint movement constraints defined in the planning process of each joint; designing a multi-objective optimization function; searching by using an NSGA-II algorithm to obtain a Pareto optimal solution set; and selecting an optimal solution according to the weight values of different objective functions to obtain the optimized trajectory of each joint. The invention provides a multi-objective optimized S-shaped speed curve trajectory planning method for solving the problems that in the single-joint trajectory planning process of a mechanical arm, parameters of a conventional S-shaped curve are fixed, planning of multi-joint cooperative movement cannot be dynamically adapted, and multi-objective requirements such as energy consumption and stability cannot be considered at the same time.

Description

Technical Field

[0001] The present invention belongs to the technical field of robot motion planning and control, and particularly relates to a multi-objective robotic arm joint trajectory optimization method based on an S-shaped velocity curve. Background Art

[0002] Existing robotic arm trajectory planning methods mostly adopt trapezoidal velocity curves or simple interpolation algorithms, which have problems such as sudden acceleration changes and uneven motion, resulting in mechanical vibrations and component wear. Although high-order interpolation methods can better generate smooth trajectory curves, they still have the problem of large computational resource requirements. For the existing trajectory planning problem of point-to-point in the robotic arm joint space, single-objective optimization cannot balance multi-objective conflicts, and it is difficult to obtain optimal parameters based on the S-shaped curve in actual engineering. To address the above problems, there is an urgent need to propose a multi-objective collaborative optimization method to achieve comprehensive optimization of time, smoothness, and energy consumption on the premise of satisfying the dynamic constraints of the robotic arm. Summary of the Invention

[0003] The main purpose of the present invention is to overcome the disadvantages and deficiencies of the fixed parameters of the conventional S-shaped curve in the process of single-joint trajectory planning of existing robotic arms, which cannot dynamically adapt to the planning of multi-joint collaborative motion and cannot take into account multi-objective requirements such as energy consumption and smoothness. The present invention provides a multi-objective robotic arm joint trajectory optimization method based on an S-shaped velocity curve, which realizes obtaining optimal parameters of the S-shaped velocity curve during trajectory planning, and on the premise of satisfying the dynamic constraints of the robotic arm, realizes comprehensive trajectory optimization of time, smoothness, and energy consumption.

[0004] To achieve the above object, the present invention adopts the following technical solutions: A multi-objective robotic arm joint trajectory optimization method based on an S-shaped velocity curve, the multi-objective robotic arm joint trajectory optimization method comprising the following steps: S1. According to the trajectory planning characteristics of the S-shaped velocity curve of the robotic arm in the joint space, respectively establish mathematical models of the seven-segment S-shaped velocity curves of each joint of the three-joint robotic arm, where the three-joint robotic arm includes a shoulder joint, an elbow joint, and a wrist joint; S2. Set the starting position and target position of each joint and the joint displacement constraints and motion constraints defined during the planning of each joint. Among them, the displacement constraint ensures that the optimized joint motion displacement is the difference between the set target position and the starting position, and the motion constraints of the joint include the maximum velocity, acceleration, and jerk during the motion of each joint; S3. Design a multi-objective optimization function, and establish three objective optimization functions with the least time, the least energy consumption, and the least impact as the optimization objectives; S4. Determine the optimization objective variables based on the characteristics of the S-shaped velocity curve trajectory planning; S5. Generate an initial population that satisfies the motion constraints using Monte Carlo sampling, and search for the Pareto optimal solution set using the NSGA-II algorithm; S6. Select the optimal solution according to the weight values of different objective functions to obtain the optimized trajectories of each joint.

[0005] Furthermore, in step S1, according to the S-shaped speed curve planning characteristics of the joint manipulator, the seven-segment S-shaped speed curves of the k = 1, 2, and 3 joints are respectively established The mathematical model descriptions are as follows:

[0006] Specifically, as Figure 2 shown, where represents time, represents different time stages, then the times of the seven stages of the S-shaped speed curve corresponding to the th joint are respectively denoted as , represents the total acceleration time of the th joint, represents the total time of the entire trajectory planning process of the th joint, represents the end velocity of the acceleration section of the th joint, represents the maximum velocity of the th joint, represents the magnitude of the jerk.

[0007] Furthermore, step S2 stipulates the displacement constraints and motion parameter constraints of each joint, and the process is as follows: Displacement constraint: The purpose of performing S-shaped speed curve planning on the manipulator is to enable the joint to move from the initial point to the target point. During the optimization process, since the displacement residual is calculated analytically, the displacement is constrained to ensure the accuracy of angle control. The technical effect of this step is to eliminate the cumulative error and ensure the accuracy of the final pose of the end effector of the manipulator.

[0008] First, set the starting position and the target position of the th joint. Then, the target displacement of the th joint is . In order to make the total displacement of the optimized trajectory accurately match the preset joint angle target displacement, the displacement constraint is expressed as:

[0009] where is the obtained during the trajectory planning process.The seven-segment S-shaped velocity curve of each joint, is the total time of the entire trajectory planning process of the th joint; Motion parameter constraints: Then, according to the parameters of each joint of the three-joint manipulator on the implementation platform, set the maximum velocity, acceleration, and jerk of each joint during the motion process to ensure the safety and reachability of the manipulator during the motion process. respectively represent the inequality constraint functions of the velocity, acceleration, and jerk of the th joint during the trajectory planning process. Then, the mathematical formula of the motion constraint is expressed as:

[0010] where respectively represent the velocity, acceleration, and jerk functions of the th joint during the trajectory planning process, are respectively the upper limit values of the velocity, acceleration, and jerk of the th joint of the manipulator during the motion process.

[0011] Furthermore, step S3 is to construct a multi-objective optimization function to achieve the collaborative optimization of the manipulator joint trajectory among time, energy consumption, and motion shock. The specific process is as follows: Establish optimization objective functions representing time, energy consumption, and shock , . Take the maximum value of the time spent among all joints as the time optimization objective function . By minimizing , using the time spent by the slowest joint trajectory as the time objective function to avoid a single joint taking too long and slowing down the overall task efficiency; taking the average acceleration value of the three joints as the evaluation function of energy consumption, that is, the optimization objective function . The optimization of energy consumption mainly targets the dynamic energy consumption of the joint motors. Using the average acceleration as the equivalent evaluation index and calculating the root mean square to smooth the instantaneous energy consumption fluctuations can more truly reflect the average power of the motors. By minimizing , the energy loss of the manipulator during the motion process can be reduced and the service time can be extended; taking the average jerk value of the three joints as the evaluation function of shock, that is, the optimization objective function . The shock characteristics are evaluated by the average jerk as the evaluation index. Minimizing can suppress high-frequency vibrations and improve the trajectory smoothness.

[0012] In summary, the expression of the multi-objective optimization function is established as follows:

[0013] where, and respectively represent the acceleration and jerk magnitudes of the th joint during the time period.

[0014] By establishing a multi-objective optimization function of time, energy consumption, and impact , , and by using a multi-objective optimization algorithm to find the Pareto optimal solution set, this solution is more suitable for selecting the optimal solution under complex working conditions compared to a single optimization objective. That is, users can select a suitable solution according to task requirements such as emergency task priority time, precision task priority impact, etc., avoiding the local optimal trap of single-objective optimization and adapting to the dynamic requirements of complex industrial scenarios.

[0015] Furthermore, the process of step S4 is as follows: According to the trajectory planning characteristics of the S-shaped velocity curve, the target variables to be optimized for the th joint are (a total of 10 dimensions), where respectively represent the maximum velocity, maximum acceleration, and maximum jerk magnitudes of the th joint. If 3 joints are considered, the target variables to be optimized are a total of 30 dimensions. The high-dimensional variables lead to a slow convergence speed of the optimization algorithm, high platform computing power requirements, and it is easy for conflicts between parameters to result in extreme solutions. Therefore, to reduce the dimension of the optimization target variables and improve the calculation efficiency, based on the S-shaped velocity curve model, the following constraints are made:

[0016] The dimension of the time variable changes from 7 dimensions to 3 dimensions , and then the final optimized target variables for the th joint are . By introducing this constraint, the acceleration section, constant-speed section, and deceleration section can be made symmetric, avoiding the imbalance in the time ratio between the constant-speed section and the acceleration / deceleration section, ensuring a smooth transition of the speed, and making the initial and final speeds, acceleration section, and jerk of the trajectory zero, reducing the residual vibration when the robotic arm reaches the position. At the same time, in the dimension reduction optimization, the planning method of the S-shaped velocity curve is maintained to ensure the smoothness and safety of the optimized trajectory.

[0017] Furthermore, step S5 uses Monte Carlo sampling to generate an initial population that satisfies the motion constraints, and uses an improved NSGA-II algorithm (Non-dominated Sorting Genetic Algorithm II, a classic multi-objective optimization algorithm) to search for the Pareto optimal solution set. The process is as follows: S5.1. Initialize the motion constraint parameters and the parameters related to NSGA-II. Set the motion constraint parameters of each joint of the robotic arm according to the actual situation of the joints, including the maximum speed , the maximum acceleration and the maximum jerk of each joint, as well as the starting position and the target position of each joint, to avoid invalid solutions caused by incorrect parameter ranges. At the same time, initialize the parameters related to the NSGA-II algorithm, including the population size , the number of generations , the crossover probability and the mutation probability . Selecting appropriate parameters can effectively improve the convergence speed and stability of the algorithm and generate an appropriate number of solution sets; S5.2. Generate the initial population. NSGA-II uses random uniform initialization. The randomly initialized population is prone to aggregation in local areas, reducing the diversity of the solution set. To improve the distribution uniformity of the initial population and enhance the global search ability, generate uniformly random maximum jerks through the Monte Carlo sampling method, generate randomly through the Monte Carlo method, and combine the displacement integral equation to reverse infer the feasible range. According to the joint displacement constraints and motion constraints defined in the planning process of each joint, filter and generate an initial population with a population size of to ensure that the initial parameters meet the kinematic constraints of the robotic arm and avoid invalid solutions; S5.3. Establish the fitness evaluation function. Define the parent population as , is the population generation number. According to the above three objective optimization functions, calculate and for each individual in as the evaluation indicators for judging the quality of individuals. At the same time, set the penalty term Penalty, and regard the values exceeding the maximum speed, maximum acceleration, and maximum jerk as physical constraints and convert them into additional costs of the objective function, effectively eliminating infeasible solutions such as overspeed and over-acceleration, and guiding the algorithm to converge in the direction of physically feasible and multi-objective balance; S5.4. Perform genetic operator operations. Perform selection, simulated binary crossover, and polynomial mutation evolution operations on the parent to generate an offspring population of comparable size; To ensure that the number of feasible solutions in the final Pareto solution set is large and evenly distributed, boundary correction is performed on the crossover offspring during simulated binary crossover. According to the physical limitations of each joint of the robotic arm, the feasible range of each parameter is predefined as low and up. If the parameters of the offspring after crossover exceed low or up, they are automatically truncated to the nearest boundary.

[0018] For each individual's decision variable in the parent generation According to the mutation probability Judge whether to perform the mutation operation. If polynomial mutation is performed, the following polynomial operator is used to calculate the new decision variable :

[0019]

[0020]

[0021] where is a uniformly random number in the range of (0, 1), represents the mutation distribution index, and are the upper and lower bounds of the decision variable respectively.

[0022] S5.5. Perform elitist retention. In multi-objective optimization, the solution set is prone to falling into local fronts or being unevenly distributed. Through the elitist retention mechanism, excellent genes are retained by combining cross-generation populations and a selection mechanism. First, construct a composite population , that is, this composite population is defined as the union of the offspring and the parent generation. Then perform environmental selection operations: perform non-dominated ranking and crowding degree calculation on and sort them in ascending order of hierarchy and descending order of crowding degree, eliminate the latter 50% of the individuals, and retain the top N elites to form the generation population , where non-dominated ranking refers to performing non-dominated sorting on , and dividing individuals into multiple levels according to the dominance relationship, such as front 1, front 2. Among them, the individuals in front 1 are not dominated by any other individuals, and the individuals in front 2 are only dominated by the individuals in front 1, and so on. Crowding degree calculation refers to sorting the individuals in the same non-dominated level according to each objective function value respectively and calculating the crowding degree of each individual; S5.6. Detect whether the current iteration number has reached the evolution number . If the termination condition is met, output the current Pareto front solution set as the Pareto optimal solution set; otherwise, loop and execute the evolution process of steps SS5.3 to SS5.5.

[0023] Furthermore, the process of step S6 is as follows: The Pareto front obtained by the above steps contains a large number of equivalent optimal solutions, and the fixed solution selection strategy cannot meet the dynamic scenarios. Therefore, according to the actual needs of different working conditions, different weight factors can be adjusted Select a set of optimal solutions from the Pareto optimal solution set As the final solution, this selection method can meet the diverse scenario requirements such as emergency tasks, energy-saving modes, and high-precision modes, where represents the th objective optimization function, and an optimized trajectory for each joint of the robotic arm is generated based on this set of solutions. The optimal solution is calculated as follows:

[0024] where , and finally, the optimal solution is substituted into the S-shaped velocity planning model to obtain the optimal trajectory.

[0025] Compared with the prior art, the present invention proposes a multi-objective trajectory optimization method based on the S-shaped velocity curve, solves the problem of fixed trajectory planning parameters of the S-shaped velocity curve, and realizes the comprehensive optimization of time, smoothness, and energy consumption on the premise of meeting the dynamic constraints of the robotic arm. It has the following advantages and beneficial effects: (1) Breaking through the limitations of traditional single-objective optimization, a multi-objective collaborative optimization model with the total motion time, root mean square (RMS) of acceleration, and RMS of jerk as the optimization objective functions. By dynamically adjusting the jerk time ( ), uniform acceleration time ( ), constant velocity time ( ), maximum velocity ( ), maximum acceleration ( ), and maximum jerk ( ) of the S-shaped curve, the balance between motion efficiency and mechanical smoothness is achieved.

[0026] (2) Combining the parametric modeling of the seven-stage S-shaped curve with the multi-objective search algorithm NSGA-II algorithm, improving the simulated binary crossover and polynomial mutation algorithms to enhance the global search ability, and effectively avoiding local optima.

[0027] (3) By generating a multi-objective Pareto optimal solution set, providing a multi-dimensional decision-making space for different application scenarios, significantly reducing mechanical vibration and energy consumption, and prolonging the service life of the equipment, which is applicable to complex scenarios such as high-precision control and precision assembly. Brief Description of the Drawings

[0028] To more clearly illustrate the technical solutions in the embodiments of the present application, the following will briefly introduce the accompanying drawings required for the description of the embodiments. Obviously, the accompanying drawings in the following description are only some embodiments of the present application. For those of ordinary skill in the art, without creative efforts, other accompanying drawings can be obtained based on these drawings.

[0029] Figure 1 It is a schematic flowchart of the multi-objective trajectory optimization method based on the S-shaped speed curve provided in the embodiments of the present invention; Figure 2 It is a schematic diagram of the seven-segment S-shaped speed planning curve in the embodiments of the present invention; Figure 3 It is a flowchart of the NSGA-II algorithm in the embodiments of the present invention; Figure 4 It is the position curve of each joint obtained by multi-objective optimization of each joint of the three-joint serial manipulator in the embodiments of the present invention; Figure 5 It is the speed curve of each joint obtained by multi-objective optimization of each joint of the three-joint serial manipulator in the embodiments of the present invention; Figure 6 It is the acceleration curve of each joint obtained by multi-objective optimization of each joint of the three-joint serial manipulator in the embodiments of the present invention; Figure 7 It is the jerk curve of each joint obtained by multi-objective optimization of each joint of the three-joint serial manipulator in the embodiments of the present invention; Detailed implementation manners

[0030] In order to enable those skilled in the art to better understand the solutions of the present application, the following will clearly and completely describe the technical solutions in the embodiments of the present application with reference to the accompanying drawings in the embodiments of the present application. Obviously, the described embodiments are only some embodiments of the present application, rather than all embodiments. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative efforts belong to the scope of protection of the present application.

[0031] In the present application, referring to "embodiment" means that the specific features, structures or characteristics described in connection with the embodiment may be included in at least one embodiment of the present application. The phrase appears in various positions in the specification does not necessarily refer to the same embodiment, nor is it an independent or alternative embodiment mutually exclusive with other embodiments. Those skilled in the art explicitly and implicitly understand that the embodiments described in the present application can be combined with other embodiments.

[0032] Embodiment 1 Figure 1The following is a schematic flowchart of the multi-objective trajectory optimization method based on the S-shaped speed curve provided in the embodiment of the present invention; the multi-objective trajectory optimization method based on the S-shaped speed curve includes the following steps: S1. According to the trajectory planning characteristics of the S-shaped speed curve of the robotic arm in the joint space, establish the mathematical models of the seven-segment S-shaped speed curves of each joint of the three-joint robotic arm respectively. The three-joint robotic arm includes a shoulder joint, an elbow joint, and a wrist joint. According to the Figure 2 seven-segment S-shaped speed curve shown in the figure, establish the S-shaped speed curve of the k = 1, 2, 3 joints The mathematical model is described as follows:

[0033] Among them, represents time, represents different time stages. Then, the times of the seven stages of the S-shaped speed curve corresponding to the th joint are respectively recorded as , represents the total acceleration time of the th joint, represents the total time of the entire trajectory planning process of the th joint, represents the end velocity of the acceleration stage of the th joint, represents the maximum velocity of the th joint, represents the magnitude of the jerk.

[0034] S2. Set the starting position and target position of each joint, as well as the joint displacement constraints and motion constraints defined during the planning process of each joint. The process is as follows: First, define the starting position of the th joint as , and the target position as . Then, the given target displacement of the th joint is . The total displacement of the trajectory exactly matches the preset joint angle target displacement. Then, the displacement constraint is expressed as:

[0035] Among them, is the seven-segment S-shaped speed curve of the th joint during the trajectory planning process, is the total time of the entire trajectory planning process of the th joint; respectively represent the The inequality constraint functions of the speed, acceleration, and jerk of each joint, then the mathematical formula of the motion constraint is expressed as:

[0036] Where respectively represent the speed, acceleration, and jerk functions of the th joint during the trajectory planning process, are respectively the upper limit values of the speed, acceleration, and jerk of the th joint of the robotic arm during the motion process.

[0037] According to the data of the three-joint robotic arm of the platform implemented in this paper, the specific values of the motion parameter constraints of each joint in the example are shown in Table 1.

[0038] Table 1. Definition Table of Motion Parameter Constraints for the Three-Joint Robotic Arm System

[0039] S3. Establish an optimization objective function with the least time, minimum energy consumption, and minimum impact as the optimization objectives. The process is as follows: Establish optimization objective functions representing time, energy consumption, and impact respectively , , take the maximum value of the time spent among all joints as the time optimization objective function , use the average acceleration value of the 3 joints as the evaluation function of energy consumption, that is, the optimization objective function , use the average jerk value of the 3 joints as the evaluation function of impact, that is, the optimization objective function , and the expression is as follows:

[0040] Where and respectively represent the acceleration and jerk magnitudes of the th joint during the time period .

[0041] S4. Determine the optimization objective variables based on the characteristics of the S-shaped velocity curve trajectory planning: According to the characteristics of the S-shaped velocity curve trajectory planning, the optimization objective variables to be optimized for the th joint are for a total of 10 dimensions. Among them respectively represent the maximum speed, maximum acceleration, and maximum jerk magnitudes of the th joint. If 3 joints are considered, the optimization objective variables to be optimized are a total of 30 dimensions. Therefore, to reduce the dimension of the optimization objective variables and improve the calculation efficiency, based on the S-shaped velocity curve model, the following constraints are made:

[0042] By introducing this constraint, the time variable dimension changes from 7 dimensions to 3 dimensions , and then the final optimized target variable of the th joint is .

[0043] S5. Generate an initial population that satisfies the motion constraints by Monte Carlo sampling, and use the improved NSGA-II algorithm to search for the Pareto optimal solution set. The process is as follows: S5.1. Initialize the motion constraint parameters and the NSGA-II related parameters. Set the motion constraint parameters of each joint of the robotic arm according to the actual situation of the joint, including the maximum speed , maximum acceleration , and maximum jerk of each joint, as well as the starting position and target position of each joint. Initialize the NSGA-II algorithm related parameters, including the population size , number of generations , crossover probability , and mutation probability ; S5.2. Generate the initial population. Generate uniformly random maximum jerk through Monte Carlo sampling, and generate an initial population with a population size of according to the joint displacement constraints and motion constraints defined in the planning process of each joint ; S5.3. Establish a fitness evaluation function. Define the parent population as , is the population generation number. According to the above three objective optimization functions, calculate for each individual in , as the evaluation indicators for judging the quality of individuals. At the same time, set the penalty term Penalty, and regard the values exceeding the maximum speed, maximum acceleration, and maximum jerk as physical constraints and convert them into additional costs of the objective function. Finally, obtain the fitness value of each individual; S5.4. Perform genetic operator operations. Perform selection, simulated binary crossover, and polynomial mutation evolution operations on the parent to generate a child population of comparable size; To ensure that the number of feasible solutions in the final Pareto solution set is large and evenly distributed, boundary correction is performed on the crossover offspring during simulated binary crossover. According to the physical limitations of each joint of the robotic arm, the feasible range of each parameter is predefined as low and up. If the parameters of the crossover offspring exceed low or up, they are automatically truncated to the nearest boundary.

[0044] For each individual's decision variable in the parent generation According to the mutation probability Determine whether to perform the mutation operation. If polynomial mutation is performed, the following polynomial operator is used to calculate the new decision variable :

[0045]

[0046]

[0047] where is a uniformly distributed random number in the range of (0, 1), represents the mutation distribution index, and are the upper and lower bounds of the decision variable respectively.

[0048] S5.5. Perform elitism retention and construct a composite population , that is, the composite population is defined as the union of the offspring and the parent generation, and the environmental selection operation is performed: for perform non-dominated ranking and crowding degree calculation and sort them in ascending order of hierarchy and descending order of crowding degree, eliminate the latter 50% of the individuals, and retain the first N elites to form the generation population , where non-dominated ranking refers to performing non-dominated sorting on , and dividing the individuals into multiple levels according to the dominance relationship, such as front 1, front 2. The individuals in front 1 are not dominated by any other individuals, and the individuals in front 2 are only dominated by the individuals in front 1, and so on. Crowding degree calculation refers to sorting the individuals in the same non-dominated level according to each objective function value respectively and calculating the crowding degree of each individual; S5.6. Detect whether the current iteration number reaches the evolution number , if the termination condition is met, output the current Pareto front solution set as the Pareto optimal solution set, otherwise loop and execute the evolution process of steps SS5.3 to SS5.5.

[0049] S6. Select a set of optimal solutions. According to the actual different working conditions, by adjusting different weight factors Select a set of optimal solutions from the Pareto optimal solution set as the final solution, and generate the optimized trajectories of each joint of the robotic arm based on this set of solutions.

[0050]

[0051]

[0052] When executing Figure 3 the NSGA-II algorithm process shown, set the population size to 400, the maximum number of iterations to 180, the crossover probability to 0.8, the mutation probability to 0.2, and the mutation distribution index to 20. After executing the improved NSGA-II algorithm, obtain the Pareto solution set. According to the weight values of different objective functions, the time, energy consumption, and impact are respectively The optimal solutions of the optimization variables are selected as shown in Table 2: Table 2. Optimal solution table of each joint of the three-joint robotic arm

[0053] Substitute the above solutions into the S-shaped velocity planning model to obtain the optimized position curves, velocity curves, acceleration curves, and jerk curves of each joint of the three-joint robotic arm, respectively, as shown in Figure 4 、 5 、6, and 7.

[0054] Substitute the above results into the established optimization objective functions 、 , and conduct result analysis. Compared with the robotic arm joint trajectories planned with the S-shaped velocity curve directly using , the index changes of the optimized trajectories are as follows: The total time of joint 1 is 5.52 s, the time increases by 33.6%, the energy consumption (acceleration RMS) decreases by 48.3%, and the impact (jerk RMS) decreases by 55.4%; The total time of joint 2 is 4.43 s, the time increases by 18.7%, the energy consumption (acceleration RMS) decreases by 32.5%, and the impact (jerk RMS) decreases by 44.5%; The total time of joint 3 is 5.79 s, the time increases by 2.8%, the energy consumption (acceleration RMS) decreases by 7.1%, and the impact (jerk RMS) decreases by 18.6%. From the above data analysis, because the two indicators of time and energy consumption impact restrict each other, that is, the three indicators cannot reach the optimal at the same time. When the trajectory is not planned using the set maximum values of velocity, acceleration, and jerk, the time will increase accordingly. However, from the calculation of energy consumption and impact of the results, each joint has a significant reduction, which proves that this method effectively realizes the trajectory optimization of the robotic arm joints for the three objectives of time, energy consumption, and impact.

[0055] Example 2 This embodiment continues to disclose the specific implementation process of the multi-objective manipulator joint trajectory optimization method based on the S-shaped speed curve. The technical means and steps used in this embodiment are the same as those in Embodiment 1, only different in parameters. This embodiment includes the following steps: Steps S1, S3, and S4 are the same as steps S1, S3, and S4 in Example 1, where the parameter values in S2, S5, and S6 are changed.

[0056] S2. Set the starting position and target position of each joint, as well as the joint displacement constraints and motion constraints defined during the planning of each joint.

[0057] According to the data of the three-joint manipulator of the platform implemented in this article, the specific values of the motion parameter constraints of each joint in the example are set as shown in Table 3.

[0058] Table 3. Definition Table of Motion Parameter Constraints for Three-Joint Manipulator System

[0059] S5. Use the improved NSGA-II algorithm to search for the Pareto optimal solution set, and the process is the same as that in Embodiment 1.

[0060] S6. Select a set of optimal solutions. According to the actual different working conditions, by adjusting different weight factors Select a set of optimal solutions from the Pareto optimal solution set as the final solution, and generate the optimized trajectory of each joint of the manipulator based on this set of solutions.

[0061] When executing this algorithm, set the population size to 300, the maximum number of iterations to 150 times, the crossover probability to 0.7, the mutation probability to 0.2, and the mutation distribution index to 20. After executing the improved NSGA-II algorithm, obtain the Pareto solution set. According to the weight values of different objective functions, time, energy consumption, and impact are respectively The optimal solutions of the optimization variables are selected as shown in Table 4: Table 4. Optimal Solution Table for Each Joint of Three-Joint Manipulator

[0062] Substitute the above results into the established optimization objective function in S3 、 and perform result analysis. Compared with directly using The joint trajectory of the robotic arm with an S-shaped velocity curve planning, and the index changes of the optimized trajectory are as follows: The total time of joint 1 is 7.23 s, with a time increase of 74.9%, the energy consumption (acceleration RMS) is reduced by 69.0%, and the jerk (jerk RMS) is reduced by 82.0%; The total time of joint 2 is 7.06 s, with a time increase of 89.1%, the energy consumption (acceleration RMS) is reduced by 74.0%, and the jerk (jerk RMS) is reduced by 78.2%; The total time of joint 3 is 6.92 s, with a time increase of 22.9%, the energy consumption (acceleration RMS) is reduced by 38.7%, and the jerk (jerk RMS) is reduced by 51.0%. From the above data analysis, because the two indicators of time and energy consumption jerk are mutually restrictive, that is, the three indicators cannot reach the optimal at the same time. When the trajectory is planned without using the set maximum values of velocity, acceleration, and jerk, the time will increase accordingly. However, from the calculation of the energy consumption and jerk of the results, each joint has a significant reduction, which proves that this method effectively realizes the trajectory optimization of the robotic arm joints for the three goals of time, energy consumption, and jerk.

[0063] The technical features of the above embodiments can be combined arbitrarily. For the sake of brevity of description, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, it should be considered as the scope recorded in this specification.

[0064] The above embodiments are the preferred embodiments of the present invention, but the embodiments of the present invention are not limited to the above embodiments. Any other changes, modifications, substitutions, combinations, and simplifications made without departing from the spirit and principle of the present invention shall be equivalent substitution methods and are all included in the protection scope of the present invention.

Claims

1. A multi-objective manipulator joint trajectory optimization method based on an S-shaped speed curve, characterized in that The multi-objective manipulator joint trajectory optimization method includes the following steps: S1. According to the S-shaped velocity curve trajectory planning characteristics of the manipulator in the joint space, establish the mathematical models of the seven-segment S-shaped velocity curves of each joint of the three-joint manipulator, where the three-joint manipulator includes a shoulder joint, an elbow joint, and a wrist joint; S2. Set the starting position and target position of each joint, as well as the joint displacement constraints and motion constraints defined during the planning of each joint. Among them, the displacement constraint ensures that the optimized joint motion displacement is the difference between the set target position and the starting position, and the motion constraints of the joint include the maximum velocity, acceleration, and jerk during the motion of each joint; S3. Design a multi-objective optimization function, and establish three objective optimization functions with the least time, the least energy consumption, and the least impact as the optimization objectives; S4. Determine the optimization objective variables based on the characteristics of the S-shaped velocity curve trajectory planning; S5. Use Monte Carlo sampling to generate an initial population that satisfies the motion constraints, and use the NSGA-II algorithm to search for the Pareto optimal solution set; S6. Select the optimal solution according to the weight values of different objective functions to obtain the optimized trajectories of each joint.

2. The multi-objective robotic arm joint trajectory optimization method based on the S-shaped speed curve according to claim 1, wherein In step S1, according to the S-shaped speed curve planning characteristics of the articulated robot, a seven-segment S-shaped speed curve of the k = 1, 2, 3 joints is established The mathematical model description is as follows: Among them, represents time, represents different time stages, then the times of the seven stages of the S-shaped speed curve corresponding to the th joint are respectively denoted as , represents the total acceleration time of the th joint, represents the total time of the movement of the entire trajectory planning process of the th joint, represents the end speed of the acceleration section of the th joint, represents the maximum speed of the th joint, represents the magnitude of the jerk.

3. The multi-objective robotic arm joint trajectory optimization method based on the S-shaped speed curve according to claim 2, wherein The process of step S2 is as follows: First, define the starting position of the th joint as , the target position as . Then, the given target displacement of the th joint is . If the total displacement of the trajectory exactly matches the preset target displacement of the joint angle, the displacement constraint is expressed as: where is the seven-segment S-shaped velocity curve of the th joint in the trajectory planning process, is the total time of the movement of the entire trajectory planning process of the respectively represent the inequality constraint functions of the velocity, acceleration, and jerk of the th joint during the trajectory planning process. Then the mathematical formula of the motion constraint is expressed as: wherein respectively represent the velocity, acceleration, and jerk functions of the th joint during the trajectory planning process, respectively being the upper limit values of the velocity, acceleration, and jerk of the th joint of the robotic arm during the movement process.

4. The multi-objective robotic arm joint trajectory optimization method based on the S-shaped speed curve according to claim 3, wherein, The process of step S3 is as follows: Establish the optimization objective functions representing time, energy consumption, and shock respectively , , take the maximum value of the time spent among all joints as the time optimization objective function , use the average acceleration value of the 3 joints as the evaluation function for energy consumption, that is, the optimization objective function , use the average jerk value of the 3 joints as the evaluation function for shock, that is, the optimization objective function , the expressions are as follows: Among them, and respectively represent the acceleration and jerk magnitudes of the th joint during the time period .

5. The multi-objective manipulator joint trajectory optimization method based on the S-shaped speed curve according to claim 4, wherein, The process of step S4 is as follows: According to the trajectory planning characteristics of the S-shaped speed curve, the target variables to be optimized for the th joint are with a total of 10 dimensions. Among them, respectively represent the maximum speed, maximum acceleration, and maximum jerk magnitude of the th joint. Based on the S-shaped speed curve model, the following constraints are made: By introducing this constraint, the time variable dimension changes from 7 dimensions to 3 dimensions , and then the th joint's final optimized target variable is .

6. The multi-objective robotic arm joint trajectory optimization method based on the S-shaped speed curve according to claim 5, characterized in that, The process of step S5 is as follows: S5.

1. Initialize the motion constraint parameters and the parameters related to NSGA-II. Set the motion constraint parameters of each joint of the robotic arm according to the actual situation of the joints, including the maximum speed of each joint , the maximum acceleration and the maximum jerk as well as the starting position of each joint and the target position . Initialize the parameters related to the NSGA-II algorithm, including the population size , the number of generations of evolution , the crossover probability and the mutation probability ; S5.

2. Generate an initial population, and generate uniformly random maximum jerk through the Monte Carlo sampling method , and generate an initial population with a population size of according to the joint displacement constraints and motion constraints defined in the planning process of each joint ; S5.

3. Establish a fitness evaluation function and define the parent population as , is the population generation. According to the above three objective optimization functions, calculate for each individual in , as the evaluation index to judge the quality of individuals. At the same time, set the penalty term Penalty, and take the values exceeding the maximum speed, maximum acceleration, and maximum jerk as physical constraints and convert them into additional costs of the objective function. Finally, obtain the fitness value of each individual;​ S5.4, perform genetic operator operations on the parent Perform selection, simulate binary crossover, and polynomial mutation evolution operations to generate offspring populations of comparable size ; S5.

5. Perform elitist retention to construct a composite population , that is, the composite population is defined as the union of the offspring and the parent generation, and perform environmental selection operations: For , perform non-dominated ranking and crowding degree calculation and sort them in ascending order of hierarchy and descending order of crowding degree, eliminate the latter 50% of the individuals, and retain the top N elites to form the generation population , where non-dominated ranking means performing non-dominated sorting on , dividing individuals into multiple levels according to the domination relationship, such as front 1, front 2. The individuals in front 1 are not dominated by any other individuals, and the individuals in front 2 are only dominated by the individuals in front 1, and so on. Crowding degree calculation means sorting the individuals in the same non-dominated level according to each objective function value respectively and calculating the crowding degree of each individual; S5.

6. Detect whether the current iteration count has reached the number of evolutions If the termination condition is met, output the current Pareto front solution set as the Pareto optimal solution set; otherwise, loop through the evolution process of steps SS5.3 to SS5.

5.

7. The multi-objective robotic arm joint trajectory planning method based on the S-shaped speed curve according to claim 6, wherein The process of step S6 is as follows: According to the actual different working conditions, by adjusting different weight factors Select a set of optimal solutions from the Pareto optimal solution set As the final solution, Denote the th objective optimization function, and generate the optimized trajectories of each joint of the robotic arm with this set of solutions. The calculation formula for the optimal solution is as follows: , where .

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