Multi-objective trajectory optimization method for joint of robot arm based on s-shaped velocity curve

Abstract

The application discloses a multi-target mechanical arm joint trajectory optimization method based on an S-shaped speed curve, and the method process is as follows: a mathematical model of a seven-section S-shaped speed curve is established; the starting position and the target position of each joint are set, and the joint motion constraint defined in the joint planning process is set; a multi-target optimization function is designed; a Pareto optimal solution set is obtained by searching using an NSGA-II algorithm; and the optimal solution is selected according to the weight value of different target functions to obtain the optimized trajectory of each joint. The application proposes a multi-target optimization S-shaped speed curve trajectory planning method aiming at the problem that the parameters of the conventional S-shaped curve are fixed in the single-joint trajectory planning process of the mechanical arm, the planning cannot dynamically adapt to the multi-joint collaborative motion, and the multi-target requirements such as energy consumption and smoothness cannot be considered.

Description

Technical Field

[0001] This invention belongs to the field of robot motion planning and control technology, specifically relating to a method for optimizing the joint trajectory of a multi-objective robotic arm based on an S-shaped velocity curve. Background Technology

[0002] Existing robotic arm trajectory planning methods mostly employ trapezoidal velocity curves or simple interpolation algorithms, which suffer from abrupt acceleration changes and uneven motion, leading to mechanical vibration and component wear. While higher-order interpolation methods can generate smoother trajectory curves, they still require significant computational resources. Existing single-objective optimization methods for point-to-point trajectory planning in robotic arm joint space cannot balance multi-objective conflicts, and obtaining optimal parameters based on S-curves in practical engineering is difficult. To address these issues, there is an urgent need to propose a multi-objective collaborative optimization method that comprehensively optimizes time, smoothness, and energy consumption while satisfying the robotic arm's dynamic constraints. Summary of the Invention

[0003] The main objective of this invention is to overcome the shortcomings and deficiencies of conventional S-curve parameters being fixed in the single-joint trajectory planning process of existing robotic arms, which cannot dynamically adapt to the planning of multi-joint coordinated motion and cannot take into account multiple objectives such as energy consumption and stability. This invention provides a multi-objective robotic arm joint trajectory optimization method based on S-curve velocity curves, which enables the S-curve velocity curves to obtain optimal parameters under trajectory planning, and achieves comprehensive trajectory optimization of time, smoothness and energy consumption under the premise of satisfying the dynamic constraints of the robotic arm.

[0004] To achieve the above objectives, the present invention adopts the following technical solution:

[0005] A method for optimizing the joint trajectory of a multi-objective robotic arm based on an S-shaped velocity curve, comprising the following steps:

[0006] S1. Based on the trajectory planning characteristics of the S-shaped velocity curve of the robotic arm in the joint space, 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 the shoulder joint, elbow joint and wrist joint.

[0007] 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. The displacement constraints ensure that the optimized joint motion displacement is the difference between the set target position and the starting position. The joint motion constraints include the maximum speed, acceleration, and jerk during the motion of each joint.

[0008] S3. Design a multi-objective optimization function, with the minimum time, minimum energy consumption, and minimum impact as the optimization objectives, and establish three objective optimization functions;

[0009] S4. Determine the target variable for optimization based on the characteristics of S-shaped velocity curve trajectory planning;

[0010] 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;

[0011] S6. Select the optimal solution based on the weight values ​​of different objective functions to obtain the optimized trajectory of each joint.

[0012] Furthermore, in step S1, based on the planning characteristics of the S-shaped velocity curve of the articulated robotic arm, seven-segment S-shaped velocity curves are established for the k=1, 2, and 3 joints respectively. The mathematical model is described as follows:

[0013]

[0014] Specifically, such as Figure 2 As shown, where, Represents time, Representing different time periods, then the first The times for the seven stages of the S-shaped velocity curve corresponding to each joint are denoted as follows: , Representing the Total acceleration time for each joint Representing the The total time of motion during the entire trajectory planning process of each joint. Representing the The final velocity of the acceleration phase of each joint, Representing the The maximum speed of each joint It represents the magnitude of jerk.

[0015] Furthermore, step S2 specifies the displacement constraints and motion parameter constraints for each joint, and the process is as follows:

[0016] Displacement constraints: The purpose of S-shaped velocity curve planning for the robotic arm is to enable the joints to move from the initial point to the target point. During optimization, displacement residuals are calculated analytically, and displacement is constrained to ensure the accuracy of angle control. This step effectively eliminates accumulated errors and guarantees the accuracy of the final pose of the robotic arm's end effector.

[0017] First, set the... Joint starting position Target location Then the first The target displacement of each joint is To ensure that the optimized total trajectory displacement accurately matches the preset joint angle target displacement, the displacement constraint is expressed as:

[0018]

[0019] in For the first step in the trajectory planning process The seven-segment S-shaped velocity curve of each joint. For the first The total time of motion for the entire trajectory planning process of each joint;

[0020] Motion parameter constraints: Then, based on the parameters of each joint of the three-joint robotic arm on the implementation platform, the maximum values ​​of speed, acceleration, and jerk during the motion process of each joint are set to ensure the safety and accessibility of the robotic arm during the motion process. These represent the first step in the trajectory planning process. Given the inequality constraint functions for the velocity, acceleration, and jerk of each joint, the mathematical formula for the motion constraint is:

[0021]

[0022] in These represent the first step in the trajectory planning process. The velocity, acceleration, and jerk function of each joint. The robotic arm is the first The upper limits of velocity, acceleration, and jerk of each joint during movement.

[0023] Furthermore, step S3, in order to achieve coordinated optimization of the robotic arm joint trajectory among time, energy consumption, and motion impact, requires the construction of a multi-objective optimization function, the specific process of which is as follows:

[0024] Establish optimization objective functions representing time, energy consumption, and impact, respectively. , The maximum time spent in all joints is taken as the time optimization objective function. By minimizing The time taken by the slowest joint trajectory is used as the time objective function to avoid excessive time spent by a single joint slowing down the overall task efficiency; the average acceleration value of the three joints is used as the energy consumption evaluation function, i.e., the optimization objective function. Energy consumption optimization primarily targets the dynamic energy consumption of the joint motor. Average acceleration is used as an equivalent evaluation index, and root mean square (RMS) calculations are employed to smooth instantaneous energy consumption fluctuations, more accurately reflecting the motor's average power. This is achieved by minimizing... This can reduce energy loss of the robotic arm during movement and extend its service life; the average jerk value of the three joints is used as the impact evaluation function, i.e., the optimization objective function. Impact characteristics are evaluated using average jerk as the metric, and minimizing... It can suppress high-frequency vibrations and improve trajectory smoothness.

[0025] In summary, the expression for the multi-objective optimization function is as follows:

[0026]

[0027] in, and Representing the first Each joint in time period The acceleration and the magnitude of the jerk.

[0028] By establishing a multi-objective optimization function considering time, energy consumption, and impact. , By using a multi-objective optimization algorithm to find the Pareto optimal solution set, this approach is more suitable for selecting the optimal solution under complex working conditions compared to a single optimization objective. In other words, users can choose the appropriate solution based on task requirements, such as prioritizing time for urgent tasks or prioritizing impact for precision tasks, thus avoiding the local optimum trap of single-objective optimization and adapting to the dynamic needs of complex industrial scenarios.

[0029] Furthermore, step S4 is as follows:

[0030] Based on the characteristics of S-shaped velocity curve trajectory planning, the first The target variable to be optimized for each joint is [ There are a total of 10 dimensions, among which They represent the first The maximum velocity, maximum acceleration, and maximum jerk magnitude of each joint are considered. If three joints are considered, the objective variables to be optimized have a total of 30 dimensions. High-dimensional variables lead to slow convergence speed of the optimization algorithm, high platform computing power requirements, and are prone to conflicts between parameters, resulting in extreme solutions. Therefore, in order to reduce the dimensionality of the objective variables and improve computational efficiency, the following constraints are made based on the S-shaped velocity curve model:

[0031]

[0032] The time variable has 7 dimensions. ] becomes 3D[ ], then we get the first The final optimization objective variable for each joint is [ By introducing this constraint, the acceleration, constant velocity, and deceleration phases are made symmetrical, which avoids an imbalance in the time ratio between the constant velocity and acceleration / deceleration phases, ensuring a smooth speed transition. Furthermore, it ensures that the initial and final velocities, acceleration phases, and jerk are all zeroed, reducing residual vibration when the robotic arm reaches its final position. Simultaneously, maintaining the S-shaped velocity curve planning method during dimensionality reduction optimization ensures the smoothness and safety of the optimized trajectory.

[0033] Further, in step S5, Monte Carlo sampling is used to generate an initial population that satisfies the motion constraints, and the improved NSGA-II algorithm (Non-dominated Sorting Genetic Algorithm II, a classic multi-objective optimization algorithm) is used to search for the Pareto optimal solution set, as follows:

[0034] S5.1 Initialize motion constraint parameters and NSGA-II related parameters. 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. Maximum acceleration and maximum jerk and the starting position of each joint and target location This avoids invalid solutions caused by incorrect parameter ranges; it also initializes relevant parameters of the NSGA-II algorithm, including population size. Evolutionary algebra Crossover probability and mutation probability Choosing appropriate parameters can effectively improve the convergence speed and stability of the algorithm and generate a suitable number of solution sets;

[0035] S5.2 Generating the initial population: NSGA-II uses random uniform initialization. However, random initial populations tend to cluster in local regions, reducing solution set diversity. To improve the uniformity of the initial population distribution and enhance global search capabilities, a uniformly random maximum jerk is generated using Monte Carlo sampling. Generated randomly by Monte Carlo By combining the displacement integral equation, we can deduce the result in reverse. Within the feasible range, based on the joint displacement constraints and motion constraints defined during the planning process of each joint, a population size of [size missing] is generated by filtering. The initial population is determined to ensure that the initial parameters satisfy the kinematic constraints of the robotic arm and to avoid invalid solutions.

[0036] S5.3 Establish a fitness evaluation function, defining the parent population as... , For the number of population generations, based on the three objective optimization functions mentioned above, for Calculate for each individual , As an evaluation index for judging the quality of individuals, a penalty term Penalty is set at the same time. The values ​​exceeding the maximum speed, maximum acceleration, and maximum jerk are taken as physical constraints and transformed into additional costs of the objective function. This effectively eliminates infeasible solutions such as overspeed and over-acceleration, and guides the algorithm to converge towards a physically feasible and multi-objective balanced direction.

[0037] S5.4 Perform genetic operator operations on the parent generation. Perform selection, simulated binary crossover, and polynomial mutation evolution operations to produce a sizable offspring population. ;

[0038] To ensure that the final Pareto solution set has a large number of feasible solutions and is evenly distributed, boundary correction is performed on the crossed offspring during simulated binary crossover. Based on the physical constraints of each joint of the robotic arm, the feasible range of each parameter is predefined as low and up. If the parameters of the crossed offspring exceed low or up, they are automatically truncated to the nearest boundary.

[0039] Decision variables for each individual in the parent generation According to mutation probability Determine whether to perform a mutation operation. If a polynomial mutation is to be performed, the following polynomial operator is used to calculate the new decision variables. :

[0040]

[0041]

[0042]

[0043] in A uniformly random number in the range (0,1). Indicates the distribution index of variation. and These are the upper and lower bounds of the decision variable, respectively.

[0044] S5.5. Elite retention is implemented because in multi-objective optimization, the solution set is prone to getting trapped in local frontiers or uneven distribution. An elite retention mechanism is used to preserve superior genes by merging cross-generational populations and employing a merit-based selection mechanism. First, a composite population is constructed. That is, the composite population is defined as the union of offspring and parents, and then an environmental selection operation is performed: for Non-dominant grading and crowding calculations are performed, and individuals are sorted in ascending order of rank and descending order of crowding. The bottom 50% of individuals are eliminated, and the top N elites are retained. Generation population Among them, non-dominated hierarchies refer to the... Non-dominated sorting is performed, and individuals are divided into multiple levels such as front surface 1 and front surface 2 according to the dominance relationship. Individuals in front surface 1 are not dominated by any other individuals, individuals in front surface 2 are dominated only by individuals in front surface 1, and so on. Crowding degree calculation refers to sorting individuals in the same non-dominated level according to each objective function value and calculating the crowding degree of each individual.

[0045] S5.6 Check if the current iteration number has reached the evolution number. If the termination condition is met, the current Pareto front solution set is output as the Pareto optimal solution set; otherwise, the evolutionary process from step SS5.3 to SS5.5 is executed repeatedly.

[0046] Furthermore, step S6 is as follows:

[0047] The Pareto front obtained through the above steps contains a large number of equivalent optimal solutions. A fixed solution selection strategy cannot meet the needs of dynamic scenarios. Therefore, different weighting factors can be adjusted according to the actual needs of different operating conditions. Select a set of optimal solutions from the Pareto optimal solution set. As a final solution, this selection method can adapt to diverse scenario requirements such as emergency tasks, energy-saving modes, and high-precision modes. Indicates the first A set of objective optimization functions is used to generate the optimized trajectories of each joint of the robotic arm, and the optimal solution is obtained. The calculation formula is as follows:

[0048]

[0049] in, Finally, the optimal solution is substituted into the S-shaped velocity planning model to obtain the optimal trajectory.

[0050] Compared with existing technologies, this invention proposes a multi-objective trajectory optimization method based on an S-shaped velocity curve, which solves the problem of fixing trajectory planning parameters for S-shaped velocity curves. Under the premise of satisfying the dynamic constraints of the robotic arm, it achieves comprehensive optimization of time, smoothness, and energy consumption, and has the following advantages and beneficial effects:

[0051] (1) Breaking the limitations of traditional single-objective optimization, a multi-objective collaborative optimization model is adopted, using total motion time, root mean square acceleration (RMS), and jerk RMS as the objective functions. This model dynamically adjusts the jerk time of the S-curve. Uniform acceleration time ( ), uniform time ( ), maximum speed ( ), maximum acceleration ( ), maximum jerk ( To achieve a balance between motion efficiency and mechanical smoothness.

[0052] (2) The seven-stage S-curve parameterization modeling is combined with the multi-objective search algorithm NSGA-II, and the simulated binary crossover and polynomial mutation algorithms are improved to enhance the global search capability and effectively avoid local optima.

[0053] (3) By generating a multi-objective Pareto optimal solution set, it provides a multi-dimensional decision space for different application scenarios, significantly reduces mechanical vibration and energy consumption, extends equipment service life, and is suitable for complex scenarios such as high-precision control and precision assembly. Attached Figure Description

[0054] To more clearly illustrate the technical solutions in the embodiments of this application, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0055] Figure 1 This is a schematic diagram of the multi-objective trajectory optimization method based on S-shaped velocity curves provided in this embodiment of the invention;

[0056] Figure 2 This is a schematic diagram of the seven-segment S-shaped velocity planning curve in an embodiment of the present invention;

[0057] Figure 3 This is a flowchart of the NSGA-II algorithm according to an embodiment of the present invention;

[0058] Figure 4 These are the joint position curves obtained from multi-objective optimization of each joint of a three-joint serial robotic arm in this embodiment of the invention.

[0059] Figure 5 These are the joint velocity curves obtained from multi-objective optimization of each joint of a three-joint serial robotic arm in this embodiment of the invention.

[0060] Figure 6 These are the acceleration curves of each joint obtained from multi-objective optimization of each joint of a three-joint serial robotic arm in this embodiment of the invention.

[0061] Figure 7 These are the acceleration curves of each joint obtained by multi-objective optimization of each joint of a three-joint serial robotic arm in this embodiment of the invention. Detailed Implementation

[0062] To enable those skilled in the art to better understand the present application, the technical solutions in the embodiments of the present application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are merely some embodiments of the present application, and not all embodiments. All other embodiments obtained by those skilled in the art based on the embodiments of the present application without creative effort are within the scope of protection of the present application.

[0063] In this application, the reference to "embodiment" means that a specific feature, structure, or characteristic described in connection with an embodiment may be included in at least one embodiment of this application. The appearance of this phrase in various places throughout the specification does not necessarily refer to the same embodiment, nor is it a mutually exclusive, independent, or alternative embodiment. It will be explicitly and implicitly understood by those skilled in the art that the embodiments described in this application can be combined with other embodiments.

[0064] Example 1

[0065] Figure 1 The diagram shown is a flowchart of a multi-objective trajectory optimization method based on an S-shaped velocity curve provided in an embodiment of the present invention. The multi-objective trajectory optimization method based on an S-shaped velocity curve includes the following steps:

[0066] S1. Based on the S-shaped velocity curve trajectory planning characteristics of the robotic arm in joint space, establish mathematical models of the seven-segment S-shaped velocity curves for each joint of the three-joint robotic arm. The three-joint robotic arm includes the shoulder joint, elbow joint, and wrist joint. According to... Figure 2 The seven-segment S-shaped velocity curve shown establishes the S-shaped velocity curves for the k=1, 2, and 3 joints. The mathematical model is described as follows:

[0067]

[0068] in, Represents time, Representing different time periods, then the first The times for the seven stages of the S-shaped velocity curve corresponding to each joint are denoted as follows: , Representing the Total acceleration time for each joint Representing the The total time of motion during the entire trajectory planning process of each joint. Representing the The final velocity of the acceleration phase of each joint, Representing the The maximum speed of each joint It represents the magnitude of jerk.

[0069] S2. Set the starting and target positions of each joint, as well as the joint displacement and motion constraints defined during the planning process. The process is as follows:

[0070] First, define the... The starting position of each joint is represented as The target location is represented as Then the first The given target displacement of each joint is If the total trajectory displacement precisely matches the preset target displacement of the joint angle, then the displacement constraint is expressed as:

[0071]

[0072] in For the first step in the trajectory planning process The seven-segment S-shaped velocity curve of each joint. For the first The total time of motion for the entire trajectory planning process of each joint;

[0073] These represent the first step in the trajectory planning process. Given the inequality constraint functions for the velocity, acceleration, and jerk of each joint, the mathematical formula for the motion constraint is:

[0074]

[0075] in These represent the first step in the trajectory planning process. The velocity, acceleration, and jerk function of each joint. The robotic arm is the first The upper limits of velocity, acceleration, and jerk of each joint during movement.

[0076] Based on the data of the three-joint robotic arm implemented in this paper, the specific values ​​of the motion parameter constraints for each joint in the example are shown in Table 1.

[0077] Table 1. Definition of Motion Parameter Constraints for a Three-Joint Robotic Arm System

[0078]

[0079] S3. Establish the optimization objective function with the goals of minimizing time, energy consumption, and impact as the optimization objectives. The process is as follows: Establish optimization objective functions representing time, energy consumption, and impact respectively. , The maximum time spent in all joints is taken as the time optimization objective function. The average acceleration value of the three joints is used as the evaluation function for energy consumption, i.e., the optimization objective function. The average jerk value of the three joints is used as the impact evaluation function, i.e., the optimization objective function. The expression is as follows:

[0080]

[0081] in, and Representing the first Each joint in time period The acceleration and the magnitude of the jerk.

[0082] S4. Determining the target variable for optimization based on the characteristics of S-shaped velocity curve trajectory planning:

[0083] Based on the characteristics of S-shaped velocity curve trajectory planning, the first The target variable to be optimized for each joint is [ There are a total of 10 dimensions, among which They represent the first The maximum velocity, maximum acceleration, and maximum jerk magnitude of each joint. If three joints are considered, the total number of objective variables to be optimized is 30. Therefore, to reduce the dimensionality of the objective variables and improve computational efficiency, the following constraints are imposed based on the S-shaped velocity curve model:

[0084]

[0085] By introducing this constraint, the dimension of the time variable is reduced from 7 dimensions. ] becomes 3D[ ], then we get the first The final optimization objective variable for each joint is [ ].

[0086] S5. An initial population satisfying the motion constraints is generated using Monte Carlo sampling. The Pareto optimal solution set is obtained using the improved NSGA-II algorithm, as follows:

[0087] S5.1 Initialize motion constraint parameters and NSGA-II related parameters. 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. Maximum acceleration and maximum jerk and the starting position of each joint and target location Initialization parameters for the NSGA-II algorithm include population size. Evolutionary algebra Crossover probability and mutation probability ;

[0088] S5.2. Generate the initial population and generate uniformly random maximum jerk using the Monte Carlo sampling method. Based on the joint displacement constraints and motion constraints defined during the planning process of each joint, a population of size is generated. initial population ;

[0089] S5.3 Establish a fitness evaluation function, defining the parent population as... , For the number of population generations, based on the three objective optimization functions mentioned above, for Calculate for each individual , As an evaluation index for judging the quality of individuals, a penalty term Penalty is set, which takes the values ​​of exceeding the maximum speed, maximum acceleration, and maximum jerk as physical constraints and transforms them into additional costs of the objective function, and finally obtains the fitness value of each individual.

[0090] S5.4 Perform genetic operator operations on the parent generation. Perform selection, simulated binary crossover, and polynomial mutation evolution operations to produce a sizable offspring population. ;

[0091] To ensure that the final Pareto solution set has a large number of feasible solutions and is evenly distributed, boundary correction is performed on the crossed offspring during simulated binary crossover. Based on the physical constraints of each joint of the robotic arm, the feasible range of each parameter is predefined as low and up. If the parameters of the crossed offspring exceed low or up, they are automatically truncated to the nearest boundary.

[0092] Decision variables for each individual in the parent generation According to mutation probability Determine whether to perform a mutation operation. If a polynomial mutation is to be performed, the following polynomial operator is used to calculate the new decision variables. :

[0093]

[0094]

[0095]

[0096] in A uniformly random number in the range (0,1). Indicates the distribution index of variation. and These are the upper and lower bounds of the decision variable, respectively.

[0097] S5.5. Preserve elites and construct composite populations. That is, the composite population is defined as the union of offspring and parents, and the environmental selection operation is performed: for Non-dominant grading and crowding calculations are performed, and individuals are sorted in ascending order of rank and descending order of crowding. The bottom 50% of individuals are eliminated, and the top N elites are retained. Generation population Among them, non-dominated hierarchies refer to the... Non-dominated sorting is performed, and individuals are divided into multiple levels such as front surface 1 and front surface 2 according to the dominance relationship. Individuals in front surface 1 are not dominated by any other individuals, individuals in front surface 2 are dominated only by individuals in front surface 1, and so on. Crowding degree calculation refers to sorting individuals in the same non-dominated level according to each objective function value and calculating the crowding degree of each individual.

[0098] S5.6 Check if the current iteration number has reached the evolution number. If the termination condition is met, the current Pareto front solution set is output as the Pareto optimal solution set; otherwise, the evolutionary process from step SS5.3 to SS5.5 is executed repeatedly.

[0099] S6. Select an optimal solution. Adjust different weighting factors according to the actual working conditions. Select a set of optimal solutions from the Pareto optimal solution set as the final solution, and use this set of solutions to generate the optimized trajectory of each joint of the robotic arm.

[0100]

[0101]

[0102] In execution Figure 3 The NSGA-II algorithm flow shown is configured with a population size of 400, a maximum number of iterations of 180, a crossover probability of 0.8, a mutation probability of 0.2, and a mutation distribution exponent of 20. After executing the improved NSGA-II algorithm, the Pareto solution set is obtained. The time, energy consumption, and impact are determined based on the weight values ​​of different objective functions. The optimal solutions for selecting the optimization variables are shown in Table 2:

[0103] Table 2. Optimal solutions for each joint of a three-joint robotic arm

[0104]

[0105] Substituting the above solution into the S-shaped velocity programming model, we obtain the optimized position curves, velocity curves, acceleration curves, and jerk curves of each joint of the three-joint robotic arm, as shown below. Figure 4 , 5 As shown in Figures 6 and 7.

[0106] Substitute the above results into the established optimization objective function. , Results analysis shows that compared to direct use The optimized trajectory of the robotic arm joints, planned using an S-curve velocity curve, shows the following changes in performance indicators: Joint 1's total time is 5.52s, an increase of 33.6%, with energy consumption (acceleration RMS) decreasing by 48.3% and impact (jerk RMS) decreasing by 55.4%; Joint 2's total time is 4.43s, an increase of 18.7%, with energy consumption (acceleration RMS) decreasing by 32.5% and impact (jerk RMS) decreasing by 44.5%; Joint 3's total time is 5.79s, an increase of 2.8%, with energy consumption (acceleration RMS) decreasing by 7.1% and impact (jerk RMS) decreasing by 18.6%. Analysis of the data shows that time, energy consumption, and impact are mutually constraining, meaning all three indicators cannot be simultaneously optimized. While not using the maximum values ​​for speed, acceleration, and jerk during trajectory planning increases time, the energy consumption and impact calculations show significant reductions for each joint. This demonstrates that the method effectively optimizes the robotic arm joint trajectory for the three objectives of time, energy consumption, and impact.

[0107] Example 2

[0108] This embodiment continues to disclose the specific implementation process of the multi-objective robotic arm joint trajectory optimization method based on S-shaped velocity curves. The technical means and steps used in this embodiment are the same as in Embodiment 1, differing only in parameters. This embodiment includes the following steps:

[0109] Steps S1, S3, and S4 are the same as steps S1, S3, and S4 in Example 1, except that the parameter values ​​in S2, S5, and S6 are changed.

[0110] S2. Set the starting and target positions of each joint, as well as the joint displacement and motion constraints defined during the joint planning process.

[0111] Based on the data of the three-joint robotic arm implemented in this paper, the specific values ​​of the motion parameter constraints for each joint in the example are shown in Table 3.

[0112] Table 3. Definition of Motion Parameter Constraints for the Three-Joint Robotic Arm System

[0113]

[0114] S5. Use the improved NSGA-II algorithm to search for the Pareto optimal solution set, the process is the same as in Example 1.

[0115] S6. Select an optimal solution. Adjust different weighting factors according to the actual working conditions. Select a set of optimal solutions from the Pareto optimal solution set as the final solution, and use this set of solutions to generate the optimized trajectory of each joint of the robotic arm.

[0116] When executing this algorithm, the population size was set to 300, the maximum number of iterations to 150, the crossover probability to 0.7, the mutation probability to 0.2, and the mutation distribution exponent to 20. After executing the improved NSGA-II algorithm, the Pareto solution set was obtained. The time, energy consumption, and impact were determined based on the weight values ​​of different objective functions. The optimal solutions for selecting the optimization variables are shown in Table 4:

[0117] Table 4. Optimal solutions for each joint of a three-joint robotic arm

[0118]

[0119] Substitute the above results into the established optimization objective function in S3. , Results analysis shows that compared to direct use The joint trajectories of the robotic arm were planned using an S-shaped velocity curve. The optimized trajectory showed the following changes in indicators: the total time for joint 1 was 7.23s, an increase of 74.9%, with energy consumption (acceleration RMS) decreasing by 69.0% and impact (jerk RMS) decreasing by 82.0%; the total time for joint 2 was 7.06s, an increase of 89.1%, with energy consumption (acceleration RMS) decreasing by 74.0% and impact (jerk RMS) decreasing by 78.2%; and the total time for joint 3 was 6.92s, an increase of 22.9%, with energy consumption (acceleration RMS) decreasing by 38.7% and impact (jerk RMS) decreasing by 51.0%. Based on the above data analysis, since time and energy consumption impact are mutually restrictive, it is impossible for all three indicators to reach their optimal values ​​simultaneously. When the maximum values ​​of the set speed, acceleration, and jerk are not used for trajectory planning, the time will increase accordingly. However, judging from the energy consumption and impact calculation results, each joint shows a significant reduction, proving that the method effectively optimizes the trajectory of the robotic arm joints for the three objectives of time, energy consumption, and impact.

[0120] The technical features of the above embodiments can be combined in any way. For the sake of brevity, 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, they should be considered to be within the scope of this specification.

[0121] The above embodiments are preferred embodiments of the present invention, but the embodiments of the present invention are not limited to the above embodiments. Any changes, modifications, substitutions, combinations, or simplifications made without departing from the spirit and principle of the present invention shall be considered equivalent substitutions and shall be included within the protection scope of the present invention.

Claims

1. A method for optimizing the joint trajectory of a multi-objective robotic arm based on an S-shaped velocity curve, characterized in that, The multi-objective robotic arm joint trajectory optimization method includes the following steps: S1. Based on the trajectory planning characteristics of the S-shaped velocity curve of the robotic arm in the joint space, 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 the shoulder joint, elbow joint and 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 process. The displacement constraints ensure that the optimized joint motion displacement is the difference between the set target position and the starting position. The joint motion constraints include the maximum speed, acceleration, and jerk during the motion of each joint. S3. Design a multi-objective optimization function, with the minimum time, minimum energy consumption, and minimum impact as the optimization objectives, and establish three objective optimization functions; S4. Determine the target variable for optimization based on the characteristics of S-shaped velocity curve trajectory planning; S5. An initial population satisfying the motion constraints is generated using Monte Carlo sampling, and the Pareto optimal solution set is obtained using the NSGA-II algorithm; the process of step S5 is as follows: S5.1 Initialize motion constraint parameters and NSGA-II related parameters. 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. Maximum acceleration and maximum jerk and the starting position of each joint and target location Initialization parameters for the NSGA-II algorithm include population size. Evolutionary algebra Crossover probability and mutation probability ; S5.

2. Generate the initial population and generate uniformly random maximum accelerometers using the Monte Carlo sampling method. Based on the joint displacement constraints and motion constraints defined during the planning process of each joint, a population of size is generated. initial population ; S5.3 Establish a fitness evaluation function, defining the parent population as... , For the number of population generations, based on the three objective optimization functions mentioned above, for Calculate for each individual , As an evaluation index for judging the quality of individuals, a penalty term Penalty is set, which takes the values ​​of exceeding the maximum speed, maximum acceleration, and maximum jerk as physical constraints and transforms them into additional costs of the objective function, and finally obtains the fitness value of each individual. S5.4 Perform genetic operator operations on the parent generation. Perform selection, simulated binary crossover, and polynomial mutation evolution operations to produce a sizable offspring population. ; The process is as follows: When performing simulated binary crossover, boundary correction is performed on the crossover offspring. Based on the physical limitations of each joint of the robotic arm, the feasible range of each parameter is predefined as low and up. If the parameter of the crossover offspring exceeds low or up, it is automatically truncated to the nearest boundary. Decision variables for each individual in the parent generation According to mutation probability Determine whether to perform a mutation operation. If a polynomial mutation is to be performed, the following polynomial operator is used to calculate the new decision variables. : in A uniformly random number in the range (0,1). Indicates the distribution index of variation. and These are the upper and lower bounds of the decision variable, respectively; S5.

5. Preserve elites and construct composite populations. That is, the composite population is defined as the union of offspring and parents, and the environmental selection operation is performed: for Non-dominant grading and crowding calculations are performed, and individuals are sorted in ascending order of rank and descending order of crowding. The bottom 50% of individuals are eliminated, and the top N elites are retained. Generation population Among them, non-dominated hierarchies refer to the... Non-dominated sorting is performed, and individuals are divided into multiple levels such as front surface 1 and front surface 2 according to the dominance relationship. Individuals in front surface 1 are not dominated by any other individuals, individuals in front surface 2 are dominated only by individuals in front surface 1, and so on. Crowding degree calculation refers to sorting individuals in the same non-dominated level according to each objective function value and calculating the crowding degree of each individual. S5.6 Check if the current iteration number has reached the evolution number. If the termination condition is met, the current Pareto front solution set is output as the Pareto optimal solution set; otherwise, the evolutionary process from step SS5.3 to SS5.5 is executed repeatedly. S6. Select the optimal solution based on the weight values ​​of different objective functions to obtain the optimized trajectory of each joint.

2. The multi-objective robotic arm joint trajectory optimization method based on S-shaped velocity curves according to claim 1, characterized in that, In step S1, based on the planning characteristics of the S-shaped velocity curve of the articulated robotic arm, a seven-segment S-shaped velocity curve is established for the k=1, 2, and 3rd joints. The mathematical model is described as follows: ， in, Represents time, Representing different time periods, then the first The times for the seven stages of the S-shaped velocity curve corresponding to each joint are denoted as follows: , Representing the Total acceleration time for each joint Representing the The total time of motion during the entire trajectory planning process of each joint. Representing the The final velocity of the acceleration phase of each joint, Representing the The maximum speed of each joint It represents the magnitude of jerk.

3. The multi-objective robotic arm joint trajectory optimization method based on S-shaped velocity curves according to claim 2, characterized in that, The process of step S2 is as follows: First, define the... The starting position of each joint is represented as The target location is represented as Then the first The given target displacement of each joint is If the total trajectory displacement precisely matches the preset target displacement of the joint angle, then the displacement constraint is expressed as: ， in For the first step in the trajectory planning process The seven-segment S-shaped velocity curve of each joint. For the first The total time of motion for the entire trajectory planning process of each joint; These represent the first step in the trajectory planning process. Given the inequality constraint functions for the velocity, acceleration, and jerk of each joint, the mathematical formula for the motion constraint is: ， in These represent the first step in the trajectory planning process. The velocity, acceleration, and jerk function of each joint. The robotic arm is the first The upper limits of velocity, acceleration, and jerk of each joint during movement.

4. The multi-objective robotic arm joint trajectory optimization method based on S-shaped velocity curves according to claim 3, characterized in that, The process of step S3 is as follows: Establish optimization objective functions representing time, energy consumption, and impact, respectively. , The maximum time spent in all joints is taken as the time optimization objective function. The average acceleration value of the three joints is used as the evaluation function for energy consumption, i.e., the optimization objective function. The average jerk value of the three joints is used as the impact evaluation function, i.e., the optimization objective function. The expression is as follows: ， in, and Representing the first Each joint in time period The acceleration and the magnitude of the jerk.

5. The multi-objective robotic arm joint trajectory optimization method based on S-shaped velocity curves according to claim 4, characterized in that, The process of step S4 is as follows: Based on the characteristics of S-shaped velocity curve trajectory planning, the first The objective variable to be optimized for each joint is [ There are a total of 10 dimensions, among which They represent the first Based on the S-curve velocity curve model, the following constraints are imposed on the maximum velocity, maximum acceleration, and maximum jerk magnitude of each joint: ， By introducing this constraint, the dimension of the time variable is reduced from 7 dimensions. ] becomes 3D[ ], then we get the first The final optimization objective variable for each joint is [ ].

6. The method for optimizing the trajectory of a multi-objective robotic arm joint based on an S-shaped velocity curve according to claim 5, characterized in that, The process of step S6 is as follows: Adjusting different weighting factors according to different actual working conditions Select a set of optimal solutions from the Pareto optimal solution set. As the final solution. Indicates the first A set of objective optimization functions is used to generate the optimized trajectories of each joint of the robotic arm, and the optimal solution is obtained. The calculation formula is as follows: ,in, .

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